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thf(sy_c_BNF__Def_OGrp_001t__FWState__Oconditional____action_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001t__FWState__Oconditional____action_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,(
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thf(sy_c_BNF__Def_OGrp_001t__FWState__Oconditional____action_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_001t__FWState__Oconditional____action_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,(
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thf(sy_c_BNF__Def_OGrp_001t__FWState__Oconditional____action_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_001t__FWState__Oconditional____action_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,(
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thf(sy_c_BNF__Def_OGrp_001t__FWState__Oconditional____action_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J_001t__FWState__Oconditional____action_It__Code____Numeral__Onatural_J,type,(
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thf(sy_c_BNF__Def_OGrp_001t__FWState__Oconditional____action_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J_001t__FWState__Oconditional____action_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,(
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thf(sy_c_BNF__Def_OGrp_001t__FWState__Oconditional____action_It__Product____Type__Oprod_It__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_J_001t__FWState__Oconditional____action_It__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_J,type,(
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thf(sy_c_BNF__Def_OGrp_001t__FWState__Oconditional____action_It__Product____Type__Oprod_It__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_J_001t__FWState__Oconditional____action_It__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J,type,(
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thf(sy_c_BNF__Def_OGrp_001t__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_001t__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J,type,(
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thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__Nat__Onat,type,(
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    bNF_pi2877385506940030580atural: ( code_natural > ( produc4972180933644002618e_term > option6357759511663192854e_term ) > $o ) > ( ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o ) > code_natural > produc7822875418678951345atural > produc4972180933644002618e_term > option6357759511663192854e_term )).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Product____Type__Oprod_It__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_001_062_It__Product____Type__Oprod_It__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_001_062_It__Product____Type__Oprod_It__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_Mt__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_J_001_062_It__Product____Type__Oprod_It__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_Mt__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_J,type,(
    bNF_re2955035557799908905_a_b_c: ( ( produc8923597787767437993list_f > produc1811169103916964701list_f ) > ( produc8923597787767437993list_f > produc1811169103916964701list_f ) > $o ) > ( ( produc8923597787767437993list_f > list_n1973683603564934665_a_b_c ) > ( produc8923597787767437993list_f > list_n1973683603564934665_a_b_c ) > $o ) > ( ( produc8923597787767437993list_f > produc1811169103916964701list_f ) > produc8923597787767437993list_f > list_n1973683603564934665_a_b_c ) > ( ( produc8923597787767437993list_f > produc1811169103916964701list_f ) > produc8923597787767437993list_f > list_n1973683603564934665_a_b_c ) > $o )).

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    bNF_re1616170189525647965_a_b_c: ( ( produc1811169103916964701list_f > list_n1973683603564934665_a_b_c ) > ( produc1811169103916964701list_f > list_n1973683603564934665_a_b_c ) > $o ) > ( ( ( produc8923597787767437993list_f > produc1811169103916964701list_f ) > produc8923597787767437993list_f > list_n1973683603564934665_a_b_c ) > ( ( produc8923597787767437993list_f > produc1811169103916964701list_f ) > produc8923597787767437993list_f > list_n1973683603564934665_a_b_c ) > $o ) > ( ( produc1811169103916964701list_f > list_n1973683603564934665_a_b_c ) > ( produc8923597787767437993list_f > produc1811169103916964701list_f ) > produc8923597787767437993list_f > list_n1973683603564934665_a_b_c ) > ( ( produc1811169103916964701list_f > list_n1973683603564934665_a_b_c ) > ( produc8923597787767437993list_f > produc1811169103916964701list_f ) > produc8923597787767437993list_f > list_n1973683603564934665_a_b_c ) > $o )).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_Eo_001_Eo,type,(
    bNF_rel_fun_o_o_o_o: ( $o > $o > $o ) > ( $o > $o > $o ) > ( $o > $o ) > ( $o > $o ) > $o )).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_001t__Product____Type__Oprod_It__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_001t__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_001t__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J,type,(
    bNF_re7641334924310632669_a_b_c: ( produc8923597787767437993list_f > produc8923597787767437993list_f > $o ) > ( list_n1973683603564934665_a_b_c > list_n1973683603564934665_a_b_c > $o ) > ( produc8923597787767437993list_f > list_n1973683603564934665_a_b_c ) > ( produc8923597787767437993list_f > list_n1973683603564934665_a_b_c ) > $o )).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_001t__Product____Type__Oprod_It__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_001t__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_001t__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J,type,(
    bNF_re6046208945862828881list_f: ( produc8923597787767437993list_f > produc8923597787767437993list_f > $o ) > ( list_n1973683603564934665_a_b_c > produc8853065311482726181list_f > $o ) > ( produc8923597787767437993list_f > list_n1973683603564934665_a_b_c ) > ( produc8923597787767437993list_f > produc8853065311482726181list_f ) > $o )).

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    bNF_re2373559492235539289_a_b_c: ( produc1811169103916964701list_f > produc1811169103916964701list_f > $o ) > ( list_n1973683603564934665_a_b_c > list_n1973683603564934665_a_b_c > $o ) > ( produc1811169103916964701list_f > list_n1973683603564934665_a_b_c ) > ( produc1811169103916964701list_f > list_n1973683603564934665_a_b_c ) > $o )).

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thf(sy_c_BNF__Def_OsndOp_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,(
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thf(sy_c_BNF__Def_OsndOp_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,(
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thf(sy_c_BNF__Def_OsndOp_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,(
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thf(sy_c_BNF__Def_OsndOp_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,(
    bNF_sn2594531154364564657nteger: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o ) > ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc1908205239877642774nteger > produc1908205239877642774nteger )).

thf(sy_c_BNF__Def_OsndOp_001_062_It__Product____Type__Oprod_It__Code____Numeral__Onatural_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Onatural_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,(
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thf(sy_c_BNF__Def_OsndOp_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,(
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thf(sy_c_BNF__Def_OsndOp_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,(
    bNF_sn7816398935535452141nt_int: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o ) > ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc2285326912895808259nt_int > produc2285326912895808259nt_int )).

thf(sy_c_BNF__Def_OsndOp_001t__Code____Numeral__Onatural_001_062_It__Product____Type__Oprod_It__Code____Numeral__Onatural_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,(
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thf(sy_c_BNF__Def_OsndOp_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,(
    bNF_sn8722645597740893163atural: ( code_natural > code_natural > $o ) > ( code_natural > produc7822875418678951345atural > $o ) > produc5835291356934675326atural > produc5835291356934675326atural )).

thf(sy_c_BNF__Def_OsndOp_001t__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_001t__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_001t__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J,type,(
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thf(sy_c_BNF__Def_OsndOp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,(
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thf(sy_c_BNF__Def_OsndOp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,(
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,(
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,(
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,(
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Onatural_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,(
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,(
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,(
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,(
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat_001t__Nat__Onat,type,(
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,(
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thf(sy_c_BNF__Wellorder__Constructions_Obsqr_001t__Nat__Onat,type,(
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thf(sy_c_BNF__Wellorder__Constructions_Obsqr_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,(
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thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Nat__Onat,type,(
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    comp_P3189659738270478212at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > ( produc859450856879609959at_nat > product_prod_nat_nat ) > produc859450856879609959at_nat > product_prod_nat_nat )).

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thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_001t__Product____Type__Oprod_It__FinFun__Ofinfun_Itf__d_Mt__List__Olist_It__FWState__Olock____action_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J_J,type,(
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thf(sy_c_Fun_Oid_001t__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J,type,(
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thf(sy_c_Fun_Oid_001t__Nat__Onat,type,(
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,(
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,(
    id_Pro3923183700162330716nt_int: produc7773217078559923341nt_int > produc7773217078559923341nt_int )).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,(
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Onatural_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,(
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,(
    id_Pro5422067691592394322nt_int: produc2285326912895808259nt_int > produc2285326912895808259nt_int )).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,(
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,(
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,(
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__List__Olist_It__FWState__Onew____thread____action_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Oconditional____action_Itf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Owait____set____action_Itf__a_Mtf__e_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__FWState__Ointerrupt____action_Itf__a_J_J_Mt__List__Olist_Itf__f_J_J_J_J_J,type,(
    id_Pro1034130279814846892list_f: produc1811169103916964701list_f > produc1811169103916964701list_f )).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,(
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thf(sy_c_Fun_Oinj__on_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,(
    inj_on8181760918054529539nteger: ( ( code_integer > option6357759511663192854e_term ) > produc8763457246119570046nteger ) > set_Co9149898834107579976e_term > $o )).

thf(sy_c_Fun_Oinj__on_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,(
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thf(sy_c_Fun_Oinj__on_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,(
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thf(sy_c_Fun_Oinj__on_001_062_It__Product____Type__Oprod_It__Code____Numeral__Onatural_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Onatural_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,(
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thf(sy_c_Fun_Oinj__on_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,(
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thf(sy_c_Fun__Def_Oin__rel_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,(
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thf(sy_c_Fun__Def_Oin__rel_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,(
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    member335077259223437328nteger: produc3334582477557816039nteger > set_Pr5052581263052047815nteger > $o )).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Onatural_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,(
    member1425327671947302374e_term: produc8392782284284844111e_term > set_Pr7460367489947184517e_term > $o )).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,(
    member3756964795865971088atural: produc6756470014200349799atural > set_Pr250711821884878663atural > $o )).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,(
    member3093737243930835296e_term: produc3832618445565238089e_term > set_Pr9020418513037773311e_term > $o )).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,(
    member7154775377335690986e_term: produc1821397396195235155e_term > set_Pr7309386651551657097e_term > $o )).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,(
    member8566619992076573584nt_int: produc1219242969750017639nt_int > set_Pr2560585780119916871nt_int > $o )).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,(
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o )).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,(
    member6033505038158179318at_nat: produc39072733474941005at_nat > set_Pr9129816769188007085at_nat > $o )).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,(
    member3048279782668807382at_nat: produc6277219514840344877at_nat > set_Pr575275573428919693at_nat > $o )).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,(
    member1995966531042493578at_nat: produc5224906263214031073at_nat > set_Pr7116486347545156417at_nat > $o )).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,(
    member3919319858682911658at_nat: produc7148259590854449153at_nat > set_Pr846279711151727201at_nat > $o )).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,(
    member8062223511168850704at_nat: produc349518998152878311at_nat > set_Pr553994874890374343at_nat > $o )).

thf(sy_c_member_001t__Typerep__Otyperep,type,(
    member_typerep: typerep > set_typerep > $o )).

thf(sy_v_jswss,type,(
    jswss: produc8853065311482726181list_f )).

thf(sy_v_ls,type,(
    ls: finfun1145668972854894571action )).

thf(sy_v_nts,type,(
    nts: list_n1973683603564934665_a_b_c )).

% Relevant facts (1117)
thf(fact_0_thr__a__def,axiom,
    ( thr_a_d_a_b_c_e_f
    = ( comp_P4767578765025054880list_f @ produc3535155421234910955list_f @ produc8710481132536983823list_f ) )).

% thr_a_def
thf(fact_1_fst__conv,axiom,(
    ! [X1: code_integer > option6357759511663192854e_term,X2: produc8923325533196201883nteger] :
      ( ( produc2702340305111172292nteger @ ( produc6137756002093451184nteger @ X1 @ X2 ) )
      = X1 ) )).

% fst_conv
thf(fact_2_fst__conv,axiom,(
    ! [X1: produc4972180933644002618e_term > option6357759511663192854e_term,X2: produc7822875418678951345atural] :
      ( ( produc3051390257598303421atural @ ( produc3831813291587773865atural @ X1 @ X2 ) )
      = X1 ) )).

% fst_conv
thf(fact_3_fst__conv,axiom,(
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X2: produc8923325533196201883nteger] :
      ( ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ X1 @ X2 ) )
      = X1 ) )).

% fst_conv
thf(fact_4_fst__conv,axiom,(
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int] :
      ( ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X1 @ X2 ) )
      = X1 ) )).

% fst_conv
thf(fact_5_fst__conv,axiom,(
    ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int] :
      ( ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X1 @ X2 ) )
      = X1 ) )).

% fst_conv
thf(fact_6_obs__a__def,axiom,
    ( obs_a_d_a_b_c_e_f
    = ( comp_P8441265476687562528list_f @ ( comp_P9196084097856615518list_f @ ( comp_P4727187950229163873list_f @ ( comp_P4844579239011320789list_f @ produc7200483066056472223list_f @ produc6347154330667827118list_f ) @ produc5461435702946143345list_f ) @ produc7287154213363835177list_f ) @ produc8710481132536983823list_f ) )).

% obs_a_def
thf(fact_7_cond__a__def,axiom,
    ( cond_a_d_a_b_c_e_f
    = ( comp_P1274227037634964381list_f @ ( comp_P5019820574536571553list_f @ produc4949502379994040371list_f @ produc7287154213363835177list_f ) @ produc8710481132536983823list_f ) )).

% cond_a_def
thf(fact_8_interrupt__a__def,axiom,
    ( interr5756268207484649009_c_e_f
    = ( comp_P1151571275026987962list_f @ ( comp_P3718283140659731780list_f @ ( comp_P1342733116751693639list_f @ ( comp_P7034530251472854715list_f @ produc3338296310991575901list_f @ produc6347154330667827118list_f ) @ produc5461435702946143345list_f ) @ produc7287154213363835177list_f ) @ produc8710481132536983823list_f ) )).

% interrupt_a_def
thf(fact_9_wset__a__def,axiom,
    ( wset_a_d_a_b_c_e_f
    = ( comp_P2261067389098190747list_f @ ( comp_P5472528119234947763list_f @ ( comp_P3272176189973965824list_f @ produc3180365725429668208list_f @ produc5461435702946143345list_f ) @ produc7287154213363835177list_f ) @ produc8710481132536983823list_f ) )).

% wset_a_def
thf(fact_10_old_Oprod_Oinject,axiom,(
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A2: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( ( produc6137756002093451184nteger @ A @ B )
        = ( produc6137756002093451184nteger @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) )).

% old.prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,(
    ! [A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural,A2: produc4972180933644002618e_term > option6357759511663192854e_term,B2: produc7822875418678951345atural] :
      ( ( ( produc3831813291587773865atural @ A @ B )
        = ( produc3831813291587773865atural @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) )).

% old.prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,(
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B )
        = ( produc8603105652947943368nteger @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) )).

% old.prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,(
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) )).

% old.prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,(
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) )).

% old.prod.inject
thf(fact_15_prod_Oinject,axiom,(
    ! [X1: code_integer > option6357759511663192854e_term,X2: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
      ( ( ( produc6137756002093451184nteger @ X1 @ X2 )
        = ( produc6137756002093451184nteger @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) )).

% prod.inject
thf(fact_16_prod_Oinject,axiom,(
    ! [X1: produc4972180933644002618e_term > option6357759511663192854e_term,X2: produc7822875418678951345atural,Y1: produc4972180933644002618e_term > option6357759511663192854e_term,Y2: produc7822875418678951345atural] :
      ( ( ( produc3831813291587773865atural @ X1 @ X2 )
        = ( produc3831813291587773865atural @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) )).

% prod.inject
thf(fact_17_prod_Oinject,axiom,(
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X2: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ X1 @ X2 )
        = ( produc8603105652947943368nteger @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) )).

% prod.inject
thf(fact_18_prod_Oinject,axiom,(
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ X1 @ X2 )
        = ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) )).

% prod.inject
thf(fact_19_prod_Oinject,axiom,(
    ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ X1 @ X2 )
        = ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) )).

% prod.inject
thf(fact_20_prod__cases3,axiom,(
    ! [Y: produc8763457246119570046nteger] :
      ~ ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C: code_integer] :
          ( Y
         != ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C ) ) ) )).

% prod_cases3
thf(fact_21_prod__cases3,axiom,(
    ! [Y: produc989692990947075319atural] :
      ~ ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: code_natural,C: code_natural] :
          ( Y
         != ( produc3831813291587773865atural @ A3 @ ( produc3574140220909816553atural @ B3 @ C ) ) ) )).

% prod_cases3
thf(fact_22_prod__cases3,axiom,(
    ! [Y: produc1908205239877642774nteger] :
      ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C: code_integer] :
          ( Y
         != ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C ) ) ) )).

% prod_cases3
thf(fact_23_prod__cases3,axiom,(
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C: int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C ) ) ) )).

% prod_cases3
thf(fact_24_prod__cases3,axiom,(
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A3: int > option6357759511663192854e_term,B3: int,C: int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C ) ) ) )).

% prod_cases3
thf(fact_25_old_Oprod_Oexhaust,axiom,(
    ! [Y: produc8763457246119570046nteger] :
      ~ ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( Y
         != ( produc6137756002093451184nteger @ A3 @ B3 ) ) )).

% old.prod.exhaust
thf(fact_26_old_Oprod_Oexhaust,axiom,(
    ! [Y: produc989692990947075319atural] :
      ~ ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural] :
          ( Y
         != ( produc3831813291587773865atural @ A3 @ B3 ) ) )).

% old.prod.exhaust
thf(fact_27_old_Oprod_Oexhaust,axiom,(
    ! [Y: produc1908205239877642774nteger] :
      ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( Y
         != ( produc8603105652947943368nteger @ A3 @ B3 ) ) )).

% old.prod.exhaust
thf(fact_28_old_Oprod_Oexhaust,axiom,(
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A3 @ B3 ) ) )).

% old.prod.exhaust
thf(fact_29_old_Oprod_Oexhaust,axiom,(
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A3 @ B3 ) ) )).

% old.prod.exhaust
thf(fact_30_lock__action_Osize_I5_J,axiom,
    ( ( size_s8978095283902022239action @ lock_Lock )
    = zero_zero_nat )).

% lock_action.size(5)
thf(fact_31_typerep__lock__action__def,axiom,
    ( typere7107042215916526635action
    = ( ^ [T: itself_lock_action] :
          ( typerep2 @ ( literal2 @ $false @ $true @ $true @ $false @ $false @ $false @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $false @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $false @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ nil_typerep ) ) )).

% typerep_lock_action_def
thf(fact_32_prod__induct3,axiom,(
    ! [P: produc8763457246119570046nteger > $o,X: produc8763457246119570046nteger] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C: code_integer] :
          ( P @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C ) ) )
     => ( P @ X ) ) )).

% prod_induct3
thf(fact_33_prod__induct3,axiom,(
    ! [P: produc989692990947075319atural > $o,X: produc989692990947075319atural] :
      ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: code_natural,C: code_natural] :
          ( P @ ( produc3831813291587773865atural @ A3 @ ( produc3574140220909816553atural @ B3 @ C ) ) )
     => ( P @ X ) ) )).

% prod_induct3
thf(fact_34_prod__induct3,axiom,(
    ! [P: produc1908205239877642774nteger > $o,X: produc1908205239877642774nteger] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C: code_integer] :
          ( P @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C ) ) )
     => ( P @ X ) ) )).

% prod_induct3
thf(fact_35_prod__induct3,axiom,(
    ! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C: int] :
          ( P @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C ) ) )
     => ( P @ X ) ) )).

% prod_induct3
thf(fact_36_prod__induct3,axiom,(
    ! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B3: int,C: int] :
          ( P @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C ) ) )
     => ( P @ X ) ) )).

% prod_induct3
thf(fact_37_old_Oprod_Oinducts,axiom,(
    ! [P: produc8763457246119570046nteger > $o,Prod: produc8763457246119570046nteger] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( P @ ( produc6137756002093451184nteger @ A3 @ B3 ) )
     => ( P @ Prod ) ) )).

% old.prod.inducts
thf(fact_38_old_Oprod_Oinducts,axiom,(
    ! [P: produc989692990947075319atural > $o,Prod: produc989692990947075319atural] :
      ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural] :
          ( P @ ( produc3831813291587773865atural @ A3 @ B3 ) )
     => ( P @ Prod ) ) )).

% old.prod.inducts
thf(fact_39_old_Oprod_Oinducts,axiom,(
    ! [P: produc1908205239877642774nteger > $o,Prod: produc1908205239877642774nteger] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( P @ ( produc8603105652947943368nteger @ A3 @ B3 ) )
     => ( P @ Prod ) ) )).

% old.prod.inducts
thf(fact_40_old_Oprod_Oinducts,axiom,(
    ! [P: produc2285326912895808259nt_int > $o,Prod: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( P @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
     => ( P @ Prod ) ) )).

% old.prod.inducts
thf(fact_41_old_Oprod_Oinducts,axiom,(
    ! [P: produc7773217078559923341nt_int > $o,Prod: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( P @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
     => ( P @ Prod ) ) )).

% old.prod.inducts
thf(fact_42_mem__Collect__eq,axiom,(
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) )).

% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,(
    ! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
      = ( P @ A ) ) )).

% mem_Collect_eq
thf(fact_44_lock__action_Osize_I6_J,axiom,
    ( ( size_s8978095283902022239action @ lock_Unlock )
    = zero_zero_nat )).

% lock_action.size(6)
thf(fact_45_Pair__inject,axiom,(
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A2: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( ( produc6137756002093451184nteger @ A @ B )
        = ( produc6137756002093451184nteger @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) )).

% Pair_inject
thf(fact_46_Pair__inject,axiom,(
    ! [A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural,A2: produc4972180933644002618e_term > option6357759511663192854e_term,B2: produc7822875418678951345atural] :
      ( ( ( produc3831813291587773865atural @ A @ B )
        = ( produc3831813291587773865atural @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) )).

% Pair_inject
thf(fact_47_Pair__inject,axiom,(
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B )
        = ( produc8603105652947943368nteger @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) )).

% Pair_inject
thf(fact_48_Pair__inject,axiom,(
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) )).

% Pair_inject
thf(fact_49_Pair__inject,axiom,(
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) )).

% Pair_inject
thf(fact_50_lock__action_Osize_I7_J,axiom,
    ( ( size_s8978095283902022239action @ lock_UnlockFail )
    = zero_zero_nat )).

% lock_action.size(7)
thf(fact_51_prod__cases,axiom,(
    ! [P: produc8763457246119570046nteger > $o,P2: produc8763457246119570046nteger] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( P @ ( produc6137756002093451184nteger @ A3 @ B3 ) )
     => ( P @ P2 ) ) )).

% prod_cases
thf(fact_52_prod__cases,axiom,(
    ! [P: produc989692990947075319atural > $o,P2: produc989692990947075319atural] :
      ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural] :
          ( P @ ( produc3831813291587773865atural @ A3 @ B3 ) )
     => ( P @ P2 ) ) )).

% prod_cases
thf(fact_53_prod__cases,axiom,(
    ! [P: produc1908205239877642774nteger > $o,P2: produc1908205239877642774nteger] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( P @ ( produc8603105652947943368nteger @ A3 @ B3 ) )
     => ( P @ P2 ) ) )).

% prod_cases
thf(fact_54_prod__cases,axiom,(
    ! [P: produc2285326912895808259nt_int > $o,P2: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( P @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
     => ( P @ P2 ) ) )).

% prod_cases
thf(fact_55_prod__cases,axiom,(
    ! [P: produc7773217078559923341nt_int > $o,P2: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( P @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
     => ( P @ P2 ) ) )).

% prod_cases
thf(fact_56_surj__pair,axiom,(
    ! [P2: produc8763457246119570046nteger] :
    ? [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
      ( P2
      = ( produc6137756002093451184nteger @ X3 @ Y3 ) ) )).

% surj_pair
thf(fact_57_surj__pair,axiom,(
    ! [P2: produc989692990947075319atural] :
    ? [X3: produc4972180933644002618e_term > option6357759511663192854e_term,Y3: produc7822875418678951345atural] :
      ( P2
      = ( produc3831813291587773865atural @ X3 @ Y3 ) ) )).

% surj_pair
thf(fact_58_surj__pair,axiom,(
    ! [P2: produc1908205239877642774nteger] :
    ? [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
      ( P2
      = ( produc8603105652947943368nteger @ X3 @ Y3 ) ) )).

% surj_pair
thf(fact_59_surj__pair,axiom,(
    ! [P2: produc2285326912895808259nt_int] :
    ? [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P2
      = ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) )).

% surj_pair
thf(fact_60_surj__pair,axiom,(
    ! [P2: produc7773217078559923341nt_int] :
    ? [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P2
      = ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) )).

% surj_pair
thf(fact_61_lock__action_Osize_I8_J,axiom,
    ( ( size_s8978095283902022239action @ lock_ReleaseAcquire )
    = zero_zero_nat )).

% lock_action.size(8)
thf(fact_62_ord__eq__le__trans,axiom,(
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) )).

% ord_eq_le_trans
thf(fact_63_comp__eq__dest,axiom,(
    ! [A: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,B: produc8923597787767437993list_f > produc1811169103916964701list_f,C2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,D: produc8923597787767437993list_f > produc1811169103916964701list_f,V: produc8923597787767437993list_f] :
      ( ( ( comp_P4767578765025054880list_f @ A @ B )
        = ( comp_P4767578765025054880list_f @ C2 @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C2 @ ( D @ V ) ) ) ) )).

% comp_eq_dest
thf(fact_64_comp__eq__dest__lhs,axiom,(
    ! [A: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,B: produc8923597787767437993list_f > produc1811169103916964701list_f,C2: produc8923597787767437993list_f > list_n1973683603564934665_a_b_c,V: produc8923597787767437993list_f] :
      ( ( ( comp_P4767578765025054880list_f @ A @ B )
        = C2 )
     => ( ( A @ ( B @ V ) )
        = ( C2 @ V ) ) ) )).

% comp_eq_dest_lhs
thf(fact_65_old_Oprod_Ocase,axiom,(
    ! [F: code_natural > code_natural > produc6756470014200349799atural,X1: code_natural,X2: code_natural] :
      ( ( produc8282080750456430313atural @ F @ ( produc3574140220909816553atural @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) )).

% old.prod.case
thf(fact_66_ssubst__Pair__rhs,axiom,(
    ! [R: nat,S: nat,R2: set_Pr1261947904930325089at_nat,S2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R2 ) ) ) )).

% ssubst_Pair_rhs
thf(fact_67_ssubst__Pair__rhs,axiom,(
    ! [R: product_prod_nat_nat,S: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,S2: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R @ S2 ) @ R2 ) ) ) )).

% ssubst_Pair_rhs
thf(fact_68_ssubst__Pair__rhs,axiom,(
    ! [R: code_integer > option6357759511663192854e_term,S: produc8923325533196201883nteger,R2: set_Pr8056137968301705908nteger,S2: produc8923325533196201883nteger] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R @ S2 ) @ R2 ) ) ) )).

% ssubst_Pair_rhs
thf(fact_69_ssubst__Pair__rhs,axiom,(
    ! [R: produc4972180933644002618e_term > option6357759511663192854e_term,S: produc7822875418678951345atural,R2: set_Pr4244404650639074861atural,S2: produc7822875418678951345atural] :
      ( ( member3245610415464309390atural @ ( produc3831813291587773865atural @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member3245610415464309390atural @ ( produc3831813291587773865atural @ R @ S2 ) @ R2 ) ) ) )).

% ssubst_Pair_rhs
thf(fact_70_ssubst__Pair__rhs,axiom,(
    ! [R: produc6241069584506657477e_term > option6357759511663192854e_term,S: produc8923325533196201883nteger,R2: set_Pr1281608226676607948nteger,S2: produc8923325533196201883nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R @ S2 ) @ R2 ) ) ) )).

% ssubst_Pair_rhs
thf(fact_71_ssubst__Pair__rhs,axiom,(
    ! [R: produc8551481072490612790e_term > option6357759511663192854e_term,S: product_prod_int_int,R2: set_Pr9222295170931077689nt_int,S2: product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R @ S2 ) @ R2 ) ) ) )).

% ssubst_Pair_rhs
thf(fact_72_ssubst__Pair__rhs,axiom,(
    ! [R: int > option6357759511663192854e_term,S: product_prod_int_int,R2: set_Pr1872883991513573699nt_int,S2: product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R @ S2 ) @ R2 ) ) ) )).

% ssubst_Pair_rhs
thf(fact_73_lock__action_Osize_I1_J,axiom,
    ( ( lock_s5444299122428672615action @ lock_Lock )
    = zero_zero_nat )).

% lock_action.size(1)
thf(fact_74_swap__simp,axiom,(
    ! [X: produc8923325533196201883nteger,Y: code_integer > option6357759511663192854e_term] :
      ( ( produc7161663723673358902e_term @ ( produc2242505987757549526e_term @ X @ Y ) )
      = ( produc6137756002093451184nteger @ Y @ X ) ) )).

% swap_simp
thf(fact_75_swap__simp,axiom,(
    ! [X: produc7822875418678951345atural,Y: produc4972180933644002618e_term > option6357759511663192854e_term] :
      ( ( produc4817134228084260137e_term @ ( produc3460696219379557065e_term @ X @ Y ) )
      = ( produc3831813291587773865atural @ Y @ X ) ) )).

% swap_simp
thf(fact_76_swap__simp,axiom,(
    ! [X: produc8923325533196201883nteger,Y: produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( produc3254817899252985374e_term @ ( produc1898379890548282302e_term @ X @ Y ) )
      = ( produc8603105652947943368nteger @ Y @ X ) ) )).

% swap_simp
thf(fact_77_swap__simp,axiom,(
    ! [X: product_prod_int_int,Y: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( produc1360913580328244269e_term @ ( produc5219906170222623181e_term @ X @ Y ) )
      = ( produc5700946648718959541nt_int @ Y @ X ) ) )).

% swap_simp
thf(fact_78_swap__simp,axiom,(
    ! [X: product_prod_int_int,Y: int > option6357759511663192854e_term] :
      ( ( produc2386993194027891363e_term @ ( produc7516973260301520963e_term @ X @ Y ) )
      = ( produc4305682042979456191nt_int @ Y @ X ) ) )).

% swap_simp
thf(fact_79_swap__simp,axiom,(
    ! [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( produc1833541701154484752nteger @ ( produc6137756002093451184nteger @ X @ Y ) )
      = ( produc2242505987757549526e_term @ Y @ X ) ) )).

% swap_simp
thf(fact_80_swap__simp,axiom,(
    ! [X: produc4972180933644002618e_term > option6357759511663192854e_term,Y: produc7822875418678951345atural] :
      ( ( produc5188251300292476937atural @ ( produc3831813291587773865atural @ X @ Y ) )
      = ( produc3460696219379557065e_term @ Y @ X ) ) )).

% swap_simp
thf(fact_81_swap__simp,axiom,(
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( produc736171624797870632nteger @ ( produc8603105652947943368nteger @ X @ Y ) )
      = ( produc1898379890548282302e_term @ Y @ X ) ) )).

% swap_simp
thf(fact_82_swap__simp,axiom,(
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc1841954058824580629nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
      = ( produc5219906170222623181e_term @ Y @ X ) ) )).

% swap_simp
thf(fact_83_swap__simp,axiom,(
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc8399074013560602399nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
      = ( produc7516973260301520963e_term @ Y @ X ) ) )).

% swap_simp
thf(fact_84_lock__action_Osize_I2_J,axiom,
    ( ( lock_s5444299122428672615action @ lock_Unlock )
    = zero_zero_nat )).

% lock_action.size(2)
thf(fact_85_curryI,axiom,(
    ! [F: produc8763457246119570046nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( F @ ( produc6137756002093451184nteger @ A @ B ) )
     => ( produc5457605928088077978eger_o @ F @ A @ B ) ) )).

% curryI
thf(fact_86_curryI,axiom,(
    ! [F: produc989692990947075319atural > $o,A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural] :
      ( ( F @ ( produc3831813291587773865atural @ A @ B ) )
     => ( produc6465694447285001697ural_o @ F @ A @ B ) ) )).

% curryI
thf(fact_87_curryI,axiom,(
    ! [F: produc1908205239877642774nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( F @ ( produc8603105652947943368nteger @ A @ B ) )
     => ( produc1217013493180205570eger_o @ F @ A @ B ) ) )).

% curryI
thf(fact_88_curryI,axiom,(
    ! [F: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ ( produc5700946648718959541nt_int @ A @ B ) )
     => ( produc730925184835016917_int_o @ F @ A @ B ) ) )).

% curryI
thf(fact_89_curryI,axiom,(
    ! [F: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ ( produc4305682042979456191nt_int @ A @ B ) )
     => ( produc9098658269643458507_int_o @ F @ A @ B ) ) )).

% curryI
thf(fact_90_rtranclp__induct3,axiom,(
    ! [R: produc8763457246119570046nteger > produc8763457246119570046nteger > $o,Ax: code_integer > option6357759511663192854e_term,Ay: code_integer,Az: code_integer,Bx: code_integer > option6357759511663192854e_term,By: code_integer,Bz: code_integer,P: ( code_integer > option6357759511663192854e_term ) > code_integer > code_integer > $o] :
      ( ( transi5788783962313184371nteger @ R @ ( produc6137756002093451184nteger @ Ax @ ( produc1086072967326762835nteger @ Ay @ Az ) ) @ ( produc6137756002093451184nteger @ Bx @ ( produc1086072967326762835nteger @ By @ Bz ) ) )
     => ( ( P @ Ax @ Ay @ Az )
       => ( ! [A3: code_integer > option6357759511663192854e_term,Aa: code_integer,B3: code_integer,Ab: code_integer > option6357759511663192854e_term,Ac: code_integer,Ba: code_integer] :
              ( ( transi5788783962313184371nteger @ R @ ( produc6137756002093451184nteger @ Ax @ ( produc1086072967326762835nteger @ Ay @ Az ) ) @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ Aa @ B3 ) ) )
             => ( ( R @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ Aa @ B3 ) ) @ ( produc6137756002093451184nteger @ Ab @ ( produc1086072967326762835nteger @ Ac @ Ba ) ) )
               => ( ( P @ A3 @ Aa @ B3 )
                 => ( P @ Ab @ Ac @ Ba ) ) ) )
         => ( P @ Bx @ By @ Bz ) ) ) ) )).

% rtranclp_induct3
thf(fact_91_rtranclp__induct3,axiom,(
    ! [R: produc989692990947075319atural > produc989692990947075319atural > $o,Ax: produc4972180933644002618e_term > option6357759511663192854e_term,Ay: code_natural,Az: code_natural,Bx: produc4972180933644002618e_term > option6357759511663192854e_term,By: code_natural,Bz: code_natural,P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > code_natural > code_natural > $o] :
      ( ( transi4816290781924639084atural @ R @ ( produc3831813291587773865atural @ Ax @ ( produc3574140220909816553atural @ Ay @ Az ) ) @ ( produc3831813291587773865atural @ Bx @ ( produc3574140220909816553atural @ By @ Bz ) ) )
     => ( ( P @ Ax @ Ay @ Az )
       => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,Aa: code_natural,B3: code_natural,Ab: produc4972180933644002618e_term > option6357759511663192854e_term,Ac: code_natural,Ba: code_natural] :
              ( ( transi4816290781924639084atural @ R @ ( produc3831813291587773865atural @ Ax @ ( produc3574140220909816553atural @ Ay @ Az ) ) @ ( produc3831813291587773865atural @ A3 @ ( produc3574140220909816553atural @ Aa @ B3 ) ) )
             => ( ( R @ ( produc3831813291587773865atural @ A3 @ ( produc3574140220909816553atural @ Aa @ B3 ) ) @ ( produc3831813291587773865atural @ Ab @ ( produc3574140220909816553atural @ Ac @ Ba ) ) )
               => ( ( P @ A3 @ Aa @ B3 )
                 => ( P @ Ab @ Ac @ Ba ) ) ) )
         => ( P @ Bx @ By @ Bz ) ) ) ) )).

% rtranclp_induct3
thf(fact_92_rtranclp__induct3,axiom,(
    ! [R: produc1908205239877642774nteger > produc1908205239877642774nteger > $o,Ax: produc6241069584506657477e_term > option6357759511663192854e_term,Ay: code_integer,Az: code_integer,Bx: produc6241069584506657477e_term > option6357759511663192854e_term,By: code_integer,Bz: code_integer,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > code_integer > code_integer > $o] :
      ( ( transi5734803030855206539nteger @ R @ ( produc8603105652947943368nteger @ Ax @ ( produc1086072967326762835nteger @ Ay @ Az ) ) @ ( produc8603105652947943368nteger @ Bx @ ( produc1086072967326762835nteger @ By @ Bz ) ) )
     => ( ( P @ Ax @ Ay @ Az )
       => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,Aa: code_integer,B3: code_integer,Ab: produc6241069584506657477e_term > option6357759511663192854e_term,Ac: code_integer,Ba: code_integer] :
              ( ( transi5734803030855206539nteger @ R @ ( produc8603105652947943368nteger @ Ax @ ( produc1086072967326762835nteger @ Ay @ Az ) ) @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ Aa @ B3 ) ) )
             => ( ( R @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ Aa @ B3 ) ) @ ( produc8603105652947943368nteger @ Ab @ ( produc1086072967326762835nteger @ Ac @ Ba ) ) )
               => ( ( P @ A3 @ Aa @ B3 )
                 => ( P @ Ab @ Ac @ Ba ) ) ) )
         => ( P @ Bx @ By @ Bz ) ) ) ) )).

% rtranclp_induct3
thf(fact_93_rtranclp__induct3,axiom,(
    ! [R: produc2285326912895808259nt_int > produc2285326912895808259nt_int > $o,Ax: produc8551481072490612790e_term > option6357759511663192854e_term,Ay: int,Az: int,Bx: produc8551481072490612790e_term > option6357759511663192854e_term,By: int,Bz: int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > int > int > $o] :
      ( ( transi1065937927385353080nt_int @ R @ ( produc5700946648718959541nt_int @ Ax @ ( product_Pair_int_int @ Ay @ Az ) ) @ ( produc5700946648718959541nt_int @ Bx @ ( product_Pair_int_int @ By @ Bz ) ) )
     => ( ( P @ Ax @ Ay @ Az )
       => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,Aa: int,B3: int,Ab: produc8551481072490612790e_term > option6357759511663192854e_term,Ac: int,Ba: int] :
              ( ( transi1065937927385353080nt_int @ R @ ( produc5700946648718959541nt_int @ Ax @ ( product_Pair_int_int @ Ay @ Az ) ) @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ Aa @ B3 ) ) )
             => ( ( R @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ Aa @ B3 ) ) @ ( produc5700946648718959541nt_int @ Ab @ ( product_Pair_int_int @ Ac @ Ba ) ) )
               => ( ( P @ A3 @ Aa @ B3 )
                 => ( P @ Ab @ Ac @ Ba ) ) ) )
         => ( P @ Bx @ By @ Bz ) ) ) ) )).

% rtranclp_induct3
thf(fact_94_rtranclp__induct3,axiom,(
    ! [R: produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o,Ax: int > option6357759511663192854e_term,Ay: int,Az: int,Bx: int > option6357759511663192854e_term,By: int,Bz: int,P: ( int > option6357759511663192854e_term ) > int > int > $o] :
      ( ( transi5629603145743041154nt_int @ R @ ( produc4305682042979456191nt_int @ Ax @ ( product_Pair_int_int @ Ay @ Az ) ) @ ( produc4305682042979456191nt_int @ Bx @ ( product_Pair_int_int @ By @ Bz ) ) )
     => ( ( P @ Ax @ Ay @ Az )
       => ( ! [A3: int > option6357759511663192854e_term,Aa: int,B3: int,Ab: int > option6357759511663192854e_term,Ac: int,Ba: int] :
              ( ( transi5629603145743041154nt_int @ R @ ( produc4305682042979456191nt_int @ Ax @ ( product_Pair_int_int @ Ay @ Az ) ) @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ Aa @ B3 ) ) )
             => ( ( R @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ Aa @ B3 ) ) @ ( produc4305682042979456191nt_int @ Ab @ ( product_Pair_int_int @ Ac @ Ba ) ) )
               => ( ( P @ A3 @ Aa @ B3 )
                 => ( P @ Ab @ Ac @ Ba ) ) ) )
         => ( P @ Bx @ By @ Bz ) ) ) ) )).

% rtranclp_induct3
thf(fact_95_lock__action_Osize_I3_J,axiom,
    ( ( lock_s5444299122428672615action @ lock_UnlockFail )
    = zero_zero_nat )).

% lock_action.size(3)
thf(fact_96_tranclp__induct3,axiom,(
    ! [R: produc8763457246119570046nteger > produc8763457246119570046nteger > $o,Ax: code_integer > option6357759511663192854e_term,Ay: code_integer,Az: code_integer,Bx: code_integer > option6357759511663192854e_term,By: code_integer,Bz: code_integer,P: ( code_integer > option6357759511663192854e_term ) > code_integer > code_integer > $o] :
      ( ( transi5760031498222123537nteger @ R @ ( produc6137756002093451184nteger @ Ax @ ( produc1086072967326762835nteger @ Ay @ Az ) ) @ ( produc6137756002093451184nteger @ Bx @ ( produc1086072967326762835nteger @ By @ Bz ) ) )
     => ( ! [A3: code_integer > option6357759511663192854e_term,Aa: code_integer,B3: code_integer] :
            ( ( R @ ( produc6137756002093451184nteger @ Ax @ ( produc1086072967326762835nteger @ Ay @ Az ) ) @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ Aa @ B3 ) ) )
           => ( P @ A3 @ Aa @ B3 ) )
       => ( ! [A3: code_integer > option6357759511663192854e_term,Aa: code_integer,B3: code_integer,Ab: code_integer > option6357759511663192854e_term,Ac: code_integer,Ba: code_integer] :
              ( ( transi5760031498222123537nteger @ R @ ( produc6137756002093451184nteger @ Ax @ ( produc1086072967326762835nteger @ Ay @ Az ) ) @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ Aa @ B3 ) ) )
             => ( ( R @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ Aa @ B3 ) ) @ ( produc6137756002093451184nteger @ Ab @ ( produc1086072967326762835nteger @ Ac @ Ba ) ) )
               => ( ( P @ A3 @ Aa @ B3 )
                 => ( P @ Ab @ Ac @ Ba ) ) ) )
         => ( P @ Bx @ By @ Bz ) ) ) ) )).

% tranclp_induct3
thf(fact_97_tranclp__induct3,axiom,(
    ! [R: produc989692990947075319atural > produc989692990947075319atural > $o,Ax: produc4972180933644002618e_term > option6357759511663192854e_term,Ay: code_natural,Az: code_natural,Bx: produc4972180933644002618e_term > option6357759511663192854e_term,By: code_natural,Bz: code_natural,P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > code_natural > code_natural > $o] :
      ( ( transi370792155324499466atural @ R @ ( produc3831813291587773865atural @ Ax @ ( produc3574140220909816553atural @ Ay @ Az ) ) @ ( produc3831813291587773865atural @ Bx @ ( produc3574140220909816553atural @ By @ Bz ) ) )
     => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,Aa: code_natural,B3: code_natural] :
            ( ( R @ ( produc3831813291587773865atural @ Ax @ ( produc3574140220909816553atural @ Ay @ Az ) ) @ ( produc3831813291587773865atural @ A3 @ ( produc3574140220909816553atural @ Aa @ B3 ) ) )
           => ( P @ A3 @ Aa @ B3 ) )
       => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,Aa: code_natural,B3: code_natural,Ab: produc4972180933644002618e_term > option6357759511663192854e_term,Ac: code_natural,Ba: code_natural] :
              ( ( transi370792155324499466atural @ R @ ( produc3831813291587773865atural @ Ax @ ( produc3574140220909816553atural @ Ay @ Az ) ) @ ( produc3831813291587773865atural @ A3 @ ( produc3574140220909816553atural @ Aa @ B3 ) ) )
             => ( ( R @ ( produc3831813291587773865atural @ A3 @ ( produc3574140220909816553atural @ Aa @ B3 ) ) @ ( produc3831813291587773865atural @ Ab @ ( produc3574140220909816553atural @ Ac @ Ba ) ) )
               => ( ( P @ A3 @ Aa @ B3 )
                 => ( P @ Ab @ Ac @ Ba ) ) ) )
         => ( P @ Bx @ By @ Bz ) ) ) ) )).

% tranclp_induct3
thf(fact_98_tranclp__induct3,axiom,(
    ! [R: produc1908205239877642774nteger > produc1908205239877642774nteger > $o,Ax: produc6241069584506657477e_term > option6357759511663192854e_term,Ay: code_integer,Az: code_integer,Bx: produc6241069584506657477e_term > option6357759511663192854e_term,By: code_integer,Bz: code_integer,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > code_integer > code_integer > $o] :
      ( ( transi1289304404255066921nteger @ R @ ( produc8603105652947943368nteger @ Ax @ ( produc1086072967326762835nteger @ Ay @ Az ) ) @ ( produc8603105652947943368nteger @ Bx @ ( produc1086072967326762835nteger @ By @ Bz ) ) )
     => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,Aa: code_integer,B3: code_integer] :
            ( ( R @ ( produc8603105652947943368nteger @ Ax @ ( produc1086072967326762835nteger @ Ay @ Az ) ) @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ Aa @ B3 ) ) )
           => ( P @ A3 @ Aa @ B3 ) )
       => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,Aa: code_integer,B3: code_integer,Ab: produc6241069584506657477e_term > option6357759511663192854e_term,Ac: code_integer,Ba: code_integer] :
              ( ( transi1289304404255066921nteger @ R @ ( produc8603105652947943368nteger @ Ax @ ( produc1086072967326762835nteger @ Ay @ Az ) ) @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ Aa @ B3 ) ) )
             => ( ( R @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ Aa @ B3 ) ) @ ( produc8603105652947943368nteger @ Ab @ ( produc1086072967326762835nteger @ Ac @ Ba ) ) )
               => ( ( P @ A3 @ Aa @ B3 )
                 => ( P @ Ab @ Ac @ Ba ) ) ) )
         => ( P @ Bx @ By @ Bz ) ) ) ) )).

% tranclp_induct3
thf(fact_99_tranclp__induct3,axiom,(
    ! [R: produc2285326912895808259nt_int > produc2285326912895808259nt_int > $o,Ax: produc8551481072490612790e_term > option6357759511663192854e_term,Ay: int,Az: int,Bx: produc8551481072490612790e_term > option6357759511663192854e_term,By: int,Bz: int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > int > int > $o] :
      ( ( transi5292154729749739542nt_int @ R @ ( produc5700946648718959541nt_int @ Ax @ ( product_Pair_int_int @ Ay @ Az ) ) @ ( produc5700946648718959541nt_int @ Bx @ ( product_Pair_int_int @ By @ Bz ) ) )
     => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,Aa: int,B3: int] :
            ( ( R @ ( produc5700946648718959541nt_int @ Ax @ ( product_Pair_int_int @ Ay @ Az ) ) @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ Aa @ B3 ) ) )
           => ( P @ A3 @ Aa @ B3 ) )
       => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,Aa: int,B3: int,Ab: produc8551481072490612790e_term > option6357759511663192854e_term,Ac: int,Ba: int] :
              ( ( transi5292154729749739542nt_int @ R @ ( produc5700946648718959541nt_int @ Ax @ ( product_Pair_int_int @ Ay @ Az ) ) @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ Aa @ B3 ) ) )
             => ( ( R @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ Aa @ B3 ) ) @ ( produc5700946648718959541nt_int @ Ab @ ( product_Pair_int_int @ Ac @ Ba ) ) )
               => ( ( P @ A3 @ Aa @ B3 )
                 => ( P @ Ab @ Ac @ Ba ) ) ) )
         => ( P @ Bx @ By @ Bz ) ) ) ) )).

% tranclp_induct3
thf(fact_100_tranclp__induct3,axiom,(
    ! [R: produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o,Ax: int > option6357759511663192854e_term,Ay: int,Az: int,Bx: int > option6357759511663192854e_term,By: int,Bz: int,P: ( int > option6357759511663192854e_term ) > int > int > $o] :
      ( ( transi7773869560681156640nt_int @ R @ ( produc4305682042979456191nt_int @ Ax @ ( product_Pair_int_int @ Ay @ Az ) ) @ ( produc4305682042979456191nt_int @ Bx @ ( product_Pair_int_int @ By @ Bz ) ) )
     => ( ! [A3: int > option6357759511663192854e_term,Aa: int,B3: int] :
            ( ( R @ ( produc4305682042979456191nt_int @ Ax @ ( product_Pair_int_int @ Ay @ Az ) ) @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ Aa @ B3 ) ) )
           => ( P @ A3 @ Aa @ B3 ) )
       => ( ! [A3: int > option6357759511663192854e_term,Aa: int,B3: int,Ab: int > option6357759511663192854e_term,Ac: int,Ba: int] :
              ( ( transi7773869560681156640nt_int @ R @ ( produc4305682042979456191nt_int @ Ax @ ( product_Pair_int_int @ Ay @ Az ) ) @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ Aa @ B3 ) ) )
             => ( ( R @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ Aa @ B3 ) ) @ ( produc4305682042979456191nt_int @ Ab @ ( product_Pair_int_int @ Ac @ Ba ) ) )
               => ( ( P @ A3 @ Aa @ B3 )
                 => ( P @ Ab @ Ac @ Ba ) ) ) )
         => ( P @ Bx @ By @ Bz ) ) ) ) )).

% tranclp_induct3
thf(fact_101_apsnd__conv,axiom,(
    ! [F: produc8923325533196201883nteger > produc8923325533196201883nteger,X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( produc1584323072462815568e_term @ F @ ( produc6137756002093451184nteger @ X @ Y ) )
      = ( produc6137756002093451184nteger @ X @ ( F @ Y ) ) ) )).

% apsnd_conv
thf(fact_102_apsnd__conv,axiom,(
    ! [F: produc7822875418678951345atural > produc7822875418678951345atural,X: produc4972180933644002618e_term > option6357759511663192854e_term,Y: produc7822875418678951345atural] :
      ( ( produc2824789715873180845e_term @ F @ ( produc3831813291587773865atural @ X @ Y ) )
      = ( produc3831813291587773865atural @ X @ ( F @ Y ) ) ) )).

% apsnd_conv
thf(fact_103_apsnd__conv,axiom,(
    ! [F: produc8923325533196201883nteger > produc8923325533196201883nteger,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( produc717174245515237944e_term @ F @ ( produc8603105652947943368nteger @ X @ Y ) )
      = ( produc8603105652947943368nteger @ X @ ( F @ Y ) ) ) )).

% apsnd_conv
thf(fact_104_apsnd__conv,axiom,(
    ! [F: product_prod_int_int > product_prod_int_int,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc4973430039190721449e_term @ F @ ( produc5700946648718959541nt_int @ X @ Y ) )
      = ( produc5700946648718959541nt_int @ X @ ( F @ Y ) ) ) )).

% apsnd_conv
thf(fact_105_apsnd__conv,axiom,(
    ! [F: product_prod_int_int > product_prod_int_int,X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc7011684061226097695e_term @ F @ ( produc4305682042979456191nt_int @ X @ Y ) )
      = ( produc4305682042979456191nt_int @ X @ ( F @ Y ) ) ) )).

% apsnd_conv
thf(fact_106_lock__action_Osize_I4_J,axiom,
    ( ( lock_s5444299122428672615action @ lock_ReleaseAcquire )
    = zero_zero_nat )).

% lock_action.size(4)
thf(fact_107_comp__apply,axiom,
    ( comp_P4767578765025054880list_f
    = ( ^ [F2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G: produc8923597787767437993list_f > produc1811169103916964701list_f,X4: produc8923597787767437993list_f] :
          ( F2 @ ( G @ X4 ) ) ) )).

% comp_apply
thf(fact_108_fst__eqD,axiom,(
    ! [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger,A: code_integer > option6357759511663192854e_term] :
      ( ( ( produc2702340305111172292nteger @ ( produc6137756002093451184nteger @ X @ Y ) )
        = A )
     => ( X = A ) ) )).

% fst_eqD
thf(fact_109_fst__eqD,axiom,(
    ! [X: produc4972180933644002618e_term > option6357759511663192854e_term,Y: produc7822875418678951345atural,A: produc4972180933644002618e_term > option6357759511663192854e_term] :
      ( ( ( produc3051390257598303421atural @ ( produc3831813291587773865atural @ X @ Y ) )
        = A )
     => ( X = A ) ) )).

% fst_eqD
thf(fact_110_fst__eqD,axiom,(
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,A: produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ X @ Y ) )
        = A )
     => ( X = A ) ) )).

% fst_eqD
thf(fact_111_fst__eqD,axiom,(
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,A: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
        = A )
     => ( X = A ) ) )).

% fst_eqD
thf(fact_112_fst__eqD,axiom,(
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,A: int > option6357759511663192854e_term] :
      ( ( ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
        = A )
     => ( X = A ) ) )).

% fst_eqD
thf(fact_113_diag__imp__eq__le,axiom,(
    ! [A4: set_Pr1261947904930325089at_nat,R2: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A4 )
         => ( R2 @ X3 @ X3 ) )
     => ! [X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ A4 )
         => ( ( member8440522571783428010at_nat @ Y4 @ A4 )
           => ( ( X5 = Y4 )
             => ( R2 @ X5 @ Y4 ) ) ) ) ) )).

% diag_imp_eq_le
thf(fact_114_diag__imp__eq__le,axiom,(
    ! [A4: set_Pr8693737435421807431at_nat,R2: produc859450856879609959at_nat > produc859450856879609959at_nat > $o] :
      ( ! [X3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A4 )
         => ( R2 @ X3 @ X3 ) )
     => ! [X5: produc859450856879609959at_nat,Y4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X5 @ A4 )
         => ( ( member8206827879206165904at_nat @ Y4 @ A4 )
           => ( ( X5 = Y4 )
             => ( R2 @ X5 @ Y4 ) ) ) ) ) )).

% diag_imp_eq_le
thf(fact_115_fst__apsnd,axiom,(
    ! [F: produc8923325533196201883nteger > produc8923325533196201883nteger,X: produc8763457246119570046nteger] :
      ( ( produc2702340305111172292nteger @ ( produc1584323072462815568e_term @ F @ X ) )
      = ( produc2702340305111172292nteger @ X ) ) )).

% fst_apsnd
thf(fact_116_fst__apsnd,axiom,(
    ! [F: produc7822875418678951345atural > produc7822875418678951345atural,X: produc989692990947075319atural] :
      ( ( produc3051390257598303421atural @ ( produc2824789715873180845e_term @ F @ X ) )
      = ( produc3051390257598303421atural @ X ) ) )).

% fst_apsnd
thf(fact_117_fst__apsnd,axiom,(
    ! [F: produc8923325533196201883nteger > produc8923325533196201883nteger,X: produc1908205239877642774nteger] :
      ( ( produc7822682618958472924nteger @ ( produc717174245515237944e_term @ F @ X ) )
      = ( produc7822682618958472924nteger @ X ) ) )).

% fst_apsnd
thf(fact_118_fst__apsnd,axiom,(
    ! [F: product_prod_int_int > product_prod_int_int,X: produc2285326912895808259nt_int] :
      ( ( produc6816164490631068361nt_int @ ( produc4973430039190721449e_term @ F @ X ) )
      = ( produc6816164490631068361nt_int @ X ) ) )).

% fst_apsnd
thf(fact_119_fst__apsnd,axiom,(
    ! [F: product_prod_int_int > product_prod_int_int,X: produc7773217078559923341nt_int] :
      ( ( produc6230002227079971283nt_int @ ( produc7011684061226097695e_term @ F @ X ) )
      = ( produc6230002227079971283nt_int @ X ) ) )).

% fst_apsnd
thf(fact_120_converse__tranclp__induct2,axiom,(
    ! [R: produc8763457246119570046nteger > produc8763457246119570046nteger > $o,Ax: code_integer > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: code_integer > option6357759511663192854e_term,By: produc8923325533196201883nteger,P: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( transi5760031498222123537nteger @ R @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ Bx @ By ) )
     => ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
            ( ( R @ ( produc6137756002093451184nteger @ A3 @ B3 ) @ ( produc6137756002093451184nteger @ Bx @ By ) )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: code_integer > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( R @ ( produc6137756002093451184nteger @ A3 @ B3 ) @ ( produc6137756002093451184nteger @ Aa @ Ba ) )
             => ( ( transi5760031498222123537nteger @ R @ ( produc6137756002093451184nteger @ Aa @ Ba ) @ ( produc6137756002093451184nteger @ Bx @ By ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_tranclp_induct2
thf(fact_121_converse__tranclp__induct2,axiom,(
    ! [R: produc989692990947075319atural > produc989692990947075319atural > $o,Ax: produc4972180933644002618e_term > option6357759511663192854e_term,Ay: produc7822875418678951345atural,Bx: produc4972180933644002618e_term > option6357759511663192854e_term,By: produc7822875418678951345atural,P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o] :
      ( ( transi370792155324499466atural @ R @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ Bx @ By ) )
     => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural] :
            ( ( R @ ( produc3831813291587773865atural @ A3 @ B3 ) @ ( produc3831813291587773865atural @ Bx @ By ) )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural,Aa: produc4972180933644002618e_term > option6357759511663192854e_term,Ba: produc7822875418678951345atural] :
              ( ( R @ ( produc3831813291587773865atural @ A3 @ B3 ) @ ( produc3831813291587773865atural @ Aa @ Ba ) )
             => ( ( transi370792155324499466atural @ R @ ( produc3831813291587773865atural @ Aa @ Ba ) @ ( produc3831813291587773865atural @ Bx @ By ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_tranclp_induct2
thf(fact_122_converse__tranclp__induct2,axiom,(
    ! [R: produc1908205239877642774nteger > produc1908205239877642774nteger > $o,Ax: produc6241069584506657477e_term > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: produc6241069584506657477e_term > option6357759511663192854e_term,By: produc8923325533196201883nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( transi1289304404255066921nteger @ R @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ Bx @ By ) )
     => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
            ( ( R @ ( produc8603105652947943368nteger @ A3 @ B3 ) @ ( produc8603105652947943368nteger @ Bx @ By ) )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: produc6241069584506657477e_term > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( R @ ( produc8603105652947943368nteger @ A3 @ B3 ) @ ( produc8603105652947943368nteger @ Aa @ Ba ) )
             => ( ( transi1289304404255066921nteger @ R @ ( produc8603105652947943368nteger @ Aa @ Ba ) @ ( produc8603105652947943368nteger @ Bx @ By ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_tranclp_induct2
thf(fact_123_converse__tranclp__induct2,axiom,(
    ! [R: produc2285326912895808259nt_int > produc2285326912895808259nt_int > $o,Ax: produc8551481072490612790e_term > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: produc8551481072490612790e_term > option6357759511663192854e_term,By: product_prod_int_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( transi5292154729749739542nt_int @ R @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ Bx @ By ) )
     => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
            ( ( R @ ( produc5700946648718959541nt_int @ A3 @ B3 ) @ ( produc5700946648718959541nt_int @ Bx @ By ) )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int,Aa: produc8551481072490612790e_term > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( R @ ( produc5700946648718959541nt_int @ A3 @ B3 ) @ ( produc5700946648718959541nt_int @ Aa @ Ba ) )
             => ( ( transi5292154729749739542nt_int @ R @ ( produc5700946648718959541nt_int @ Aa @ Ba ) @ ( produc5700946648718959541nt_int @ Bx @ By ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_tranclp_induct2
thf(fact_124_converse__tranclp__induct2,axiom,(
    ! [R: produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o,Ax: int > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: int > option6357759511663192854e_term,By: product_prod_int_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( transi7773869560681156640nt_int @ R @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ Bx @ By ) )
     => ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
            ( ( R @ ( produc4305682042979456191nt_int @ A3 @ B3 ) @ ( produc4305682042979456191nt_int @ Bx @ By ) )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int,Aa: int > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( R @ ( produc4305682042979456191nt_int @ A3 @ B3 ) @ ( produc4305682042979456191nt_int @ Aa @ Ba ) )
             => ( ( transi7773869560681156640nt_int @ R @ ( produc4305682042979456191nt_int @ Aa @ Ba ) @ ( produc4305682042979456191nt_int @ Bx @ By ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_tranclp_induct2
thf(fact_125_order__less__irrefl,axiom,(
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) )).

% order_less_irrefl
thf(fact_126_curryD,axiom,(
    ! [F: produc8763457246119570046nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc5457605928088077978eger_o @ F @ A @ B )
     => ( F @ ( produc6137756002093451184nteger @ A @ B ) ) ) )).

% curryD
thf(fact_127_curryD,axiom,(
    ! [F: produc989692990947075319atural > $o,A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural] :
      ( ( produc6465694447285001697ural_o @ F @ A @ B )
     => ( F @ ( produc3831813291587773865atural @ A @ B ) ) ) )).

% curryD
thf(fact_128_curryD,axiom,(
    ! [F: produc1908205239877642774nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc1217013493180205570eger_o @ F @ A @ B )
     => ( F @ ( produc8603105652947943368nteger @ A @ B ) ) ) )).

% curryD
thf(fact_129_curryD,axiom,(
    ! [F: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc730925184835016917_int_o @ F @ A @ B )
     => ( F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) )).

% curryD
thf(fact_130_curryD,axiom,(
    ! [F: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc9098658269643458507_int_o @ F @ A @ B )
     => ( F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) )).

% curryD
thf(fact_131_comp__def,axiom,
    ( comp_P4767578765025054880list_f
    = ( ^ [F2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G: produc8923597787767437993list_f > produc1811169103916964701list_f,X4: produc8923597787767437993list_f] :
          ( F2 @ ( G @ X4 ) ) ) )).

% comp_def
thf(fact_132_curryE,axiom,(
    ! [F: produc8763457246119570046nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc5457605928088077978eger_o @ F @ A @ B )
     => ( F @ ( produc6137756002093451184nteger @ A @ B ) ) ) )).

% curryE
thf(fact_133_curryE,axiom,(
    ! [F: produc989692990947075319atural > $o,A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural] :
      ( ( produc6465694447285001697ural_o @ F @ A @ B )
     => ( F @ ( produc3831813291587773865atural @ A @ B ) ) ) )).

% curryE
thf(fact_134_curryE,axiom,(
    ! [F: produc1908205239877642774nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc1217013493180205570eger_o @ F @ A @ B )
     => ( F @ ( produc8603105652947943368nteger @ A @ B ) ) ) )).

% curryE
thf(fact_135_curryE,axiom,(
    ! [F: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc730925184835016917_int_o @ F @ A @ B )
     => ( F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) )).

% curryE
thf(fact_136_curryE,axiom,(
    ! [F: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc9098658269643458507_int_o @ F @ A @ B )
     => ( F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) )).

% curryE
thf(fact_137_snd__conv,axiom,(
    ! [X1: code_integer > option6357759511663192854e_term,X2: produc8923325533196201883nteger] :
      ( ( produc6514156286822040066nteger @ ( produc6137756002093451184nteger @ X1 @ X2 ) )
      = X2 ) )).

% snd_conv
thf(fact_138_snd__conv,axiom,(
    ! [X1: produc4972180933644002618e_term > option6357759511663192854e_term,X2: produc7822875418678951345atural] :
      ( ( produc3085575039554877691atural @ ( produc3831813291587773865atural @ X1 @ X2 ) )
      = X2 ) )).

% snd_conv
thf(fact_139_snd__conv,axiom,(
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X2: produc8923325533196201883nteger] :
      ( ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ X1 @ X2 ) )
      = X2 ) )).

% snd_conv
thf(fact_140_snd__conv,axiom,(
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int] :
      ( ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X1 @ X2 ) )
      = X2 ) )).

% snd_conv
thf(fact_141_snd__conv,axiom,(
    ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int] :
      ( ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X1 @ X2 ) )
      = X2 ) )).

% snd_conv
thf(fact_142_snd__conv,axiom,(
    ! [X1: code_natural,X2: produc7822875418678951345atural] :
      ( ( produc6591912806276919810atural @ ( produc6639722614265839536atural @ X1 @ X2 ) )
      = X2 ) )).

% snd_conv
thf(fact_143_snd__conv,axiom,(
    ! [X1: finfun1145668972854894571action,X2: produc1811169103916964701list_f] :
      ( ( produc8710481132536983823list_f @ ( produc8331056147679645153list_f @ X1 @ X2 ) )
      = X2 ) )).

% snd_conv
thf(fact_144_lock__action_Odistinct_I1_J,axiom,(
    lock_Lock != lock_Unlock )).

% lock_action.distinct(1)
thf(fact_145_antisym,axiom,(
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) )).

% antisym
thf(fact_146_Collect__mem__eq,axiom,(
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( member8440522571783428010at_nat @ X4 @ A4 ) )
      = A4 ) )).

% Collect_mem_eq
thf(fact_147_Collect__mem__eq,axiom,(
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] :
            ( member8206827879206165904at_nat @ X4 @ A4 ) )
      = A4 ) )).

% Collect_mem_eq
thf(fact_148_lock__action_Odistinct_I10_J,axiom,(
    lock_ReleaseAcquire != lock_Unlock )).

% lock_action.distinct(10)
thf(fact_149_internal__case__prod__def,axiom,(
    produc6882689234361595027atural = produc8282080750456430313atural )).

% internal_case_prod_def
thf(fact_150_tranclp__induct2,axiom,(
    ! [R: produc8763457246119570046nteger > produc8763457246119570046nteger > $o,Ax: code_integer > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: code_integer > option6357759511663192854e_term,By: produc8923325533196201883nteger,P: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( transi5760031498222123537nteger @ R @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ Bx @ By ) )
     => ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
            ( ( R @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ A3 @ B3 ) )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: code_integer > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( transi5760031498222123537nteger @ R @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ A3 @ B3 ) )
             => ( ( R @ ( produc6137756002093451184nteger @ A3 @ B3 ) @ ( produc6137756002093451184nteger @ Aa @ Ba ) )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% tranclp_induct2
thf(fact_151_tranclp__induct2,axiom,(
    ! [R: produc989692990947075319atural > produc989692990947075319atural > $o,Ax: produc4972180933644002618e_term > option6357759511663192854e_term,Ay: produc7822875418678951345atural,Bx: produc4972180933644002618e_term > option6357759511663192854e_term,By: produc7822875418678951345atural,P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o] :
      ( ( transi370792155324499466atural @ R @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ Bx @ By ) )
     => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural] :
            ( ( R @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ A3 @ B3 ) )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural,Aa: produc4972180933644002618e_term > option6357759511663192854e_term,Ba: produc7822875418678951345atural] :
              ( ( transi370792155324499466atural @ R @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ A3 @ B3 ) )
             => ( ( R @ ( produc3831813291587773865atural @ A3 @ B3 ) @ ( produc3831813291587773865atural @ Aa @ Ba ) )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% tranclp_induct2
thf(fact_152_tranclp__induct2,axiom,(
    ! [R: produc1908205239877642774nteger > produc1908205239877642774nteger > $o,Ax: produc6241069584506657477e_term > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: produc6241069584506657477e_term > option6357759511663192854e_term,By: produc8923325533196201883nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( transi1289304404255066921nteger @ R @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ Bx @ By ) )
     => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
            ( ( R @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ A3 @ B3 ) )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: produc6241069584506657477e_term > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( transi1289304404255066921nteger @ R @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ A3 @ B3 ) )
             => ( ( R @ ( produc8603105652947943368nteger @ A3 @ B3 ) @ ( produc8603105652947943368nteger @ Aa @ Ba ) )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% tranclp_induct2
thf(fact_153_tranclp__induct2,axiom,(
    ! [R: produc2285326912895808259nt_int > produc2285326912895808259nt_int > $o,Ax: produc8551481072490612790e_term > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: produc8551481072490612790e_term > option6357759511663192854e_term,By: product_prod_int_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( transi5292154729749739542nt_int @ R @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ Bx @ By ) )
     => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
            ( ( R @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int,Aa: produc8551481072490612790e_term > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( transi5292154729749739542nt_int @ R @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
             => ( ( R @ ( produc5700946648718959541nt_int @ A3 @ B3 ) @ ( produc5700946648718959541nt_int @ Aa @ Ba ) )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% tranclp_induct2
thf(fact_154_tranclp__induct2,axiom,(
    ! [R: produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o,Ax: int > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: int > option6357759511663192854e_term,By: product_prod_int_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( transi7773869560681156640nt_int @ R @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ Bx @ By ) )
     => ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
            ( ( R @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int,Aa: int > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( transi7773869560681156640nt_int @ R @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
             => ( ( R @ ( produc4305682042979456191nt_int @ A3 @ B3 ) @ ( produc4305682042979456191nt_int @ Aa @ Ba ) )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% tranclp_induct2
thf(fact_155_rtranclp__induct2,axiom,(
    ! [R: produc8763457246119570046nteger > produc8763457246119570046nteger > $o,Ax: code_integer > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: code_integer > option6357759511663192854e_term,By: produc8923325533196201883nteger,P: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( transi5788783962313184371nteger @ R @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ Bx @ By ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: code_integer > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( transi5788783962313184371nteger @ R @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ A3 @ B3 ) )
             => ( ( R @ ( produc6137756002093451184nteger @ A3 @ B3 ) @ ( produc6137756002093451184nteger @ Aa @ Ba ) )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% rtranclp_induct2
thf(fact_156_rtranclp__induct2,axiom,(
    ! [R: produc989692990947075319atural > produc989692990947075319atural > $o,Ax: produc4972180933644002618e_term > option6357759511663192854e_term,Ay: produc7822875418678951345atural,Bx: produc4972180933644002618e_term > option6357759511663192854e_term,By: produc7822875418678951345atural,P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o] :
      ( ( transi4816290781924639084atural @ R @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ Bx @ By ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural,Aa: produc4972180933644002618e_term > option6357759511663192854e_term,Ba: produc7822875418678951345atural] :
              ( ( transi4816290781924639084atural @ R @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ A3 @ B3 ) )
             => ( ( R @ ( produc3831813291587773865atural @ A3 @ B3 ) @ ( produc3831813291587773865atural @ Aa @ Ba ) )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% rtranclp_induct2
thf(fact_157_rtranclp__induct2,axiom,(
    ! [R: produc1908205239877642774nteger > produc1908205239877642774nteger > $o,Ax: produc6241069584506657477e_term > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: produc6241069584506657477e_term > option6357759511663192854e_term,By: produc8923325533196201883nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( transi5734803030855206539nteger @ R @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ Bx @ By ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: produc6241069584506657477e_term > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( transi5734803030855206539nteger @ R @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ A3 @ B3 ) )
             => ( ( R @ ( produc8603105652947943368nteger @ A3 @ B3 ) @ ( produc8603105652947943368nteger @ Aa @ Ba ) )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% rtranclp_induct2
thf(fact_158_rtranclp__induct2,axiom,(
    ! [R: produc2285326912895808259nt_int > produc2285326912895808259nt_int > $o,Ax: produc8551481072490612790e_term > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: produc8551481072490612790e_term > option6357759511663192854e_term,By: product_prod_int_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( transi1065937927385353080nt_int @ R @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ Bx @ By ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int,Aa: produc8551481072490612790e_term > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( transi1065937927385353080nt_int @ R @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
             => ( ( R @ ( produc5700946648718959541nt_int @ A3 @ B3 ) @ ( produc5700946648718959541nt_int @ Aa @ Ba ) )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% rtranclp_induct2
thf(fact_159_rtranclp__induct2,axiom,(
    ! [R: produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o,Ax: int > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: int > option6357759511663192854e_term,By: product_prod_int_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( transi5629603145743041154nt_int @ R @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ Bx @ By ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int,Aa: int > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( transi5629603145743041154nt_int @ R @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
             => ( ( R @ ( produc4305682042979456191nt_int @ A3 @ B3 ) @ ( produc4305682042979456191nt_int @ Aa @ Ba ) )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% rtranclp_induct2
thf(fact_160_case__prod__conv,axiom,(
    ! [F: code_natural > code_natural > produc6756470014200349799atural,A: code_natural,B: code_natural] :
      ( ( produc8282080750456430313atural @ F @ ( produc3574140220909816553atural @ A @ B ) )
      = ( F @ A @ B ) ) )).

% case_prod_conv
thf(fact_161_lock__action_Odistinct_I11_J,axiom,(
    lock_UnlockFail != lock_ReleaseAcquire )).

% lock_action.distinct(11)
thf(fact_162_lock__action_Odistinct_I12_J,axiom,(
    lock_ReleaseAcquire != lock_UnlockFail )).

% lock_action.distinct(12)
thf(fact_163_lock__action_Odistinct_I2_J,axiom,(
    lock_Unlock != lock_Lock )).

% lock_action.distinct(2)
thf(fact_164_converse__rtranclpE2,axiom,(
    ! [R: produc8763457246119570046nteger > produc8763457246119570046nteger > $o,Xa: code_integer > option6357759511663192854e_term,Xb: produc8923325533196201883nteger,Za: code_integer > option6357759511663192854e_term,Zb: produc8923325533196201883nteger] :
      ( ( transi5788783962313184371nteger @ R @ ( produc6137756002093451184nteger @ Xa @ Xb ) @ ( produc6137756002093451184nteger @ Za @ Zb ) )
     => ( ( ( produc6137756002093451184nteger @ Xa @ Xb )
         != ( produc6137756002093451184nteger @ Za @ Zb ) )
       => ~ ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
              ( ( R @ ( produc6137756002093451184nteger @ Xa @ Xb ) @ ( produc6137756002093451184nteger @ A3 @ B3 ) )
             => ~ ( transi5788783962313184371nteger @ R @ ( produc6137756002093451184nteger @ A3 @ B3 ) @ ( produc6137756002093451184nteger @ Za @ Zb ) ) ) ) ) )).

% converse_rtranclpE2
thf(fact_165_converse__rtranclpE2,axiom,(
    ! [R: produc989692990947075319atural > produc989692990947075319atural > $o,Xa: produc4972180933644002618e_term > option6357759511663192854e_term,Xb: produc7822875418678951345atural,Za: produc4972180933644002618e_term > option6357759511663192854e_term,Zb: produc7822875418678951345atural] :
      ( ( transi4816290781924639084atural @ R @ ( produc3831813291587773865atural @ Xa @ Xb ) @ ( produc3831813291587773865atural @ Za @ Zb ) )
     => ( ( ( produc3831813291587773865atural @ Xa @ Xb )
         != ( produc3831813291587773865atural @ Za @ Zb ) )
       => ~ ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural] :
              ( ( R @ ( produc3831813291587773865atural @ Xa @ Xb ) @ ( produc3831813291587773865atural @ A3 @ B3 ) )
             => ~ ( transi4816290781924639084atural @ R @ ( produc3831813291587773865atural @ A3 @ B3 ) @ ( produc3831813291587773865atural @ Za @ Zb ) ) ) ) ) )).

% converse_rtranclpE2
thf(fact_166_converse__rtranclpE2,axiom,(
    ! [R: produc1908205239877642774nteger > produc1908205239877642774nteger > $o,Xa: produc6241069584506657477e_term > option6357759511663192854e_term,Xb: produc8923325533196201883nteger,Za: produc6241069584506657477e_term > option6357759511663192854e_term,Zb: produc8923325533196201883nteger] :
      ( ( transi5734803030855206539nteger @ R @ ( produc8603105652947943368nteger @ Xa @ Xb ) @ ( produc8603105652947943368nteger @ Za @ Zb ) )
     => ( ( ( produc8603105652947943368nteger @ Xa @ Xb )
         != ( produc8603105652947943368nteger @ Za @ Zb ) )
       => ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
              ( ( R @ ( produc8603105652947943368nteger @ Xa @ Xb ) @ ( produc8603105652947943368nteger @ A3 @ B3 ) )
             => ~ ( transi5734803030855206539nteger @ R @ ( produc8603105652947943368nteger @ A3 @ B3 ) @ ( produc8603105652947943368nteger @ Za @ Zb ) ) ) ) ) )).

% converse_rtranclpE2
thf(fact_167_converse__rtranclpE2,axiom,(
    ! [R: produc2285326912895808259nt_int > produc2285326912895808259nt_int > $o,Xa: produc8551481072490612790e_term > option6357759511663192854e_term,Xb: product_prod_int_int,Za: produc8551481072490612790e_term > option6357759511663192854e_term,Zb: product_prod_int_int] :
      ( ( transi1065937927385353080nt_int @ R @ ( produc5700946648718959541nt_int @ Xa @ Xb ) @ ( produc5700946648718959541nt_int @ Za @ Zb ) )
     => ( ( ( produc5700946648718959541nt_int @ Xa @ Xb )
         != ( produc5700946648718959541nt_int @ Za @ Zb ) )
       => ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
              ( ( R @ ( produc5700946648718959541nt_int @ Xa @ Xb ) @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
             => ~ ( transi1065937927385353080nt_int @ R @ ( produc5700946648718959541nt_int @ A3 @ B3 ) @ ( produc5700946648718959541nt_int @ Za @ Zb ) ) ) ) ) )).

% converse_rtranclpE2
thf(fact_168_converse__rtranclpE2,axiom,(
    ! [R: produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o,Xa: int > option6357759511663192854e_term,Xb: product_prod_int_int,Za: int > option6357759511663192854e_term,Zb: product_prod_int_int] :
      ( ( transi5629603145743041154nt_int @ R @ ( produc4305682042979456191nt_int @ Xa @ Xb ) @ ( produc4305682042979456191nt_int @ Za @ Zb ) )
     => ( ( ( produc4305682042979456191nt_int @ Xa @ Xb )
         != ( produc4305682042979456191nt_int @ Za @ Zb ) )
       => ~ ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
              ( ( R @ ( produc4305682042979456191nt_int @ Xa @ Xb ) @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
             => ~ ( transi5629603145743041154nt_int @ R @ ( produc4305682042979456191nt_int @ A3 @ B3 ) @ ( produc4305682042979456191nt_int @ Za @ Zb ) ) ) ) ) )).

% converse_rtranclpE2
thf(fact_169_image__eqI,axiom,(
    ! [B: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member8440522571783428010at_nat @ X @ A4 )
       => ( member8440522571783428010at_nat @ B @ ( image_5168914502847457605at_nat @ F @ A4 ) ) ) ) )).

% image_eqI
thf(fact_170_image__eqI,axiom,(
    ! [B: produc859450856879609959at_nat,F: product_prod_nat_nat > produc859450856879609959at_nat,X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member8440522571783428010at_nat @ X @ A4 )
       => ( member8206827879206165904at_nat @ B @ ( image_1119892238107900587at_nat @ F @ A4 ) ) ) ) )).

% image_eqI
thf(fact_171_image__eqI,axiom,(
    ! [B: product_prod_nat_nat,F: produc859450856879609959at_nat > product_prod_nat_nat,X: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member8206827879206165904at_nat @ X @ A4 )
       => ( member8440522571783428010at_nat @ B @ ( image_6706873702294942047at_nat @ F @ A4 ) ) ) ) )).

% image_eqI
thf(fact_172_image__eqI,axiom,(
    ! [B: produc859450856879609959at_nat,F: produc859450856879609959at_nat > produc859450856879609959at_nat,X: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member8206827879206165904at_nat @ X @ A4 )
       => ( member8206827879206165904at_nat @ B @ ( image_6321802852832981957at_nat @ F @ A4 ) ) ) ) )).

% image_eqI
thf(fact_173_lock__action_Odistinct_I3_J,axiom,(
    lock_Lock != lock_UnlockFail )).

% lock_action.distinct(3)
thf(fact_174_converse__rtranclp__induct2,axiom,(
    ! [R: produc8763457246119570046nteger > produc8763457246119570046nteger > $o,Ax: code_integer > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: code_integer > option6357759511663192854e_term,By: produc8923325533196201883nteger,P: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( transi5788783962313184371nteger @ R @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ Bx @ By ) )
     => ( ( P @ Bx @ By )
       => ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: code_integer > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( R @ ( produc6137756002093451184nteger @ A3 @ B3 ) @ ( produc6137756002093451184nteger @ Aa @ Ba ) )
             => ( ( transi5788783962313184371nteger @ R @ ( produc6137756002093451184nteger @ Aa @ Ba ) @ ( produc6137756002093451184nteger @ Bx @ By ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_rtranclp_induct2
thf(fact_175_converse__rtranclp__induct2,axiom,(
    ! [R: produc989692990947075319atural > produc989692990947075319atural > $o,Ax: produc4972180933644002618e_term > option6357759511663192854e_term,Ay: produc7822875418678951345atural,Bx: produc4972180933644002618e_term > option6357759511663192854e_term,By: produc7822875418678951345atural,P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o] :
      ( ( transi4816290781924639084atural @ R @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ Bx @ By ) )
     => ( ( P @ Bx @ By )
       => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural,Aa: produc4972180933644002618e_term > option6357759511663192854e_term,Ba: produc7822875418678951345atural] :
              ( ( R @ ( produc3831813291587773865atural @ A3 @ B3 ) @ ( produc3831813291587773865atural @ Aa @ Ba ) )
             => ( ( transi4816290781924639084atural @ R @ ( produc3831813291587773865atural @ Aa @ Ba ) @ ( produc3831813291587773865atural @ Bx @ By ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_rtranclp_induct2
thf(fact_176_converse__rtranclp__induct2,axiom,(
    ! [R: produc1908205239877642774nteger > produc1908205239877642774nteger > $o,Ax: produc6241069584506657477e_term > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: produc6241069584506657477e_term > option6357759511663192854e_term,By: produc8923325533196201883nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( transi5734803030855206539nteger @ R @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ Bx @ By ) )
     => ( ( P @ Bx @ By )
       => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: produc6241069584506657477e_term > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( R @ ( produc8603105652947943368nteger @ A3 @ B3 ) @ ( produc8603105652947943368nteger @ Aa @ Ba ) )
             => ( ( transi5734803030855206539nteger @ R @ ( produc8603105652947943368nteger @ Aa @ Ba ) @ ( produc8603105652947943368nteger @ Bx @ By ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_rtranclp_induct2
thf(fact_177_converse__rtranclp__induct2,axiom,(
    ! [R: produc2285326912895808259nt_int > produc2285326912895808259nt_int > $o,Ax: produc8551481072490612790e_term > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: produc8551481072490612790e_term > option6357759511663192854e_term,By: product_prod_int_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( transi1065937927385353080nt_int @ R @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ Bx @ By ) )
     => ( ( P @ Bx @ By )
       => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int,Aa: produc8551481072490612790e_term > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( R @ ( produc5700946648718959541nt_int @ A3 @ B3 ) @ ( produc5700946648718959541nt_int @ Aa @ Ba ) )
             => ( ( transi1065937927385353080nt_int @ R @ ( produc5700946648718959541nt_int @ Aa @ Ba ) @ ( produc5700946648718959541nt_int @ Bx @ By ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_rtranclp_induct2
thf(fact_178_converse__rtranclp__induct2,axiom,(
    ! [R: produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o,Ax: int > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: int > option6357759511663192854e_term,By: product_prod_int_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( transi5629603145743041154nt_int @ R @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ Bx @ By ) )
     => ( ( P @ Bx @ By )
       => ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int,Aa: int > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( R @ ( produc4305682042979456191nt_int @ A3 @ B3 ) @ ( produc4305682042979456191nt_int @ Aa @ Ba ) )
             => ( ( transi5629603145743041154nt_int @ R @ ( produc4305682042979456191nt_int @ Aa @ Ba ) @ ( produc4305682042979456191nt_int @ Bx @ By ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_rtranclp_induct2
thf(fact_179_lock__action_Odistinct_I4_J,axiom,(
    lock_UnlockFail != lock_Lock )).

% lock_action.distinct(4)
thf(fact_180_lock__action_Odistinct_I5_J,axiom,(
    lock_Lock != lock_ReleaseAcquire )).

% lock_action.distinct(5)
thf(fact_181_lock__action_Odistinct_I6_J,axiom,(
    lock_ReleaseAcquire != lock_Lock )).

% lock_action.distinct(6)
thf(fact_182_prod_Ocollapse,axiom,(
    ! [Prod: produc8763457246119570046nteger] :
      ( ( produc6137756002093451184nteger @ ( produc2702340305111172292nteger @ Prod ) @ ( produc6514156286822040066nteger @ Prod ) )
      = Prod ) )).

% prod.collapse
thf(fact_183_prod_Ocollapse,axiom,(
    ! [Prod: produc989692990947075319atural] :
      ( ( produc3831813291587773865atural @ ( produc3051390257598303421atural @ Prod ) @ ( produc3085575039554877691atural @ Prod ) )
      = Prod ) )).

% prod.collapse
thf(fact_184_prod_Ocollapse,axiom,(
    ! [Prod: produc1908205239877642774nteger] :
      ( ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ Prod ) @ ( produc7856867400915047194nteger @ Prod ) )
      = Prod ) )).

% prod.collapse
thf(fact_185_prod_Ocollapse,axiom,(
    ! [Prod: produc2285326912895808259nt_int] :
      ( ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ Prod ) @ ( produc7328097813583171335nt_int @ Prod ) )
      = Prod ) )).

% prod.collapse
thf(fact_186_prod_Ocollapse,axiom,(
    ! [Prod: produc7773217078559923341nt_int] :
      ( ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ Prod ) @ ( produc3162348030201620241nt_int @ Prod ) )
      = Prod ) )).

% prod.collapse
thf(fact_187_prod_Ocollapse,axiom,(
    ! [Prod: produc5835291356934675326atural] :
      ( ( produc6639722614265839536atural @ ( produc497848011034438852atural @ Prod ) @ ( produc6591912806276919810atural @ Prod ) )
      = Prod ) )).

% prod.collapse
thf(fact_188_prod_Ocollapse,axiom,(
    ! [Prod: produc8923597787767437993list_f] :
      ( ( produc8331056147679645153list_f @ ( produc3089225533250117581list_f @ Prod ) @ ( produc8710481132536983823list_f @ Prod ) )
      = Prod ) )).

% prod.collapse
thf(fact_189_bspec,axiom,(
    ! [A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A4 )
         => ( P @ X3 ) )
     => ( ( member8440522571783428010at_nat @ X @ A4 )
       => ( P @ X ) ) ) )).

% bspec
thf(fact_190_bspec,axiom,(
    ! [A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
      ( ! [X3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A4 )
         => ( P @ X3 ) )
     => ( ( member8206827879206165904at_nat @ X @ A4 )
       => ( P @ X ) ) ) )).

% bspec
thf(fact_191_lock__action_Odistinct_I7_J,axiom,(
    lock_Unlock != lock_UnlockFail )).

% lock_action.distinct(7)
thf(fact_192_lock__action_Odistinct_I8_J,axiom,(
    lock_UnlockFail != lock_Unlock )).

% lock_action.distinct(8)
thf(fact_193_lock__action_Odistinct_I9_J,axiom,(
    lock_Unlock != lock_ReleaseAcquire )).

% lock_action.distinct(9)
thf(fact_194_imageE,axiom,(
    ! [B: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_5168914502847457605at_nat @ F @ A4 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member8440522571783428010at_nat @ X3 @ A4 ) ) ) )).

% imageE
thf(fact_195_imageE,axiom,(
    ! [B: product_prod_nat_nat,F: produc859450856879609959at_nat > product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_6706873702294942047at_nat @ F @ A4 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member8206827879206165904at_nat @ X3 @ A4 ) ) ) )).

% imageE
thf(fact_196_imageE,axiom,(
    ! [B: produc859450856879609959at_nat,F: product_prod_nat_nat > produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ B @ ( image_1119892238107900587at_nat @ F @ A4 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member8440522571783428010at_nat @ X3 @ A4 ) ) ) )).

% imageE
thf(fact_197_imageE,axiom,(
    ! [B: produc859450856879609959at_nat,F: produc859450856879609959at_nat > produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ B @ ( image_6321802852832981957at_nat @ F @ A4 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member8206827879206165904at_nat @ X3 @ A4 ) ) ) )).

% imageE
thf(fact_198_eq__fst__iff,axiom,(
    ! [A: code_integer > option6357759511663192854e_term,P2: produc8763457246119570046nteger] :
      ( ( A
        = ( produc2702340305111172292nteger @ P2 ) )
      = ( ? [B4: produc8923325533196201883nteger] :
            ( P2
            = ( produc6137756002093451184nteger @ A @ B4 ) ) ) ) )).

% eq_fst_iff
thf(fact_199_eq__fst__iff,axiom,(
    ! [A: produc4972180933644002618e_term > option6357759511663192854e_term,P2: produc989692990947075319atural] :
      ( ( A
        = ( produc3051390257598303421atural @ P2 ) )
      = ( ? [B4: produc7822875418678951345atural] :
            ( P2
            = ( produc3831813291587773865atural @ A @ B4 ) ) ) ) )).

% eq_fst_iff
thf(fact_200_eq__fst__iff,axiom,(
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,P2: produc1908205239877642774nteger] :
      ( ( A
        = ( produc7822682618958472924nteger @ P2 ) )
      = ( ? [B4: produc8923325533196201883nteger] :
            ( P2
            = ( produc8603105652947943368nteger @ A @ B4 ) ) ) ) )).

% eq_fst_iff
thf(fact_201_eq__fst__iff,axiom,(
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,P2: produc2285326912895808259nt_int] :
      ( ( A
        = ( produc6816164490631068361nt_int @ P2 ) )
      = ( ? [B4: product_prod_int_int] :
            ( P2
            = ( produc5700946648718959541nt_int @ A @ B4 ) ) ) ) )).

% eq_fst_iff
thf(fact_202_eq__fst__iff,axiom,(
    ! [A: int > option6357759511663192854e_term,P2: produc7773217078559923341nt_int] :
      ( ( A
        = ( produc6230002227079971283nt_int @ P2 ) )
      = ( ? [B4: product_prod_int_int] :
            ( P2
            = ( produc4305682042979456191nt_int @ A @ B4 ) ) ) ) )).

% eq_fst_iff
thf(fact_203_fstI,axiom,(
    ! [X: produc8763457246119570046nteger,Y: code_integer > option6357759511663192854e_term,Z: produc8923325533196201883nteger] :
      ( ( X
        = ( produc6137756002093451184nteger @ Y @ Z ) )
     => ( ( produc2702340305111172292nteger @ X )
        = Y ) ) )).

% fstI
thf(fact_204_fstI,axiom,(
    ! [X: produc989692990947075319atural,Y: produc4972180933644002618e_term > option6357759511663192854e_term,Z: produc7822875418678951345atural] :
      ( ( X
        = ( produc3831813291587773865atural @ Y @ Z ) )
     => ( ( produc3051390257598303421atural @ X )
        = Y ) ) )).

% fstI
thf(fact_205_fstI,axiom,(
    ! [X: produc1908205239877642774nteger,Y: produc6241069584506657477e_term > option6357759511663192854e_term,Z: produc8923325533196201883nteger] :
      ( ( X
        = ( produc8603105652947943368nteger @ Y @ Z ) )
     => ( ( produc7822682618958472924nteger @ X )
        = Y ) ) )).

% fstI
thf(fact_206_fstI,axiom,(
    ! [X: produc2285326912895808259nt_int,Y: produc8551481072490612790e_term > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc5700946648718959541nt_int @ Y @ Z ) )
     => ( ( produc6816164490631068361nt_int @ X )
        = Y ) ) )).

% fstI
thf(fact_207_fstI,axiom,(
    ! [X: produc7773217078559923341nt_int,Y: int > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc4305682042979456191nt_int @ Y @ Z ) )
     => ( ( produc6230002227079971283nt_int @ X )
        = Y ) ) )).

% fstI
thf(fact_208_subsetI,axiom,(
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A4 )
         => ( member8440522571783428010at_nat @ X3 @ B5 ) )
     => ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) )).

% subsetI
thf(fact_209_subsetI,axiom,(
    ! [A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ! [X3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A4 )
         => ( member8206827879206165904at_nat @ X3 @ B5 ) )
     => ( ord_le3000389064537975527at_nat @ A4 @ B5 ) ) )).

% subsetI
thf(fact_210_lock__action_Oexhaust,axiom,(
    ! [Y: lock_action] :
      ( ( Y != lock_Lock )
     => ( ( Y != lock_Unlock )
       => ( ( Y != lock_UnlockFail )
         => ( Y = lock_ReleaseAcquire ) ) ) ) )).

% lock_action.exhaust
thf(fact_211_lock__action_Oinduct,axiom,(
    ! [P: lock_action > $o,Lock_action: lock_action] :
      ( ( P @ lock_Lock )
     => ( ( P @ lock_Unlock )
       => ( ( P @ lock_UnlockFail )
         => ( ( P @ lock_ReleaseAcquire )
           => ( P @ Lock_action ) ) ) ) ) )).

% lock_action.induct
thf(fact_212_case__prod__beta,axiom,
    ( produc8282080750456430313atural
    = ( ^ [F2: code_natural > code_natural > produc6756470014200349799atural,P3: produc7822875418678951345atural] :
          ( F2 @ ( produc1773691148792264405atural @ P3 ) @ ( produc8662200840462671639atural @ P3 ) ) ) )).

% case_prod_beta
thf(fact_213_prod_Ocase__eq__if,axiom,
    ( produc8282080750456430313atural
    = ( ^ [F2: code_natural > code_natural > produc6756470014200349799atural,Prod2: produc7822875418678951345atural] :
          ( F2 @ ( produc1773691148792264405atural @ Prod2 ) @ ( produc8662200840462671639atural @ Prod2 ) ) ) )).

% prod.case_eq_if
thf(fact_214_sndI,axiom,(
    ! [X: produc8763457246119570046nteger,Y: code_integer > option6357759511663192854e_term,Z: produc8923325533196201883nteger] :
      ( ( X
        = ( produc6137756002093451184nteger @ Y @ Z ) )
     => ( ( produc6514156286822040066nteger @ X )
        = Z ) ) )).

% sndI
thf(fact_215_sndI,axiom,(
    ! [X: produc989692990947075319atural,Y: produc4972180933644002618e_term > option6357759511663192854e_term,Z: produc7822875418678951345atural] :
      ( ( X
        = ( produc3831813291587773865atural @ Y @ Z ) )
     => ( ( produc3085575039554877691atural @ X )
        = Z ) ) )).

% sndI
thf(fact_216_sndI,axiom,(
    ! [X: produc1908205239877642774nteger,Y: produc6241069584506657477e_term > option6357759511663192854e_term,Z: produc8923325533196201883nteger] :
      ( ( X
        = ( produc8603105652947943368nteger @ Y @ Z ) )
     => ( ( produc7856867400915047194nteger @ X )
        = Z ) ) )).

% sndI
thf(fact_217_sndI,axiom,(
    ! [X: produc2285326912895808259nt_int,Y: produc8551481072490612790e_term > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc5700946648718959541nt_int @ Y @ Z ) )
     => ( ( produc7328097813583171335nt_int @ X )
        = Z ) ) )).

% sndI
thf(fact_218_sndI,axiom,(
    ! [X: produc7773217078559923341nt_int,Y: int > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc4305682042979456191nt_int @ Y @ Z ) )
     => ( ( produc3162348030201620241nt_int @ X )
        = Z ) ) )).

% sndI
thf(fact_219_sndI,axiom,(
    ! [X: produc5835291356934675326atural,Y: code_natural,Z: produc7822875418678951345atural] :
      ( ( X
        = ( produc6639722614265839536atural @ Y @ Z ) )
     => ( ( produc6591912806276919810atural @ X )
        = Z ) ) )).

% sndI
thf(fact_220_sndI,axiom,(
    ! [X: produc8923597787767437993list_f,Y: finfun1145668972854894571action,Z: produc1811169103916964701list_f] :
      ( ( X
        = ( produc8331056147679645153list_f @ Y @ Z ) )
     => ( ( produc8710481132536983823list_f @ X )
        = Z ) ) )).

% sndI
thf(fact_221_list_Odistinct_I1_J,axiom,(
    ! [X21: typerep,X22: list_typerep] :
      ( nil_typerep
     != ( cons_typerep @ X21 @ X22 ) ) )).

% list.distinct(1)
thf(fact_222_prod_Oswap__def,axiom,
    ( produc850777735424141840atural
    = ( ^ [P3: produc5835291356934675326atural] :
          ( produc1272419604612767874atural @ ( produc6591912806276919810atural @ P3 ) @ ( produc497848011034438852atural @ P3 ) ) ) )).

% prod.swap_def
thf(fact_223_prod_Oswap__def,axiom,
    ( produc7161663723673358902e_term
    = ( ^ [P3: produc1017113440723519068e_term] :
          ( produc6137756002093451184nteger @ ( produc2618906272486138408e_term @ P3 ) @ ( produc8030462327630046442e_term @ P3 ) ) ) )).

% prod.swap_def
thf(fact_224_prod_Oswap__def,axiom,
    ( produc2386993194027891363e_term
    = ( ^ [P3: produc3832618445565238089e_term] :
          ( produc4305682042979456191nt_int @ ( produc6373639247523685013e_term @ P3 ) @ ( produc217921407547260247e_term @ P3 ) ) ) )).

% prod.swap_def
thf(fact_225_prod_Oswap__def,axiom,
    ( produc1833541701154484752nteger
    = ( ^ [P3: produc8763457246119570046nteger] :
          ( produc2242505987757549526e_term @ ( produc6514156286822040066nteger @ P3 ) @ ( produc2702340305111172292nteger @ P3 ) ) ) )).

% prod.swap_def
thf(fact_226_prod_Oswap__def,axiom,
    ( produc8399074013560602399nt_int
    = ( ^ [P3: produc7773217078559923341nt_int] :
          ( produc7516973260301520963e_term @ ( produc3162348030201620241nt_int @ P3 ) @ ( produc6230002227079971283nt_int @ P3 ) ) ) )).

% prod.swap_def
thf(fact_227_prod_Oswap__def,axiom,
    ( produc4817134228084260137e_term
    = ( ^ [P3: produc8392782284284844111e_term] :
          ( produc3831813291587773865atural @ ( produc2714457967346660891e_term @ P3 ) @ ( produc2680273185390086621e_term @ P3 ) ) ) )).

% prod.swap_def
thf(fact_228_prod_Oswap__def,axiom,
    ( produc3254817899252985374e_term
    = ( ^ [P3: produc998928222018024388e_term] :
          ( produc8603105652947943368nteger @ ( produc1152141638515386128e_term @ P3 ) @ ( produc1117956856558811858e_term @ P3 ) ) ) )).

% prod.swap_def
thf(fact_229_prod_Oswap__def,axiom,
    ( produc1360913580328244269e_term
    = ( ^ [P3: produc1821397396195235155e_term] :
          ( produc5700946648718959541nt_int @ ( produc6847057335086834975e_term @ P3 ) @ ( produc6335124012134732001e_term @ P3 ) ) ) )).

% prod.swap_def
thf(fact_230_prod_Oswap__def,axiom,
    ( produc5188251300292476937atural
    = ( ^ [P3: produc989692990947075319atural] :
          ( produc3460696219379557065e_term @ ( produc3085575039554877691atural @ P3 ) @ ( produc3051390257598303421atural @ P3 ) ) ) )).

% prod.swap_def
thf(fact_231_prod_Oswap__def,axiom,
    ( produc736171624797870632nteger
    = ( ^ [P3: produc1908205239877642774nteger] :
          ( produc1898379890548282302e_term @ ( produc7856867400915047194nteger @ P3 ) @ ( produc7822682618958472924nteger @ P3 ) ) ) )).

% prod.swap_def
thf(fact_232_eq__snd__iff,axiom,(
    ! [B: produc8923325533196201883nteger,P2: produc8763457246119570046nteger] :
      ( ( B
        = ( produc6514156286822040066nteger @ P2 ) )
      = ( ? [A5: code_integer > option6357759511663192854e_term] :
            ( P2
            = ( produc6137756002093451184nteger @ A5 @ B ) ) ) ) )).

% eq_snd_iff
thf(fact_233_eq__snd__iff,axiom,(
    ! [B: produc7822875418678951345atural,P2: produc989692990947075319atural] :
      ( ( B
        = ( produc3085575039554877691atural @ P2 ) )
      = ( ? [A5: produc4972180933644002618e_term > option6357759511663192854e_term] :
            ( P2
            = ( produc3831813291587773865atural @ A5 @ B ) ) ) ) )).

% eq_snd_iff
thf(fact_234_eq__snd__iff,axiom,(
    ! [B: produc8923325533196201883nteger,P2: produc1908205239877642774nteger] :
      ( ( B
        = ( produc7856867400915047194nteger @ P2 ) )
      = ( ? [A5: produc6241069584506657477e_term > option6357759511663192854e_term] :
            ( P2
            = ( produc8603105652947943368nteger @ A5 @ B ) ) ) ) )).

% eq_snd_iff
thf(fact_235_eq__snd__iff,axiom,(
    ! [B: product_prod_int_int,P2: produc2285326912895808259nt_int] :
      ( ( B
        = ( produc7328097813583171335nt_int @ P2 ) )
      = ( ? [A5: produc8551481072490612790e_term > option6357759511663192854e_term] :
            ( P2
            = ( produc5700946648718959541nt_int @ A5 @ B ) ) ) ) )).

% eq_snd_iff
thf(fact_236_eq__snd__iff,axiom,(
    ! [B: product_prod_int_int,P2: produc7773217078559923341nt_int] :
      ( ( B
        = ( produc3162348030201620241nt_int @ P2 ) )
      = ( ? [A5: int > option6357759511663192854e_term] :
            ( P2
            = ( produc4305682042979456191nt_int @ A5 @ B ) ) ) ) )).

% eq_snd_iff
thf(fact_237_eq__snd__iff,axiom,(
    ! [B: produc7822875418678951345atural,P2: produc5835291356934675326atural] :
      ( ( B
        = ( produc6591912806276919810atural @ P2 ) )
      = ( ? [A5: code_natural] :
            ( P2
            = ( produc6639722614265839536atural @ A5 @ B ) ) ) ) )).

% eq_snd_iff
thf(fact_238_eq__snd__iff,axiom,(
    ! [B: produc1811169103916964701list_f,P2: produc8923597787767437993list_f] :
      ( ( B
        = ( produc8710481132536983823list_f @ P2 ) )
      = ( ? [A5: finfun1145668972854894571action] :
            ( P2
            = ( produc8331056147679645153list_f @ A5 @ B ) ) ) ) )).

% eq_snd_iff
thf(fact_239_snd__swap,axiom,(
    ! [X: produc8763457246119570046nteger] :
      ( ( produc2618906272486138408e_term @ ( produc1833541701154484752nteger @ X ) )
      = ( produc2702340305111172292nteger @ X ) ) )).

% snd_swap
thf(fact_240_snd__swap,axiom,(
    ! [X: produc989692990947075319atural] :
      ( ( produc2714457967346660891e_term @ ( produc5188251300292476937atural @ X ) )
      = ( produc3051390257598303421atural @ X ) ) )).

% snd_swap
thf(fact_241_snd__swap,axiom,(
    ! [X: produc1908205239877642774nteger] :
      ( ( produc1152141638515386128e_term @ ( produc736171624797870632nteger @ X ) )
      = ( produc7822682618958472924nteger @ X ) ) )).

% snd_swap
thf(fact_242_snd__swap,axiom,(
    ! [X: produc2285326912895808259nt_int] :
      ( ( produc6847057335086834975e_term @ ( produc1841954058824580629nt_int @ X ) )
      = ( produc6816164490631068361nt_int @ X ) ) )).

% snd_swap
thf(fact_243_snd__swap,axiom,(
    ! [X: produc7773217078559923341nt_int] :
      ( ( produc6373639247523685013e_term @ ( produc8399074013560602399nt_int @ X ) )
      = ( produc6230002227079971283nt_int @ X ) ) )).

% snd_swap
thf(fact_244_snd__swap,axiom,(
    ! [X: produc3981471736659609608atural] :
      ( ( produc6591912806276919810atural @ ( produc4706846762625845986atural @ X ) )
      = ( produc4353917038236142998atural @ X ) ) )).

% snd_swap
thf(fact_245_snd__swap,axiom,(
    ! [X: produc682957911814224041action] :
      ( ( produc8710481132536983823list_f @ ( produc2793610610439131777action @ X ) )
      = ( produc904728346056738509action @ X ) ) )).

% snd_swap
thf(fact_246_fst__swap,axiom,(
    ! [X: produc1017113440723519068e_term] :
      ( ( produc2702340305111172292nteger @ ( produc7161663723673358902e_term @ X ) )
      = ( produc2618906272486138408e_term @ X ) ) )).

% fst_swap
thf(fact_247_fst__swap,axiom,(
    ! [X: produc8392782284284844111e_term] :
      ( ( produc3051390257598303421atural @ ( produc4817134228084260137e_term @ X ) )
      = ( produc2714457967346660891e_term @ X ) ) )).

% fst_swap
thf(fact_248_fst__swap,axiom,(
    ! [X: produc998928222018024388e_term] :
      ( ( produc7822682618958472924nteger @ ( produc3254817899252985374e_term @ X ) )
      = ( produc1152141638515386128e_term @ X ) ) )).

% fst_swap
thf(fact_249_fst__swap,axiom,(
    ! [X: produc1821397396195235155e_term] :
      ( ( produc6816164490631068361nt_int @ ( produc1360913580328244269e_term @ X ) )
      = ( produc6847057335086834975e_term @ X ) ) )).

% fst_swap
thf(fact_250_fst__swap,axiom,(
    ! [X: produc3832618445565238089e_term] :
      ( ( produc6230002227079971283nt_int @ ( produc2386993194027891363e_term @ X ) )
      = ( produc6373639247523685013e_term @ X ) ) )).

% fst_swap
thf(fact_251_fst__swap,axiom,(
    ! [X: produc5835291356934675326atural] :
      ( ( produc4353917038236142998atural @ ( produc850777735424141840atural @ X ) )
      = ( produc6591912806276919810atural @ X ) ) )).

% fst_swap
thf(fact_252_fst__swap,axiom,(
    ! [X: produc8923597787767437993list_f] :
      ( ( produc904728346056738509action @ ( produc4978107797632510849list_f @ X ) )
      = ( produc8710481132536983823list_f @ X ) ) )).

% fst_swap
thf(fact_253_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc8763457246119570046nteger,Z2: produc8763457246119570046nteger] : ( Y5 = Z2 ) )
    = ( ^ [S3: produc8763457246119570046nteger,T2: produc8763457246119570046nteger] :
          ( ( ( produc2702340305111172292nteger @ S3 )
            = ( produc2702340305111172292nteger @ T2 ) )
          & ( ( produc6514156286822040066nteger @ S3 )
            = ( produc6514156286822040066nteger @ T2 ) ) ) ) )).

% prod_eq_iff
thf(fact_254_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc989692990947075319atural,Z2: produc989692990947075319atural] : ( Y5 = Z2 ) )
    = ( ^ [S3: produc989692990947075319atural,T2: produc989692990947075319atural] :
          ( ( ( produc3051390257598303421atural @ S3 )
            = ( produc3051390257598303421atural @ T2 ) )
          & ( ( produc3085575039554877691atural @ S3 )
            = ( produc3085575039554877691atural @ T2 ) ) ) ) )).

% prod_eq_iff
thf(fact_255_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc1908205239877642774nteger,Z2: produc1908205239877642774nteger] : ( Y5 = Z2 ) )
    = ( ^ [S3: produc1908205239877642774nteger,T2: produc1908205239877642774nteger] :
          ( ( ( produc7822682618958472924nteger @ S3 )
            = ( produc7822682618958472924nteger @ T2 ) )
          & ( ( produc7856867400915047194nteger @ S3 )
            = ( produc7856867400915047194nteger @ T2 ) ) ) ) )).

% prod_eq_iff
thf(fact_256_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc2285326912895808259nt_int,Z2: produc2285326912895808259nt_int] : ( Y5 = Z2 ) )
    = ( ^ [S3: produc2285326912895808259nt_int,T2: produc2285326912895808259nt_int] :
          ( ( ( produc6816164490631068361nt_int @ S3 )
            = ( produc6816164490631068361nt_int @ T2 ) )
          & ( ( produc7328097813583171335nt_int @ S3 )
            = ( produc7328097813583171335nt_int @ T2 ) ) ) ) )).

% prod_eq_iff
thf(fact_257_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc7773217078559923341nt_int,Z2: produc7773217078559923341nt_int] : ( Y5 = Z2 ) )
    = ( ^ [S3: produc7773217078559923341nt_int,T2: produc7773217078559923341nt_int] :
          ( ( ( produc6230002227079971283nt_int @ S3 )
            = ( produc6230002227079971283nt_int @ T2 ) )
          & ( ( produc3162348030201620241nt_int @ S3 )
            = ( produc3162348030201620241nt_int @ T2 ) ) ) ) )).

% prod_eq_iff
thf(fact_258_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc5835291356934675326atural,Z2: produc5835291356934675326atural] : ( Y5 = Z2 ) )
    = ( ^ [S3: produc5835291356934675326atural,T2: produc5835291356934675326atural] :
          ( ( ( produc497848011034438852atural @ S3 )
            = ( produc497848011034438852atural @ T2 ) )
          & ( ( produc6591912806276919810atural @ S3 )
            = ( produc6591912806276919810atural @ T2 ) ) ) ) )).

% prod_eq_iff
thf(fact_259_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc8923597787767437993list_f,Z2: produc8923597787767437993list_f] : ( Y5 = Z2 ) )
    = ( ^ [S3: produc8923597787767437993list_f,T2: produc8923597787767437993list_f] :
          ( ( ( produc3089225533250117581list_f @ S3 )
            = ( produc3089225533250117581list_f @ T2 ) )
          & ( ( produc8710481132536983823list_f @ S3 )
            = ( produc8710481132536983823list_f @ T2 ) ) ) ) )).

% prod_eq_iff
thf(fact_260_snd__eqD,axiom,(
    ! [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger,A: produc8923325533196201883nteger] :
      ( ( ( produc6514156286822040066nteger @ ( produc6137756002093451184nteger @ X @ Y ) )
        = A )
     => ( Y = A ) ) )).

% snd_eqD
thf(fact_261_snd__eqD,axiom,(
    ! [X: produc4972180933644002618e_term > option6357759511663192854e_term,Y: produc7822875418678951345atural,A: produc7822875418678951345atural] :
      ( ( ( produc3085575039554877691atural @ ( produc3831813291587773865atural @ X @ Y ) )
        = A )
     => ( Y = A ) ) )).

% snd_eqD
thf(fact_262_snd__eqD,axiom,(
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,A: produc8923325533196201883nteger] :
      ( ( ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ X @ Y ) )
        = A )
     => ( Y = A ) ) )).

% snd_eqD
thf(fact_263_snd__eqD,axiom,(
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,A: product_prod_int_int] :
      ( ( ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
        = A )
     => ( Y = A ) ) )).

% snd_eqD
thf(fact_264_snd__eqD,axiom,(
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,A: product_prod_int_int] :
      ( ( ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
        = A )
     => ( Y = A ) ) )).

% snd_eqD
thf(fact_265_snd__eqD,axiom,(
    ! [X: code_natural,Y: produc7822875418678951345atural,A: produc7822875418678951345atural] :
      ( ( ( produc6591912806276919810atural @ ( produc6639722614265839536atural @ X @ Y ) )
        = A )
     => ( Y = A ) ) )).

% snd_eqD
thf(fact_266_snd__eqD,axiom,(
    ! [X: finfun1145668972854894571action,Y: produc1811169103916964701list_f,A: produc1811169103916964701list_f] :
      ( ( ( produc8710481132536983823list_f @ ( produc8331056147679645153list_f @ X @ Y ) )
        = A )
     => ( Y = A ) ) )).

% snd_eqD
thf(fact_267_pair__in__swap__image,axiom,(
    ! [Y: nat,X: nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A4 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A4 ) ) )).

% pair_in_swap_image
thf(fact_268_pair__in__swap__image,axiom,(
    ! [Y: product_prod_nat_nat,X: product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ ( image_6321802852832981957at_nat @ produc7225117575323628663at_nat @ A4 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ A4 ) ) )).

% pair_in_swap_image
thf(fact_269_pair__in__swap__image,axiom,(
    ! [Y: produc8923325533196201883nteger,X: code_integer > option6357759511663192854e_term,A4: set_Pr8056137968301705908nteger] :
      ( ( member4545690668652318835e_term @ ( produc2242505987757549526e_term @ Y @ X ) @ ( image_6949034436205712993e_term @ produc1833541701154484752nteger @ A4 ) )
      = ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ A4 ) ) )).

% pair_in_swap_image
thf(fact_270_pair__in__swap__image,axiom,(
    ! [Y: product_prod_int_int,X: int > option6357759511663192854e_term,A4: set_Pr1872883991513573699nt_int] :
      ( ( member3093737243930835296e_term @ ( produc7516973260301520963e_term @ Y @ X ) @ ( image_463623704579739357e_term @ produc8399074013560602399nt_int @ A4 ) )
      = ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ A4 ) ) )).

% pair_in_swap_image
thf(fact_271_pair__in__swap__image,axiom,(
    ! [Y: code_integer > option6357759511663192854e_term,X: produc8923325533196201883nteger,A4: set_Pr5868096553338476434e_term] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ Y @ X ) @ ( image_1021663378620967265nteger @ produc7161663723673358902e_term @ A4 ) )
      = ( member4545690668652318835e_term @ ( produc2242505987757549526e_term @ X @ Y ) @ A4 ) ) )).

% pair_in_swap_image
thf(fact_272_pair__in__swap__image,axiom,(
    ! [Y: int > option6357759511663192854e_term,X: product_prod_int_int,A4: set_Pr9020418513037773311e_term] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ Y @ X ) @ ( image_1292712855239066845nt_int @ produc2386993194027891363e_term @ A4 ) )
      = ( member3093737243930835296e_term @ ( produc7516973260301520963e_term @ X @ Y ) @ A4 ) ) )).

% pair_in_swap_image
thf(fact_273_pair__in__swap__image,axiom,(
    ! [Y: produc7822875418678951345atural,X: produc4972180933644002618e_term > option6357759511663192854e_term,A4: set_Pr4244404650639074861atural] :
      ( ( member1425327671947302374e_term @ ( produc3460696219379557065e_term @ Y @ X ) @ ( image_9140756557244149581e_term @ produc5188251300292476937atural @ A4 ) )
      = ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X @ Y ) @ A4 ) ) )).

% pair_in_swap_image
thf(fact_274_pair__in__swap__image,axiom,(
    ! [Y: produc8923325533196201883nteger,X: produc6241069584506657477e_term > option6357759511663192854e_term,A4: set_Pr1281608226676607948nteger] :
      ( ( member3254845646535258459e_term @ ( produc1898379890548282302e_term @ Y @ X ) @ ( image_610307673866561761e_term @ produc736171624797870632nteger @ A4 ) )
      = ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ A4 ) ) )).

% pair_in_swap_image
thf(fact_275_pair__in__swap__image,axiom,(
    ! [Y: product_prod_int_int,X: produc8551481072490612790e_term > option6357759511663192854e_term,A4: set_Pr9222295170931077689nt_int] :
      ( ( member7154775377335690986e_term @ ( produc5219906170222623181e_term @ Y @ X ) @ ( image_3490000813295636061e_term @ produc1841954058824580629nt_int @ A4 ) )
      = ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ A4 ) ) )).

% pair_in_swap_image
thf(fact_276_pair__in__swap__image,axiom,(
    ! [Y: produc4972180933644002618e_term > option6357759511663192854e_term,X: produc7822875418678951345atural,A4: set_Pr7460367489947184517e_term] :
      ( ( member3245610415464309390atural @ ( produc3831813291587773865atural @ Y @ X ) @ ( image_2288693727274440525atural @ produc4817134228084260137e_term @ A4 ) )
      = ( member1425327671947302374e_term @ ( produc3460696219379557065e_term @ X @ Y ) @ A4 ) ) )).

% pair_in_swap_image
thf(fact_277_fst__comp__apsnd,axiom,(
    ! [F: produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( comp_P1446425635098721907nteger @ produc2702340305111172292nteger @ ( produc1584323072462815568e_term @ F ) )
      = produc2702340305111172292nteger ) )).

% fst_comp_apsnd
thf(fact_278_fst__comp__apsnd,axiom,(
    ! [F: produc7822875418678951345atural > produc7822875418678951345atural] :
      ( ( comp_P6380929843994337986atural @ produc3051390257598303421atural @ ( produc2824789715873180845e_term @ F ) )
      = produc3051390257598303421atural ) )).

% fst_comp_apsnd
thf(fact_279_fst__comp__apsnd,axiom,(
    ! [F: produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( comp_P336008677415481483nteger @ produc7822682618958472924nteger @ ( produc717174245515237944e_term @ F ) )
      = produc7822682618958472924nteger ) )).

% fst_comp_apsnd
thf(fact_280_fst__comp__apsnd,axiom,(
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( comp_P9041276550952352214nt_int @ produc6816164490631068361nt_int @ ( produc4973430039190721449e_term @ F ) )
      = produc6816164490631068361nt_int ) )).

% fst_comp_apsnd
thf(fact_281_fst__comp__apsnd,axiom,(
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( comp_P5815951645081071072nt_int @ produc6230002227079971283nt_int @ ( produc7011684061226097695e_term @ F ) )
      = produc6230002227079971283nt_int ) )).

% fst_comp_apsnd
thf(fact_282_rec__prod__split,axiom,(
    produc2276705271987446478atural = produc8282080750456430313atural )).

% rec_prod_split
thf(fact_283_Ball__def,axiom,
    ( ball_P8096063237992195499at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,P4: product_prod_nat_nat > $o] :
        ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A6 )
         => ( P4 @ X4 ) ) ) )).

% Ball_def
thf(fact_284_Ball__def,axiom,
    ( ball_P2135918875936212497at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,P4: produc859450856879609959at_nat > $o] :
        ! [X4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ A6 )
         => ( P4 @ X4 ) ) ) )).

% Ball_def
thf(fact_285_snd__apsnd,axiom,(
    ! [F: produc7822875418678951345atural > produc7822875418678951345atural,X: produc5835291356934675326atural] :
      ( ( produc6591912806276919810atural @ ( produc4464772112783624294atural @ F @ X ) )
      = ( F @ ( produc6591912806276919810atural @ X ) ) ) )).

% snd_apsnd
thf(fact_286_snd__apsnd,axiom,(
    ! [F: produc1811169103916964701list_f > produc1811169103916964701list_f,X: produc8923597787767437993list_f] :
      ( ( produc8710481132536983823list_f @ ( produc3341670575178904229action @ F @ X ) )
      = ( F @ ( produc8710481132536983823list_f @ X ) ) ) )).

% snd_apsnd
thf(fact_287_case__prod__curry,axiom,(
    ! [F: produc7822875418678951345atural > produc6756470014200349799atural] :
      ( ( produc8282080750456430313atural @ ( produc3440786442043642122atural @ F ) )
      = F ) )).

% case_prod_curry
thf(fact_288_curry__case__prod,axiom,(
    ! [F: code_natural > code_natural > produc6756470014200349799atural] :
      ( ( produc3440786442043642122atural @ ( produc8282080750456430313atural @ F ) )
      = F ) )).

% curry_case_prod
thf(fact_289_exI__realizer,axiom,(
    ! [P: produc8923325533196201883nteger > ( code_integer > option6357759511663192854e_term ) > $o,Y: produc8923325533196201883nteger,X: code_integer > option6357759511663192854e_term] :
      ( ( P @ Y @ X )
     => ( P @ ( produc6514156286822040066nteger @ ( produc6137756002093451184nteger @ X @ Y ) ) @ ( produc2702340305111172292nteger @ ( produc6137756002093451184nteger @ X @ Y ) ) ) ) )).

% exI_realizer
thf(fact_290_exI__realizer,axiom,(
    ! [P: produc7822875418678951345atural > ( produc4972180933644002618e_term > option6357759511663192854e_term ) > $o,Y: produc7822875418678951345atural,X: produc4972180933644002618e_term > option6357759511663192854e_term] :
      ( ( P @ Y @ X )
     => ( P @ ( produc3085575039554877691atural @ ( produc3831813291587773865atural @ X @ Y ) ) @ ( produc3051390257598303421atural @ ( produc3831813291587773865atural @ X @ Y ) ) ) ) )).

% exI_realizer
thf(fact_291_exI__realizer,axiom,(
    ! [P: produc8923325533196201883nteger > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,Y: produc8923325533196201883nteger,X: produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( P @ Y @ X )
     => ( P @ ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ X @ Y ) ) @ ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ X @ Y ) ) ) ) )).

% exI_realizer
thf(fact_292_exI__realizer,axiom,(
    ! [P: product_prod_int_int > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,Y: product_prod_int_int,X: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( P @ Y @ X )
     => ( P @ ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) ) @ ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) ) ) ) )).

% exI_realizer
thf(fact_293_exI__realizer,axiom,(
    ! [P: product_prod_int_int > ( int > option6357759511663192854e_term ) > $o,Y: product_prod_int_int,X: int > option6357759511663192854e_term] :
      ( ( P @ Y @ X )
     => ( P @ ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) ) @ ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) ) ) ) )).

% exI_realizer
thf(fact_294_exI__realizer,axiom,(
    ! [P: produc7822875418678951345atural > code_natural > $o,Y: produc7822875418678951345atural,X: code_natural] :
      ( ( P @ Y @ X )
     => ( P @ ( produc6591912806276919810atural @ ( produc6639722614265839536atural @ X @ Y ) ) @ ( produc497848011034438852atural @ ( produc6639722614265839536atural @ X @ Y ) ) ) ) )).

% exI_realizer
thf(fact_295_exI__realizer,axiom,(
    ! [P: produc1811169103916964701list_f > finfun1145668972854894571action > $o,Y: produc1811169103916964701list_f,X: finfun1145668972854894571action] :
      ( ( P @ Y @ X )
     => ( P @ ( produc8710481132536983823list_f @ ( produc8331056147679645153list_f @ X @ Y ) ) @ ( produc3089225533250117581list_f @ ( produc8331056147679645153list_f @ X @ Y ) ) ) ) )).

% exI_realizer
thf(fact_296_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,(
    ! [P: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger,A: produc8763457246119570046nteger] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc6137756002093451184nteger @ X @ Y ) )
       => ( P @ ( produc2702340305111172292nteger @ A ) @ ( produc6514156286822040066nteger @ A ) ) ) ) )).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_297_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,(
    ! [P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o,X: produc4972180933644002618e_term > option6357759511663192854e_term,Y: produc7822875418678951345atural,A: produc989692990947075319atural] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc3831813291587773865atural @ X @ Y ) )
       => ( P @ ( produc3051390257598303421atural @ A ) @ ( produc3085575039554877691atural @ A ) ) ) ) )).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_298_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,(
    ! [P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,A: produc1908205239877642774nteger] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc8603105652947943368nteger @ X @ Y ) )
       => ( P @ ( produc7822682618958472924nteger @ A ) @ ( produc7856867400915047194nteger @ A ) ) ) ) )).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_299_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,(
    ! [P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,A: produc2285326912895808259nt_int] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc5700946648718959541nt_int @ X @ Y ) )
       => ( P @ ( produc6816164490631068361nt_int @ A ) @ ( produc7328097813583171335nt_int @ A ) ) ) ) )).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_300_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,(
    ! [P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,X: int > option6357759511663192854e_term,Y: product_prod_int_int,A: produc7773217078559923341nt_int] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc4305682042979456191nt_int @ X @ Y ) )
       => ( P @ ( produc6230002227079971283nt_int @ A ) @ ( produc3162348030201620241nt_int @ A ) ) ) ) )).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_301_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,(
    ! [P: code_natural > produc7822875418678951345atural > $o,X: code_natural,Y: produc7822875418678951345atural,A: produc5835291356934675326atural] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc6639722614265839536atural @ X @ Y ) )
       => ( P @ ( produc497848011034438852atural @ A ) @ ( produc6591912806276919810atural @ A ) ) ) ) )).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_302_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,(
    ! [P: finfun1145668972854894571action > produc1811169103916964701list_f > $o,X: finfun1145668972854894571action,Y: produc1811169103916964701list_f,A: produc8923597787767437993list_f] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc8331056147679645153list_f @ X @ Y ) )
       => ( P @ ( produc3089225533250117581list_f @ A ) @ ( produc8710481132536983823list_f @ A ) ) ) ) )).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_303_conjI__realizer,axiom,(
    ! [P: ( code_integer > option6357759511663192854e_term ) > $o,P2: code_integer > option6357759511663192854e_term,Q: produc8923325533196201883nteger > $o,Q2: produc8923325533196201883nteger] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc2702340305111172292nteger @ ( produc6137756002093451184nteger @ P2 @ Q2 ) ) )
          & ( Q @ ( produc6514156286822040066nteger @ ( produc6137756002093451184nteger @ P2 @ Q2 ) ) ) ) ) ) )).

% conjI_realizer
thf(fact_304_conjI__realizer,axiom,(
    ! [P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > $o,P2: produc4972180933644002618e_term > option6357759511663192854e_term,Q: produc7822875418678951345atural > $o,Q2: produc7822875418678951345atural] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc3051390257598303421atural @ ( produc3831813291587773865atural @ P2 @ Q2 ) ) )
          & ( Q @ ( produc3085575039554877691atural @ ( produc3831813291587773865atural @ P2 @ Q2 ) ) ) ) ) ) )).

% conjI_realizer
thf(fact_305_conjI__realizer,axiom,(
    ! [P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,P2: produc6241069584506657477e_term > option6357759511663192854e_term,Q: produc8923325533196201883nteger > $o,Q2: produc8923325533196201883nteger] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ P2 @ Q2 ) ) )
          & ( Q @ ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ P2 @ Q2 ) ) ) ) ) ) )).

% conjI_realizer
thf(fact_306_conjI__realizer,axiom,(
    ! [P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,P2: produc8551481072490612790e_term > option6357759511663192854e_term,Q: product_prod_int_int > $o,Q2: product_prod_int_int] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ P2 @ Q2 ) ) )
          & ( Q @ ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ P2 @ Q2 ) ) ) ) ) ) )).

% conjI_realizer
thf(fact_307_conjI__realizer,axiom,(
    ! [P: ( int > option6357759511663192854e_term ) > $o,P2: int > option6357759511663192854e_term,Q: product_prod_int_int > $o,Q2: product_prod_int_int] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ P2 @ Q2 ) ) )
          & ( Q @ ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ P2 @ Q2 ) ) ) ) ) ) )).

% conjI_realizer
thf(fact_308_conjI__realizer,axiom,(
    ! [P: code_natural > $o,P2: code_natural,Q: produc7822875418678951345atural > $o,Q2: produc7822875418678951345atural] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc497848011034438852atural @ ( produc6639722614265839536atural @ P2 @ Q2 ) ) )
          & ( Q @ ( produc6591912806276919810atural @ ( produc6639722614265839536atural @ P2 @ Q2 ) ) ) ) ) ) )).

% conjI_realizer
thf(fact_309_conjI__realizer,axiom,(
    ! [P: finfun1145668972854894571action > $o,P2: finfun1145668972854894571action,Q: produc1811169103916964701list_f > $o,Q2: produc1811169103916964701list_f] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc3089225533250117581list_f @ ( produc8331056147679645153list_f @ P2 @ Q2 ) ) )
          & ( Q @ ( produc8710481132536983823list_f @ ( produc8331056147679645153list_f @ P2 @ Q2 ) ) ) ) ) ) )).

% conjI_realizer
thf(fact_310_snd__comp__apsnd,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ( comp_P4457869461070014998list_f @ produc2867122274176885975_a_b_c @ ( produc265111499003474649action @ F ) )
      = ( comp_P4767578765025054880list_f @ F @ produc8710481132536983823list_f ) ) )).

% snd_comp_apsnd
thf(fact_311_snd__comp__apsnd,axiom,(
    ! [F: produc7822875418678951345atural > produc7822875418678951345atural] :
      ( ( comp_P7610810914149544718atural @ produc6591912806276919810atural @ ( produc4464772112783624294atural @ F ) )
      = ( comp_P3067963646784605345atural @ F @ produc6591912806276919810atural ) ) )).

% snd_comp_apsnd
thf(fact_312_snd__comp__apsnd,axiom,(
    ! [F: produc1811169103916964701list_f > produc1811169103916964701list_f] :
      ( ( comp_P4267156721827287172list_f @ produc8710481132536983823list_f @ ( produc3341670575178904229action @ F ) )
      = ( comp_P2673361230361438154list_f @ F @ produc8710481132536983823list_f ) ) )).

% snd_comp_apsnd
thf(fact_313_case__prod__unfold,axiom,
    ( produc8282080750456430313atural
    = ( ^ [C3: code_natural > code_natural > produc6756470014200349799atural,P3: produc7822875418678951345atural] :
          ( C3 @ ( produc1773691148792264405atural @ P3 ) @ ( produc8662200840462671639atural @ P3 ) ) ) )).

% case_prod_unfold
thf(fact_314_comp__apply__eq,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc8923597787767437993list_f > produc1811169103916964701list_f,X: produc8923597787767437993list_f,H: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,K: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( ( F @ ( G2 @ X ) )
        = ( H @ ( K @ X ) ) )
     => ( ( comp_P4767578765025054880list_f @ F @ G2 @ X )
        = ( comp_P4767578765025054880list_f @ H @ K @ X ) ) ) )).

% comp_apply_eq
thf(fact_315_exhaustive__int_H_Ocases,axiom,(
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F3: int > option6357759511663192854e_term,D2: int,I: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F3 @ ( product_Pair_int_int @ D2 @ I ) ) ) )).

% exhaustive_int'.cases
thf(fact_316_Product__Type_OCollect__case__prodD,axiom,(
    ! [X: product_prod_nat_nat,A4: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A4 ) ) )
     => ( A4 @ ( product_fst_nat_nat @ X ) @ ( product_snd_nat_nat @ X ) ) ) )).

% Product_Type.Collect_case_prodD
thf(fact_317_Product__Type_OCollect__case__prodD,axiom,(
    ! [X: produc859450856879609959at_nat,A4: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ X @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ A4 ) ) )
     => ( A4 @ ( produc3213797794245857475at_nat @ X ) @ ( produc6408287024330202629at_nat @ X ) ) ) )).

% Product_Type.Collect_case_prodD
thf(fact_318_Product__Type_OCollect__case__prodD,axiom,(
    ! [X: produc8763457246119570046nteger,A4: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member3068662437193594005nteger @ X @ ( collec6350289969311926995nteger @ ( produc127349428274296955eger_o @ A4 ) ) )
     => ( A4 @ ( produc2702340305111172292nteger @ X ) @ ( produc6514156286822040066nteger @ X ) ) ) )).

% Product_Type.Collect_case_prodD
thf(fact_319_Product__Type_OCollect__case__prodD,axiom,(
    ! [X: produc989692990947075319atural,A4: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o] :
      ( ( member3245610415464309390atural @ X @ ( collec6032437096461363148atural @ ( produc2278936416606902338ural_o @ A4 ) ) )
     => ( A4 @ ( produc3051390257598303421atural @ X ) @ ( produc3085575039554877691atural @ X ) ) ) )).

% Product_Type.Collect_case_prodD
thf(fact_320_Product__Type_OCollect__case__prodD,axiom,(
    ! [X: produc1908205239877642774nteger,A4: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member4164122664394876845nteger @ X @ ( collec6950949345391930603nteger @ ( produc6253627499356882019eger_o @ A4 ) ) )
     => ( A4 @ ( produc7822682618958472924nteger @ X ) @ ( produc7856867400915047194nteger @ X ) ) ) )).

% Product_Type.Collect_case_prodD
thf(fact_321_Product__Type_OCollect__case__prodD,axiom,(
    ! [X: produc2285326912895808259nt_int,A4: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7618704894036264090nt_int @ X @ ( collec1790188477890212312nt_int @ ( produc1573362020775583542_int_o @ A4 ) ) )
     => ( A4 @ ( produc6816164490631068361nt_int @ X ) @ ( produc7328097813583171335nt_int @ X ) ) ) )).

% Product_Type.Collect_case_prodD
thf(fact_322_Product__Type_OCollect__case__prodD,axiom,(
    ! [X: produc7773217078559923341nt_int,A4: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7034335876925520548nt_int @ X @ ( collec506566255779805410nt_int @ ( produc2558449545302689196_int_o @ A4 ) ) )
     => ( A4 @ ( produc6230002227079971283nt_int @ X ) @ ( produc3162348030201620241nt_int @ X ) ) ) )).

% Product_Type.Collect_case_prodD
thf(fact_323_Product__Type_OCollect__case__prodD,axiom,(
    ! [X: produc5835291356934675326atural,A4: code_natural > produc7822875418678951345atural > $o] :
      ( ( member2108296798382401941atural @ X @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ A4 ) ) )
     => ( A4 @ ( produc497848011034438852atural @ X ) @ ( produc6591912806276919810atural @ X ) ) ) )).

% Product_Type.Collect_case_prodD
thf(fact_324_Product__Type_OCollect__case__prodD,axiom,(
    ! [X: produc8923597787767437993list_f,A4: finfun1145668972854894571action > produc1811169103916964701list_f > $o] :
      ( ( member1992181931100238290list_f @ X @ ( collec2941370454899091988list_f @ ( produc7065790152434098210st_f_o @ A4 ) ) )
     => ( A4 @ ( produc3089225533250117581list_f @ X ) @ ( produc8710481132536983823list_f @ X ) ) ) )).

% Product_Type.Collect_case_prodD
thf(fact_325_full__exhaustive__int_H_Ocases,axiom,(
    ! [X: produc2285326912895808259nt_int] :
      ~ ! [F3: produc8551481072490612790e_term > option6357759511663192854e_term,D2: int,I: int] :
          ( X
         != ( produc5700946648718959541nt_int @ F3 @ ( product_Pair_int_int @ D2 @ I ) ) ) )).

% full_exhaustive_int'.cases
thf(fact_326_empty__iff,axiom,(
    ! [C2: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ C2 @ bot_bo2099793752762293965at_nat ) )).

% empty_iff
thf(fact_327_empty__iff,axiom,(
    ! [C2: produc859450856879609959at_nat] :
      ~ ( member8206827879206165904at_nat @ C2 @ bot_bo5327735625951526323at_nat ) )).

% empty_iff
thf(fact_328_insert__iff,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member8440522571783428010at_nat @ A @ A4 ) ) ) )).

% insert_iff
thf(fact_329_insert__iff,axiom,(
    ! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ ( insert5050368324300391991at_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member8206827879206165904at_nat @ A @ A4 ) ) ) )).

% insert_iff
thf(fact_330_prod_Osplit__sel,axiom,(
    ! [P: produc6756470014200349799atural > $o,F: code_natural > code_natural > produc6756470014200349799atural,Prod: produc7822875418678951345atural] :
      ( ( P @ ( produc8282080750456430313atural @ F @ Prod ) )
      = ( ( Prod
          = ( produc3574140220909816553atural @ ( produc1773691148792264405atural @ Prod ) @ ( produc8662200840462671639atural @ Prod ) ) )
       => ( P @ ( F @ ( produc1773691148792264405atural @ Prod ) @ ( produc8662200840462671639atural @ Prod ) ) ) ) ) )).

% prod.split_sel
thf(fact_331_UNIV__I,axiom,(
    ! [X: product_prod_nat_nat] :
      ( member8440522571783428010at_nat @ X @ top_to4669805908274784177at_nat ) )).

% UNIV_I
thf(fact_332_UNIV__I,axiom,(
    ! [X: produc859450856879609959at_nat] :
      ( member8206827879206165904at_nat @ X @ top_to8454300755403694231at_nat ) )).

% UNIV_I
thf(fact_333_prod_Osplit__sel__asm,axiom,(
    ! [P: produc6756470014200349799atural > $o,F: code_natural > code_natural > produc6756470014200349799atural,Prod: produc7822875418678951345atural] :
      ( ( P @ ( produc8282080750456430313atural @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc3574140220909816553atural @ ( produc1773691148792264405atural @ Prod ) @ ( produc8662200840462671639atural @ Prod ) ) )
            & ~ ( P @ ( F @ ( produc1773691148792264405atural @ Prod ) @ ( produc8662200840462671639atural @ Prod ) ) ) ) ) ) )).

% prod.split_sel_asm
thf(fact_334_in__lex__prod,axiom,(
    ! [A: nat,B: product_prod_nat_nat,A2: nat,B2: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member3048279782668807382at_nat @ ( produc2653010282673554021at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc487386426758144856at_nat @ A2 @ B2 ) ) @ ( lex_pr8029265285556086080at_nat @ R @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B2 ) @ S ) ) ) ) )).

% in_lex_prod
thf(fact_335_in__lex__prod,axiom,(
    ! [A: product_prod_nat_nat,B: nat,A2: product_prod_nat_nat,B2: nat,R: set_Pr8693737435421807431at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member1995966531042493578at_nat @ ( produc7904928797850150681at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc6350711070570205562at_nat @ A2 @ B2 ) ) @ ( lex_pr4669217892513370978at_nat @ R @ S ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S ) ) ) ) )).

% in_lex_prod
thf(fact_336_in__lex__prod,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc6161850002892822231at_nat @ A2 @ B2 ) ) @ ( lex_pr8801849515957261039at_nat @ R @ S ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B2 ) @ S ) ) ) ) )).

% in_lex_prod
thf(fact_337_in__lex__prod,axiom,(
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A2: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger,R: set_Pr6140516264755836645e_term,S: set_Pr5052581263052047815nteger] :
      ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ A @ B ) @ ( produc6137756002093451184nteger @ A2 @ B2 ) ) @ ( lex_pr1455869853123938712nteger @ R @ S ) )
      = ( ( member1138991337916034606e_term @ ( produc2189037350439096765e_term @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member335077259223437328nteger @ ( produc4911887526399440471nteger @ B @ B2 ) @ S ) ) ) ) )).

% in_lex_prod
thf(fact_338_in__lex__prod,axiom,(
    ! [A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural,A2: produc4972180933644002618e_term > option6357759511663192854e_term,B2: produc7822875418678951345atural,R: set_Pr8830603011318012703e_term,S: set_Pr250711821884878663atural] :
      ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ A @ B ) @ ( produc3831813291587773865atural @ A2 @ B2 ) ) @ ( lex_pr8216006927760890769atural @ R @ S ) )
      = ( ( member6437180440544822120e_term @ ( produc734250632046660983e_term @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member3756964795865971088atural @ ( produc4480994950612372183atural @ B @ B2 ) @ S ) ) ) ) )).

% in_lex_prod
thf(fact_339_in__lex__prod,axiom,(
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,R: set_Pr6418322821503574069e_term,S: set_Pr5052581263052047815nteger] :
      ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ A @ B ) @ ( produc8603105652947943368nteger @ A2 @ B2 ) ) @ ( lex_pr3763927252266284464nteger @ R @ S ) )
      = ( ( member6786453453753086846e_term @ ( produc6915061378690470669e_term @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member335077259223437328nteger @ ( produc4911887526399440471nteger @ B @ B2 ) @ S ) ) ) ) )).

% in_lex_prod
thf(fact_340_in__lex__prod,axiom,(
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,R: set_Pr4675205556198584599e_term,S: set_Pr2560585780119916871nt_int] :
      ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ ( produc5700946648718959541nt_int @ A2 @ B2 ) ) @ ( lex_pr4548132154224949661nt_int @ R @ S ) )
      = ( ( member8796044733138646368e_term @ ( produc8124735558134971759e_term @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B @ B2 ) @ S ) ) ) ) )).

% in_lex_prod
thf(fact_341_in__lex__prod,axiom,(
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int,R: set_Pr7140365069852192003e_term,S: set_Pr2560585780119916871nt_int] :
      ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ ( produc4305682042979456191nt_int @ A2 @ B2 ) ) @ ( lex_pr7234932505866811047nt_int @ R @ S ) )
      = ( ( member6538663773838357324e_term @ ( produc4510790414113558107e_term @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B @ B2 ) @ S ) ) ) ) )).

% in_lex_prod
thf(fact_342_in__lex__prod,axiom,(
    ! [A: nat,B: nat,A2: nat,B2: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A2 @ B2 ) ) @ ( lex_prod_nat_nat @ R @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S ) ) ) ) )).

% in_lex_prod
thf(fact_343_exhaustive__natural_H_Ocases,axiom,(
    ! [X: produc8731074985263844745atural] :
      ~ ! [F3: code_natural > option6357759511663192854e_term,D2: code_natural,I: code_natural] :
          ( X
         != ( produc2252593628808123835atural @ F3 @ ( produc3574140220909816553atural @ D2 @ I ) ) ) )).

% exhaustive_natural'.cases
thf(fact_344_mem__case__prodE,axiom,(
    ! [Z: product_prod_nat_nat,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_Pr1261947904930325089at_nat,P2: produc8763457246119570046nteger] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc4551263766066629700at_nat @ C2 @ P2 ) )
     => ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ( ( P2
              = ( produc6137756002093451184nteger @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) )).

% mem_case_prodE
thf(fact_345_mem__case__prodE,axiom,(
    ! [Z: produc859450856879609959at_nat,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_Pr8693737435421807431at_nat,P2: produc8763457246119570046nteger] :
      ( ( member8206827879206165904at_nat @ Z @ ( produc5744607925856648362at_nat @ C2 @ P2 ) )
     => ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ( ( P2
              = ( produc6137756002093451184nteger @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) )).

% mem_case_prodE
thf(fact_346_mem__case__prodE,axiom,(
    ! [Z: product_prod_nat_nat,C2: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > set_Pr1261947904930325089at_nat,P2: produc989692990947075319atural] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc5359040883261003019at_nat @ C2 @ P2 ) )
     => ~ ! [X3: produc4972180933644002618e_term > option6357759511663192854e_term,Y3: produc7822875418678951345atural] :
            ( ( P2
              = ( produc3831813291587773865atural @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) )).

% mem_case_prodE
thf(fact_347_mem__case__prodE,axiom,(
    ! [Z: produc859450856879609959at_nat,C2: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > set_Pr8693737435421807431at_nat,P2: produc989692990947075319atural] :
      ( ( member8206827879206165904at_nat @ Z @ ( produc5211177361811169009at_nat @ C2 @ P2 ) )
     => ~ ! [X3: produc4972180933644002618e_term > option6357759511663192854e_term,Y3: produc7822875418678951345atural] :
            ( ( P2
              = ( produc3831813291587773865atural @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) )).

% mem_case_prodE
thf(fact_348_mem__case__prodE,axiom,(
    ! [Z: product_prod_nat_nat,C2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_Pr1261947904930325089at_nat,P2: produc1908205239877642774nteger] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc6029229664795570220at_nat @ C2 @ P2 ) )
     => ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ( ( P2
              = ( produc8603105652947943368nteger @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) )).

% mem_case_prodE
thf(fact_349_mem__case__prodE,axiom,(
    ! [Z: produc859450856879609959at_nat,C2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_Pr8693737435421807431at_nat,P2: produc1908205239877642774nteger] :
      ( ( member8206827879206165904at_nat @ Z @ ( produc2704644331147814034at_nat @ C2 @ P2 ) )
     => ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ( ( P2
              = ( produc8603105652947943368nteger @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) )).

% mem_case_prodE
thf(fact_350_mem__case__prodE,axiom,(
    ! [Z: product_prod_nat_nat,C2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_Pr1261947904930325089at_nat,P2: produc2285326912895808259nt_int] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc2642701577560500223at_nat @ C2 @ P2 ) )
     => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P2
              = ( produc5700946648718959541nt_int @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) )).

% mem_case_prodE
thf(fact_351_mem__case__prodE,axiom,(
    ! [Z: produc859450856879609959at_nat,C2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_Pr8693737435421807431at_nat,P2: produc2285326912895808259nt_int] :
      ( ( member8206827879206165904at_nat @ Z @ ( produc517896604054025189at_nat @ C2 @ P2 ) )
     => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P2
              = ( produc5700946648718959541nt_int @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) )).

% mem_case_prodE
thf(fact_352_mem__case__prodE,axiom,(
    ! [Z: product_prod_nat_nat,C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_Pr1261947904930325089at_nat,P2: produc7773217078559923341nt_int] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc5154663461364529269at_nat @ C2 @ P2 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P2
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) )).

% mem_case_prodE
thf(fact_353_mem__case__prodE,axiom,(
    ! [Z: produc859450856879609959at_nat,C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_Pr8693737435421807431at_nat,P2: produc7773217078559923341nt_int] :
      ( ( member8206827879206165904at_nat @ Z @ ( produc2114697605475793755at_nat @ C2 @ P2 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P2
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) )).

% mem_case_prodE
thf(fact_354_prod__eqI,axiom,(
    ! [P2: produc8763457246119570046nteger,Q2: produc8763457246119570046nteger] :
      ( ( ( produc2702340305111172292nteger @ P2 )
        = ( produc2702340305111172292nteger @ Q2 ) )
     => ( ( ( produc6514156286822040066nteger @ P2 )
          = ( produc6514156286822040066nteger @ Q2 ) )
       => ( P2 = Q2 ) ) ) )).

% prod_eqI
thf(fact_355_prod__eqI,axiom,(
    ! [P2: produc989692990947075319atural,Q2: produc989692990947075319atural] :
      ( ( ( produc3051390257598303421atural @ P2 )
        = ( produc3051390257598303421atural @ Q2 ) )
     => ( ( ( produc3085575039554877691atural @ P2 )
          = ( produc3085575039554877691atural @ Q2 ) )
       => ( P2 = Q2 ) ) ) )).

% prod_eqI
thf(fact_356_prod__eqI,axiom,(
    ! [P2: produc1908205239877642774nteger,Q2: produc1908205239877642774nteger] :
      ( ( ( produc7822682618958472924nteger @ P2 )
        = ( produc7822682618958472924nteger @ Q2 ) )
     => ( ( ( produc7856867400915047194nteger @ P2 )
          = ( produc7856867400915047194nteger @ Q2 ) )
       => ( P2 = Q2 ) ) ) )).

% prod_eqI
thf(fact_357_prod__eqI,axiom,(
    ! [P2: produc2285326912895808259nt_int,Q2: produc2285326912895808259nt_int] :
      ( ( ( produc6816164490631068361nt_int @ P2 )
        = ( produc6816164490631068361nt_int @ Q2 ) )
     => ( ( ( produc7328097813583171335nt_int @ P2 )
          = ( produc7328097813583171335nt_int @ Q2 ) )
       => ( P2 = Q2 ) ) ) )).

% prod_eqI
thf(fact_358_prod__eqI,axiom,(
    ! [P2: produc7773217078559923341nt_int,Q2: produc7773217078559923341nt_int] :
      ( ( ( produc6230002227079971283nt_int @ P2 )
        = ( produc6230002227079971283nt_int @ Q2 ) )
     => ( ( ( produc3162348030201620241nt_int @ P2 )
          = ( produc3162348030201620241nt_int @ Q2 ) )
       => ( P2 = Q2 ) ) ) )).

% prod_eqI
thf(fact_359_prod__eqI,axiom,(
    ! [P2: produc5835291356934675326atural,Q2: produc5835291356934675326atural] :
      ( ( ( produc497848011034438852atural @ P2 )
        = ( produc497848011034438852atural @ Q2 ) )
     => ( ( ( produc6591912806276919810atural @ P2 )
          = ( produc6591912806276919810atural @ Q2 ) )
       => ( P2 = Q2 ) ) ) )).

% prod_eqI
thf(fact_360_prod__eqI,axiom,(
    ! [P2: produc8923597787767437993list_f,Q2: produc8923597787767437993list_f] :
      ( ( ( produc3089225533250117581list_f @ P2 )
        = ( produc3089225533250117581list_f @ Q2 ) )
     => ( ( ( produc8710481132536983823list_f @ P2 )
          = ( produc8710481132536983823list_f @ Q2 ) )
       => ( P2 = Q2 ) ) ) )).

% prod_eqI
thf(fact_361_prod_Oexpand,axiom,(
    ! [Prod: produc8763457246119570046nteger,Prod3: produc8763457246119570046nteger] :
      ( ( ( ( produc2702340305111172292nteger @ Prod )
          = ( produc2702340305111172292nteger @ Prod3 ) )
        & ( ( produc6514156286822040066nteger @ Prod )
          = ( produc6514156286822040066nteger @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) )).

% prod.expand
thf(fact_362_prod_Oexpand,axiom,(
    ! [Prod: produc989692990947075319atural,Prod3: produc989692990947075319atural] :
      ( ( ( ( produc3051390257598303421atural @ Prod )
          = ( produc3051390257598303421atural @ Prod3 ) )
        & ( ( produc3085575039554877691atural @ Prod )
          = ( produc3085575039554877691atural @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) )).

% prod.expand
thf(fact_363_prod_Oexpand,axiom,(
    ! [Prod: produc1908205239877642774nteger,Prod3: produc1908205239877642774nteger] :
      ( ( ( ( produc7822682618958472924nteger @ Prod )
          = ( produc7822682618958472924nteger @ Prod3 ) )
        & ( ( produc7856867400915047194nteger @ Prod )
          = ( produc7856867400915047194nteger @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) )).

% prod.expand
thf(fact_364_prod_Oexpand,axiom,(
    ! [Prod: produc2285326912895808259nt_int,Prod3: produc2285326912895808259nt_int] :
      ( ( ( ( produc6816164490631068361nt_int @ Prod )
          = ( produc6816164490631068361nt_int @ Prod3 ) )
        & ( ( produc7328097813583171335nt_int @ Prod )
          = ( produc7328097813583171335nt_int @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) )).

% prod.expand
thf(fact_365_prod_Oexpand,axiom,(
    ! [Prod: produc7773217078559923341nt_int,Prod3: produc7773217078559923341nt_int] :
      ( ( ( ( produc6230002227079971283nt_int @ Prod )
          = ( produc6230002227079971283nt_int @ Prod3 ) )
        & ( ( produc3162348030201620241nt_int @ Prod )
          = ( produc3162348030201620241nt_int @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) )).

% prod.expand
thf(fact_366_prod_Oexpand,axiom,(
    ! [Prod: produc5835291356934675326atural,Prod3: produc5835291356934675326atural] :
      ( ( ( ( produc497848011034438852atural @ Prod )
          = ( produc497848011034438852atural @ Prod3 ) )
        & ( ( produc6591912806276919810atural @ Prod )
          = ( produc6591912806276919810atural @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) )).

% prod.expand
thf(fact_367_prod_Oexpand,axiom,(
    ! [Prod: produc8923597787767437993list_f,Prod3: produc8923597787767437993list_f] :
      ( ( ( ( produc3089225533250117581list_f @ Prod )
          = ( produc3089225533250117581list_f @ Prod3 ) )
        & ( ( produc8710481132536983823list_f @ Prod )
          = ( produc8710481132536983823list_f @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) )).

% prod.expand
thf(fact_368_exhaustive__integer_H_Ocases,axiom,(
    ! [X: produc8763457246119570046nteger] :
      ~ ! [F3: code_integer > option6357759511663192854e_term,D2: code_integer,I: code_integer] :
          ( X
         != ( produc6137756002093451184nteger @ F3 @ ( produc1086072967326762835nteger @ D2 @ I ) ) ) )).

% exhaustive_integer'.cases
thf(fact_369_dropWhile__eq__Nil__conv,axiom,(
    ! [P: typerep > $o,Xs: list_typerep] :
      ( ( ( dropWhile_typerep @ P @ Xs )
        = nil_typerep )
      = ( ! [X4: typerep] :
            ( ( member_typerep @ X4 @ ( set_typerep2 @ Xs ) )
           => ( P @ X4 ) ) ) ) )).

% dropWhile_eq_Nil_conv
thf(fact_370_UnCI,axiom,(
    ! [C2: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member8440522571783428010at_nat @ C2 @ B5 )
       => ( member8440522571783428010at_nat @ C2 @ A4 ) )
     => ( member8440522571783428010at_nat @ C2 @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) ) ) )).

% UnCI
thf(fact_371_UnCI,axiom,(
    ! [C2: produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( ~ ( member8206827879206165904at_nat @ C2 @ B5 )
       => ( member8206827879206165904at_nat @ C2 @ A4 ) )
     => ( member8206827879206165904at_nat @ C2 @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) ) )).

% UnCI
thf(fact_372_Rel__def,axiom,
    ( rel_o_o
    = ( ^ [R3: $o > $o > $o] : R3 ) )).

% Rel_def
thf(fact_373_surjective__pairing,axiom,(
    ! [T3: produc8763457246119570046nteger] :
      ( T3
      = ( produc6137756002093451184nteger @ ( produc2702340305111172292nteger @ T3 ) @ ( produc6514156286822040066nteger @ T3 ) ) ) )).

% surjective_pairing
thf(fact_374_surjective__pairing,axiom,(
    ! [T3: produc989692990947075319atural] :
      ( T3
      = ( produc3831813291587773865atural @ ( produc3051390257598303421atural @ T3 ) @ ( produc3085575039554877691atural @ T3 ) ) ) )).

% surjective_pairing
thf(fact_375_surjective__pairing,axiom,(
    ! [T3: produc1908205239877642774nteger] :
      ( T3
      = ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ T3 ) @ ( produc7856867400915047194nteger @ T3 ) ) ) )).

% surjective_pairing
thf(fact_376_surjective__pairing,axiom,(
    ! [T3: produc2285326912895808259nt_int] :
      ( T3
      = ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ T3 ) @ ( produc7328097813583171335nt_int @ T3 ) ) ) )).

% surjective_pairing
thf(fact_377_surjective__pairing,axiom,(
    ! [T3: produc7773217078559923341nt_int] :
      ( T3
      = ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ T3 ) @ ( produc3162348030201620241nt_int @ T3 ) ) ) )).

% surjective_pairing
thf(fact_378_surjective__pairing,axiom,(
    ! [T3: produc5835291356934675326atural] :
      ( T3
      = ( produc6639722614265839536atural @ ( produc497848011034438852atural @ T3 ) @ ( produc6591912806276919810atural @ T3 ) ) ) )).

% surjective_pairing
thf(fact_379_surjective__pairing,axiom,(
    ! [T3: produc8923597787767437993list_f] :
      ( T3
      = ( produc8331056147679645153list_f @ ( produc3089225533250117581list_f @ T3 ) @ ( produc8710481132536983823list_f @ T3 ) ) ) )).

% surjective_pairing
thf(fact_380_prod_Oexhaust__sel,axiom,(
    ! [Prod: produc8763457246119570046nteger] :
      ( Prod
      = ( produc6137756002093451184nteger @ ( produc2702340305111172292nteger @ Prod ) @ ( produc6514156286822040066nteger @ Prod ) ) ) )).

% prod.exhaust_sel
thf(fact_381_prod_Oexhaust__sel,axiom,(
    ! [Prod: produc989692990947075319atural] :
      ( Prod
      = ( produc3831813291587773865atural @ ( produc3051390257598303421atural @ Prod ) @ ( produc3085575039554877691atural @ Prod ) ) ) )).

% prod.exhaust_sel
thf(fact_382_prod_Oexhaust__sel,axiom,(
    ! [Prod: produc1908205239877642774nteger] :
      ( Prod
      = ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ Prod ) @ ( produc7856867400915047194nteger @ Prod ) ) ) )).

% prod.exhaust_sel
thf(fact_383_prod_Oexhaust__sel,axiom,(
    ! [Prod: produc2285326912895808259nt_int] :
      ( Prod
      = ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ Prod ) @ ( produc7328097813583171335nt_int @ Prod ) ) ) )).

% prod.exhaust_sel
thf(fact_384_prod_Oexhaust__sel,axiom,(
    ! [Prod: produc7773217078559923341nt_int] :
      ( Prod
      = ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ Prod ) @ ( produc3162348030201620241nt_int @ Prod ) ) ) )).

% prod.exhaust_sel
thf(fact_385_prod_Oexhaust__sel,axiom,(
    ! [Prod: produc5835291356934675326atural] :
      ( Prod
      = ( produc6639722614265839536atural @ ( produc497848011034438852atural @ Prod ) @ ( produc6591912806276919810atural @ Prod ) ) ) )).

% prod.exhaust_sel
thf(fact_386_prod_Oexhaust__sel,axiom,(
    ! [Prod: produc8923597787767437993list_f] :
      ( Prod
      = ( produc8331056147679645153list_f @ ( produc3089225533250117581list_f @ Prod ) @ ( produc8710481132536983823list_f @ Prod ) ) ) )).

% prod.exhaust_sel
thf(fact_387_apfst__conv,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( produc6157546124166486841nteger @ F @ ( produc6137756002093451184nteger @ X @ Y ) )
      = ( produc6137756002093451184nteger @ ( F @ X ) @ Y ) ) )).

% apfst_conv
thf(fact_388_apfst__conv,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc6241069584506657477e_term > option6357759511663192854e_term,X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( produc2291436665917485009nteger @ F @ ( produc6137756002093451184nteger @ X @ Y ) )
      = ( produc8603105652947943368nteger @ ( F @ X ) @ Y ) ) )).

% apfst_conv
thf(fact_389_apfst__conv,axiom,(
    ! [F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc4972180933644002618e_term > option6357759511663192854e_term,X: produc4972180933644002618e_term > option6357759511663192854e_term,Y: produc7822875418678951345atural] :
      ( ( produc7227253073565913365atural @ F @ ( produc3831813291587773865atural @ X @ Y ) )
      = ( produc3831813291587773865atural @ ( F @ X ) @ Y ) ) )).

% apfst_conv
thf(fact_390_apfst__conv,axiom,(
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( produc1843087649659261265nteger @ F @ ( produc8603105652947943368nteger @ X @ Y ) )
      = ( produc6137756002093451184nteger @ ( F @ X ) @ Y ) ) )).

% apfst_conv
thf(fact_391_apfst__conv,axiom,(
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc6241069584506657477e_term > option6357759511663192854e_term,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( produc2515316116081422825nteger @ F @ ( produc8603105652947943368nteger @ X @ Y ) )
      = ( produc8603105652947943368nteger @ ( F @ X ) @ Y ) ) )).

% apfst_conv
thf(fact_392_apfst__conv,axiom,(
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc7277921532047535397nt_int @ F @ ( produc5700946648718959541nt_int @ X @ Y ) )
      = ( produc5700946648718959541nt_int @ ( F @ X ) @ Y ) ) )).

% apfst_conv
thf(fact_393_apfst__conv,axiom,(
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > int > option6357759511663192854e_term,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc6919319362177967663nt_int @ F @ ( produc5700946648718959541nt_int @ X @ Y ) )
      = ( produc4305682042979456191nt_int @ ( F @ X ) @ Y ) ) )).

% apfst_conv
thf(fact_394_apfst__conv,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term,X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc6067946493426971951nt_int @ F @ ( produc4305682042979456191nt_int @ X @ Y ) )
      = ( produc5700946648718959541nt_int @ ( F @ X ) @ Y ) ) )).

% apfst_conv
thf(fact_395_apfst__conv,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > int > option6357759511663192854e_term,X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc7131893314732144953nt_int @ F @ ( produc4305682042979456191nt_int @ X @ Y ) )
      = ( produc4305682042979456191nt_int @ ( F @ X ) @ Y ) ) )).

% apfst_conv
thf(fact_396_split__cong,axiom,(
    ! [Q2: produc7822875418678951345atural,F: code_natural > code_natural > produc6756470014200349799atural,G2: code_natural > code_natural > produc6756470014200349799atural,P2: produc7822875418678951345atural] :
      ( ! [X3: code_natural,Y3: code_natural] :
          ( ( ( produc3574140220909816553atural @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G2 @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc8282080750456430313atural @ F @ P2 )
          = ( produc8282080750456430313atural @ G2 @ Q2 ) ) ) ) )).

% split_cong
thf(fact_397_full__exhaustive__natural_H_Ocases,axiom,(
    ! [X: produc989692990947075319atural] :
      ~ ! [F3: produc4972180933644002618e_term > option6357759511663192854e_term,D2: code_natural,I: code_natural] :
          ( X
         != ( produc3831813291587773865atural @ F3 @ ( produc3574140220909816553atural @ D2 @ I ) ) ) )).

% full_exhaustive_natural'.cases
thf(fact_398_Un__iff,axiom,(
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ C2 @ A4 )
        | ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) )).

% Un_iff
thf(fact_399_Un__iff,axiom,(
    ! [C2: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) )
      = ( ( member8206827879206165904at_nat @ C2 @ A4 )
        | ( member8206827879206165904at_nat @ C2 @ B5 ) ) ) )).

% Un_iff
thf(fact_400_full__exhaustive__integer_H_Ocases,axiom,(
    ! [X: produc1908205239877642774nteger] :
      ~ ! [F3: produc6241069584506657477e_term > option6357759511663192854e_term,D2: code_integer,I: code_integer] :
          ( X
         != ( produc8603105652947943368nteger @ F3 @ ( produc1086072967326762835nteger @ D2 @ I ) ) ) )).

% full_exhaustive_integer'.cases
thf(fact_401_rev__image__eqI,axiom,(
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member8440522571783428010at_nat @ B @ ( image_5168914502847457605at_nat @ F @ A4 ) ) ) ) )).

% rev_image_eqI
thf(fact_402_rev__image__eqI,axiom,(
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: produc859450856879609959at_nat,F: product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member8206827879206165904at_nat @ B @ ( image_1119892238107900587at_nat @ F @ A4 ) ) ) ) )).

% rev_image_eqI
thf(fact_403_rev__image__eqI,axiom,(
    ! [X: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B: product_prod_nat_nat,F: produc859450856879609959at_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member8440522571783428010at_nat @ B @ ( image_6706873702294942047at_nat @ F @ A4 ) ) ) ) )).

% rev_image_eqI
thf(fact_404_rev__image__eqI,axiom,(
    ! [X: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B: produc859450856879609959at_nat,F: produc859450856879609959at_nat > produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member8206827879206165904at_nat @ B @ ( image_6321802852832981957at_nat @ F @ A4 ) ) ) ) )).

% rev_image_eqI
thf(fact_405_zip__assoc,axiom,(
    ! [Xs: list_C878401137130745250e_term,Ys: list_Code_integer,Zs: list_Code_integer] :
      ( ( zip_Co8729459035503499408nteger @ Xs @ ( zip_Co3543743374963494515nteger @ Ys @ Zs ) )
      = ( map_Pr3058416263758923424nteger
        @ ( produc705019733726621697nteger
          @ ( produc5029926423600508311nteger
            @ ^ [X4: code_integer > option6357759511663192854e_term,Y6: code_integer,Z3: code_integer] :
                ( produc6137756002093451184nteger @ X4 @ ( produc1086072967326762835nteger @ Y6 @ Z3 ) ) ) )
        @ ( zip_Pr5968401763255418038nteger @ ( zip_Co2624754328424008104nteger @ Xs @ Ys ) @ Zs ) ) ) )).

% zip_assoc
thf(fact_406_zip__assoc,axiom,(
    ! [Xs: list_P5726480267095810431e_term,Ys: list_Code_natural,Zs: list_Code_natural] :
      ( ( zip_Pr3521053968933873289atural @ Xs @ ( zip_Co6031810628546548233atural @ Ys @ Zs ) )
      = ( map_Pr5379678695501496844atural
        @ ( produc5983314677236137325atural
          @ ( produc7496289516936076547atural
            @ ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term,Y6: code_natural,Z3: code_natural] :
                ( produc3831813291587773865atural @ X4 @ ( produc3574140220909816553atural @ Y6 @ Z3 ) ) ) )
        @ ( zip_Pr4029875937752541993atural @ ( zip_Pr7521778153925083984atural @ Xs @ Ys ) @ Zs ) ) ) )).

% zip_assoc
thf(fact_407_zip__assoc,axiom,(
    ! [Xs: list_P1316552470764441098e_term,Ys: list_Code_integer,Zs: list_Code_integer] :
      ( ( zip_Pr8292346330294042792nteger @ Xs @ ( zip_Co3543743374963494515nteger @ Ys @ Zs ) )
      = ( map_Pr5734261727927366560nteger
        @ ( produc5736310350991833473nteger
          @ ( produc1025208726644532247nteger
            @ ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: code_integer,Z3: code_integer] :
                ( produc8603105652947943368nteger @ X4 @ ( produc1086072967326762835nteger @ Y6 @ Z3 ) ) ) )
        @ ( zip_Pr2462386327729532702nteger @ ( zip_Pr1380701361066033040nteger @ Xs @ Ys ) @ Zs ) ) ) )).

% zip_assoc
thf(fact_408_zip__assoc,axiom,(
    ! [Xs: list_P1743416141875011707e_term,Ys: list_int,Zs: list_int] :
      ( ( zip_Pr4168994715204986005nt_int @ Xs @ ( zip_int_int @ Ys @ Zs ) )
      = ( map_Pr2636829087904227612nt_int
        @ ( produc3012078809480436349nt_int
          @ ( produc5776623027272482323nt_int
            @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: int,Z3: int] :
                ( produc5700946648718959541nt_int @ X4 @ ( product_Pair_int_int @ Y6 @ Z3 ) ) ) )
        @ ( zip_Pr8329346017641158189nt_int @ ( zip_Pr7008984055488428112rm_int @ Xs @ Ys ) @ Zs ) ) ) )).

% zip_assoc
thf(fact_409_zip__assoc,axiom,(
    ! [Xs: list_i8448526496819171953e_term,Ys: list_int,Zs: list_int] :
      ( ( zip_in8766932505889695135nt_int @ Xs @ ( zip_int_int @ Ys @ Zs ) )
      = ( map_Pr1488936844085459484nt_int
        @ ( produc6062533421992007421nt_int
          @ ( produc1688802057988075411nt_int
            @ ^ [X4: int > option6357759511663192854e_term,Y6: int,Z3: int] :
                ( produc4305682042979456191nt_int @ X4 @ ( product_Pair_int_int @ Y6 @ Z3 ) ) ) )
        @ ( zip_Pr2852282319877772579nt_int @ ( zip_in1054202108563164102rm_int @ Xs @ Ys ) @ Zs ) ) ) )).

% zip_assoc
thf(fact_410_UnE,axiom,(
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
     => ( ~ ( member8440522571783428010at_nat @ C2 @ A4 )
       => ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) )).

% UnE
thf(fact_411_UnE,axiom,(
    ! [C2: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) )
     => ( ~ ( member8206827879206165904at_nat @ C2 @ A4 )
       => ( member8206827879206165904at_nat @ C2 @ B5 ) ) ) )).

% UnE
thf(fact_412_Rel__abs,axiom,(
    ! [A4: $o > $o > $o,B5: $o > $o > $o,F: $o > $o,G2: $o > $o] :
      ( ! [X3: $o,Y3: $o] :
          ( ( rel_o_o @ A4 @ X3 @ Y3 )
         => ( rel_o_o @ B5 @ ( F @ X3 ) @ ( G2 @ Y3 ) ) )
     => ( rel_o_o_o_o @ ( bNF_rel_fun_o_o_o_o @ A4 @ B5 ) @ F @ G2 ) ) )).

% Rel_abs
thf(fact_413_zip__left__commute,axiom,(
    ! [Xs: list_C878401137130745250e_term,Ys: list_Code_integer,Zs: list_Code_integer] :
      ( ( zip_Co8729459035503499408nteger @ Xs @ ( zip_Co3543743374963494515nteger @ Ys @ Zs ) )
      = ( map_Pr6440346534341297260nteger
        @ ( produc4616540482307369605nteger
          @ ^ [Y6: code_integer] :
              ( produc377316013938373531nteger
              @ ^ [X4: code_integer > option6357759511663192854e_term,Z3: code_integer] :
                  ( produc6137756002093451184nteger @ X4 @ ( produc1086072967326762835nteger @ Y6 @ Z3 ) ) ) )
        @ ( zip_Co1879396967077854522nteger @ Ys @ ( zip_Co2624754328424008104nteger @ Xs @ Zs ) ) ) ) )).

% zip_left_commute
thf(fact_414_zip__left__commute,axiom,(
    ! [Xs: list_P5726480267095810431e_term,Ys: list_Code_natural,Zs: list_Code_natural] :
      ( ( zip_Pr3521053968933873289atural @ Xs @ ( zip_Co6031810628546548233atural @ Ys @ Zs ) )
      = ( map_Pr5208547534834374018atural
        @ ( produc2461152820012758555atural
          @ ^ [Y6: code_natural] :
              ( produc650483176397606972atural
              @ ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term,Z3: code_natural] :
                  ( produc3831813291587773865atural @ X4 @ ( produc3574140220909816553atural @ Y6 @ Z3 ) ) ) )
        @ ( zip_Co4321876045980164567atural @ Ys @ ( zip_Pr7521778153925083984atural @ Xs @ Zs ) ) ) ) )).

% zip_left_commute
thf(fact_415_zip__left__commute,axiom,(
    ! [Xs: list_P1316552470764441098e_term,Ys: list_Code_integer,Zs: list_Code_integer] :
      ( ( zip_Pr8292346330294042792nteger @ Xs @ ( zip_Co3543743374963494515nteger @ Ys @ Zs ) )
      = ( map_Pr375869290728506988nteger
        @ ( produc3008082790229755653nteger
          @ ^ [Y6: code_integer] :
              ( produc2753938625573523867nteger
              @ ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Z3: code_integer] :
                  ( produc8603105652947943368nteger @ X4 @ ( produc1086072967326762835nteger @ Y6 @ Z3 ) ) ) )
        @ ( zip_Co1027725496633450658nteger @ Ys @ ( zip_Pr1380701361066033040nteger @ Xs @ Zs ) ) ) ) )).

% zip_left_commute
thf(fact_416_zip__left__commute,axiom,(
    ! [Xs: list_P1743416141875011707e_term,Ys: list_int,Zs: list_int] :
      ( ( zip_Pr4168994715204986005nt_int @ Xs @ ( zip_int_int @ Ys @ Zs ) )
      = ( map_Pr9151314201186384778nt_int
        @ ( produc2730073284378773539nt_int
          @ ^ [Y6: int] :
              ( produc2771024828437622600nt_int
              @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Z3: int] :
                  ( produc5700946648718959541nt_int @ X4 @ ( product_Pair_int_int @ Y6 @ Z3 ) ) ) )
        @ ( zip_in1519018847819706835rm_int @ Ys @ ( zip_Pr7008984055488428112rm_int @ Xs @ Zs ) ) ) ) )).

% zip_left_commute
thf(fact_417_zip__left__commute,axiom,(
    ! [Xs: list_i8448526496819171953e_term,Ys: list_int,Zs: list_int] :
      ( ( zip_in8766932505889695135nt_int @ Xs @ ( zip_int_int @ Ys @ Zs ) )
      = ( map_Pr8730504324269258506nt_int
        @ ( produc7790648554307352611nt_int
          @ ^ [Y6: int] :
              ( produc1017238998450650824nt_int
              @ ^ [X4: int > option6357759511663192854e_term,Z3: int] :
                  ( produc4305682042979456191nt_int @ X4 @ ( product_Pair_int_int @ Y6 @ Z3 ) ) ) )
        @ ( zip_in3046706307806627657rm_int @ Ys @ ( zip_in1054202108563164102rm_int @ Xs @ Zs ) ) ) ) )).

% zip_left_commute
thf(fact_418_takeWhile__append1,axiom,(
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,Ys: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ~ ( P @ X )
       => ( ( takeWh6395431351003862016at_nat @ P @ ( append985823374593552924at_nat @ Xs @ Ys ) )
          = ( takeWh6395431351003862016at_nat @ P @ Xs ) ) ) ) )).

% takeWhile_append1
thf(fact_419_takeWhile__append1,axiom,(
    ! [X: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o,Ys: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ~ ( P @ X )
       => ( ( takeWh1554011562171007206at_nat @ P @ ( append8751754712269456642at_nat @ Xs @ Ys ) )
          = ( takeWh1554011562171007206at_nat @ P @ Xs ) ) ) ) )).

% takeWhile_append1
thf(fact_420_Rel__app,axiom,(
    ! [A4: $o > $o > $o,B5: $o > $o > $o,F: $o > $o,G2: $o > $o,X: $o,Y: $o] :
      ( ( rel_o_o_o_o @ ( bNF_rel_fun_o_o_o_o @ A4 @ B5 ) @ F @ G2 )
     => ( ( rel_o_o @ A4 @ X @ Y )
       => ( rel_o_o @ B5 @ ( F @ X ) @ ( G2 @ Y ) ) ) ) )).

% Rel_app
thf(fact_421_exE__realizer_H,axiom,(
    ! [P: produc8923325533196201883nteger > ( code_integer > option6357759511663192854e_term ) > $o,P2: produc8763457246119570046nteger] :
      ( ( P @ ( produc6514156286822040066nteger @ P2 ) @ ( produc2702340305111172292nteger @ P2 ) )
     => ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ~ ( P @ Y3 @ X3 ) ) )).

% exE_realizer'
thf(fact_422_exE__realizer_H,axiom,(
    ! [P: produc7822875418678951345atural > ( produc4972180933644002618e_term > option6357759511663192854e_term ) > $o,P2: produc989692990947075319atural] :
      ( ( P @ ( produc3085575039554877691atural @ P2 ) @ ( produc3051390257598303421atural @ P2 ) )
     => ~ ! [X3: produc4972180933644002618e_term > option6357759511663192854e_term,Y3: produc7822875418678951345atural] :
            ~ ( P @ Y3 @ X3 ) ) )).

% exE_realizer'
thf(fact_423_exE__realizer_H,axiom,(
    ! [P: produc8923325533196201883nteger > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,P2: produc1908205239877642774nteger] :
      ( ( P @ ( produc7856867400915047194nteger @ P2 ) @ ( produc7822682618958472924nteger @ P2 ) )
     => ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ~ ( P @ Y3 @ X3 ) ) )).

% exE_realizer'
thf(fact_424_exE__realizer_H,axiom,(
    ! [P: product_prod_int_int > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,P2: produc2285326912895808259nt_int] :
      ( ( P @ ( produc7328097813583171335nt_int @ P2 ) @ ( produc6816164490631068361nt_int @ P2 ) )
     => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ~ ( P @ Y3 @ X3 ) ) )).

% exE_realizer'
thf(fact_425_exE__realizer_H,axiom,(
    ! [P: product_prod_int_int > ( int > option6357759511663192854e_term ) > $o,P2: produc7773217078559923341nt_int] :
      ( ( P @ ( produc3162348030201620241nt_int @ P2 ) @ ( produc6230002227079971283nt_int @ P2 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ~ ( P @ Y3 @ X3 ) ) )).

% exE_realizer'
thf(fact_426_exE__realizer_H,axiom,(
    ! [P: produc7822875418678951345atural > code_natural > $o,P2: produc5835291356934675326atural] :
      ( ( P @ ( produc6591912806276919810atural @ P2 ) @ ( produc497848011034438852atural @ P2 ) )
     => ~ ! [X3: code_natural,Y3: produc7822875418678951345atural] :
            ~ ( P @ Y3 @ X3 ) ) )).

% exE_realizer'
thf(fact_427_exE__realizer_H,axiom,(
    ! [P: produc1811169103916964701list_f > finfun1145668972854894571action > $o,P2: produc8923597787767437993list_f] :
      ( ( P @ ( produc8710481132536983823list_f @ P2 ) @ ( produc3089225533250117581list_f @ P2 ) )
     => ~ ! [X3: finfun1145668972854894571action,Y3: produc1811169103916964701list_f] :
            ~ ( P @ Y3 @ X3 ) ) )).

% exE_realizer'
thf(fact_428_map__prod__simp,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc8923325533196201883nteger,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc291189458620429660nteger @ F @ G2 @ ( produc6137756002093451184nteger @ A @ B ) )
      = ( produc6137756002093451184nteger @ ( F @ A ) @ ( G2 @ B ) ) ) )).

% map_prod_simp
thf(fact_429_map__prod__simp,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > int > option6357759511663192854e_term,G2: produc8923325533196201883nteger > product_prod_int_int,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc4848173044407040841nt_int @ F @ G2 @ ( produc6137756002093451184nteger @ A @ B ) )
      = ( produc4305682042979456191nt_int @ ( F @ A ) @ ( G2 @ B ) ) ) )).

% map_prod_simp
thf(fact_430_map__prod__simp,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: product_prod_int_int > produc8923325533196201883nteger,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc4564144299210792333nteger @ F @ G2 @ ( produc4305682042979456191nt_int @ A @ B ) )
      = ( produc6137756002093451184nteger @ ( F @ A ) @ ( G2 @ B ) ) ) )).

% map_prod_simp
thf(fact_431_map__prod__simp,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > int > option6357759511663192854e_term,G2: product_prod_int_int > product_prod_int_int,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc7905387574911623034nt_int @ F @ G2 @ ( produc4305682042979456191nt_int @ A @ B ) )
      = ( produc4305682042979456191nt_int @ ( F @ A ) @ ( G2 @ B ) ) ) )).

% map_prod_simp
thf(fact_432_map__prod__simp,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc4972180933644002618e_term > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc7822875418678951345atural,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc2114834699021641935atural @ F @ G2 @ ( produc6137756002093451184nteger @ A @ B ) )
      = ( produc3831813291587773865atural @ ( F @ A ) @ ( G2 @ B ) ) ) )).

% map_prod_simp
thf(fact_433_map__prod__simp,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc6241069584506657477e_term > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc8923325533196201883nteger,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc4923703365104367684nteger @ F @ G2 @ ( produc6137756002093451184nteger @ A @ B ) )
      = ( produc8603105652947943368nteger @ ( F @ A ) @ ( G2 @ B ) ) ) )).

% map_prod_simp
thf(fact_434_map__prod__simp,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term,G2: produc8923325533196201883nteger > product_prod_int_int,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc7771412575674646995nt_int @ F @ G2 @ ( produc6137756002093451184nteger @ A @ B ) )
      = ( produc5700946648718959541nt_int @ ( F @ A ) @ ( G2 @ B ) ) ) )).

% map_prod_simp
thf(fact_435_map__prod__simp,axiom,(
    ! [F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: produc7822875418678951345atural > produc8923325533196201883nteger,A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural] :
      ( ( produc1485061364871261091nteger @ F @ G2 @ ( produc3831813291587773865atural @ A @ B ) )
      = ( produc6137756002093451184nteger @ ( F @ A ) @ ( G2 @ B ) ) ) )).

% map_prod_simp
thf(fact_436_map__prod__simp,axiom,(
    ! [F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > int > option6357759511663192854e_term,G2: produc7822875418678951345atural > product_prod_int_int,A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural] :
      ( ( produc5159889016986197392nt_int @ F @ G2 @ ( produc3831813291587773865atural @ A @ B ) )
      = ( produc4305682042979456191nt_int @ ( F @ A ) @ ( G2 @ B ) ) ) )).

% map_prod_simp
thf(fact_437_map__prod__simp,axiom,(
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc8923325533196201883nteger,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc5027624217247135428nteger @ F @ G2 @ ( produc8603105652947943368nteger @ A @ B ) )
      = ( produc6137756002093451184nteger @ ( F @ A ) @ ( G2 @ B ) ) ) )).

% map_prod_simp
thf(fact_438_equalityCE,axiom,(
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
      ( ( A4 = B5 )
     => ( ( ( member8440522571783428010at_nat @ C2 @ A4 )
         => ~ ( member8440522571783428010at_nat @ C2 @ B5 ) )
       => ~ ( ~ ( member8440522571783428010at_nat @ C2 @ A4 )
           => ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) ) )).

% equalityCE
thf(fact_439_equalityCE,axiom,(
    ! [A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,C2: produc859450856879609959at_nat] :
      ( ( A4 = B5 )
     => ( ( ( member8206827879206165904at_nat @ C2 @ A4 )
         => ~ ( member8206827879206165904at_nat @ C2 @ B5 ) )
       => ~ ( ~ ( member8206827879206165904at_nat @ C2 @ A4 )
           => ( member8206827879206165904at_nat @ C2 @ B5 ) ) ) ) )).

% equalityCE
thf(fact_440_less__than__iff,axiom,(
    ! [X: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
      = ( ord_less_nat @ X @ Y ) ) )).

% less_than_iff
thf(fact_441_insertCI,axiom,(
    ! [A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ~ ( member8440522571783428010at_nat @ A @ B5 )
       => ( A = B ) )
     => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B5 ) ) ) )).

% insertCI
thf(fact_442_insertCI,axiom,(
    ! [A: produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat,B: produc859450856879609959at_nat] :
      ( ( ~ ( member8206827879206165904at_nat @ A @ B5 )
       => ( A = B ) )
     => ( member8206827879206165904at_nat @ A @ ( insert5050368324300391991at_nat @ B @ B5 ) ) ) )).

% insertCI
thf(fact_443_apsnd__apfst,axiom,(
    ! [F: produc7822875418678951345atural > produc8923325533196201883nteger,G2: code_natural > code_integer > option6357759511663192854e_term,X: produc5835291356934675326atural] :
      ( ( produc2451119032623472570e_term @ F @ ( produc4988284218490272569atural @ G2 @ X ) )
      = ( produc6137756002093451184nteger @ ( G2 @ ( produc497848011034438852atural @ X ) ) @ ( F @ ( produc6591912806276919810atural @ X ) ) ) ) )).

% apsnd_apfst
thf(fact_444_apsnd__apfst,axiom,(
    ! [F: produc7822875418678951345atural > product_prod_int_int,G2: code_natural > int > option6357759511663192854e_term,X: produc5835291356934675326atural] :
      ( ( produc7977970570655791143e_term @ F @ ( produc8323928115967317034atural @ G2 @ X ) )
      = ( produc4305682042979456191nt_int @ ( G2 @ ( produc497848011034438852atural @ X ) ) @ ( F @ ( produc6591912806276919810atural @ X ) ) ) ) )).

% apsnd_apfst
thf(fact_445_apsnd__apfst,axiom,(
    ! [F: produc7822875418678951345atural > produc7822875418678951345atural,G2: code_natural > produc4972180933644002618e_term > option6357759511663192854e_term,X: produc5835291356934675326atural] :
      ( ( produc2824789715873180845e_term @ F @ ( produc5541935546810125468atural @ G2 @ X ) )
      = ( produc3831813291587773865atural @ ( G2 @ ( produc497848011034438852atural @ X ) ) @ ( F @ ( produc6591912806276919810atural @ X ) ) ) ) )).

% apsnd_apfst
thf(fact_446_apsnd__apfst,axiom,(
    ! [F: produc7822875418678951345atural > produc8923325533196201883nteger,G2: code_natural > produc6241069584506657477e_term > option6357759511663192854e_term,X: produc5835291356934675326atural] :
      ( ( produc1262473387041906082e_term @ F @ ( produc9212777793653044433atural @ G2 @ X ) )
      = ( produc8603105652947943368nteger @ ( G2 @ ( produc497848011034438852atural @ X ) ) @ ( F @ ( produc6591912806276919810atural @ X ) ) ) ) )).

% apsnd_apfst
thf(fact_447_apsnd__apfst,axiom,(
    ! [F: produc7822875418678951345atural > product_prod_int_int,G2: code_natural > produc8551481072490612790e_term > option6357759511663192854e_term,X: produc5835291356934675326atural] :
      ( ( produc1638435738411217841e_term @ F @ ( produc1694427877005358240atural @ G2 @ X ) )
      = ( produc5700946648718959541nt_int @ ( G2 @ ( produc497848011034438852atural @ X ) ) @ ( F @ ( produc6591912806276919810atural @ X ) ) ) ) )).

% apsnd_apfst
thf(fact_448_apsnd__apfst,axiom,(
    ! [F: produc8923325533196201883nteger > produc8923325533196201883nteger,G2: ( code_integer > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,X: produc8763457246119570046nteger] :
      ( ( produc1584323072462815568e_term @ F @ ( produc6157546124166486841nteger @ G2 @ X ) )
      = ( produc6137756002093451184nteger @ ( G2 @ ( produc2702340305111172292nteger @ X ) ) @ ( F @ ( produc6514156286822040066nteger @ X ) ) ) ) )).

% apsnd_apfst
thf(fact_449_apsnd__apfst,axiom,(
    ! [F: produc8923325533196201883nteger > product_prod_int_int,G2: ( code_integer > option6357759511663192854e_term ) > int > option6357759511663192854e_term,X: produc8763457246119570046nteger] :
      ( ( produc5563627732817600957e_term @ F @ ( produc3697415957282278442nteger @ G2 @ X ) )
      = ( produc4305682042979456191nt_int @ ( G2 @ ( produc2702340305111172292nteger @ X ) ) @ ( F @ ( produc6514156286822040066nteger @ X ) ) ) ) )).

% apsnd_apfst
thf(fact_450_apsnd__apfst,axiom,(
    ! [F: product_prod_int_int > produc8923325533196201883nteger,G2: ( int > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,X: produc7773217078559923341nt_int] :
      ( ( produc7785557238454018994e_term @ F @ ( produc7187738725762595912nt_int @ G2 @ X ) )
      = ( produc6137756002093451184nteger @ ( G2 @ ( produc6230002227079971283nt_int @ X ) ) @ ( F @ ( produc3162348030201620241nt_int @ X ) ) ) ) )).

% apsnd_apfst
thf(fact_451_apsnd__apfst,axiom,(
    ! [F: product_prod_int_int > product_prod_int_int,G2: ( int > option6357759511663192854e_term ) > int > option6357759511663192854e_term,X: produc7773217078559923341nt_int] :
      ( ( produc7011684061226097695e_term @ F @ ( produc7131893314732144953nt_int @ G2 @ X ) )
      = ( produc4305682042979456191nt_int @ ( G2 @ ( produc6230002227079971283nt_int @ X ) ) @ ( F @ ( produc3162348030201620241nt_int @ X ) ) ) ) )).

% apsnd_apfst
thf(fact_452_apsnd__apfst,axiom,(
    ! [F: produc8923325533196201883nteger > produc7822875418678951345atural,G2: ( code_integer > option6357759511663192854e_term ) > produc4972180933644002618e_term > option6357759511663192854e_term,X: produc8763457246119570046nteger] :
      ( ( produc2279490574346512707e_term @ F @ ( produc7843966455929341852nteger @ G2 @ X ) )
      = ( produc3831813291587773865atural @ ( G2 @ ( produc2702340305111172292nteger @ X ) ) @ ( F @ ( produc6514156286822040066nteger @ X ) ) ) ) )).

% apsnd_apfst
thf(fact_453_apfst__apsnd,axiom,(
    ! [F: code_natural > code_integer > option6357759511663192854e_term,G2: produc7822875418678951345atural > produc8923325533196201883nteger,X: produc5835291356934675326atural] :
      ( ( produc6088734333007523107nteger @ F @ ( produc8287249056265611856atural @ G2 @ X ) )
      = ( produc6137756002093451184nteger @ ( F @ ( produc497848011034438852atural @ X ) ) @ ( G2 @ ( produc6591912806276919810atural @ X ) ) ) ) )).

% apfst_apsnd
thf(fact_454_apfst__apsnd,axiom,(
    ! [F: code_natural > int > option6357759511663192854e_term,G2: produc7822875418678951345atural > product_prod_int_int,X: produc5835291356934675326atural] :
      ( ( produc7347430915296828978nt_int @ F @ ( produc5167862320620071534atural @ G2 @ X ) )
      = ( produc4305682042979456191nt_int @ ( F @ ( produc497848011034438852atural @ X ) ) @ ( G2 @ ( produc6591912806276919810atural @ X ) ) ) ) )).

% apfst_apsnd
thf(fact_455_apfst__apsnd,axiom,(
    ! [F: code_natural > produc4972180933644002618e_term > option6357759511663192854e_term,G2: produc7822875418678951345atural > produc7822875418678951345atural,X: produc5835291356934675326atural] :
      ( ( produc5541935546810125468atural @ F @ ( produc4464772112783624294atural @ G2 @ X ) )
      = ( produc3831813291587773865atural @ ( F @ ( produc497848011034438852atural @ X ) ) @ ( G2 @ ( produc6591912806276919810atural @ X ) ) ) ) )).

% apfst_apsnd
thf(fact_456_apfst__apsnd,axiom,(
    ! [F: code_natural > produc6241069584506657477e_term > option6357759511663192854e_term,G2: produc7822875418678951345atural > produc8923325533196201883nteger,X: produc5835291356934675326atural] :
      ( ( produc1089855871315519163nteger @ F @ ( produc8287249056265611856atural @ G2 @ X ) )
      = ( produc8603105652947943368nteger @ ( F @ ( produc497848011034438852atural @ X ) ) @ ( G2 @ ( produc6591912806276919810atural @ X ) ) ) ) )).

% apfst_apsnd
thf(fact_457_apfst__apsnd,axiom,(
    ! [F: code_natural > produc8551481072490612790e_term > option6357759511663192854e_term,G2: produc7822875418678951345atural > product_prod_int_int,X: produc5835291356934675326atural] :
      ( ( produc1939560113476300456nt_int @ F @ ( produc5167862320620071534atural @ G2 @ X ) )
      = ( produc5700946648718959541nt_int @ ( F @ ( produc497848011034438852atural @ X ) ) @ ( G2 @ ( produc6591912806276919810atural @ X ) ) ) ) )).

% apfst_apsnd
thf(fact_458_apfst__apsnd,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc8923325533196201883nteger,X: produc8763457246119570046nteger] :
      ( ( produc6157546124166486841nteger @ F @ ( produc1584323072462815568e_term @ G2 @ X ) )
      = ( produc6137756002093451184nteger @ ( F @ ( produc2702340305111172292nteger @ X ) ) @ ( G2 @ ( produc6514156286822040066nteger @ X ) ) ) ) )).

% apfst_apsnd
thf(fact_459_apfst__apsnd,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > int > option6357759511663192854e_term,G2: produc8923325533196201883nteger > product_prod_int_int,X: produc8763457246119570046nteger] :
      ( ( produc1044911711615390792nt_int @ F @ ( produc4747959564457519470e_term @ G2 @ X ) )
      = ( produc4305682042979456191nt_int @ ( F @ ( produc2702340305111172292nteger @ X ) ) @ ( G2 @ ( produc6514156286822040066nteger @ X ) ) ) ) )).

% apfst_apsnd
thf(fact_460_apfst__apsnd,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: product_prod_int_int > produc8923325533196201883nteger,X: produc7773217078559923341nt_int] :
      ( ( produc7871700312893338666nteger @ F @ ( produc185712198560197633e_term @ G2 @ X ) )
      = ( produc6137756002093451184nteger @ ( F @ ( produc6230002227079971283nt_int @ X ) ) @ ( G2 @ ( produc3162348030201620241nt_int @ X ) ) ) ) )).

% apfst_apsnd
thf(fact_461_apfst__apsnd,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > int > option6357759511663192854e_term,G2: product_prod_int_int > product_prod_int_int,X: produc7773217078559923341nt_int] :
      ( ( produc7131893314732144953nt_int @ F @ ( produc7011684061226097695e_term @ G2 @ X ) )
      = ( produc4305682042979456191nt_int @ ( F @ ( produc6230002227079971283nt_int @ X ) ) @ ( G2 @ ( produc3162348030201620241nt_int @ X ) ) ) ) )).

% apfst_apsnd
thf(fact_462_apfst__apsnd,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc4972180933644002618e_term > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc7822875418678951345atural,X: produc8763457246119570046nteger] :
      ( ( produc6743516341412091314atural @ F @ ( produc5989066703516284774e_term @ G2 @ X ) )
      = ( produc3831813291587773865atural @ ( F @ ( produc2702340305111172292nteger @ X ) ) @ ( G2 @ ( produc6514156286822040066nteger @ X ) ) ) ) )).

% apfst_apsnd
thf(fact_463_snd__apfst,axiom,(
    ! [F: code_natural > code_natural,X: produc5835291356934675326atural] :
      ( ( produc6591912806276919810atural @ ( produc4623092060929046819atural @ F @ X ) )
      = ( produc6591912806276919810atural @ X ) ) )).

% snd_apfst
thf(fact_464_snd__apfst,axiom,(
    ! [F: finfun1145668972854894571action > finfun1145668972854894571action,X: produc8923597787767437993list_f] :
      ( ( produc8710481132536983823list_f @ ( produc5173547218785033201list_f @ F @ X ) )
      = ( produc8710481132536983823list_f @ X ) ) )).

% snd_apfst
thf(fact_465_fst__apfst,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,X: produc8763457246119570046nteger] :
      ( ( produc2702340305111172292nteger @ ( produc6157546124166486841nteger @ F @ X ) )
      = ( F @ ( produc2702340305111172292nteger @ X ) ) ) )).

% fst_apfst
thf(fact_466_fst__apfst,axiom,(
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,X: produc1908205239877642774nteger] :
      ( ( produc2702340305111172292nteger @ ( produc1843087649659261265nteger @ F @ X ) )
      = ( F @ ( produc7822682618958472924nteger @ X ) ) ) )).

% fst_apfst
thf(fact_467_fst__apfst,axiom,(
    ! [F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc4972180933644002618e_term > option6357759511663192854e_term,X: produc989692990947075319atural] :
      ( ( produc3051390257598303421atural @ ( produc7227253073565913365atural @ F @ X ) )
      = ( F @ ( produc3051390257598303421atural @ X ) ) ) )).

% fst_apfst
thf(fact_468_fst__apfst,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc6241069584506657477e_term > option6357759511663192854e_term,X: produc8763457246119570046nteger] :
      ( ( produc7822682618958472924nteger @ ( produc2291436665917485009nteger @ F @ X ) )
      = ( F @ ( produc2702340305111172292nteger @ X ) ) ) )).

% fst_apfst
thf(fact_469_fst__apfst,axiom,(
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc6241069584506657477e_term > option6357759511663192854e_term,X: produc1908205239877642774nteger] :
      ( ( produc7822682618958472924nteger @ ( produc2515316116081422825nteger @ F @ X ) )
      = ( F @ ( produc7822682618958472924nteger @ X ) ) ) )).

% fst_apfst
thf(fact_470_fst__apfst,axiom,(
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term,X: produc2285326912895808259nt_int] :
      ( ( produc6816164490631068361nt_int @ ( produc7277921532047535397nt_int @ F @ X ) )
      = ( F @ ( produc6816164490631068361nt_int @ X ) ) ) )).

% fst_apfst
thf(fact_471_fst__apfst,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term,X: produc7773217078559923341nt_int] :
      ( ( produc6816164490631068361nt_int @ ( produc6067946493426971951nt_int @ F @ X ) )
      = ( F @ ( produc6230002227079971283nt_int @ X ) ) ) )).

% fst_apfst
thf(fact_472_fst__apfst,axiom,(
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > int > option6357759511663192854e_term,X: produc2285326912895808259nt_int] :
      ( ( produc6230002227079971283nt_int @ ( produc6919319362177967663nt_int @ F @ X ) )
      = ( F @ ( produc6816164490631068361nt_int @ X ) ) ) )).

% fst_apfst
thf(fact_473_fst__apfst,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > int > option6357759511663192854e_term,X: produc7773217078559923341nt_int] :
      ( ( produc6230002227079971283nt_int @ ( produc7131893314732144953nt_int @ F @ X ) )
      = ( F @ ( produc6230002227079971283nt_int @ X ) ) ) )).

% fst_apfst
thf(fact_474_rtrancl3p__induct3,axiom,(
    ! [R: ( code_integer > option6357759511663192854e_term ) > code_integer > code_integer > typerep > ( code_integer > option6357759511663192854e_term ) > code_integer > code_integer > $o,Ax: code_integer > option6357759511663192854e_term,Ay: code_integer,Az: code_integer,X2a: list_typerep,Cx: code_integer > option6357759511663192854e_term,Cy: code_integer,Cz: code_integer,P: ( code_integer > option6357759511663192854e_term ) > code_integer > code_integer > list_typerep > ( code_integer > option6357759511663192854e_term ) > code_integer > code_integer > $o] :
      ( ( rtranc8414796551611986717yperep
        @ ^ [V2: produc8763457246119570046nteger,W: typerep] :
            ( produc127349428274296955eger_o
            @ ^ [X4: code_integer > option6357759511663192854e_term] :
                ( produc2066375834425727024eger_o
                @ ( produc8655822042727976503eger_o
                  @ ^ [Y6: code_integer > option6357759511663192854e_term] :
                      ( produc808496206219636204eger_o @ ( R @ Y6 ) )
                  @ V2
                  @ W
                  @ X4 ) ) )
        @ ( produc6137756002093451184nteger @ Ax @ ( produc1086072967326762835nteger @ Ay @ Az ) )
        @ X2a
        @ ( produc6137756002093451184nteger @ Cx @ ( produc1086072967326762835nteger @ Cy @ Cz ) ) )
     => ( ! [A3: code_integer > option6357759511663192854e_term,Aa: code_integer,B3: code_integer] :
            ( P @ A3 @ Aa @ B3 @ nil_typerep @ A3 @ Aa @ B3 )
       => ( ! [A3: code_integer > option6357759511663192854e_term,Aa: code_integer,B3: code_integer,Bs: list_typerep,Ab: code_integer > option6357759511663192854e_term,Ac: code_integer,Ba: code_integer,Bb: typerep,Ad: code_integer > option6357759511663192854e_term,Ae: code_integer,Bc: code_integer] :
              ( ( rtranc8414796551611986717yperep
                @ ^ [V2: produc8763457246119570046nteger,W: typerep] :
                    ( produc127349428274296955eger_o
                    @ ^ [X4: code_integer > option6357759511663192854e_term] :
                        ( produc2066375834425727024eger_o
                        @ ( produc8655822042727976503eger_o
                          @ ^ [Y6: code_integer > option6357759511663192854e_term] :
                              ( produc808496206219636204eger_o @ ( R @ Y6 ) )
                          @ V2
                          @ W
                          @ X4 ) ) )
                @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ Aa @ B3 ) )
                @ Bs
                @ ( produc6137756002093451184nteger @ Ab @ ( produc1086072967326762835nteger @ Ac @ Ba ) ) )
             => ( ( P @ A3 @ Aa @ B3 @ Bs @ Ab @ Ac @ Ba )
               => ( ( R @ Ab @ Ac @ Ba @ Bb @ Ad @ Ae @ Bc )
                 => ( P @ A3 @ Aa @ B3 @ ( append_typerep @ Bs @ ( cons_typerep @ Bb @ nil_typerep ) ) @ Ad @ Ae @ Bc ) ) ) )
         => ( P @ Ax @ Ay @ Az @ X2a @ Cx @ Cy @ Cz ) ) ) ) )).

% rtrancl3p_induct3
thf(fact_475_rtrancl3p__induct3,axiom,(
    ! [R: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > code_natural > code_natural > typerep > ( produc4972180933644002618e_term > option6357759511663192854e_term ) > code_natural > code_natural > $o,Ax: produc4972180933644002618e_term > option6357759511663192854e_term,Ay: code_natural,Az: code_natural,X2a: list_typerep,Cx: produc4972180933644002618e_term > option6357759511663192854e_term,Cy: code_natural,Cz: code_natural,P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > code_natural > code_natural > list_typerep > ( produc4972180933644002618e_term > option6357759511663192854e_term ) > code_natural > code_natural > $o] :
      ( ( rtranc672236675657768086yperep
        @ ^ [V2: produc989692990947075319atural,W: typerep] :
            ( produc2278936416606902338ural_o
            @ ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term] :
                ( produc8374411701426027290ural_o
                @ ( produc166806319883733707ural_o
                  @ ^ [Y6: produc4972180933644002618e_term > option6357759511663192854e_term] :
                      ( produc3244309023640426915ural_o @ ( R @ Y6 ) )
                  @ V2
                  @ W
                  @ X4 ) ) )
        @ ( produc3831813291587773865atural @ Ax @ ( produc3574140220909816553atural @ Ay @ Az ) )
        @ X2a
        @ ( produc3831813291587773865atural @ Cx @ ( produc3574140220909816553atural @ Cy @ Cz ) ) )
     => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,Aa: code_natural,B3: code_natural] :
            ( P @ A3 @ Aa @ B3 @ nil_typerep @ A3 @ Aa @ B3 )
       => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,Aa: code_natural,B3: code_natural,Bs: list_typerep,Ab: produc4972180933644002618e_term > option6357759511663192854e_term,Ac: code_natural,Ba: code_natural,Bb: typerep,Ad: produc4972180933644002618e_term > option6357759511663192854e_term,Ae: code_natural,Bc: code_natural] :
              ( ( rtranc672236675657768086yperep
                @ ^ [V2: produc989692990947075319atural,W: typerep] :
                    ( produc2278936416606902338ural_o
                    @ ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term] :
                        ( produc8374411701426027290ural_o
                        @ ( produc166806319883733707ural_o
                          @ ^ [Y6: produc4972180933644002618e_term > option6357759511663192854e_term] :
                              ( produc3244309023640426915ural_o @ ( R @ Y6 ) )
                          @ V2
                          @ W
                          @ X4 ) ) )
                @ ( produc3831813291587773865atural @ A3 @ ( produc3574140220909816553atural @ Aa @ B3 ) )
                @ Bs
                @ ( produc3831813291587773865atural @ Ab @ ( produc3574140220909816553atural @ Ac @ Ba ) ) )
             => ( ( P @ A3 @ Aa @ B3 @ Bs @ Ab @ Ac @ Ba )
               => ( ( R @ Ab @ Ac @ Ba @ Bb @ Ad @ Ae @ Bc )
                 => ( P @ A3 @ Aa @ B3 @ ( append_typerep @ Bs @ ( cons_typerep @ Bb @ nil_typerep ) ) @ Ad @ Ae @ Bc ) ) ) )
         => ( P @ Ax @ Ay @ Az @ X2a @ Cx @ Cy @ Cz ) ) ) ) )).

% rtrancl3p_induct3
thf(fact_476_rtrancl3p__induct3,axiom,(
    ! [R: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > code_integer > code_integer > typerep > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > code_integer > code_integer > $o,Ax: produc6241069584506657477e_term > option6357759511663192854e_term,Ay: code_integer,Az: code_integer,X2a: list_typerep,Cx: produc6241069584506657477e_term > option6357759511663192854e_term,Cy: code_integer,Cz: code_integer,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > code_integer > code_integer > list_typerep > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > code_integer > code_integer > $o] :
      ( ( rtranc7906171019283146677yperep
        @ ^ [V2: produc1908205239877642774nteger,W: typerep] :
            ( produc6253627499356882019eger_o
            @ ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term] :
                ( produc2066375834425727024eger_o
                @ ( produc5172121922217842743eger_o
                  @ ^ [Y6: produc6241069584506657477e_term > option6357759511663192854e_term] :
                      ( produc4206498124266244740eger_o @ ( R @ Y6 ) )
                  @ V2
                  @ W
                  @ X4 ) ) )
        @ ( produc8603105652947943368nteger @ Ax @ ( produc1086072967326762835nteger @ Ay @ Az ) )
        @ X2a
        @ ( produc8603105652947943368nteger @ Cx @ ( produc1086072967326762835nteger @ Cy @ Cz ) ) )
     => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,Aa: code_integer,B3: code_integer] :
            ( P @ A3 @ Aa @ B3 @ nil_typerep @ A3 @ Aa @ B3 )
       => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,Aa: code_integer,B3: code_integer,Bs: list_typerep,Ab: produc6241069584506657477e_term > option6357759511663192854e_term,Ac: code_integer,Ba: code_integer,Bb: typerep,Ad: produc6241069584506657477e_term > option6357759511663192854e_term,Ae: code_integer,Bc: code_integer] :
              ( ( rtranc7906171019283146677yperep
                @ ^ [V2: produc1908205239877642774nteger,W: typerep] :
                    ( produc6253627499356882019eger_o
                    @ ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term] :
                        ( produc2066375834425727024eger_o
                        @ ( produc5172121922217842743eger_o
                          @ ^ [Y6: produc6241069584506657477e_term > option6357759511663192854e_term] :
                              ( produc4206498124266244740eger_o @ ( R @ Y6 ) )
                          @ V2
                          @ W
                          @ X4 ) ) )
                @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ Aa @ B3 ) )
                @ Bs
                @ ( produc8603105652947943368nteger @ Ab @ ( produc1086072967326762835nteger @ Ac @ Ba ) ) )
             => ( ( P @ A3 @ Aa @ B3 @ Bs @ Ab @ Ac @ Ba )
               => ( ( R @ Ab @ Ac @ Ba @ Bb @ Ad @ Ae @ Bc )
                 => ( P @ A3 @ Aa @ B3 @ ( append_typerep @ Bs @ ( cons_typerep @ Bb @ nil_typerep ) ) @ Ad @ Ae @ Bc ) ) ) )
         => ( P @ Ax @ Ay @ Az @ X2a @ Cx @ Cy @ Cz ) ) ) ) )).

% rtrancl3p_induct3
thf(fact_477_rtrancl3p__induct3,axiom,(
    ! [R: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > int > int > typerep > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > int > int > $o,Ax: produc8551481072490612790e_term > option6357759511663192854e_term,Ay: int,Az: int,X2a: list_typerep,Cx: produc8551481072490612790e_term > option6357759511663192854e_term,Cy: int,Cz: int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > int > int > list_typerep > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > int > int > $o] :
      ( ( rtranc3619583098232052386yperep
        @ ^ [V2: produc2285326912895808259nt_int,W: typerep] :
            ( produc1573362020775583542_int_o
            @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term] :
                ( produc4947309494688390418_int_o
                @ ( produc8603098794475515835_int_o
                  @ ^ [Y6: produc8551481072490612790e_term > option6357759511663192854e_term] :
                      ( produc9047590124310039447_int_o @ ( R @ Y6 ) )
                  @ V2
                  @ W
                  @ X4 ) ) )
        @ ( produc5700946648718959541nt_int @ Ax @ ( product_Pair_int_int @ Ay @ Az ) )
        @ X2a
        @ ( produc5700946648718959541nt_int @ Cx @ ( product_Pair_int_int @ Cy @ Cz ) ) )
     => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,Aa: int,B3: int] :
            ( P @ A3 @ Aa @ B3 @ nil_typerep @ A3 @ Aa @ B3 )
       => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,Aa: int,B3: int,Bs: list_typerep,Ab: produc8551481072490612790e_term > option6357759511663192854e_term,Ac: int,Ba: int,Bb: typerep,Ad: produc8551481072490612790e_term > option6357759511663192854e_term,Ae: int,Bc: int] :
              ( ( rtranc3619583098232052386yperep
                @ ^ [V2: produc2285326912895808259nt_int,W: typerep] :
                    ( produc1573362020775583542_int_o
                    @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term] :
                        ( produc4947309494688390418_int_o
                        @ ( produc8603098794475515835_int_o
                          @ ^ [Y6: produc8551481072490612790e_term > option6357759511663192854e_term] :
                              ( produc9047590124310039447_int_o @ ( R @ Y6 ) )
                          @ V2
                          @ W
                          @ X4 ) ) )
                @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ Aa @ B3 ) )
                @ Bs
                @ ( produc5700946648718959541nt_int @ Ab @ ( product_Pair_int_int @ Ac @ Ba ) ) )
             => ( ( P @ A3 @ Aa @ B3 @ Bs @ Ab @ Ac @ Ba )
               => ( ( R @ Ab @ Ac @ Ba @ Bb @ Ad @ Ae @ Bc )
                 => ( P @ A3 @ Aa @ B3 @ ( append_typerep @ Bs @ ( cons_typerep @ Bb @ nil_typerep ) ) @ Ad @ Ae @ Bc ) ) ) )
         => ( P @ Ax @ Ay @ Az @ X2a @ Cx @ Cy @ Cz ) ) ) ) )).

% rtrancl3p_induct3
thf(fact_478_rtrancl3p__induct3,axiom,(
    ! [R: ( int > option6357759511663192854e_term ) > int > int > typerep > ( int > option6357759511663192854e_term ) > int > int > $o,Ax: int > option6357759511663192854e_term,Ay: int,Az: int,X2a: list_typerep,Cx: int > option6357759511663192854e_term,Cy: int,Cz: int,P: ( int > option6357759511663192854e_term ) > int > int > list_typerep > ( int > option6357759511663192854e_term ) > int > int > $o] :
      ( ( rtranc4852721826211545900yperep
        @ ^ [V2: produc7773217078559923341nt_int,W: typerep] :
            ( produc2558449545302689196_int_o
            @ ^ [X4: int > option6357759511663192854e_term] :
                ( produc4947309494688390418_int_o
                @ ( produc7553314257896368827_int_o
                  @ ^ [Y6: int > option6357759511663192854e_term] :
                      ( produc4739070747422591521_int_o @ ( R @ Y6 ) )
                  @ V2
                  @ W
                  @ X4 ) ) )
        @ ( produc4305682042979456191nt_int @ Ax @ ( product_Pair_int_int @ Ay @ Az ) )
        @ X2a
        @ ( produc4305682042979456191nt_int @ Cx @ ( product_Pair_int_int @ Cy @ Cz ) ) )
     => ( ! [A3: int > option6357759511663192854e_term,Aa: int,B3: int] :
            ( P @ A3 @ Aa @ B3 @ nil_typerep @ A3 @ Aa @ B3 )
       => ( ! [A3: int > option6357759511663192854e_term,Aa: int,B3: int,Bs: list_typerep,Ab: int > option6357759511663192854e_term,Ac: int,Ba: int,Bb: typerep,Ad: int > option6357759511663192854e_term,Ae: int,Bc: int] :
              ( ( rtranc4852721826211545900yperep
                @ ^ [V2: produc7773217078559923341nt_int,W: typerep] :
                    ( produc2558449545302689196_int_o
                    @ ^ [X4: int > option6357759511663192854e_term] :
                        ( produc4947309494688390418_int_o
                        @ ( produc7553314257896368827_int_o
                          @ ^ [Y6: int > option6357759511663192854e_term] :
                              ( produc4739070747422591521_int_o @ ( R @ Y6 ) )
                          @ V2
                          @ W
                          @ X4 ) ) )
                @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ Aa @ B3 ) )
                @ Bs
                @ ( produc4305682042979456191nt_int @ Ab @ ( product_Pair_int_int @ Ac @ Ba ) ) )
             => ( ( P @ A3 @ Aa @ B3 @ Bs @ Ab @ Ac @ Ba )
               => ( ( R @ Ab @ Ac @ Ba @ Bb @ Ad @ Ae @ Bc )
                 => ( P @ A3 @ Aa @ B3 @ ( append_typerep @ Bs @ ( cons_typerep @ Bb @ nil_typerep ) ) @ Ad @ Ae @ Bc ) ) ) )
         => ( P @ Ax @ Ay @ Az @ X2a @ Cx @ Cy @ Cz ) ) ) ) )).

% rtrancl3p_induct3
thf(fact_479_snd__comp__apfst,axiom,(
    ! [F: code_natural > code_natural] :
      ( ( comp_P7610810914149544718atural @ produc6591912806276919810atural @ ( produc4623092060929046819atural @ F ) )
      = produc6591912806276919810atural ) )).

% snd_comp_apfst
thf(fact_480_snd__comp__apfst,axiom,(
    ! [F: finfun1145668972854894571action > finfun1145668972854894571action] :
      ( ( comp_P4267156721827287172list_f @ produc8710481132536983823list_f @ ( produc5173547218785033201list_f @ F ) )
      = produc8710481132536983823list_f ) )).

% snd_comp_apfst
thf(fact_481_fst__comp__apfst,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term] :
      ( ( comp_P1446425635098721907nteger @ produc2702340305111172292nteger @ ( produc6157546124166486841nteger @ F ) )
      = ( comp_C6219200931300381191nteger @ F @ produc2702340305111172292nteger ) ) )).

% fst_comp_apfst
thf(fact_482_fst__comp__apfst,axiom,(
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term] :
      ( ( comp_P8141014749814354059nteger @ produc2702340305111172292nteger @ ( produc1843087649659261265nteger @ F ) )
      = ( comp_P6703551259509254535nteger @ F @ produc7822682618958472924nteger ) ) )).

% fst_comp_apfst
thf(fact_483_fst__comp__apfst,axiom,(
    ! [F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc4972180933644002618e_term > option6357759511663192854e_term] :
      ( ( comp_P6380929843994337986atural @ produc3051390257598303421atural @ ( produc7227253073565913365atural @ F ) )
      = ( comp_P9178335274882037178atural @ F @ produc3051390257598303421atural ) ) )).

% fst_comp_apfst
thf(fact_484_fst__comp__apfst,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( comp_P6133064115347390067nteger @ produc7822682618958472924nteger @ ( produc2291436665917485009nteger @ F ) )
      = ( comp_C1695206475658992367nteger @ F @ produc2702340305111172292nteger ) ) )).

% fst_comp_apfst
thf(fact_485_fst__comp__apfst,axiom,(
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( comp_P336008677415481483nteger @ produc7822682618958472924nteger @ ( produc2515316116081422825nteger @ F ) )
      = ( comp_P7860067074541754991nteger @ F @ produc7822682618958472924nteger ) ) )).

% fst_comp_apfst
thf(fact_486_fst__comp__apfst,axiom,(
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( comp_P9041276550952352214nt_int @ produc6816164490631068361nt_int @ ( produc7277921532047535397nt_int @ F ) )
      = ( comp_P2617176247610644414nt_int @ F @ produc6816164490631068361nt_int ) ) )).

% fst_comp_apfst
thf(fact_487_fst__comp__apfst,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( comp_P4788939897393574368nt_int @ produc6816164490631068361nt_int @ ( produc6067946493426971951nt_int @ F ) )
      = ( comp_i3547140843593382846nt_int @ F @ produc6230002227079971283nt_int ) ) )).

% fst_comp_apfst
thf(fact_488_fst__comp__apfst,axiom,(
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > int > option6357759511663192854e_term] :
      ( ( comp_P758448786518603222nt_int @ produc6230002227079971283nt_int @ ( produc6919319362177967663nt_int @ F ) )
      = ( comp_P4200961762685571508nt_int @ F @ produc6816164490631068361nt_int ) ) )).

% fst_comp_apfst
thf(fact_489_fst__comp__apfst,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > int > option6357759511663192854e_term] :
      ( ( comp_P5815951645081071072nt_int @ produc6230002227079971283nt_int @ ( produc7131893314732144953nt_int @ F ) )
      = ( comp_i1330301680410484660nt_int @ F @ produc6230002227079971283nt_int ) ) )).

% fst_comp_apfst
thf(fact_490_imageI,axiom,(
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X @ A4 )
     => ( member8440522571783428010at_nat @ ( F @ X ) @ ( image_5168914502847457605at_nat @ F @ A4 ) ) ) )).

% imageI
thf(fact_491_imageI,axiom,(
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A4 )
     => ( member8206827879206165904at_nat @ ( F @ X ) @ ( image_1119892238107900587at_nat @ F @ A4 ) ) ) )).

% imageI
thf(fact_492_imageI,axiom,(
    ! [X: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ X @ A4 )
     => ( member8440522571783428010at_nat @ ( F @ X ) @ ( image_6706873702294942047at_nat @ F @ A4 ) ) ) )).

% imageI
thf(fact_493_imageI,axiom,(
    ! [X: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ X @ A4 )
     => ( member8206827879206165904at_nat @ ( F @ X ) @ ( image_6321802852832981957at_nat @ F @ A4 ) ) ) )).

% imageI
thf(fact_494_rtrancl__induct,axiom,(
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( ( P @ A )
       => ( ! [Y3: nat,Z4: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi2905341329935302413cl_nat @ R ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z4 ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z4 ) ) ) )
         => ( P @ B ) ) ) ) )).

% rtrancl_induct
thf(fact_495_rtrancl__induct,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ R ) )
     => ( ( P @ A )
       => ( ! [Y3: product_prod_nat_nat,Z4: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ Y3 ) @ ( transi8609417484261754244at_nat @ R ) )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z4 ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z4 ) ) ) )
         => ( P @ B ) ) ) ) )).

% rtrancl_induct
thf(fact_496_comp__cong,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc8923597787767437993list_f > produc1811169103916964701list_f,X: produc8923597787767437993list_f,F4: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G3: produc8923597787767437993list_f > produc1811169103916964701list_f,X6: produc8923597787767437993list_f] :
      ( ( ( F @ ( G2 @ X ) )
        = ( F4 @ ( G3 @ X6 ) ) )
     => ( ( comp_P4767578765025054880list_f @ F @ G2 @ X )
        = ( comp_P4767578765025054880list_f @ F4 @ G3 @ X6 ) ) ) )).

% comp_cong
thf(fact_497_comp__eq__elim,axiom,(
    ! [A: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,B: produc8923597787767437993list_f > produc1811169103916964701list_f,C2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,D: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( ( comp_P4767578765025054880list_f @ A @ B )
        = ( comp_P4767578765025054880list_f @ C2 @ D ) )
     => ! [V3: produc8923597787767437993list_f] :
          ( ( A @ ( B @ V3 ) )
          = ( C2 @ ( D @ V3 ) ) ) ) )).

% comp_eq_elim
thf(fact_498_singletonD,axiom,(
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
     => ( B = A ) ) )).

% singletonD
thf(fact_499_singletonD,axiom,(
    ! [B: produc859450856879609959at_nat,A: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ B @ ( insert5050368324300391991at_nat @ A @ bot_bo5327735625951526323at_nat ) )
     => ( B = A ) ) )).

% singletonD
thf(fact_500_comp__assoc,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc8923597787767437993list_f > produc1811169103916964701list_f,H: produc8923597787767437993list_f > produc8923597787767437993list_f] :
      ( ( comp_P1644921750638218854list_f @ ( comp_P4767578765025054880list_f @ F @ G2 ) @ H )
      = ( comp_P4767578765025054880list_f @ F @ ( comp_P4267156721827287172list_f @ G2 @ H ) ) ) )).

% comp_assoc
thf(fact_501_comp__assoc,axiom,(
    ! [F: list_n1973683603564934665_a_b_c > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,H: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( comp_P4767578765025054880list_f @ ( comp_l6492537726383095040list_f @ F @ G2 ) @ H )
      = ( comp_l8432902307485339590list_f @ F @ ( comp_P4767578765025054880list_f @ G2 @ H ) ) ) )).

% comp_assoc
thf(fact_502_comp__assoc,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > produc1811169103916964701list_f,H: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( comp_P4767578765025054880list_f @ ( comp_P5271726855292207526list_f @ F @ G2 ) @ H )
      = ( comp_P4767578765025054880list_f @ F @ ( comp_P2673361230361438154list_f @ G2 @ H ) ) ) )).

% comp_assoc
thf(fact_503_subrelI,axiom,(
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R @ S ) ) )).

% subrelI
thf(fact_504_subrelI,axiom,(
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3000389064537975527at_nat @ R @ S ) ) )).

% subrelI
thf(fact_505_subrelI,axiom,(
    ! [R: set_Pr8056137968301705908nteger,S: set_Pr8056137968301705908nteger] :
      ( ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
          ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ R )
         => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ S ) )
     => ( ord_le3216752416896350996nteger @ R @ S ) ) )).

% subrelI
thf(fact_506_subrelI,axiom,(
    ! [R: set_Pr4244404650639074861atural,S: set_Pr4244404650639074861atural] :
      ( ! [X3: produc4972180933644002618e_term > option6357759511663192854e_term,Y3: produc7822875418678951345atural] :
          ( ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X3 @ Y3 ) @ R )
         => ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X3 @ Y3 ) @ S ) )
     => ( ord_le3616440322383431309atural @ R @ S ) ) )).

% subrelI
thf(fact_507_subrelI,axiom,(
    ! [R: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
      ( ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
          ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R )
         => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ S ) )
     => ( ord_le653643898420964396nteger @ R @ S ) ) )).

% subrelI
thf(fact_508_subrelI,axiom,(
    ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
          ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R )
         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S ) )
     => ( ord_le8725513860283290265nt_int @ R @ S ) ) )).

% subrelI
thf(fact_509_subrelI,axiom,(
    ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
          ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S ) )
     => ( ord_le135402666524580259nt_int @ R @ S ) ) )).

% subrelI
thf(fact_510_cond__case__prod__eta,axiom,(
    ! [F: code_natural > code_natural > produc6756470014200349799atural,G2: produc7822875418678951345atural > produc6756470014200349799atural] :
      ( ! [X3: code_natural,Y3: code_natural] :
          ( ( F @ X3 @ Y3 )
          = ( G2 @ ( produc3574140220909816553atural @ X3 @ Y3 ) ) )
     => ( ( produc8282080750456430313atural @ F )
        = G2 ) ) )).

% cond_case_prod_eta
thf(fact_511_pair__lessI1,axiom,(
    ! [A: nat,B: nat,S: nat,T3: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_less ) ) )).

% pair_lessI1
thf(fact_512_image__eq__imp__comp,axiom,(
    ! [F: produc8923597787767437993list_f > produc1811169103916964701list_f,A4: set_Pr8454954044441558665list_f,G2: produc8923597787767437993list_f > produc1811169103916964701list_f,B5: set_Pr8454954044441558665list_f,H: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ( ( image_4621308865571145409list_f @ F @ A4 )
        = ( image_4621308865571145409list_f @ G2 @ B5 ) )
     => ( ( image_6608088440629302181_a_b_c @ ( comp_P4767578765025054880list_f @ H @ F ) @ A4 )
        = ( image_6608088440629302181_a_b_c @ ( comp_P4767578765025054880list_f @ H @ G2 ) @ B5 ) ) ) )).

% image_eq_imp_comp
thf(fact_513_pair__leqI2,axiom,(
    ! [A: nat,B: nat,S: nat,T3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S @ T3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_leq ) ) ) )).

% pair_leqI2
thf(fact_514_image__comp,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc8923597787767437993list_f > produc1811169103916964701list_f,R: set_Pr8454954044441558665list_f] :
      ( ( image_6684545758539136421_a_b_c @ F @ ( image_4621308865571145409list_f @ G2 @ R ) )
      = ( image_6608088440629302181_a_b_c @ ( comp_P4767578765025054880list_f @ F @ G2 ) @ R ) ) )).

% image_comp
thf(fact_515_pair__leqI1,axiom,(
    ! [A: nat,B: nat,S: nat,T3: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_leq ) ) )).

% pair_leqI1
thf(fact_516_Sup_OSUP__image,axiom,(
    ! [Sup: set_li1323980635854540777_a_b_c > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,F: produc8923597787767437993list_f > produc1811169103916964701list_f,A4: set_Pr8454954044441558665list_f] :
      ( ( Sup @ ( image_6684545758539136421_a_b_c @ G2 @ ( image_4621308865571145409list_f @ F @ A4 ) ) )
      = ( Sup @ ( image_6608088440629302181_a_b_c @ ( comp_P4767578765025054880list_f @ G2 @ F ) @ A4 ) ) ) )).

% Sup.SUP_image
thf(fact_517_Inf_OINF__image,axiom,(
    ! [Inf: set_li1323980635854540777_a_b_c > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,F: produc8923597787767437993list_f > produc1811169103916964701list_f,A4: set_Pr8454954044441558665list_f] :
      ( ( Inf @ ( image_6684545758539136421_a_b_c @ G2 @ ( image_4621308865571145409list_f @ F @ A4 ) ) )
      = ( Inf @ ( image_6608088440629302181_a_b_c @ ( comp_P4767578765025054880list_f @ G2 @ F ) @ A4 ) ) ) )).

% Inf.INF_image
thf(fact_518_fst__fstOp,axiom,(
    ! [P: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,Q: produc8923325533196201883nteger > produc8923325533196201883nteger > $o] :
      ( produc2702340305111172292nteger
      = ( comp_P1446425635098721907nteger @ produc2702340305111172292nteger @ ( bNF_fs7749581648564785796nteger @ P @ Q ) ) ) )).

% fst_fstOp
thf(fact_519_fst__fstOp,axiom,(
    ! [P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o,Q: produc7822875418678951345atural > produc7822875418678951345atural > $o] :
      ( produc3051390257598303421atural
      = ( comp_P6380929843994337986atural @ produc3051390257598303421atural @ ( bNF_fs632686752235921505atural @ P @ Q ) ) ) )).

% fst_fstOp
thf(fact_520_fst__fstOp,axiom,(
    ! [P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,Q: produc8923325533196201883nteger > produc8923325533196201883nteger > $o] :
      ( produc7822682618958472924nteger
      = ( comp_P336008677415481483nteger @ produc7822682618958472924nteger @ ( bNF_fs2771997879588465004nteger @ P @ Q ) ) ) )).

% fst_fstOp
thf(fact_521_fst__fstOp,axiom,(
    ! [P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,Q: product_prod_int_int > product_prod_int_int > $o] :
      ( produc6816164490631068361nt_int
      = ( comp_P9041276550952352214nt_int @ produc6816164490631068361nt_int @ ( bNF_fs7693636466261331293nt_int @ P @ Q ) ) ) )).

% fst_fstOp
thf(fact_522_fst__fstOp,axiom,(
    ! [P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,Q: product_prod_int_int > product_prod_int_int > $o] :
      ( produc6230002227079971283nt_int
      = ( comp_P5815951645081071072nt_int @ produc6230002227079971283nt_int @ ( bNF_fs296160452230587091nt_int @ P @ Q ) ) ) )).

% fst_fstOp
thf(fact_523_pair__lessI2,axiom,(
    ! [A: nat,B: nat,S: nat,T3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S @ T3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_less ) ) ) )).

% pair_lessI2
thf(fact_524_snd__sndOp,axiom,(
    ! [P: code_natural > code_natural > $o,Q: code_natural > produc7822875418678951345atural > $o] :
      ( produc6591912806276919810atural
      = ( comp_P7610810914149544718atural @ produc6591912806276919810atural @ ( bNF_sn8722645597740893163atural @ P @ Q ) ) ) )).

% snd_sndOp
thf(fact_525_snd__sndOp,axiom,(
    ! [P: finfun1145668972854894571action > finfun1145668972854894571action > $o,Q: finfun1145668972854894571action > produc1811169103916964701list_f > $o] :
      ( produc8710481132536983823list_f
      = ( comp_P4267156721827287172list_f @ produc8710481132536983823list_f @ ( bNF_sn101281456809739049list_f @ P @ Q ) ) ) )).

% snd_sndOp
thf(fact_526_Collect__split__mono__strong,axiom,(
    ! [X7: set_Co9149898834107579976e_term,A4: set_Pr8056137968301705908nteger,Y7: set_Pr4811707699266497531nteger,P: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,Q: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( X7
        = ( image_4933254100216391447e_term @ produc2702340305111172292nteger @ A4 ) )
     => ( ( Y7
          = ( image_3972216771474090454nteger @ produc6514156286822040066nteger @ A4 ) )
       => ( ! [X3: code_integer > option6357759511663192854e_term] :
              ( ( member1535805642427569193e_term @ X3 @ X7 )
             => ! [Xa2: produc8923325533196201883nteger] :
                  ( ( member157494554546826820nteger @ Xa2 @ Y7 )
                 => ( ( P @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le3216752416896350996nteger @ A4 @ ( collec6350289969311926995nteger @ ( produc127349428274296955eger_o @ P ) ) )
           => ( ord_le3216752416896350996nteger @ A4 @ ( collec6350289969311926995nteger @ ( produc127349428274296955eger_o @ Q ) ) ) ) ) ) ) )).

% Collect_split_mono_strong
thf(fact_527_Collect__split__mono__strong,axiom,(
    ! [X7: set_Pr2791530082921190693e_term,A4: set_Pr4244404650639074861atural,Y7: set_Pr7564037697178849041atural,P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o,Q: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o] :
      ( ( X7
        = ( image_5638616612462892909e_term @ produc3051390257598303421atural @ A4 ) )
     => ( ( Y7
          = ( image_5725007063562324787atural @ produc3085575039554877691atural @ A4 ) )
       => ( ! [X3: produc4972180933644002618e_term > option6357759511663192854e_term] :
              ( ( member2820825060907577222e_term @ X3 @ X7 )
             => ! [Xa2: produc7822875418678951345atural] :
                  ( ( member8280416476884352090atural @ Xa2 @ Y7 )
                 => ( ( P @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le3616440322383431309atural @ A4 @ ( collec6032437096461363148atural @ ( produc2278936416606902338ural_o @ P ) ) )
           => ( ord_le3616440322383431309atural @ A4 @ ( collec6032437096461363148atural @ ( produc2278936416606902338ural_o @ Q ) ) ) ) ) ) ) )).

% Collect_split_mono_strong
thf(fact_528_Collect__split__mono__strong,axiom,(
    ! [X7: set_Pr7604974323444597168e_term,A4: set_Pr1281608226676607948nteger,Y7: set_Pr4811707699266497531nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,Q: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( X7
        = ( image_4436781500471753239e_term @ produc7822682618958472924nteger @ A4 ) )
     => ( ( Y7
          = ( image_3141030398089242942nteger @ produc7856867400915047194nteger @ A4 ) )
       => ( ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term] :
              ( ( member4242434998011752849e_term @ X3 @ X7 )
             => ! [Xa2: produc8923325533196201883nteger] :
                  ( ( member157494554546826820nteger @ Xa2 @ Y7 )
                 => ( ( P @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le653643898420964396nteger @ A4 @ ( collec6950949345391930603nteger @ ( produc6253627499356882019eger_o @ P ) ) )
           => ( ord_le653643898420964396nteger @ A4 @ ( collec6950949345391930603nteger @ ( produc6253627499356882019eger_o @ Q ) ) ) ) ) ) ) )).

% Collect_split_mono_strong
thf(fact_529_Collect__split__mono__strong,axiom,(
    ! [X7: set_Pr3642885161833720865e_term,A4: set_Pr9222295170931077689nt_int,Y7: set_Pr958786334691620121nt_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,Q: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( X7
        = ( image_142517975987106933e_term @ produc6816164490631068361nt_int @ A4 ) )
     => ( ( Y7
          = ( image_3915291023682402351nt_int @ produc7328097813583171335nt_int @ A4 ) )
       => ( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term] :
              ( ( member3222579708246209666e_term @ X3 @ X7 )
             => ! [Xa2: product_prod_int_int] :
                  ( ( member5262025264175285858nt_int @ Xa2 @ Y7 )
                 => ( ( P @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le8725513860283290265nt_int @ A4 @ ( collec1790188477890212312nt_int @ ( produc1573362020775583542_int_o @ P ) ) )
           => ( ord_le8725513860283290265nt_int @ A4 @ ( collec1790188477890212312nt_int @ ( produc1573362020775583542_int_o @ Q ) ) ) ) ) ) ) )).

% Collect_split_mono_strong
thf(fact_530_Collect__split__mono__strong,axiom,(
    ! [X7: set_in3461395444621081367e_term,A4: set_Pr1872883991513573699nt_int,Y7: set_Pr958786334691620121nt_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,Q: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( X7
        = ( image_4217645046469565813e_term @ produc6230002227079971283nt_int @ A4 ) )
     => ( ( Y7
          = ( image_3108077152341812517nt_int @ produc3162348030201620241nt_int @ A4 ) )
       => ( ! [X3: int > option6357759511663192854e_term] :
              ( ( member8845023287901829240e_term @ X3 @ X7 )
             => ! [Xa2: product_prod_int_int] :
                  ( ( member5262025264175285858nt_int @ Xa2 @ Y7 )
                 => ( ( P @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le135402666524580259nt_int @ A4 @ ( collec506566255779805410nt_int @ ( produc2558449545302689196_int_o @ P ) ) )
           => ( ord_le135402666524580259nt_int @ A4 @ ( collec506566255779805410nt_int @ ( produc2558449545302689196_int_o @ Q ) ) ) ) ) ) ) )).

% Collect_split_mono_strong
thf(fact_531_Collect__split__mono__strong,axiom,(
    ! [X7: set_Code_natural,A4: set_Pr2352657477744078772atural,Y7: set_Pr7564037697178849041atural,P: code_natural > produc7822875418678951345atural > $o,Q: code_natural > produc7822875418678951345atural > $o] :
      ( ( X7
        = ( image_8194678751871465133atural @ produc497848011034438852atural @ A4 ) )
     => ( ( Y7
          = ( image_3837677093542026988atural @ produc6591912806276919810atural @ A4 ) )
       => ( ! [X3: code_natural] :
              ( ( member_Code_natural @ X3 @ X7 )
             => ! [Xa2: produc7822875418678951345atural] :
                  ( ( member8280416476884352090atural @ Xa2 @ Y7 )
                 => ( ( P @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le2202040463148982292atural @ A4 @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ P ) ) )
           => ( ord_le2202040463148982292atural @ A4 @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ Q ) ) ) ) ) ) ) )).

% Collect_split_mono_strong
thf(fact_532_Collect__split__mono__strong,axiom,(
    ! [X7: set_fi1428165280859657121action,A4: set_Pr8454954044441558665list_f,Y7: set_Pr4157219595312563347list_f,P: finfun1145668972854894571action > produc1811169103916964701list_f > $o,Q: finfun1145668972854894571action > produc1811169103916964701list_f > $o] :
      ( ( X7
        = ( image_6251736531144035663action @ produc3089225533250117581list_f @ A4 ) )
     => ( ( Y7
          = ( image_4621308865571145409list_f @ produc8710481132536983823list_f @ A4 ) )
       => ( ! [X3: finfun1145668972854894571action] :
              ( ( member4650752750499974146action @ X3 @ X7 )
             => ! [Xa2: produc1811169103916964701list_f] :
                  ( ( member6304641671555014644list_f @ Xa2 @ Y7 )
                 => ( ( P @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le3852096572277380649list_f @ A4 @ ( collec2941370454899091988list_f @ ( produc7065790152434098210st_f_o @ P ) ) )
           => ( ord_le3852096572277380649list_f @ A4 @ ( collec2941370454899091988list_f @ ( produc7065790152434098210st_f_o @ Q ) ) ) ) ) ) ) )).

% Collect_split_mono_strong
thf(fact_533_case__prodE,axiom,(
    ! [C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P2: produc8763457246119570046nteger] :
      ( ( produc127349428274296955eger_o @ C2 @ P2 )
     => ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ( ( P2
              = ( produc6137756002093451184nteger @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) )).

% case_prodE
thf(fact_534_case__prodE,axiom,(
    ! [C2: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o,P2: produc989692990947075319atural] :
      ( ( produc2278936416606902338ural_o @ C2 @ P2 )
     => ~ ! [X3: produc4972180933644002618e_term > option6357759511663192854e_term,Y3: produc7822875418678951345atural] :
            ( ( P2
              = ( produc3831813291587773865atural @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) )).

% case_prodE
thf(fact_535_case__prodE,axiom,(
    ! [C2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P2: produc1908205239877642774nteger] :
      ( ( produc6253627499356882019eger_o @ C2 @ P2 )
     => ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ( ( P2
              = ( produc8603105652947943368nteger @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) )).

% case_prodE
thf(fact_536_case__prodE,axiom,(
    ! [C2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P2: produc2285326912895808259nt_int] :
      ( ( produc1573362020775583542_int_o @ C2 @ P2 )
     => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P2
              = ( produc5700946648718959541nt_int @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) )).

% case_prodE
thf(fact_537_case__prodE,axiom,(
    ! [C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P2: produc7773217078559923341nt_int] :
      ( ( produc2558449545302689196_int_o @ C2 @ P2 )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P2
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) )).

% case_prodE
thf(fact_538_pair__less__iff1,axiom,(
    ! [X: nat,Y: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z ) ) )).

% pair_less_iff1
thf(fact_539_Collect__case__prod__Grp__eqD,axiom,(
    ! [Z: product_prod_nat_nat,A4: set_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A4 @ F ) ) ) )
     => ( ( comp_n2711240574518091068at_nat @ F @ product_fst_nat_nat @ Z )
        = ( product_snd_nat_nat @ Z ) ) ) )).

% Collect_case_prod_Grp_eqD
thf(fact_540_Collect__case__prod__Grp__eqD,axiom,(
    ! [Z: produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ ( bNF_Gr7275288873855648526at_nat @ A4 @ F ) ) ) )
     => ( ( comp_P3189659738270478212at_nat @ F @ produc3213797794245857475at_nat @ Z )
        = ( produc6408287024330202629at_nat @ Z ) ) ) )).

% Collect_case_prod_Grp_eqD
thf(fact_541_Collect__case__prod__Grp__eqD,axiom,(
    ! [Z: produc8763457246119570046nteger,A4: set_Co9149898834107579976e_term,F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger] :
      ( ( member3068662437193594005nteger @ Z @ ( collec6350289969311926995nteger @ ( produc127349428274296955eger_o @ ( bNF_Gr6578204635149681721nteger @ A4 @ F ) ) ) )
     => ( ( comp_C2113120126217125220nteger @ F @ produc2702340305111172292nteger @ Z )
        = ( produc6514156286822040066nteger @ Z ) ) ) )).

% Collect_case_prod_Grp_eqD
thf(fact_542_Collect__case__prod__Grp__eqD,axiom,(
    ! [Z: produc989692990947075319atural,A4: set_Pr2791530082921190693e_term,F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural] :
      ( ( member3245610415464309390atural @ Z @ ( collec6032437096461363148atural @ ( produc2278936416606902338ural_o @ ( bNF_Gr8806722549218754226atural @ A4 @ F ) ) ) )
     => ( ( comp_P2069287802696329814atural @ F @ produc3051390257598303421atural @ Z )
        = ( produc3085575039554877691atural @ Z ) ) ) )).

% Collect_case_prod_Grp_eqD
thf(fact_543_Collect__case__prod__Grp__eqD,axiom,(
    ! [Z: produc1908205239877642774nteger,A4: set_Pr7604974323444597168e_term,F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger] :
      ( ( member4164122664394876845nteger @ Z @ ( collec6950949345391930603nteger @ ( produc6253627499356882019eger_o @ ( bNF_Gr4354642873724147921nteger @ A4 @ F ) ) ) )
     => ( ( comp_P7281769421547479700nteger @ F @ produc7822682618958472924nteger @ Z )
        = ( produc7856867400915047194nteger @ Z ) ) ) )).

% Collect_case_prod_Grp_eqD
thf(fact_544_Collect__case__prod__Grp__eqD,axiom,(
    ! [Z: produc2285326912895808259nt_int,A4: set_Pr3642885161833720865e_term,F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ Z @ ( collec1790188477890212312nt_int @ ( produc1573362020775583542_int_o @ ( bNF_Gr5420098529801562302nt_int @ A4 @ F ) ) ) )
     => ( ( comp_P7575764690606639726nt_int @ F @ produc6816164490631068361nt_int @ Z )
        = ( produc7328097813583171335nt_int @ Z ) ) ) )).

% Collect_case_prod_Grp_eqD
thf(fact_545_Collect__case__prod__Grp__eqD,axiom,(
    ! [Z: produc7773217078559923341nt_int,A4: set_in3461395444621081367e_term,F: ( int > option6357759511663192854e_term ) > product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ Z @ ( collec506566255779805410nt_int @ ( produc2558449545302689196_int_o @ ( bNF_Gr6359101500228959560nt_int @ A4 @ F ) ) ) )
     => ( ( comp_i8411724616380820994nt_int @ F @ produc6230002227079971283nt_int @ Z )
        = ( produc3162348030201620241nt_int @ Z ) ) ) )).

% Collect_case_prod_Grp_eqD
thf(fact_546_Collect__case__prod__Grp__eqD,axiom,(
    ! [Z: produc5835291356934675326atural,A4: set_Code_natural,F: code_natural > produc7822875418678951345atural] :
      ( ( member2108296798382401941atural @ Z @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ ( bNF_Gr1183324042980456505atural @ A4 @ F ) ) ) )
     => ( ( comp_C3540846879249570404atural @ F @ produc497848011034438852atural @ Z )
        = ( produc6591912806276919810atural @ Z ) ) ) )).

% Collect_case_prod_Grp_eqD
thf(fact_547_Collect__case__prod__Grp__eqD,axiom,(
    ! [Z: produc8923597787767437993list_f,A4: set_fi1428165280859657121action,F: finfun1145668972854894571action > produc1811169103916964701list_f] :
      ( ( member1992181931100238290list_f @ Z @ ( collec2941370454899091988list_f @ ( produc7065790152434098210st_f_o @ ( bNF_Gr6286777782477588504list_f @ A4 @ F ) ) ) )
     => ( ( comp_f5362652388678882620list_f @ F @ produc3089225533250117581list_f @ Z )
        = ( produc8710481132536983823list_f @ Z ) ) ) )).

% Collect_case_prod_Grp_eqD
thf(fact_548_fst__comp__map__prod,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( comp_P1446425635098721907nteger @ produc2702340305111172292nteger @ ( produc291189458620429660nteger @ F @ G2 ) )
      = ( comp_C6219200931300381191nteger @ F @ produc2702340305111172292nteger ) ) )).

% fst_comp_map_prod
thf(fact_549_fst__comp__map__prod,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: product_prod_int_int > produc8923325533196201883nteger] :
      ( ( comp_P2774341336710753410nt_int @ produc2702340305111172292nteger @ ( produc4564144299210792333nteger @ F @ G2 ) )
      = ( comp_i4808050162669647333nt_int @ F @ produc6230002227079971283nt_int ) ) )).

% fst_comp_map_prod
thf(fact_550_fst__comp__map__prod,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > int > option6357759511663192854e_term,G2: produc8923325533196201883nteger > product_prod_int_int] :
      ( ( comp_P7563362646663813073nteger @ produc6230002227079971283nt_int @ ( produc4848173044407040841nt_int @ F @ G2 ) )
      = ( comp_C6263816306611391446nteger @ F @ produc2702340305111172292nteger ) ) )).

% fst_comp_map_prod
thf(fact_551_fst__comp__map__prod,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > int > option6357759511663192854e_term,G2: product_prod_int_int > product_prod_int_int] :
      ( ( comp_P5815951645081071072nt_int @ produc6230002227079971283nt_int @ ( produc7905387574911623034nt_int @ F @ G2 ) )
      = ( comp_i1330301680410484660nt_int @ F @ produc6230002227079971283nt_int ) ) )).

% fst_comp_map_prod
thf(fact_552_fst__comp__map__prod,axiom,(
    ! [F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: produc7822875418678951345atural > produc8923325533196201883nteger] :
      ( ( comp_P7222502500883786604atural @ produc2702340305111172292nteger @ ( produc1485061364871261091nteger @ F @ G2 ) )
      = ( comp_P1244703571580363101atural @ F @ produc3051390257598303421atural ) ) )).

% fst_comp_map_prod
thf(fact_553_fst__comp__map__prod,axiom,(
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( comp_P8141014749814354059nteger @ produc2702340305111172292nteger @ ( produc5027624217247135428nteger @ F @ G2 ) )
      = ( comp_P6703551259509254535nteger @ F @ produc7822682618958472924nteger ) ) )).

% fst_comp_map_prod
thf(fact_554_fst__comp__map__prod,axiom,(
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: product_prod_int_int > produc8923325533196201883nteger] :
      ( ( comp_P3682586444822787448nt_int @ produc2702340305111172292nteger @ ( produc1916341625465281175nteger @ F @ G2 ) )
      = ( comp_P195263689248665701nt_int @ F @ produc6816164490631068361nt_int ) ) )).

% fst_comp_map_prod
thf(fact_555_fst__comp__map__prod,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc4972180933644002618e_term > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc7822875418678951345atural] :
      ( ( comp_P1885529236250819273nteger @ produc3051390257598303421atural @ ( produc2114834699021641935atural @ F @ G2 ) )
      = ( comp_C1035002832859404644nteger @ F @ produc2702340305111172292nteger ) ) )).

% fst_comp_map_prod
thf(fact_556_fst__comp__map__prod,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > produc4972180933644002618e_term > option6357759511663192854e_term,G2: product_prod_int_int > produc7822875418678951345atural] :
      ( ( comp_P8584773556875604184nt_int @ produc3051390257598303421atural @ ( produc2222610939478984832atural @ F @ G2 ) )
      = ( comp_i7942217246757202114nt_int @ F @ produc6230002227079971283nt_int ) ) )).

% fst_comp_map_prod
thf(fact_557_fst__comp__map__prod,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc6241069584506657477e_term > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( comp_P6133064115347390067nteger @ produc7822682618958472924nteger @ ( produc4923703365104367684nteger @ F @ G2 ) )
      = ( comp_C1695206475658992367nteger @ F @ produc2702340305111172292nteger ) ) )).

% fst_comp_map_prod
thf(fact_558_snd__comp__map__prod,axiom,(
    ! [F: code_natural > code_natural,G2: produc7822875418678951345atural > produc7822875418678951345atural] :
      ( ( comp_P7610810914149544718atural @ produc6591912806276919810atural @ ( produc6172530018443213064atural @ F @ G2 ) )
      = ( comp_P3067963646784605345atural @ G2 @ produc6591912806276919810atural ) ) )).

% snd_comp_map_prod
thf(fact_559_snd__comp__map__prod,axiom,(
    ! [F: finfun1145668972854894571action > code_natural,G2: produc1811169103916964701list_f > produc7822875418678951345atural] :
      ( ( comp_P754933049744969497list_f @ produc6591912806276919810atural @ ( produc6433193301019619911atural @ F @ G2 ) )
      = ( comp_P4126295255191388280list_f @ G2 @ produc8710481132536983823list_f ) ) )).

% snd_comp_map_prod
thf(fact_560_snd__comp__map__prod,axiom,(
    ! [F: code_natural > finfun1145668972854894571action,G2: produc7822875418678951345atural > produc1811169103916964701list_f] :
      ( ( comp_P3912863319257206115atural @ produc8710481132536983823list_f @ ( produc9125308813900113731list_f @ F @ G2 ) )
      = ( comp_P894719457104077291atural @ G2 @ produc6591912806276919810atural ) ) )).

% snd_comp_map_prod
thf(fact_561_snd__comp__map__prod,axiom,(
    ! [F: finfun1145668972854894571action > finfun1145668972854894571action,G2: produc1811169103916964701list_f > produc1811169103916964701list_f] :
      ( ( comp_P4267156721827287172list_f @ produc8710481132536983823list_f @ ( produc1848554412176970122list_f @ F @ G2 ) )
      = ( comp_P2673361230361438154list_f @ G2 @ produc8710481132536983823list_f ) ) )).

% snd_comp_map_prod
thf(fact_562_fstOp__def,axiom,
    ( bNF_fs7749581648564785796nteger
    = ( ^ [P4: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,Q3: produc8923325533196201883nteger > produc8923325533196201883nteger > $o,Ac2: produc8763457246119570046nteger] :
          ( produc6137756002093451184nteger @ ( produc2702340305111172292nteger @ Ac2 ) @ ( bNF_pi839428095174260662nteger @ P4 @ Q3 @ ( produc2702340305111172292nteger @ Ac2 ) @ ( produc6514156286822040066nteger @ Ac2 ) ) ) ) )).

% fstOp_def
thf(fact_563_fstOp__def,axiom,
    ( bNF_fs632686752235921505atural
    = ( ^ [P4: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o,Q3: produc7822875418678951345atural > produc7822875418678951345atural > $o,Ac2: produc989692990947075319atural] :
          ( produc3831813291587773865atural @ ( produc3051390257598303421atural @ Ac2 ) @ ( bNF_pi2113054283598696083atural @ P4 @ Q3 @ ( produc3051390257598303421atural @ Ac2 ) @ ( produc3085575039554877691atural @ Ac2 ) ) ) ) )).

% fstOp_def
thf(fact_564_fstOp__def,axiom,
    ( bNF_fs2771997879588465004nteger
    = ( ^ [P4: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,Q3: produc8923325533196201883nteger > produc8923325533196201883nteger > $o,Ac2: produc1908205239877642774nteger] :
          ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ Ac2 ) @ ( bNF_pi4252365410951239582nteger @ P4 @ Q3 @ ( produc7822682618958472924nteger @ Ac2 ) @ ( produc7856867400915047194nteger @ Ac2 ) ) ) ) )).

% fstOp_def
thf(fact_565_fstOp__def,axiom,
    ( bNF_fs7693636466261331293nt_int
    = ( ^ [P4: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,Q3: product_prod_int_int > product_prod_int_int > $o,Ac2: produc2285326912895808259nt_int] :
          ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ Ac2 ) @ ( bNF_pi5969911828475470735nt_int @ P4 @ Q3 @ ( produc6816164490631068361nt_int @ Ac2 ) @ ( produc7328097813583171335nt_int @ Ac2 ) ) ) ) )).

% fstOp_def
thf(fact_566_fstOp__def,axiom,
    ( bNF_fs296160452230587091nt_int
    = ( ^ [P4: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,Q3: product_prod_int_int > product_prod_int_int > $o,Ac2: produc7773217078559923341nt_int] :
          ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ Ac2 ) @ ( bNF_pi6237251963727065093nt_int @ P4 @ Q3 @ ( produc6230002227079971283nt_int @ Ac2 ) @ ( produc3162348030201620241nt_int @ Ac2 ) ) ) ) )).

% fstOp_def
thf(fact_567_case__prodI,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( F @ A @ B )
     => ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) ) ) )).

% case_prodI
thf(fact_568_case__prodI,axiom,(
    ! [F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o,A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural] :
      ( ( F @ A @ B )
     => ( produc2278936416606902338ural_o @ F @ ( produc3831813291587773865atural @ A @ B ) ) ) )).

% case_prodI
thf(fact_569_case__prodI,axiom,(
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( F @ A @ B )
     => ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) ) ) )).

% case_prodI
thf(fact_570_case__prodI,axiom,(
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ A @ B )
     => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) )).

% case_prodI
thf(fact_571_case__prodI,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ A @ B )
     => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) )).

% case_prodI
thf(fact_572_rtrancl__induct2,axiom,(
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A3: nat,B3: nat,Aa: nat,Ba: nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A3 @ B3 ) ) @ ( transi8609417484261754244at_nat @ R ) )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% rtrancl_induct2
thf(fact_573_rtrancl__induct2,axiom,(
    ! [Ax: code_integer > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: code_integer > option6357759511663192854e_term,By: produc8923325533196201883nteger,R: set_Pr5477269989976292413nteger,P: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ Bx @ By ) ) @ ( transi4577376780619456059nteger @ R ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: code_integer > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ A3 @ B3 ) ) @ ( transi4577376780619456059nteger @ R ) )
             => ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ A3 @ B3 ) @ ( produc6137756002093451184nteger @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% rtrancl_induct2
thf(fact_574_rtrancl__induct2,axiom,(
    ! [Ax: produc4972180933644002618e_term > option6357759511663192854e_term,Ay: produc7822875418678951345atural,Bx: produc4972180933644002618e_term > option6357759511663192854e_term,By: produc7822875418678951345atural,R: set_Pr3316353890524442159atural,P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o] :
      ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ Bx @ By ) ) @ ( transi3001887490979966772atural @ R ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural,Aa: produc4972180933644002618e_term > option6357759511663192854e_term,Ba: produc7822875418678951345atural] :
              ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ A3 @ B3 ) ) @ ( transi3001887490979966772atural @ R ) )
             => ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ A3 @ B3 ) @ ( produc3831813291587773865atural @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% rtrancl_induct2
thf(fact_575_rtrancl__induct2,axiom,(
    ! [Ax: produc6241069584506657477e_term > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: produc6241069584506657477e_term > option6357759511663192854e_term,By: produc8923325533196201883nteger,R: set_Pr1722956316086891245nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ Bx @ By ) ) @ ( transi3920399739910534227nteger @ R ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: produc6241069584506657477e_term > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ A3 @ B3 ) ) @ ( transi3920399739910534227nteger @ R ) )
             => ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ A3 @ B3 ) @ ( produc8603105652947943368nteger @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% rtrancl_induct2
thf(fact_576_rtrancl__induct2,axiom,(
    ! [Ax: produc8551481072490612790e_term > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: produc8551481072490612790e_term > option6357759511663192854e_term,By: product_prod_int_int,R: set_Pr4708930517165415495nt_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ Bx @ By ) ) @ ( transi2741729202006808896nt_int @ R ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int,Aa: produc8551481072490612790e_term > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ A3 @ B3 ) ) @ ( transi2741729202006808896nt_int @ R ) )
             => ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ A3 @ B3 ) @ ( produc5700946648718959541nt_int @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% rtrancl_induct2
thf(fact_577_rtrancl__induct2,axiom,(
    ! [Ax: int > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: int > option6357759511663192854e_term,By: product_prod_int_int,R: set_Pr8634505666381077339nt_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ Bx @ By ) ) @ ( transi8659768982470569034nt_int @ R ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int,Aa: int > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ A3 @ B3 ) ) @ ( transi8659768982470569034nt_int @ R ) )
             => ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ A3 @ B3 ) @ ( produc4305682042979456191nt_int @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% rtrancl_induct2
thf(fact_578_pair__less__def,axiom,
    ( fun_pair_less
    = ( lex_prod_nat_nat @ less_than @ less_than ) )).

% pair_less_def
thf(fact_579_le__sup__iff,axiom,(
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        & ( ord_less_eq_nat @ Y @ Z ) ) ) )).

% le_sup_iff
thf(fact_580_sup_Obounded__iff,axiom,(
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C2 @ A ) ) ) )).

% sup.bounded_iff
thf(fact_581_in__rel__def,axiom,
    ( fun_in_rel_nat_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat,X4: nat,Y6: nat] :
          ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y6 ) @ R4 ) ) )).

% in_rel_def
thf(fact_582_in__rel__def,axiom,
    ( fun_in3916733394112622082at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat,X4: product_prod_nat_nat,Y6: product_prod_nat_nat] :
          ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y6 ) @ R4 ) ) )).

% in_rel_def
thf(fact_583_in__rel__def,axiom,
    ( fun_in8327221098041113157nteger
    = ( ^ [R4: set_Pr8056137968301705908nteger,X4: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
          ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y6 ) @ R4 ) ) )).

% in_rel_def
thf(fact_584_in__rel__def,axiom,
    ( fun_in7102582974668839614atural
    = ( ^ [R4: set_Pr4244404650639074861atural,X4: produc4972180933644002618e_term > option6357759511663192854e_term,Y6: produc7822875418678951345atural] :
          ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X4 @ Y6 ) @ R4 ) ) )).

% in_rel_def
thf(fact_585_in__rel__def,axiom,
    ( fun_in2650503299174233309nteger
    = ( ^ [R4: set_Pr1281608226676607948nteger,X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
          ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y6 ) @ R4 ) ) )).

% in_rel_def
thf(fact_586_in__rel__def,axiom,
    ( fun_in6676002345827390666nt_int
    = ( ^ [R4: set_Pr9222295170931077689nt_int,X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
          ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y6 ) @ R4 ) ) )).

% in_rel_def
thf(fact_587_in__rel__def,axiom,
    ( fun_in612320021611004756nt_int
    = ( ^ [R4: set_Pr1872883991513573699nt_int,X4: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
          ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y6 ) @ R4 ) ) )).

% in_rel_def
thf(fact_588_rtrancl__tab_Oequation_I3_J,axiom,
    ( transi8137332151331134886at_nat
    = ( ^ [R3: pred_P40958870688161405at_nat,Xa3: list_P6011104703257516679at_nat,Xb2: product_prod_nat_nat,Xc: product_prod_nat_nat] :
          ( sup_su2981425912110785871t_unit
          @ ( bind_P7133822938289884078t_unit @ ( single5216280447982286995at_nat @ ( produc3454366016999414071at_nat @ Xa3 @ ( produc6161850002892822231at_nat @ Xb2 @ Xc ) ) )
            @ ( produc7613932700751717743t_unit
              @ ^ [Xs2: list_P6011104703257516679at_nat] :
                  ( produc4288560984398973711t_unit
                  @ ^ [X8: product_prod_nat_nat,Xa4: product_prod_nat_nat] :
                      ( if_pred_Product_unit @ ( X8 = Xa4 ) @ ( single_Product_unit @ product_Unity ) @ bot_bo2717538794563056311t_unit ) ) ) )
          @ ( bind_P7133822938289884078t_unit @ ( single5216280447982286995at_nat @ ( produc3454366016999414071at_nat @ Xa3 @ ( produc6161850002892822231at_nat @ Xb2 @ Xc ) ) )
            @ ( produc7613932700751717743t_unit
              @ ^ [Xs2: list_P6011104703257516679at_nat] :
                  ( produc4288560984398973711t_unit
                  @ ^ [X8: product_prod_nat_nat,Z5: product_prod_nat_nat] :
                      ( bind_P7128388325418652648t_unit
                      @ ( if_pred
                        @ ~ ( member8440522571783428010at_nat @ X8 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) )
                      @ ( produc1731598583106934569t_unit
                        @ ( bind_P8554545925139817076t_unit @ ( transi882309894880223404at_nat @ R3 @ ( cons_P6512896166579812791at_nat @ X8 @ Xs2 ) @ Z5 )
                          @ ^ [Xd: product_prod_nat_nat] :
                              ( bind_P1979233827931663630t_unit @ R3
                              @ ( produc4288560984398973711t_unit
                                @ ^ [Xa4: product_prod_nat_nat,Ya: product_prod_nat_nat] :
                                    ( if_pred_Product_unit
                                    @ ( ( X8 = Xa4 )
                                      & ( Xd = Ya ) )
                                    @ ( single_Product_unit @ product_Unity )
                                    @ bot_bo2717538794563056311t_unit ) ) ) ) ) ) ) ) ) ) ) )).

% rtrancl_tab.equation(3)
thf(fact_589_rtrancl__tab_Oequation_I3_J,axiom,
    ( transi5680681622043916684at_nat
    = ( ^ [R3: pred_P8888073063311132157at_nat,Xa3: list_P8469869581646625389at_nat,Xb2: produc859450856879609959at_nat,Xc: produc859450856879609959at_nat] :
          ( sup_su2981425912110785871t_unit
          @ ( bind_P4922376816526270484t_unit @ ( single2428174337336239661at_nat @ ( produc7781176426044631377at_nat @ Xa3 @ ( produc4662710985925991255at_nat @ Xb2 @ Xc ) ) )
            @ ( produc3209168420654000521t_unit
              @ ^ [Xs2: list_P8469869581646625389at_nat] :
                  ( produc5963772262439867279t_unit
                  @ ^ [X8: produc859450856879609959at_nat,Xa4: produc859450856879609959at_nat] :
                      ( if_pred_Product_unit @ ( X8 = Xa4 ) @ ( single_Product_unit @ product_Unity ) @ bot_bo2717538794563056311t_unit ) ) ) )
          @ ( bind_P4922376816526270484t_unit @ ( single2428174337336239661at_nat @ ( produc7781176426044631377at_nat @ Xa3 @ ( produc4662710985925991255at_nat @ Xb2 @ Xc ) ) )
            @ ( produc3209168420654000521t_unit
              @ ^ [Xs2: list_P8469869581646625389at_nat] :
                  ( produc5963772262439867279t_unit
                  @ ^ [X8: produc859450856879609959at_nat,Z5: produc859450856879609959at_nat] :
                      ( bind_P7128388325418652648t_unit
                      @ ( if_pred
                        @ ~ ( member8206827879206165904at_nat @ X8 @ ( set_Pr5518436109238095868at_nat @ Xs2 ) ) )
                      @ ( produc1731598583106934569t_unit
                        @ ( bind_P1979233827931663630t_unit @ ( transi8068056145586740114at_nat @ R3 @ ( cons_P8732206157123786781at_nat @ X8 @ Xs2 ) @ Z5 )
                          @ ^ [Xd: produc859450856879609959at_nat] :
                              ( bind_P5740067618745592206t_unit @ R3
                              @ ( produc5963772262439867279t_unit
                                @ ^ [Xa4: produc859450856879609959at_nat,Ya: produc859450856879609959at_nat] :
                                    ( if_pred_Product_unit
                                    @ ( ( X8 = Xa4 )
                                      & ( Xd = Ya ) )
                                    @ ( single_Product_unit @ product_Unity )
                                    @ bot_bo2717538794563056311t_unit ) ) ) ) ) ) ) ) ) ) ) )).

% rtrancl_tab.equation(3)
thf(fact_590_snd__map__prod,axiom,(
    ! [F: code_natural > code_natural,G2: produc7822875418678951345atural > produc7822875418678951345atural,X: produc5835291356934675326atural] :
      ( ( produc6591912806276919810atural @ ( produc6172530018443213064atural @ F @ G2 @ X ) )
      = ( G2 @ ( produc6591912806276919810atural @ X ) ) ) )).

% snd_map_prod
thf(fact_591_snd__map__prod,axiom,(
    ! [F: finfun1145668972854894571action > code_natural,G2: produc1811169103916964701list_f > produc7822875418678951345atural,X: produc8923597787767437993list_f] :
      ( ( produc6591912806276919810atural @ ( produc6433193301019619911atural @ F @ G2 @ X ) )
      = ( G2 @ ( produc8710481132536983823list_f @ X ) ) ) )).

% snd_map_prod
thf(fact_592_snd__map__prod,axiom,(
    ! [F: code_natural > finfun1145668972854894571action,G2: produc7822875418678951345atural > produc1811169103916964701list_f,X: produc5835291356934675326atural] :
      ( ( produc8710481132536983823list_f @ ( produc9125308813900113731list_f @ F @ G2 @ X ) )
      = ( G2 @ ( produc6591912806276919810atural @ X ) ) ) )).

% snd_map_prod
thf(fact_593_snd__map__prod,axiom,(
    ! [F: finfun1145668972854894571action > finfun1145668972854894571action,G2: produc1811169103916964701list_f > produc1811169103916964701list_f,X: produc8923597787767437993list_f] :
      ( ( produc8710481132536983823list_f @ ( produc1848554412176970122list_f @ F @ G2 @ X ) )
      = ( G2 @ ( produc8710481132536983823list_f @ X ) ) ) )).

% snd_map_prod
thf(fact_594_min__def,axiom,
    ( ord_min_nat
    = ( ^ [A5: nat,B4: nat] :
          ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ A5 @ B4 ) ) )).

% min_def
thf(fact_595_fst__map__prod,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc8923325533196201883nteger,X: produc8763457246119570046nteger] :
      ( ( produc2702340305111172292nteger @ ( produc291189458620429660nteger @ F @ G2 @ X ) )
      = ( F @ ( produc2702340305111172292nteger @ X ) ) ) )).

% fst_map_prod
thf(fact_596_fst__map__prod,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: product_prod_int_int > produc8923325533196201883nteger,X: produc7773217078559923341nt_int] :
      ( ( produc2702340305111172292nteger @ ( produc4564144299210792333nteger @ F @ G2 @ X ) )
      = ( F @ ( produc6230002227079971283nt_int @ X ) ) ) )).

% fst_map_prod
thf(fact_597_fst__map__prod,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > int > option6357759511663192854e_term,G2: produc8923325533196201883nteger > product_prod_int_int,X: produc8763457246119570046nteger] :
      ( ( produc6230002227079971283nt_int @ ( produc4848173044407040841nt_int @ F @ G2 @ X ) )
      = ( F @ ( produc2702340305111172292nteger @ X ) ) ) )).

% fst_map_prod
thf(fact_598_fst__map__prod,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > int > option6357759511663192854e_term,G2: product_prod_int_int > product_prod_int_int,X: produc7773217078559923341nt_int] :
      ( ( produc6230002227079971283nt_int @ ( produc7905387574911623034nt_int @ F @ G2 @ X ) )
      = ( F @ ( produc6230002227079971283nt_int @ X ) ) ) )).

% fst_map_prod
thf(fact_599_fst__map__prod,axiom,(
    ! [F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: produc7822875418678951345atural > produc8923325533196201883nteger,X: produc989692990947075319atural] :
      ( ( produc2702340305111172292nteger @ ( produc1485061364871261091nteger @ F @ G2 @ X ) )
      = ( F @ ( produc3051390257598303421atural @ X ) ) ) )).

% fst_map_prod
thf(fact_600_fst__map__prod,axiom,(
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc8923325533196201883nteger,X: produc1908205239877642774nteger] :
      ( ( produc2702340305111172292nteger @ ( produc5027624217247135428nteger @ F @ G2 @ X ) )
      = ( F @ ( produc7822682618958472924nteger @ X ) ) ) )).

% fst_map_prod
thf(fact_601_fst__map__prod,axiom,(
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > code_integer > option6357759511663192854e_term,G2: product_prod_int_int > produc8923325533196201883nteger,X: produc2285326912895808259nt_int] :
      ( ( produc2702340305111172292nteger @ ( produc1916341625465281175nteger @ F @ G2 @ X ) )
      = ( F @ ( produc6816164490631068361nt_int @ X ) ) ) )).

% fst_map_prod
thf(fact_602_fst__map__prod,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc4972180933644002618e_term > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc7822875418678951345atural,X: produc8763457246119570046nteger] :
      ( ( produc3051390257598303421atural @ ( produc2114834699021641935atural @ F @ G2 @ X ) )
      = ( F @ ( produc2702340305111172292nteger @ X ) ) ) )).

% fst_map_prod
thf(fact_603_fst__map__prod,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > produc4972180933644002618e_term > option6357759511663192854e_term,G2: product_prod_int_int > produc7822875418678951345atural,X: produc7773217078559923341nt_int] :
      ( ( produc3051390257598303421atural @ ( produc2222610939478984832atural @ F @ G2 @ X ) )
      = ( F @ ( produc6230002227079971283nt_int @ X ) ) ) )).

% fst_map_prod
thf(fact_604_fst__map__prod,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc6241069584506657477e_term > option6357759511663192854e_term,G2: produc8923325533196201883nteger > produc8923325533196201883nteger,X: produc8763457246119570046nteger] :
      ( ( produc7822682618958472924nteger @ ( produc4923703365104367684nteger @ F @ G2 @ X ) )
      = ( F @ ( produc2702340305111172292nteger @ X ) ) ) )).

% fst_map_prod
thf(fact_605_rtrancl__tab_Oequation_I1_J,axiom,
    ( transi7776283402896926368at_nat
    = ( ^ [R3: product_prod_nat_nat > pred_P7117560977466480279at_nat,Xa3: list_P6011104703257516679at_nat,Xb2: product_prod_nat_nat,Xc: product_prod_nat_nat] :
          ( sup_su2981425912110785871t_unit
          @ ( bind_P7133822938289884078t_unit @ ( single5216280447982286995at_nat @ ( produc3454366016999414071at_nat @ Xa3 @ ( produc6161850002892822231at_nat @ Xb2 @ Xc ) ) )
            @ ( produc7613932700751717743t_unit
              @ ^ [Xs2: list_P6011104703257516679at_nat] :
                  ( produc4288560984398973711t_unit
                  @ ^ [X8: product_prod_nat_nat,Xa4: product_prod_nat_nat] :
                      ( if_pred_Product_unit @ ( X8 = Xa4 ) @ ( single_Product_unit @ product_Unity ) @ bot_bo2717538794563056311t_unit ) ) ) )
          @ ( bind_P7133822938289884078t_unit @ ( single5216280447982286995at_nat @ ( produc3454366016999414071at_nat @ Xa3 @ ( produc6161850002892822231at_nat @ Xb2 @ Xc ) ) )
            @ ( produc7613932700751717743t_unit
              @ ^ [Xs2: list_P6011104703257516679at_nat] :
                  ( produc4288560984398973711t_unit
                  @ ^ [X8: product_prod_nat_nat,Z5: product_prod_nat_nat] :
                      ( bind_P7128388325418652648t_unit
                      @ ( if_pred
                        @ ~ ( member8440522571783428010at_nat @ X8 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) )
                      @ ( produc1731598583106934569t_unit
                        @ ( bind_P8554545925139817076t_unit @ ( R3 @ X8 )
                          @ ^ [Xd: product_prod_nat_nat] :
                              ( bind_P7128388325418652648t_unit @ ( transi7776283402896926368at_nat @ R3 @ ( cons_P6512896166579812791at_nat @ X8 @ Xs2 ) @ Xd @ Z5 ) @ ( produc1731598583106934569t_unit @ ( single_Product_unit @ product_Unity ) ) ) ) ) ) ) ) ) ) ) )).

% rtrancl_tab.equation(1)
thf(fact_606_rtrancl__tab_Oequation_I1_J,axiom,
    ( transi4223599960602944390at_nat
    = ( ^ [R3: produc859450856879609959at_nat > pred_P40958870688161405at_nat,Xa3: list_P8469869581646625389at_nat,Xb2: produc859450856879609959at_nat,Xc: produc859450856879609959at_nat] :
          ( sup_su2981425912110785871t_unit
          @ ( bind_P4922376816526270484t_unit @ ( single2428174337336239661at_nat @ ( produc7781176426044631377at_nat @ Xa3 @ ( produc4662710985925991255at_nat @ Xb2 @ Xc ) ) )
            @ ( produc3209168420654000521t_unit
              @ ^ [Xs2: list_P8469869581646625389at_nat] :
                  ( produc5963772262439867279t_unit
                  @ ^ [X8: produc859450856879609959at_nat,Xa4: produc859450856879609959at_nat] :
                      ( if_pred_Product_unit @ ( X8 = Xa4 ) @ ( single_Product_unit @ product_Unity ) @ bot_bo2717538794563056311t_unit ) ) ) )
          @ ( bind_P4922376816526270484t_unit @ ( single2428174337336239661at_nat @ ( produc7781176426044631377at_nat @ Xa3 @ ( produc4662710985925991255at_nat @ Xb2 @ Xc ) ) )
            @ ( produc3209168420654000521t_unit
              @ ^ [Xs2: list_P8469869581646625389at_nat] :
                  ( produc5963772262439867279t_unit
                  @ ^ [X8: produc859450856879609959at_nat,Z5: produc859450856879609959at_nat] :
                      ( bind_P7128388325418652648t_unit
                      @ ( if_pred
                        @ ~ ( member8206827879206165904at_nat @ X8 @ ( set_Pr5518436109238095868at_nat @ Xs2 ) ) )
                      @ ( produc1731598583106934569t_unit
                        @ ( bind_P1979233827931663630t_unit @ ( R3 @ X8 )
                          @ ^ [Xd: produc859450856879609959at_nat] :
                              ( bind_P7128388325418652648t_unit @ ( transi4223599960602944390at_nat @ R3 @ ( cons_P8732206157123786781at_nat @ X8 @ Xs2 ) @ Xd @ Z5 ) @ ( produc1731598583106934569t_unit @ ( single_Product_unit @ product_Unity ) ) ) ) ) ) ) ) ) ) ) )).

% rtrancl_tab.equation(1)
thf(fact_607_map__prod__imageI,axiom,(
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: nat > nat,G2: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G2 @ B ) ) @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G2 ) @ R2 ) ) ) )).

% map_prod_imageI
thf(fact_608_map__prod__imageI,axiom,(
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: nat > product_prod_nat_nat,G2: nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ A ) @ ( G2 @ B ) ) @ ( image_1119892238107900587at_nat @ ( produc1153188318382123958at_nat @ F @ G2 ) @ R2 ) ) ) )).

% map_prod_imageI
thf(fact_609_map__prod__imageI,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > nat,G2: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G2 @ B ) ) @ ( image_6706873702294942047at_nat @ ( produc5156754000680454454at_nat @ F @ G2 ) @ R2 ) ) ) )).

% map_prod_imageI
thf(fact_610_map__prod__imageI,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > product_prod_nat_nat,G2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ A ) @ ( G2 @ B ) ) @ ( image_6321802852832981957at_nat @ ( produc7147751436741673334at_nat @ F @ G2 ) @ R2 ) ) ) )).

% map_prod_imageI
thf(fact_611_map__prod__imageI,axiom,(
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: nat > code_integer > option6357759511663192854e_term,G2: nat > produc8923325533196201883nteger] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ ( F @ A ) @ ( G2 @ B ) ) @ ( image_7586372328468041018nteger @ ( produc6729254856488394607nteger @ F @ G2 ) @ R2 ) ) ) )).

% map_prod_imageI
thf(fact_612_map__prod__imageI,axiom,(
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: nat > int > option6357759511663192854e_term,G2: nat > product_prod_int_int] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ ( F @ A ) @ ( G2 @ B ) ) @ ( image_1715401223925157705nt_int @ ( produc3178275182335418494nt_int @ F @ G2 ) @ R2 ) ) ) )).

% map_prod_imageI
thf(fact_613_map__prod__imageI,axiom,(
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,R2: set_Pr8056137968301705908nteger,F: ( code_integer > option6357759511663192854e_term ) > nat,G2: produc8923325533196201883nteger > nat] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G2 @ B ) ) @ ( image_9094096100532249404at_nat @ ( produc6033578056857175441er_nat @ F @ G2 ) @ R2 ) ) ) )).

% map_prod_imageI
thf(fact_614_map__prod__imageI,axiom,(
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,R2: set_Pr1872883991513573699nt_int,F: ( int > option6357759511663192854e_term ) > nat,G2: product_prod_int_int > nat] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G2 @ B ) ) @ ( image_6286574459949954669at_nat @ ( produc5394751608776890016nt_nat @ F @ G2 ) @ R2 ) ) ) )).

% map_prod_imageI
thf(fact_615_map__prod__imageI,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > code_integer > option6357759511663192854e_term,G2: product_prod_nat_nat > produc8923325533196201883nteger] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
     => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ ( F @ A ) @ ( G2 @ B ) ) @ ( image_7378368697092541600nteger @ ( produc4759713928480622455nteger @ F @ G2 ) @ R2 ) ) ) )).

% map_prod_imageI
thf(fact_616_map__prod__imageI,axiom,(
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: nat > produc4972180933644002618e_term > option6357759511663192854e_term,G2: nat > produc7822875418678951345atural] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member3245610415464309390atural @ ( produc3831813291587773865atural @ ( F @ A ) @ ( G2 @ B ) ) @ ( image_5310278726981273011atural @ ( produc161022182982199656atural @ F @ G2 ) @ R2 ) ) ) )).

% map_prod_imageI
thf(fact_617_finite__induct,axiom,(
    ! [F5: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F5 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X3: product_prod_nat_nat,F6: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F6 )
             => ( ~ ( member8440522571783428010at_nat @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert8211810215607154385at_nat @ X3 @ F6 ) ) ) ) )
         => ( P @ F5 ) ) ) ) )).

% finite_induct
thf(fact_618_finite__induct,axiom,(
    ! [F5: set_Pr8693737435421807431at_nat,P: set_Pr8693737435421807431at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ F5 )
     => ( ( P @ bot_bo5327735625951526323at_nat )
       => ( ! [X3: produc859450856879609959at_nat,F6: set_Pr8693737435421807431at_nat] :
              ( ( finite4392333629123659920at_nat @ F6 )
             => ( ~ ( member8206827879206165904at_nat @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert5050368324300391991at_nat @ X3 @ F6 ) ) ) ) )
         => ( P @ F5 ) ) ) ) )).

% finite_induct
thf(fact_619_trancl__induct,axiom,(
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ! [Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: nat,Z4: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi6264000038957366511cl_nat @ R ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z4 ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z4 ) ) ) )
         => ( P @ B ) ) ) ) )).

% trancl_induct
thf(fact_620_trancl__induct,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi243908449541399842at_nat @ R ) )
     => ( ! [Y3: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ Y3 ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: product_prod_nat_nat,Z4: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ Y3 ) @ ( transi243908449541399842at_nat @ R ) )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z4 ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z4 ) ) ) )
         => ( P @ B ) ) ) ) )).

% trancl_induct
thf(fact_621_case__prod__Pair,axiom,
    ( ( produc3906647086178084059nteger @ produc6137756002093451184nteger )
    = id_Pro7099404743814044237nteger )).

% case_prod_Pair
thf(fact_622_case__prod__Pair,axiom,
    ( ( produc1300469243244976333atural @ produc3831813291587773865atural )
    = id_Pro460758741513610310atural )).

% case_prod_Pair
thf(fact_623_case__prod__Pair,axiom,
    ( ( produc6512950862096126219nteger @ produc8603105652947943368nteger )
    = id_Pro1379270990444177765nteger )).

% case_prod_Pair
thf(fact_624_case__prod__Pair,axiom,
    ( ( produc8492565224438309093nt_int @ produc5700946648718959541nt_int )
    = id_Pro5422067691592394322nt_int )).

% case_prod_Pair
thf(fact_625_case__prod__Pair,axiom,
    ( ( produc5122537100556696953nt_int @ produc4305682042979456191nt_int )
    = id_Pro3923183700162330716nt_int )).

% case_prod_Pair
thf(fact_626_mem__Sigma__iff,axiom,(
    ! [A: product_prod_nat_nat,B: produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B ) @ ( produc7465995907834253422at_nat @ A4 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ A @ A4 )
        & ( member8206827879206165904at_nat @ B @ ( B5 @ A ) ) ) ) )).

% mem_Sigma_iff
thf(fact_627_mem__Sigma__iff,axiom,(
    ! [A: produc859450856879609959at_nat,B: product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ A @ B ) @ ( produc3829605335166519074at_nat @ A4 @ B5 ) )
      = ( ( member8206827879206165904at_nat @ A @ A4 )
        & ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) )).

% mem_Sigma_iff
thf(fact_628_mem__Sigma__iff,axiom,(
    ! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ ( produc18557253517588744at_nat @ A4 @ B5 ) )
      = ( ( member8206827879206165904at_nat @ A @ A4 )
        & ( member8206827879206165904at_nat @ B @ ( B5 @ A ) ) ) ) )).

% mem_Sigma_iff
thf(fact_629_mem__Sigma__iff,axiom,(
    ! [A: nat,B: nat,A4: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
      = ( ( member_nat @ A @ A4 )
        & ( member_nat @ B @ ( B5 @ A ) ) ) ) )).

% mem_Sigma_iff
thf(fact_630_mem__Sigma__iff,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A4 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ A @ A4 )
        & ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) )).

% mem_Sigma_iff
thf(fact_631_mem__Sigma__iff,axiom,(
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: set_Co9149898834107579976e_term,B5: ( code_integer > option6357759511663192854e_term ) > set_Pr4811707699266497531nteger] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ A @ B ) @ ( produc4741912807555180607nteger @ A4 @ B5 ) )
      = ( ( member1535805642427569193e_term @ A @ A4 )
        & ( member157494554546826820nteger @ B @ ( B5 @ A ) ) ) ) )).

% mem_Sigma_iff
thf(fact_632_mem__Sigma__iff,axiom,(
    ! [A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural,A4: set_Pr2791530082921190693e_term,B5: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > set_Pr7564037697178849041atural] :
      ( ( member3245610415464309390atural @ ( produc3831813291587773865atural @ A @ B ) @ ( produc2485531694476998072atural @ A4 @ B5 ) )
      = ( ( member2820825060907577222e_term @ A @ A4 )
        & ( member8280416476884352090atural @ B @ ( B5 @ A ) ) ) ) )).

% mem_Sigma_iff
thf(fact_633_mem__Sigma__iff,axiom,(
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: set_Pr7604974323444597168e_term,B5: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr4811707699266497531nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ ( produc7256824055837167575nteger @ A4 @ B5 ) )
      = ( ( member4242434998011752849e_term @ A @ A4 )
        & ( member157494554546826820nteger @ B @ ( B5 @ A ) ) ) ) )).

% mem_Sigma_iff
thf(fact_634_mem__Sigma__iff,axiom,(
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A4: set_Pr3642885161833720865e_term,B5: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ ( produc5808185664642557892nt_int @ A4 @ B5 ) )
      = ( ( member3222579708246209666e_term @ A @ A4 )
        & ( member5262025264175285858nt_int @ B @ ( B5 @ A ) ) ) ) )).

% mem_Sigma_iff
thf(fact_635_mem__Sigma__iff,axiom,(
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: set_in3461395444621081367e_term,B5: ( int > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ ( produc1318478031590598990nt_int @ A4 @ B5 ) )
      = ( ( member8845023287901829240e_term @ A @ A4 )
        & ( member5262025264175285858nt_int @ B @ ( B5 @ A ) ) ) ) )).

% mem_Sigma_iff
thf(fact_636_trancl__induct2,axiom,(
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi243908449541399842at_nat @ R ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A3 @ B3 ) ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: nat,B3: nat,Aa: nat,Ba: nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A3 @ B3 ) ) @ ( transi243908449541399842at_nat @ R ) )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% trancl_induct2
thf(fact_637_trancl__induct2,axiom,(
    ! [Ax: code_integer > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: code_integer > option6357759511663192854e_term,By: produc8923325533196201883nteger,R: set_Pr5477269989976292413nteger,P: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ Bx @ By ) ) @ ( transi7147165069543534877nteger @ R ) )
     => ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
            ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ A3 @ B3 ) ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: code_integer > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ A3 @ B3 ) ) @ ( transi7147165069543534877nteger @ R ) )
             => ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ A3 @ B3 ) @ ( produc6137756002093451184nteger @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% trancl_induct2
thf(fact_638_trancl__induct2,axiom,(
    ! [Ax: produc4972180933644002618e_term > option6357759511663192854e_term,Ay: produc7822875418678951345atural,Bx: produc4972180933644002618e_term > option6357759511663192854e_term,By: produc7822875418678951345atural,R: set_Pr3316353890524442159atural,P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o] :
      ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ Bx @ By ) ) @ ( transi5310228724877665046atural @ R ) )
     => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural] :
            ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ A3 @ B3 ) ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural,Aa: produc4972180933644002618e_term > option6357759511663192854e_term,Ba: produc7822875418678951345atural] :
              ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ A3 @ B3 ) ) @ ( transi5310228724877665046atural @ R ) )
             => ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ A3 @ B3 ) @ ( produc3831813291587773865atural @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% trancl_induct2
thf(fact_639_trancl__induct2,axiom,(
    ! [Ax: produc6241069584506657477e_term > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: produc6241069584506657477e_term > option6357759511663192854e_term,By: produc8923325533196201883nteger,R: set_Pr1722956316086891245nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ Bx @ By ) ) @ ( transi6228740973808232501nteger @ R ) )
     => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
            ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ A3 @ B3 ) ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: produc6241069584506657477e_term > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ A3 @ B3 ) ) @ ( transi6228740973808232501nteger @ R ) )
             => ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ A3 @ B3 ) @ ( produc8603105652947943368nteger @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% trancl_induct2
thf(fact_640_trancl__induct2,axiom,(
    ! [Ax: produc8551481072490612790e_term > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: produc8551481072490612790e_term > option6357759511663192854e_term,By: product_prod_int_int,R: set_Pr4708930517165415495nt_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ Bx @ By ) ) @ ( transi7434818091516421410nt_int @ R ) )
     => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
            ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ A3 @ B3 ) ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int,Aa: produc8551481072490612790e_term > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ A3 @ B3 ) ) @ ( transi7434818091516421410nt_int @ R ) )
             => ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ A3 @ B3 ) @ ( produc5700946648718959541nt_int @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% trancl_induct2
thf(fact_641_trancl__induct2,axiom,(
    ! [Ax: int > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: int > option6357759511663192854e_term,By: product_prod_int_int,R: set_Pr8634505666381077339nt_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ Bx @ By ) ) @ ( transi4533545184864474924nt_int @ R ) )
     => ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
            ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ A3 @ B3 ) ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int,Aa: int > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ A3 @ B3 ) ) @ ( transi4533545184864474924nt_int @ R ) )
             => ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ A3 @ B3 ) @ ( produc4305682042979456191nt_int @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) )).

% trancl_induct2
thf(fact_642_converse__rtrancl__induct2,axiom,(
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R ) )
     => ( ( P @ Bx @ By )
       => ( ! [A3: nat,B3: nat,Aa: nat,Ba: nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_rtrancl_induct2
thf(fact_643_converse__rtrancl__induct2,axiom,(
    ! [Ax: code_integer > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: code_integer > option6357759511663192854e_term,By: produc8923325533196201883nteger,R: set_Pr5477269989976292413nteger,P: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ Ax @ Ay ) @ ( produc6137756002093451184nteger @ Bx @ By ) ) @ ( transi4577376780619456059nteger @ R ) )
     => ( ( P @ Bx @ By )
       => ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: code_integer > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ A3 @ B3 ) @ ( produc6137756002093451184nteger @ Aa @ Ba ) ) @ R )
             => ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ Aa @ Ba ) @ ( produc6137756002093451184nteger @ Bx @ By ) ) @ ( transi4577376780619456059nteger @ R ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_rtrancl_induct2
thf(fact_644_converse__rtrancl__induct2,axiom,(
    ! [Ax: produc4972180933644002618e_term > option6357759511663192854e_term,Ay: produc7822875418678951345atural,Bx: produc4972180933644002618e_term > option6357759511663192854e_term,By: produc7822875418678951345atural,R: set_Pr3316353890524442159atural,P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o] :
      ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ Ax @ Ay ) @ ( produc3831813291587773865atural @ Bx @ By ) ) @ ( transi3001887490979966772atural @ R ) )
     => ( ( P @ Bx @ By )
       => ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural,Aa: produc4972180933644002618e_term > option6357759511663192854e_term,Ba: produc7822875418678951345atural] :
              ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ A3 @ B3 ) @ ( produc3831813291587773865atural @ Aa @ Ba ) ) @ R )
             => ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ Aa @ Ba ) @ ( produc3831813291587773865atural @ Bx @ By ) ) @ ( transi3001887490979966772atural @ R ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_rtrancl_induct2
thf(fact_645_converse__rtrancl__induct2,axiom,(
    ! [Ax: produc6241069584506657477e_term > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: produc6241069584506657477e_term > option6357759511663192854e_term,By: produc8923325533196201883nteger,R: set_Pr1722956316086891245nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ Bx @ By ) ) @ ( transi3920399739910534227nteger @ R ) )
     => ( ( P @ Bx @ By )
       => ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger,Aa: produc6241069584506657477e_term > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ A3 @ B3 ) @ ( produc8603105652947943368nteger @ Aa @ Ba ) ) @ R )
             => ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ Aa @ Ba ) @ ( produc8603105652947943368nteger @ Bx @ By ) ) @ ( transi3920399739910534227nteger @ R ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_rtrancl_induct2
thf(fact_646_converse__rtrancl__induct2,axiom,(
    ! [Ax: produc8551481072490612790e_term > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: produc8551481072490612790e_term > option6357759511663192854e_term,By: product_prod_int_int,R: set_Pr4708930517165415495nt_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ Bx @ By ) ) @ ( transi2741729202006808896nt_int @ R ) )
     => ( ( P @ Bx @ By )
       => ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int,Aa: produc8551481072490612790e_term > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ A3 @ B3 ) @ ( produc5700946648718959541nt_int @ Aa @ Ba ) ) @ R )
             => ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ Aa @ Ba ) @ ( produc5700946648718959541nt_int @ Bx @ By ) ) @ ( transi2741729202006808896nt_int @ R ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_rtrancl_induct2
thf(fact_647_converse__rtrancl__induct2,axiom,(
    ! [Ax: int > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: int > option6357759511663192854e_term,By: product_prod_int_int,R: set_Pr8634505666381077339nt_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ Bx @ By ) ) @ ( transi8659768982470569034nt_int @ R ) )
     => ( ( P @ Bx @ By )
       => ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int,Aa: int > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ A3 @ B3 ) @ ( produc4305682042979456191nt_int @ Aa @ Ba ) ) @ R )
             => ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ Aa @ Ba ) @ ( produc4305682042979456191nt_int @ Bx @ By ) ) @ ( transi8659768982470569034nt_int @ R ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A3 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) )).

% converse_rtrancl_induct2
thf(fact_648_snd__o__Pair__conv__id,axiom,(
    ! [A: code_integer > option6357759511663192854e_term] :
      ( ( comp_P3738837344884965537nteger @ produc6514156286822040066nteger @ ( produc6137756002093451184nteger @ A ) )
      = id_Pro2309048762231606924nteger ) )).

% snd_o_Pair_conv_id
thf(fact_649_snd__o__Pair__conv__id,axiom,(
    ! [A: produc4972180933644002618e_term > option6357759511663192854e_term] :
      ( ( comp_P6750729840792639386atural @ produc3085575039554877691atural @ ( produc3831813291587773865atural @ A ) )
      = id_Pro1208598647714356386atural ) )).

% snd_o_Pair_conv_id
thf(fact_650_snd__o__Pair__conv__id,axiom,(
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( comp_P6417705687081997753nteger @ produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ A ) )
      = id_Pro2309048762231606924nteger ) )).

% snd_o_Pair_conv_id
thf(fact_651_snd__o__Pair__conv__id,axiom,(
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( comp_P465354750842043814nt_int @ produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ A ) )
      = id_Pro4381231610383771306nt_int ) )).

% snd_o_Pair_conv_id
thf(fact_652_snd__o__Pair__conv__id,axiom,(
    ! [A: int > option6357759511663192854e_term] :
      ( ( comp_P3461176452523280048nt_int @ produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ A ) )
      = id_Pro4381231610383771306nt_int ) )).

% snd_o_Pair_conv_id
thf(fact_653_snd__o__Pair__conv__id,axiom,(
    ! [A: code_natural] :
      ( ( comp_P5917919802303437345atural @ produc6591912806276919810atural @ ( produc6639722614265839536atural @ A ) )
      = id_Pro1208598647714356386atural ) )).

% snd_o_Pair_conv_id
thf(fact_654_snd__o__Pair__conv__id,axiom,(
    ! [A: finfun1145668972854894571action] :
      ( ( comp_P5708296701725499586list_f @ produc8710481132536983823list_f @ ( produc8331056147679645153list_f @ A ) )
      = id_Pro1034130279814846892list_f ) )).

% snd_o_Pair_conv_id
thf(fact_655_insertE,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member8440522571783428010at_nat @ A @ A4 ) ) ) )).

% insertE
thf(fact_656_insertE,axiom,(
    ! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ ( insert5050368324300391991at_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member8206827879206165904at_nat @ A @ A4 ) ) ) )).

% insertE
thf(fact_657_converse__rtrancl__induct,axiom,(
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( ( P @ B )
       => ( ! [Y3: nat,Z4: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z4 ) @ R )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z4 @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
               => ( ( P @ Z4 )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) )).

% converse_rtrancl_induct
thf(fact_658_converse__rtrancl__induct,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ R ) )
     => ( ( P @ B )
       => ( ! [Y3: product_prod_nat_nat,Z4: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z4 ) @ R )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z4 @ B ) @ ( transi8609417484261754244at_nat @ R ) )
               => ( ( P @ Z4 )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) )).

% converse_rtrancl_induct
thf(fact_659_UnI2,axiom,(
    ! [C2: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ B5 )
     => ( member8440522571783428010at_nat @ C2 @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) ) ) )).

% UnI2
thf(fact_660_UnI2,axiom,(
    ! [C2: produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ B5 )
     => ( member8206827879206165904at_nat @ C2 @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) ) )).

% UnI2
thf(fact_661_UnI1,axiom,(
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ A4 )
     => ( member8440522571783428010at_nat @ C2 @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) ) ) )).

% UnI1
thf(fact_662_UnI1,axiom,(
    ! [C2: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ A4 )
     => ( member8206827879206165904at_nat @ C2 @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) ) )).

% UnI1
thf(fact_663_lex__prod__def,axiom,
    ( lex_pr8029265285556086080at_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec3253044014973990424at_nat
          @ ( produc2869316064735525022_nat_o
            @ ( produc7480869177107369060_nat_o
              @ ^ [A5: nat,B4: product_prod_nat_nat] :
                  ( produc5864757623865647827_nat_o
                  @ ^ [A7: nat,B6: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A7 ) @ Ra )
                      | ( ( A5 = A7 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) )).

% lex_prod_def
thf(fact_664_lex__prod__def,axiom,
    ( lex_pr4669217892513370978at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec2200730763347676620at_nat
          @ ( produc3600351907215576810_nat_o
            @ ( produc6198819664464039784_nat_o
              @ ^ [A5: product_prod_nat_nat,B4: nat] :
                  ( produc8758269395934548017_nat_o
                  @ ^ [A7: product_prod_nat_nat,B6: nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ A7 ) @ Ra )
                      | ( ( A5 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) )).

% lex_prod_def
thf(fact_665_lex__prod__def,axiom,
    ( lex_pr8801849515957261039at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec7334067512558549330at_nat
          @ ( produc3603361296207914284_nat_o
            @ ( produc5077327844045112642_nat_o
              @ ^ [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
                  ( produc6590410687421337004_nat_o
                  @ ^ [A7: product_prod_nat_nat,B6: product_prod_nat_nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ A7 ) @ Ra )
                      | ( ( A5 = A7 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) )).

% lex_prod_def
thf(fact_666_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A5: nat,B4: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A7: nat,B6: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A7 ) @ Ra )
                      | ( ( A5 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) )).

% lex_prod_def
thf(fact_667_map__prod_Ocompositionality,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,H: produc8923597787767437993list_f > produc1811169103916964701list_f,I2: produc8923597787767437993list_f > produc1811169103916964701list_f,Prod: produc6644936158991452263list_f] :
      ( ( produc4841148139688540918_a_b_c @ F @ G2 @ ( produc5505274914705194486list_f @ H @ I2 @ Prod ) )
      = ( produc8198191332872049718_a_b_c @ ( comp_P4767578765025054880list_f @ F @ H ) @ ( comp_P4767578765025054880list_f @ G2 @ I2 ) @ Prod ) ) )).

% map_prod.compositionality
thf(fact_668_map__prod__compose,axiom,(
    ! [F1: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,F22: produc8923597787767437993list_f > produc1811169103916964701list_f,G1: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G22: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( produc8198191332872049718_a_b_c @ ( comp_P4767578765025054880list_f @ F1 @ F22 ) @ ( comp_P4767578765025054880list_f @ G1 @ G22 ) )
      = ( comp_P7472824288426439864list_f @ ( produc4841148139688540918_a_b_c @ F1 @ G1 ) @ ( produc5505274914705194486list_f @ F22 @ G22 ) ) ) )).

% map_prod_compose
thf(fact_669_map__prod_Ocomp,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,H: produc8923597787767437993list_f > produc1811169103916964701list_f,I2: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( comp_P7472824288426439864list_f @ ( produc4841148139688540918_a_b_c @ F @ G2 ) @ ( produc5505274914705194486list_f @ H @ I2 ) )
      = ( produc8198191332872049718_a_b_c @ ( comp_P4767578765025054880list_f @ F @ H ) @ ( comp_P4767578765025054880list_f @ G2 @ I2 ) ) ) )).

% map_prod.comp
thf(fact_670_case__prodI2,axiom,(
    ! [P2: produc8763457246119570046nteger,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( ( P2
            = ( produc6137756002093451184nteger @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc127349428274296955eger_o @ C2 @ P2 ) ) )).

% case_prodI2
thf(fact_671_case__prodI2,axiom,(
    ! [P2: produc989692990947075319atural,C2: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o] :
      ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural] :
          ( ( P2
            = ( produc3831813291587773865atural @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc2278936416606902338ural_o @ C2 @ P2 ) ) )).

% case_prodI2
thf(fact_672_case__prodI2,axiom,(
    ! [P2: produc1908205239877642774nteger,C2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( ( P2
            = ( produc8603105652947943368nteger @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc6253627499356882019eger_o @ C2 @ P2 ) ) )).

% case_prodI2
thf(fact_673_case__prodI2,axiom,(
    ! [P2: produc2285326912895808259nt_int,C2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( ( P2
            = ( produc5700946648718959541nt_int @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc1573362020775583542_int_o @ C2 @ P2 ) ) )).

% case_prodI2
thf(fact_674_case__prodI2,axiom,(
    ! [P2: produc7773217078559923341nt_int,C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( ( P2
            = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc2558449545302689196_int_o @ C2 @ P2 ) ) )).

% case_prodI2
thf(fact_675_same__fstI,axiom,(
    ! [P: nat > $o,X: nat,Y8: nat,Y: nat,R2: nat > set_Pr1261947904930325089at_nat] :
      ( ( P @ X )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y8 @ Y ) @ ( R2 @ X ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y8 ) @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( same_fst_nat_nat @ P @ R2 ) ) ) ) )).

% same_fstI
thf(fact_676_same__fstI,axiom,(
    ! [P: ( code_integer > option6357759511663192854e_term ) > $o,X: code_integer > option6357759511663192854e_term,Y8: produc8923325533196201883nteger,Y: produc8923325533196201883nteger,R2: ( code_integer > option6357759511663192854e_term ) > set_Pr5052581263052047815nteger] :
      ( ( P @ X )
     => ( ( member335077259223437328nteger @ ( produc4911887526399440471nteger @ Y8 @ Y ) @ ( R2 @ X ) )
       => ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ X @ Y8 ) @ ( produc6137756002093451184nteger @ X @ Y ) ) @ ( same_f4869244030753856539nteger @ P @ R2 ) ) ) ) )).

% same_fstI
thf(fact_677_same__fstI,axiom,(
    ! [P: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > $o,X: produc4972180933644002618e_term > option6357759511663192854e_term,Y8: produc7822875418678951345atural,Y: produc7822875418678951345atural,R2: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > set_Pr250711821884878663atural] :
      ( ( P @ X )
     => ( ( member3756964795865971088atural @ ( produc4480994950612372183atural @ Y8 @ Y ) @ ( R2 @ X ) )
       => ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ X @ Y8 ) @ ( produc3831813291587773865atural @ X @ Y ) ) @ ( same_f3267511052797212564atural @ P @ R2 ) ) ) ) )).

% same_fstI
thf(fact_678_same__fstI,axiom,(
    ! [P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y8: produc8923325533196201883nteger,Y: produc8923325533196201883nteger,R2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr5052581263052047815nteger] :
      ( ( P @ X )
     => ( ( member335077259223437328nteger @ ( produc4911887526399440471nteger @ Y8 @ Y ) @ ( R2 @ X ) )
       => ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ X @ Y8 ) @ ( produc8603105652947943368nteger @ X @ Y ) ) @ ( same_f8038803414157382067nteger @ P @ R2 ) ) ) ) )).

% same_fstI
thf(fact_679_same__fstI,axiom,(
    ! [P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y8: product_prod_int_int,Y: product_prod_int_int,R2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int] :
      ( ( P @ X )
     => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y8 @ Y ) @ ( R2 @ X ) )
       => ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ X @ Y8 ) @ ( produc5700946648718959541nt_int @ X @ Y ) ) @ ( same_f5472592420709775776nt_int @ P @ R2 ) ) ) ) )).

% same_fstI
thf(fact_680_same__fstI,axiom,(
    ! [P: ( int > option6357759511663192854e_term ) > $o,X: int > option6357759511663192854e_term,Y8: product_prod_int_int,Y: product_prod_int_int,R2: ( int > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int] :
      ( ( P @ X )
     => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y8 @ Y ) @ ( R2 @ X ) )
       => ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ X @ Y8 ) @ ( produc4305682042979456191nt_int @ X @ Y ) ) @ ( same_f6662725367016992042nt_int @ P @ R2 ) ) ) ) )).

% same_fstI
thf(fact_681_image__subsetI,axiom,(
    ! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A4 )
         => ( member8440522571783428010at_nat @ ( F @ X3 ) @ B5 ) )
     => ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ F @ A4 ) @ B5 ) ) )).

% image_subsetI
thf(fact_682_image__subsetI,axiom,(
    ! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A4 )
         => ( member8206827879206165904at_nat @ ( F @ X3 ) @ B5 ) )
     => ( ord_le3000389064537975527at_nat @ ( image_1119892238107900587at_nat @ F @ A4 ) @ B5 ) ) )).

% image_subsetI
thf(fact_683_image__subsetI,axiom,(
    ! [A4: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [X3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A4 )
         => ( member8440522571783428010at_nat @ ( F @ X3 ) @ B5 ) )
     => ( ord_le3146513528884898305at_nat @ ( image_6706873702294942047at_nat @ F @ A4 ) @ B5 ) ) )).

% image_subsetI
thf(fact_684_image__subsetI,axiom,(
    ! [A4: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ! [X3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A4 )
         => ( member8206827879206165904at_nat @ ( F @ X3 ) @ B5 ) )
     => ( ord_le3000389064537975527at_nat @ ( image_6321802852832981957at_nat @ F @ A4 ) @ B5 ) ) )).

% image_subsetI
thf(fact_685_converse__rtranclE2,axiom,(
    ! [Xa: nat,Xb: nat,Za: nat,Zb: nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R ) )
     => ( ( ( product_Pair_nat_nat @ Xa @ Xb )
         != ( product_Pair_nat_nat @ Za @ Zb ) )
       => ~ ! [A3: nat,B3: nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ A3 @ B3 ) ) @ R )
             => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R ) ) ) ) ) )).

% converse_rtranclE2
thf(fact_686_converse__rtranclE2,axiom,(
    ! [Xa: code_integer > option6357759511663192854e_term,Xb: produc8923325533196201883nteger,Za: code_integer > option6357759511663192854e_term,Zb: produc8923325533196201883nteger,R: set_Pr5477269989976292413nteger] :
      ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ Xa @ Xb ) @ ( produc6137756002093451184nteger @ Za @ Zb ) ) @ ( transi4577376780619456059nteger @ R ) )
     => ( ( ( produc6137756002093451184nteger @ Xa @ Xb )
         != ( produc6137756002093451184nteger @ Za @ Zb ) )
       => ~ ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
              ( ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ Xa @ Xb ) @ ( produc6137756002093451184nteger @ A3 @ B3 ) ) @ R )
             => ~ ( member6711696910736417670nteger @ ( produc4135541594033497365nteger @ ( produc6137756002093451184nteger @ A3 @ B3 ) @ ( produc6137756002093451184nteger @ Za @ Zb ) ) @ ( transi4577376780619456059nteger @ R ) ) ) ) ) )).

% converse_rtranclE2
thf(fact_687_converse__rtranclE2,axiom,(
    ! [Xa: produc4972180933644002618e_term > option6357759511663192854e_term,Xb: produc7822875418678951345atural,Za: produc4972180933644002618e_term > option6357759511663192854e_term,Zb: produc7822875418678951345atural,R: set_Pr3316353890524442159atural] :
      ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ Xa @ Xb ) @ ( produc3831813291587773865atural @ Za @ Zb ) ) @ ( transi3001887490979966772atural @ R ) )
     => ( ( ( produc3831813291587773865atural @ Xa @ Xb )
         != ( produc3831813291587773865atural @ Za @ Zb ) )
       => ~ ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural] :
              ( ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ Xa @ Xb ) @ ( produc3831813291587773865atural @ A3 @ B3 ) ) @ R )
             => ~ ( member6545400748642591352atural @ ( produc8172598920672188551atural @ ( produc3831813291587773865atural @ A3 @ B3 ) @ ( produc3831813291587773865atural @ Za @ Zb ) ) @ ( transi3001887490979966772atural @ R ) ) ) ) ) )).

% converse_rtranclE2
thf(fact_688_converse__rtranclE2,axiom,(
    ! [Xa: produc6241069584506657477e_term > option6357759511663192854e_term,Xb: produc8923325533196201883nteger,Za: produc6241069584506657477e_term > option6357759511663192854e_term,Zb: produc8923325533196201883nteger,R: set_Pr1722956316086891245nteger] :
      ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ Xa @ Xb ) @ ( produc8603105652947943368nteger @ Za @ Zb ) ) @ ( transi3920399739910534227nteger @ R ) )
     => ( ( ( produc8603105652947943368nteger @ Xa @ Xb )
         != ( produc8603105652947943368nteger @ Za @ Zb ) )
       => ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
              ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ Xa @ Xb ) @ ( produc8603105652947943368nteger @ A3 @ B3 ) ) @ R )
             => ~ ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ A3 @ B3 ) @ ( produc8603105652947943368nteger @ Za @ Zb ) ) @ ( transi3920399739910534227nteger @ R ) ) ) ) ) )).

% converse_rtranclE2
thf(fact_689_converse__rtranclE2,axiom,(
    ! [Xa: produc8551481072490612790e_term > option6357759511663192854e_term,Xb: product_prod_int_int,Za: produc8551481072490612790e_term > option6357759511663192854e_term,Zb: product_prod_int_int,R: set_Pr4708930517165415495nt_int] :
      ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ Xa @ Xb ) @ ( produc5700946648718959541nt_int @ Za @ Zb ) ) @ ( transi2741729202006808896nt_int @ R ) )
     => ( ( ( produc5700946648718959541nt_int @ Xa @ Xb )
         != ( produc5700946648718959541nt_int @ Za @ Zb ) )
       => ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
              ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ Xa @ Xb ) @ ( produc5700946648718959541nt_int @ A3 @ B3 ) ) @ R )
             => ~ ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ A3 @ B3 ) @ ( produc5700946648718959541nt_int @ Za @ Zb ) ) @ ( transi2741729202006808896nt_int @ R ) ) ) ) ) )).

% converse_rtranclE2
thf(fact_690_converse__rtranclE2,axiom,(
    ! [Xa: int > option6357759511663192854e_term,Xb: product_prod_int_int,Za: int > option6357759511663192854e_term,Zb: product_prod_int_int,R: set_Pr8634505666381077339nt_int] :
      ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ Xa @ Xb ) @ ( produc4305682042979456191nt_int @ Za @ Zb ) ) @ ( transi8659768982470569034nt_int @ R ) )
     => ( ( ( produc4305682042979456191nt_int @ Xa @ Xb )
         != ( produc4305682042979456191nt_int @ Za @ Zb ) )
       => ~ ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
              ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ Xa @ Xb ) @ ( produc4305682042979456191nt_int @ A3 @ B3 ) ) @ R )
             => ~ ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ A3 @ B3 ) @ ( produc4305682042979456191nt_int @ Za @ Zb ) ) @ ( transi8659768982470569034nt_int @ R ) ) ) ) ) )).

% converse_rtranclE2
thf(fact_691_inf__sup__ord_I4_J,axiom,(
    ! [Y: nat,X: nat] :
      ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) )).

% inf_sup_ord(4)
thf(fact_692_pred__prod__inject,axiom,(
    ! [P1: ( code_integer > option6357759511663192854e_term ) > $o,P22: produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( basic_8544396027437066004nteger @ P1 @ P22 @ ( produc6137756002093451184nteger @ A @ B ) )
      = ( ( P1 @ A )
        & ( P22 @ B ) ) ) )).

% pred_prod_inject
thf(fact_693_pred__prod__inject,axiom,(
    ! [P1: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > $o,P22: produc7822875418678951345atural > $o,A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural] :
      ( ( basic_2435318813636073229atural @ P1 @ P22 @ ( produc3831813291587773865atural @ A @ B ) )
      = ( ( P1 @ A )
        & ( P22 @ B ) ) ) )).

% pred_prod_inject
thf(fact_694_pred__prod__inject,axiom,(
    ! [P1: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,P22: produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( basic_7206611174996242732nteger @ P1 @ P22 @ ( produc8603105652947943368nteger @ A @ B ) )
      = ( ( P1 @ A )
        & ( P22 @ B ) ) ) )).

% pred_prod_inject
thf(fact_695_pred__prod__inject,axiom,(
    ! [P1: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,P22: product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( basic_5149139111306888473nt_int @ P1 @ P22 @ ( produc5700946648718959541nt_int @ A @ B ) )
      = ( ( P1 @ A )
        & ( P22 @ B ) ) ) )).

% pred_prod_inject
thf(fact_696_pred__prod__inject,axiom,(
    ! [P1: ( int > option6357759511663192854e_term ) > $o,P22: product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( basic_3085415318199739427nt_int @ P1 @ P22 @ ( produc4305682042979456191nt_int @ A @ B ) )
      = ( ( P1 @ A )
        & ( P22 @ B ) ) ) )).

% pred_prod_inject
thf(fact_697_inf__sup__ord_I3_J,axiom,(
    ! [X: nat,Y: nat] :
      ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) )).

% inf_sup_ord(3)
thf(fact_698_le__supE,axiom,(
    ! [A: nat,B: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_nat @ A @ X )
         => ~ ( ord_less_eq_nat @ B @ X ) ) ) )).

% le_supE
thf(fact_699_converse__rtranclE,axiom,(
    ! [X: nat,Z: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( ( X != Z )
       => ~ ! [Y3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ) )).

% converse_rtranclE
thf(fact_700_converse__rtranclE,axiom,(
    ! [X: product_prod_nat_nat,Z: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z ) @ ( transi8609417484261754244at_nat @ R ) )
     => ( ( X != Z )
       => ~ ! [Y3: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R )
             => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z ) @ ( transi8609417484261754244at_nat @ R ) ) ) ) ) )).

% converse_rtranclE
thf(fact_701_le__supI,axiom,(
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ( ord_less_eq_nat @ B @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) )).

% le_supI
thf(fact_702_Id__onI,axiom,(
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A4 ) ) ) )).

% Id_onI
thf(fact_703_Id__onI,axiom,(
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( id_on_2554058798563519774at_nat @ A4 ) ) ) )).

% Id_onI
thf(fact_704_Id__onI,axiom,(
    ! [A: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A4 )
     => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ A ) @ ( id_on_7917584626360399108at_nat @ A4 ) ) ) )).

% Id_onI
thf(fact_705_sup__ge1,axiom,(
    ! [X: nat,Y: nat] :
      ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) )).

% sup_ge1
thf(fact_706_sup__ge2,axiom,(
    ! [Y: nat,X: nat] :
      ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) )).

% sup_ge2
thf(fact_707_max_Ocobounded1,axiom,(
    ! [A: nat,B: nat] :
      ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) )).

% max.cobounded1
thf(fact_708_le__supI1,axiom,(
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) )).

% le_supI1
thf(fact_709_le__supI2,axiom,(
    ! [X: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ X @ B )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) )).

% le_supI2
thf(fact_710_sup_Omono,axiom,(
    ! [C2: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) )).

% sup.mono
thf(fact_711_sup__mono,axiom,(
    ! [A: nat,C2: nat,B: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ D )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) )).

% sup_mono
thf(fact_712_sup__least,axiom,(
    ! [Y: nat,X: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) )).

% sup_least
thf(fact_713_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y6: nat] :
          ( ( sup_sup_nat @ X4 @ Y6 )
          = Y6 ) ) )).

% le_iff_sup
thf(fact_714_case__prod__eta,axiom,(
    ! [F: produc7822875418678951345atural > produc6756470014200349799atural] :
      ( ( produc8282080750456430313atural
        @ ^ [X4: code_natural,Y6: code_natural] :
            ( F @ ( produc3574140220909816553atural @ X4 @ Y6 ) ) )
      = F ) )).

% case_prod_eta
thf(fact_715_max_Ocobounded2,axiom,(
    ! [B: nat,A: nat] :
      ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) )).

% max.cobounded2
thf(fact_716_sup_OorderE,axiom,(
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( sup_sup_nat @ A @ B ) ) ) )).

% sup.orderE
thf(fact_717_pair__imageI,axiom,(
    ! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F: nat > nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A4 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_5168914502847457605at_nat @ ( produc2626176000494625587at_nat @ F ) @ A4 ) ) ) )).

% pair_imageI
thf(fact_718_pair__imageI,axiom,(
    ! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F: nat > nat > produc859450856879609959at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A4 )
     => ( member8206827879206165904at_nat @ ( F @ A @ B ) @ ( image_1119892238107900587at_nat @ ( produc2382474379627796377at_nat @ F ) @ A4 ) ) ) )).

% pair_imageI
thf(fact_719_pair__imageI,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ A4 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_6706873702294942047at_nat @ ( produc2982569246487150357at_nat @ F ) @ A4 ) ) ) )).

% pair_imageI
thf(fact_720_pair__imageI,axiom,(
    ! [A: code_natural,B: code_natural,A4: set_Pr7564037697178849041atural,F: code_natural > code_natural > produc6756470014200349799atural] :
      ( ( member8280416476884352090atural @ ( produc3574140220909816553atural @ A @ B ) @ A4 )
     => ( member3756964795865971088atural @ ( F @ A @ B ) @ ( image_561289856197260795atural @ ( produc8282080750456430313atural @ F ) @ A4 ) ) ) )).

% pair_imageI
thf(fact_721_pair__imageI,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ A4 )
     => ( member8206827879206165904at_nat @ ( F @ A @ B ) @ ( image_6321802852832981957at_nat @ ( produc6744312701629110395at_nat @ F ) @ A4 ) ) ) )).

% pair_imageI
thf(fact_722_pair__imageI,axiom,(
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: set_Pr8056137968301705908nteger,F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > product_prod_nat_nat] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ A @ B ) @ A4 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_9094096100532249404at_nat @ ( produc6240753075098568420at_nat @ F ) @ A4 ) ) ) )).

% pair_imageI
thf(fact_723_pair__imageI,axiom,(
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: set_Pr1872883991513573699nt_int,F: ( int > option6357759511663192854e_term ) > product_prod_int_int > product_prod_nat_nat] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ A4 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_6286574459949954669at_nat @ ( produc5637086384513480469at_nat @ F ) @ A4 ) ) ) )).

% pair_imageI
thf(fact_724_pair__imageI,axiom,(
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: set_Pr8056137968301705908nteger,F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc859450856879609959at_nat] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ A @ B ) @ A4 )
     => ( member8206827879206165904at_nat @ ( F @ A @ B ) @ ( image_3866311809354581538at_nat @ ( produc6013025977344789962at_nat @ F ) @ A4 ) ) ) )).

% pair_imageI
thf(fact_725_pair__imageI,axiom,(
    ! [A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural,A4: set_Pr4244404650639074861atural,F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > product_prod_nat_nat] :
      ( ( member3245610415464309390atural @ ( produc3831813291587773865atural @ A @ B ) @ A4 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_3500444250886205059at_nat @ ( produc4745849324656339883at_nat @ F ) @ A4 ) ) ) )).

% pair_imageI
thf(fact_726_pair__imageI,axiom,(
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: set_Pr1281608226676607948nteger,F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > product_prod_nat_nat] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ A4 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_8838613453202580132at_nat @ ( produc3275924862804298444at_nat @ F ) @ A4 ) ) ) )).

% pair_imageI
thf(fact_727_case__prodE2,axiom,(
    ! [Q: produc6756470014200349799atural > $o,P: code_natural > code_natural > produc6756470014200349799atural,Z: produc7822875418678951345atural] :
      ( ( Q @ ( produc8282080750456430313atural @ P @ Z ) )
     => ~ ! [X3: code_natural,Y3: code_natural] :
            ( ( Z
              = ( produc3574140220909816553atural @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) )).

% case_prodE2
thf(fact_728_extract__Cons__code,axiom,(
    ! [P: typerep > $o,X: typerep,Xs: list_typerep] :
      ( ( ( P @ X )
       => ( ( extract_typerep @ P @ ( cons_typerep @ X @ Xs ) )
          = ( some_P4742658530298911237yperep @ ( produc2255752811254757084yperep @ nil_typerep @ ( produc8398273619233227473yperep @ X @ Xs ) ) ) ) )
      & ( ~ ( P @ X )
       => ( ( extract_typerep @ P @ ( cons_typerep @ X @ Xs ) )
          = ( case_o7657976496830776113yperep @ none_P7642766325015749769yperep
            @ ( produc362396003296346243yperep
              @ ^ [Ys2: list_typerep] :
                  ( produc7628607474913760352yperep
                  @ ^ [Y6: typerep,Zs2: list_typerep] :
                      ( some_P4742658530298911237yperep @ ( produc2255752811254757084yperep @ ( cons_typerep @ X @ Ys2 ) @ ( produc8398273619233227473yperep @ Y6 @ Zs2 ) ) ) ) )
            @ ( extract_typerep @ P @ Xs ) ) ) ) ) )).

% extract_Cons_code
thf(fact_729_sup_OorderI,axiom,(
    ! [A: nat,B: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) )).

% sup.orderI
thf(fact_730_eventually__prod__filter,axiom,(
    ! [P: produc8763457246119570046nteger > $o,F5: filter7395430969833108130e_term,G4: filter5292368968559144737nteger] :
      ( ( eventu2468618571050227062nteger @ P @ ( prod_f4507364902793478996nteger @ F5 @ G4 ) )
      = ( ? [Pf: ( code_integer > option6357759511663192854e_term ) > $o,Pg: produc8923325533196201883nteger > $o] :
            ( ( eventu2762411978193089162e_term @ Pf @ F5 )
            & ( eventu3177686391277443747nteger @ Pg @ G4 )
            & ! [X4: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
                ( ( Pf @ X4 )
               => ( ( Pg @ Y6 )
                 => ( P @ ( produc6137756002093451184nteger @ X4 @ Y6 ) ) ) ) ) ) ) )).

% eventually_prod_filter
thf(fact_731_eventually__prod__filter,axiom,(
    ! [P: produc989692990947075319atural > $o,F5: filter4526729644159477375e_term,G4: filter8044698966471496247atural] :
      ( ( eventu4166999816575306735atural @ P @ ( prod_f7592866989393777485atural @ F5 @ G4 ) )
      = ( ? [Pf: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > $o,Pg: produc7822875418678951345atural > $o] :
            ( ( eventu6388207738291344999e_term @ Pf @ F5 )
            & ( eventu2077236276760193209atural @ Pg @ G4 )
            & ! [X4: produc4972180933644002618e_term > option6357759511663192854e_term,Y6: produc7822875418678951345atural] :
                ( ( Pf @ X4 )
               => ( ( Pg @ Y6 )
                 => ( P @ ( produc3831813291587773865atural @ X4 @ Y6 ) ) ) ) ) ) ) )).

% eventually_prod_filter
thf(fact_732_eventually__prod__filter,axiom,(
    ! [P: produc1908205239877642774nteger > $o,F5: filter116801847828108042e_term,G4: filter5292368968559144737nteger] :
      ( ( eventu5085512065505874190nteger @ P @ ( prod_f3140787313899171180nteger @ F5 @ G4 ) )
      = ( ? [Pf: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,Pg: produc8923325533196201883nteger > $o] :
            ( ( eventu7809817675395520626e_term @ Pf @ F5 )
            & ( eventu3177686391277443747nteger @ Pg @ G4 )
            & ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
                ( ( Pf @ X4 )
               => ( ( Pg @ Y6 )
                 => ( P @ ( produc8603105652947943368nteger @ X4 @ Y6 ) ) ) ) ) ) ) )).

% eventually_prod_filter
thf(fact_733_eventually__prod__filter,axiom,(
    ! [P: produc2285326912895808259nt_int > $o,F5: filter5615094840926397819e_term,G4: filter938913474090903615nt_int] :
      ( ( eventu4547920219520219643nt_int @ P @ ( prod_f111972166261014873nt_int @ F5 @ G4 ) )
      = ( ? [Pf: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,Pg: product_prod_int_int > $o] :
            ( ( eventu5410056986863673187e_term @ Pf @ F5 )
            & ( eventu7082874808314849985nt_int @ Pg @ G4 )
            & ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
                ( ( Pf @ X4 )
               => ( ( Pg @ Y6 )
                 => ( P @ ( produc5700946648718959541nt_int @ X4 @ Y6 ) ) ) ) ) ) ) )).

% eventually_prod_filter
thf(fact_734_eventually__prod__filter,axiom,(
    ! [P: produc7773217078559923341nt_int > $o,F5: filter8849932399892420465e_term,G4: filter938913474090903615nt_int] :
      ( ( eventu8011441516627224453nt_int @ P @ ( prod_f5158093079382032483nt_int @ F5 @ G4 ) )
      = ( ? [Pf: ( int > option6357759511663192854e_term ) > $o,Pg: product_prod_int_int > $o] :
            ( ( eventu5275552764359891673e_term @ Pf @ F5 )
            & ( eventu7082874808314849985nt_int @ Pg @ G4 )
            & ! [X4: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
                ( ( Pf @ X4 )
               => ( ( Pg @ Y6 )
                 => ( P @ ( produc4305682042979456191nt_int @ X4 @ Y6 ) ) ) ) ) ) ) )).

% eventually_prod_filter
thf(fact_735_sup__unique,axiom,(
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ord_less_eq_nat @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: nat,Y3: nat,Z4: nat] :
              ( ( ord_less_eq_nat @ Y3 @ X3 )
             => ( ( ord_less_eq_nat @ Z4 @ X3 )
               => ( ord_less_eq_nat @ ( F @ Y3 @ Z4 ) @ X3 ) ) )
         => ( ( sup_sup_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) )).

% sup_unique
thf(fact_736_sup_Oabsorb1,axiom,(
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) )).

% sup.absorb1
thf(fact_737_sup_Oabsorb2,axiom,(
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) )).

% sup.absorb2
thf(fact_738_mem__case__prodI,axiom,(
    ! [Z: product_prod_nat_nat,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_Pr1261947904930325089at_nat,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member8440522571783428010at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc4551263766066629700at_nat @ C2 @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) )).

% mem_case_prodI
thf(fact_739_mem__case__prodI,axiom,(
    ! [Z: produc859450856879609959at_nat,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_Pr8693737435421807431at_nat,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member8206827879206165904at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc5744607925856648362at_nat @ C2 @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) )).

% mem_case_prodI
thf(fact_740_mem__case__prodI,axiom,(
    ! [Z: product_prod_nat_nat,C2: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > set_Pr1261947904930325089at_nat,A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural] :
      ( ( member8440522571783428010at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc5359040883261003019at_nat @ C2 @ ( produc3831813291587773865atural @ A @ B ) ) ) ) )).

% mem_case_prodI
thf(fact_741_mem__case__prodI,axiom,(
    ! [Z: produc859450856879609959at_nat,C2: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > set_Pr8693737435421807431at_nat,A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural] :
      ( ( member8206827879206165904at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc5211177361811169009at_nat @ C2 @ ( produc3831813291587773865atural @ A @ B ) ) ) ) )).

% mem_case_prodI
thf(fact_742_mem__case__prodI,axiom,(
    ! [Z: product_prod_nat_nat,C2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_Pr1261947904930325089at_nat,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member8440522571783428010at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc6029229664795570220at_nat @ C2 @ ( produc8603105652947943368nteger @ A @ B ) ) ) ) )).

% mem_case_prodI
thf(fact_743_mem__case__prodI,axiom,(
    ! [Z: produc859450856879609959at_nat,C2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_Pr8693737435421807431at_nat,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member8206827879206165904at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc2704644331147814034at_nat @ C2 @ ( produc8603105652947943368nteger @ A @ B ) ) ) ) )).

% mem_case_prodI
thf(fact_744_mem__case__prodI,axiom,(
    ! [Z: product_prod_nat_nat,C2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_Pr1261947904930325089at_nat,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member8440522571783428010at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc2642701577560500223at_nat @ C2 @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ) )).

% mem_case_prodI
thf(fact_745_mem__case__prodI,axiom,(
    ! [Z: produc859450856879609959at_nat,C2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_Pr8693737435421807431at_nat,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member8206827879206165904at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc517896604054025189at_nat @ C2 @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ) )).

% mem_case_prodI
thf(fact_746_mem__case__prodI,axiom,(
    ! [Z: product_prod_nat_nat,C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_Pr1261947904930325089at_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member8440522571783428010at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc5154663461364529269at_nat @ C2 @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) )).

% mem_case_prodI
thf(fact_747_mem__case__prodI,axiom,(
    ! [Z: produc859450856879609959at_nat,C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_Pr8693737435421807431at_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member8206827879206165904at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc2114697605475793755at_nat @ C2 @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) )).

% mem_case_prodI
thf(fact_748_sup__absorb1,axiom,(
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) )).

% sup_absorb1
thf(fact_749_sup__absorb2,axiom,(
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) )).

% sup_absorb2
thf(fact_750_sup_OboundedE,axiom,(
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C2 @ A ) ) ) )).

% sup.boundedE
thf(fact_751_snd__diag__snd,axiom,
    ( ( comp_P230220204662055659atural @ produc7455300194457530885atural
      @ ( comp_P4943825737035310295atural
        @ ^ [X4: produc7822875418678951345atural] :
            ( produc4480994950612372183atural @ X4 @ X4 )
        @ produc6591912806276919810atural ) )
    = produc6591912806276919810atural )).

% snd_diag_snd
thf(fact_752_snd__diag__snd,axiom,
    ( ( comp_P2690847479783168146list_f @ produc5351416724170879361list_f
      @ ( comp_P8324358310098339598list_f
        @ ^ [X4: produc1811169103916964701list_f] :
            ( produc7497196274192653651list_f @ X4 @ X4 )
        @ produc8710481132536983823list_f ) )
    = produc8710481132536983823list_f )).

% snd_diag_snd
thf(fact_753_fst__diag__fst,axiom,
    ( ( comp_P662548419035233652nteger @ produc3955429057510563241e_term
      @ ( comp_C5262483154866357838nteger
        @ ^ [X4: code_integer > option6357759511663192854e_term] :
            ( produc2189037350439096765e_term @ X4 @ X4 )
        @ produc2702340305111172292nteger ) )
    = produc2702340305111172292nteger )).

% fst_diag_fst
thf(fact_754_fst__diag__fst,axiom,
    ( ( comp_P5776439315172467588atural @ produc7419616383666439523e_term
      @ ( comp_P1227397885194180452atural
        @ ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term] :
            ( produc734250632046660983e_term @ X4 @ X4 )
        @ produc3051390257598303421atural ) )
    = produc3051390257598303421atural )).

% fst_diag_fst
thf(fact_755_fst__diag__fst,axiom,
    ( ( comp_P9158442376665023684nteger @ produc4377055093455473401e_term
      @ ( comp_P1612206844032044750nteger
        @ ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term] :
            ( produc6915061378690470669e_term @ X4 @ X4 )
        @ produc7822682618958472924nteger ) )
    = produc7822682618958472924nteger )).

% fst_diag_fst
thf(fact_756_fst__diag__fst,axiom,
    ( ( comp_P1627152983954642052nt_int @ produc4524392572497827163e_term
      @ ( comp_P4947044428546866028nt_int
        @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term] :
            ( produc8124735558134971759e_term @ X4 @ X4 )
        @ produc6816164490631068361nt_int ) )
    = produc6816164490631068361nt_int )).

% fst_diag_fst
thf(fact_757_fst__diag__fst,axiom,
    ( ( comp_P607695144065014128nt_int @ produc2261262461388061255e_term
      @ ( comp_i2761858172409570284nt_int
        @ ^ [X4: int > option6357759511663192854e_term] :
            ( produc4510790414113558107e_term @ X4 @ X4 )
        @ produc6230002227079971283nt_int ) )
    = produc6230002227079971283nt_int )).

% fst_diag_fst
thf(fact_758_sup_OboundedI,axiom,(
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A ) ) ) )).

% sup.boundedI
thf(fact_759_case__prodD,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) )
     => ( F @ A @ B ) ) )).

% case_prodD
thf(fact_760_case__prodD,axiom,(
    ! [F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o,A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural] :
      ( ( produc2278936416606902338ural_o @ F @ ( produc3831813291587773865atural @ A @ B ) )
     => ( F @ A @ B ) ) )).

% case_prodD
thf(fact_761_case__prodD,axiom,(
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) )
     => ( F @ A @ B ) ) )).

% case_prodD
thf(fact_762_case__prodD,axiom,(
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
     => ( F @ A @ B ) ) )).

% case_prodD
thf(fact_763_case__prodD,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
     => ( F @ A @ B ) ) )).

% case_prodD
thf(fact_764_case__prod__Pair__iden,axiom,(
    ! [P2: produc8763457246119570046nteger] :
      ( ( produc3906647086178084059nteger @ produc6137756002093451184nteger @ P2 )
      = P2 ) )).

% case_prod_Pair_iden
thf(fact_765_case__prod__Pair__iden,axiom,(
    ! [P2: produc989692990947075319atural] :
      ( ( produc1300469243244976333atural @ produc3831813291587773865atural @ P2 )
      = P2 ) )).

% case_prod_Pair_iden
thf(fact_766_case__prod__Pair__iden,axiom,(
    ! [P2: produc1908205239877642774nteger] :
      ( ( produc6512950862096126219nteger @ produc8603105652947943368nteger @ P2 )
      = P2 ) )).

% case_prod_Pair_iden
thf(fact_767_case__prod__Pair__iden,axiom,(
    ! [P2: produc2285326912895808259nt_int] :
      ( ( produc8492565224438309093nt_int @ produc5700946648718959541nt_int @ P2 )
      = P2 ) )).

% case_prod_Pair_iden
thf(fact_768_case__prod__Pair__iden,axiom,(
    ! [P2: produc7773217078559923341nt_int] :
      ( ( produc5122537100556696953nt_int @ produc4305682042979456191nt_int @ P2 )
      = P2 ) )).

% case_prod_Pair_iden
thf(fact_769_top__empty__eq2,axiom,
    ( top_top_nat_nat_o
    = ( ^ [X4: nat,Y6: nat] :
          ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y6 ) @ top_to4669805908274784177at_nat ) ) )).

% top_empty_eq2
thf(fact_770_top__empty__eq2,axiom,
    ( top_to2659251575160290430_nat_o
    = ( ^ [X4: product_prod_nat_nat,Y6: product_prod_nat_nat] :
          ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y6 ) @ top_to8454300755403694231at_nat ) ) )).

% top_empty_eq2
thf(fact_771_top__empty__eq2,axiom,
    ( top_to8383973604011232643eger_o
    = ( ^ [X4: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
          ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y6 ) @ top_to2348597233010745700nteger ) ) )).

% top_empty_eq2
thf(fact_772_top__empty__eq2,axiom,
    ( top_to1945756744135967306ural_o
    = ( ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term,Y6: produc7822875418678951345atural] :
          ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X4 @ Y6 ) @ top_to1252183740382221533atural ) ) )).

% top_empty_eq2
thf(fact_773_top__empty__eq2,axiom,
    ( top_to8112647782992986859eger_o
    = ( ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
          ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y6 ) @ top_to7512759353274530428nteger ) ) )).

% top_empty_eq2
thf(fact_774_top__empty__eq2,axiom,
    ( top_to6261655714344447806_int_o
    = ( ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
          ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y6 ) @ top_to3181862456159035625nt_int ) ) )).

% top_empty_eq2
thf(fact_775_top__empty__eq2,axiom,
    ( top_to6513043852502318900_int_o
    = ( ^ [X4: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
          ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y6 ) @ top_to2069137843433766899nt_int ) ) )).

% top_empty_eq2
thf(fact_776_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( A5
          = ( sup_sup_nat @ A5 @ B4 ) ) ) )).

% sup.order_iff
thf(fact_777_sup_Ocobounded1,axiom,(
    ! [A: nat,B: nat] :
      ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) )).

% sup.cobounded1
thf(fact_778_sup_Ocobounded2,axiom,(
    ! [B: nat,A: nat] :
      ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) )).

% sup.cobounded2
thf(fact_779_mem__case__prodI2,axiom,(
    ! [P2: produc8763457246119570046nteger,Z: product_prod_nat_nat,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_Pr1261947904930325089at_nat] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( ( P2
            = ( produc6137756002093451184nteger @ A3 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc4551263766066629700at_nat @ C2 @ P2 ) ) ) )).

% mem_case_prodI2
thf(fact_780_mem__case__prodI2,axiom,(
    ! [P2: produc8763457246119570046nteger,Z: produc859450856879609959at_nat,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_Pr8693737435421807431at_nat] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( ( P2
            = ( produc6137756002093451184nteger @ A3 @ B3 ) )
         => ( member8206827879206165904at_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc5744607925856648362at_nat @ C2 @ P2 ) ) ) )).

% mem_case_prodI2
thf(fact_781_mem__case__prodI2,axiom,(
    ! [P2: produc989692990947075319atural,Z: product_prod_nat_nat,C2: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > set_Pr1261947904930325089at_nat] :
      ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural] :
          ( ( P2
            = ( produc3831813291587773865atural @ A3 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc5359040883261003019at_nat @ C2 @ P2 ) ) ) )).

% mem_case_prodI2
thf(fact_782_mem__case__prodI2,axiom,(
    ! [P2: produc989692990947075319atural,Z: produc859450856879609959at_nat,C2: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > set_Pr8693737435421807431at_nat] :
      ( ! [A3: produc4972180933644002618e_term > option6357759511663192854e_term,B3: produc7822875418678951345atural] :
          ( ( P2
            = ( produc3831813291587773865atural @ A3 @ B3 ) )
         => ( member8206827879206165904at_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc5211177361811169009at_nat @ C2 @ P2 ) ) ) )).

% mem_case_prodI2
thf(fact_783_mem__case__prodI2,axiom,(
    ! [P2: produc1908205239877642774nteger,Z: product_prod_nat_nat,C2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_Pr1261947904930325089at_nat] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( ( P2
            = ( produc8603105652947943368nteger @ A3 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc6029229664795570220at_nat @ C2 @ P2 ) ) ) )).

% mem_case_prodI2
thf(fact_784_mem__case__prodI2,axiom,(
    ! [P2: produc1908205239877642774nteger,Z: produc859450856879609959at_nat,C2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_Pr8693737435421807431at_nat] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( ( P2
            = ( produc8603105652947943368nteger @ A3 @ B3 ) )
         => ( member8206827879206165904at_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc2704644331147814034at_nat @ C2 @ P2 ) ) ) )).

% mem_case_prodI2
thf(fact_785_mem__case__prodI2,axiom,(
    ! [P2: produc2285326912895808259nt_int,Z: product_prod_nat_nat,C2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_Pr1261947904930325089at_nat] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( ( P2
            = ( produc5700946648718959541nt_int @ A3 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc2642701577560500223at_nat @ C2 @ P2 ) ) ) )).

% mem_case_prodI2
thf(fact_786_mem__case__prodI2,axiom,(
    ! [P2: produc2285326912895808259nt_int,Z: produc859450856879609959at_nat,C2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_Pr8693737435421807431at_nat] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( ( P2
            = ( produc5700946648718959541nt_int @ A3 @ B3 ) )
         => ( member8206827879206165904at_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc517896604054025189at_nat @ C2 @ P2 ) ) ) )).

% mem_case_prodI2
thf(fact_787_mem__case__prodI2,axiom,(
    ! [P2: produc7773217078559923341nt_int,Z: product_prod_nat_nat,C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_Pr1261947904930325089at_nat] :
      ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( ( P2
            = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc5154663461364529269at_nat @ C2 @ P2 ) ) ) )).

% mem_case_prodI2
thf(fact_788_mem__case__prodI2,axiom,(
    ! [P2: produc7773217078559923341nt_int,Z: produc859450856879609959at_nat,C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_Pr8693737435421807431at_nat] :
      ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( ( P2
            = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
         => ( member8206827879206165904at_nat @ Z @ ( C2 @ A3 @ B3 ) ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc2114697605475793755at_nat @ C2 @ P2 ) ) ) )).

% mem_case_prodI2
thf(fact_789_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( sup_sup_nat @ A5 @ B4 )
          = A5 ) ) )).

% sup.absorb_iff1
thf(fact_790_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( sup_sup_nat @ A5 @ B4 )
          = B4 ) ) )).

% sup.absorb_iff2
thf(fact_791_pred__equals__eq2,axiom,(
    ! [R2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X4: nat,Y6: nat] :
              ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y6 ) @ R2 ) )
        = ( ^ [X4: nat,Y6: nat] :
              ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y6 ) @ S4 ) ) )
      = ( R2 = S4 ) ) )).

% pred_equals_eq2
thf(fact_792_pred__equals__eq2,axiom,(
    ! [R2: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X4: product_prod_nat_nat,Y6: product_prod_nat_nat] :
              ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y6 ) @ R2 ) )
        = ( ^ [X4: product_prod_nat_nat,Y6: product_prod_nat_nat] :
              ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y6 ) @ S4 ) ) )
      = ( R2 = S4 ) ) )).

% pred_equals_eq2
thf(fact_793_pred__equals__eq2,axiom,(
    ! [R2: set_Pr8056137968301705908nteger,S4: set_Pr8056137968301705908nteger] :
      ( ( ( ^ [X4: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
              ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y6 ) @ R2 ) )
        = ( ^ [X4: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
              ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y6 ) @ S4 ) ) )
      = ( R2 = S4 ) ) )).

% pred_equals_eq2
thf(fact_794_pred__equals__eq2,axiom,(
    ! [R2: set_Pr4244404650639074861atural,S4: set_Pr4244404650639074861atural] :
      ( ( ( ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term,Y6: produc7822875418678951345atural] :
              ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X4 @ Y6 ) @ R2 ) )
        = ( ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term,Y6: produc7822875418678951345atural] :
              ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X4 @ Y6 ) @ S4 ) ) )
      = ( R2 = S4 ) ) )).

% pred_equals_eq2
thf(fact_795_pred__equals__eq2,axiom,(
    ! [R2: set_Pr1281608226676607948nteger,S4: set_Pr1281608226676607948nteger] :
      ( ( ( ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
              ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y6 ) @ R2 ) )
        = ( ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
              ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y6 ) @ S4 ) ) )
      = ( R2 = S4 ) ) )).

% pred_equals_eq2
thf(fact_796_pred__equals__eq2,axiom,(
    ! [R2: set_Pr9222295170931077689nt_int,S4: set_Pr9222295170931077689nt_int] :
      ( ( ( ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
              ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y6 ) @ R2 ) )
        = ( ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
              ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y6 ) @ S4 ) ) )
      = ( R2 = S4 ) ) )).

% pred_equals_eq2
thf(fact_797_pred__equals__eq2,axiom,(
    ! [R2: set_Pr1872883991513573699nt_int,S4: set_Pr1872883991513573699nt_int] :
      ( ( ( ^ [X4: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
              ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y6 ) @ R2 ) )
        = ( ^ [X4: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
              ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y6 ) @ S4 ) ) )
      = ( R2 = S4 ) ) )).

% pred_equals_eq2
thf(fact_798_sup_OcoboundedI1,axiom,(
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) )).

% sup.coboundedI1
thf(fact_799_iterate_Opelims,axiom,(
    ! [X: code_natural,Xa: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger,Xb: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger > produc8763457246119570046nteger] :
      ( ( ( iterat2138450126151725500nteger @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P7628380106888448002e_term @ iterat5011629852884125657nteger @ ( produc397372062499391921e_term @ X @ ( produc2900363947226785770e_term @ Xa @ Xb ) ) )
       => ~ ( ( ( ( X = zero_z2226904508553997617atural )
               => ( Y
                  = ( produc6137756002093451184nteger @ Xb ) ) )
              & ( ( X != zero_z2226904508553997617atural )
               => ( Y
                  = ( produc1129086632781781489nteger @ ( Xa @ Xb ) @ ( iterat2138450126151725500nteger @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) )
           => ~ ( accp_P7628380106888448002e_term @ iterat5011629852884125657nteger @ ( produc397372062499391921e_term @ X @ ( produc2900363947226785770e_term @ Xa @ Xb ) ) ) ) ) ) )).

% iterate.pelims
thf(fact_800_iterate_Opelims,axiom,(
    ! [X: code_natural,Xa: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > produc989692990947075319atural,Xb: produc4972180933644002618e_term > option6357759511663192854e_term,Y: produc7822875418678951345atural > produc989692990947075319atural] :
      ( ( ( iterat325776542273385909atural @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P7667005037297283793e_term @ iterat580925020412187218atural @ ( produc7413936711424066304e_term @ X @ ( produc6122309266285285817e_term @ Xa @ Xb ) ) )
       => ~ ( ( ( ( X = zero_z2226904508553997617atural )
               => ( Y
                  = ( produc3831813291587773865atural @ Xb ) ) )
              & ( ( X != zero_z2226904508553997617atural )
               => ( Y
                  = ( produc4386088583431089485atural @ ( Xa @ Xb ) @ ( iterat325776542273385909atural @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) )
           => ~ ( accp_P7667005037297283793e_term @ iterat580925020412187218atural @ ( produc7413936711424066304e_term @ X @ ( produc6122309266285285817e_term @ Xa @ Xb ) ) ) ) ) ) )).

% iterate.pelims
thf(fact_801_iterate_Opelims,axiom,(
    ! [X: code_natural,Xa: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger,Xb: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger > produc1908205239877642774nteger] :
      ( ( ( iterat5097068903633555412nteger @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P4207847519495620250e_term @ iterat5352217381772356721nteger @ ( produc761156837658176329e_term @ X @ ( produc5301067125954940802e_term @ Xa @ Xb ) ) )
       => ~ ( ( ( ( X = zero_z2226904508553997617atural )
               => ( Y
                  = ( produc8603105652947943368nteger @ Xb ) ) )
              & ( ( X != zero_z2226904508553997617atural )
               => ( Y
                  = ( produc2548262910190615585nteger @ ( Xa @ Xb ) @ ( iterat5097068903633555412nteger @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) )
           => ~ ( accp_P4207847519495620250e_term @ iterat5352217381772356721nteger @ ( produc761156837658176329e_term @ X @ ( produc5301067125954940802e_term @ Xa @ Xb ) ) ) ) ) ) )).

% iterate.pelims
thf(fact_802_iterate_Opelims,axiom,(
    ! [X: code_natural,Xa: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int,Xb: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int > produc2285326912895808259nt_int] :
      ( ( ( iterat1751201420036670401nt_int @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8005579812532215269e_term @ iterat4509646233257557086nt_int @ ( produc2815140415168450580e_term @ X @ ( produc8963182620248894669e_term @ Xa @ Xb ) ) )
       => ~ ( ( ( ( X = zero_z2226904508553997617atural )
               => ( Y
                  = ( produc5700946648718959541nt_int @ Xb ) ) )
              & ( ( X != zero_z2226904508553997617atural )
               => ( Y
                  = ( produc8058474613819325789nt_int @ ( Xa @ Xb ) @ ( iterat1751201420036670401nt_int @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) )
           => ~ ( accp_P8005579812532215269e_term @ iterat4509646233257557086nt_int @ ( produc2815140415168450580e_term @ X @ ( produc8963182620248894669e_term @ Xa @ Xb ) ) ) ) ) ) )).

% iterate.pelims
thf(fact_803_iterate_Opelims,axiom,(
    ! [X: code_natural,Xa: ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int,Xb: int > option6357759511663192854e_term,Y: product_prod_int_int > produc7773217078559923341nt_int] :
      ( ( ( iterat3386830680617422539nt_int @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P4367507133575706479e_term @ iterat6458141894343033064nt_int @ ( produc1588918054225147806e_term @ X @ ( produc4578464555131395415e_term @ Xa @ Xb ) ) )
       => ~ ( ( ( ( X = zero_z2226904508553997617atural )
               => ( Y
                  = ( produc4305682042979456191nt_int @ Xb ) ) )
              & ( ( X != zero_z2226904508553997617atural )
               => ( Y
                  = ( produc3400328326638805489nt_int @ ( Xa @ Xb ) @ ( iterat3386830680617422539nt_int @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) )
           => ~ ( accp_P4367507133575706479e_term @ iterat6458141894343033064nt_int @ ( produc1588918054225147806e_term @ X @ ( produc4578464555131395415e_term @ Xa @ Xb ) ) ) ) ) ) )).

% iterate.pelims
thf(fact_804_sup_OcoboundedI2,axiom,(
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ B )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) )).

% sup.coboundedI2
thf(fact_805_Collect__case__prod__Grp__in,axiom,(
    ! [Z: product_prod_nat_nat,A4: set_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A4 @ F ) ) ) )
     => ( member_nat @ ( product_fst_nat_nat @ Z ) @ A4 ) ) )).

% Collect_case_prod_Grp_in
thf(fact_806_Collect__case__prod__Grp__in,axiom,(
    ! [Z: produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ ( bNF_Gr7275288873855648526at_nat @ A4 @ F ) ) ) )
     => ( member8440522571783428010at_nat @ ( produc3213797794245857475at_nat @ Z ) @ A4 ) ) )).

% Collect_case_prod_Grp_in
thf(fact_807_Collect__case__prod__Grp__in,axiom,(
    ! [Z: produc8763457246119570046nteger,A4: set_Co9149898834107579976e_term,F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger] :
      ( ( member3068662437193594005nteger @ Z @ ( collec6350289969311926995nteger @ ( produc127349428274296955eger_o @ ( bNF_Gr6578204635149681721nteger @ A4 @ F ) ) ) )
     => ( member1535805642427569193e_term @ ( produc2702340305111172292nteger @ Z ) @ A4 ) ) )).

% Collect_case_prod_Grp_in
thf(fact_808_Collect__case__prod__Grp__in,axiom,(
    ! [Z: produc989692990947075319atural,A4: set_Pr2791530082921190693e_term,F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural] :
      ( ( member3245610415464309390atural @ Z @ ( collec6032437096461363148atural @ ( produc2278936416606902338ural_o @ ( bNF_Gr8806722549218754226atural @ A4 @ F ) ) ) )
     => ( member2820825060907577222e_term @ ( produc3051390257598303421atural @ Z ) @ A4 ) ) )).

% Collect_case_prod_Grp_in
thf(fact_809_Collect__case__prod__Grp__in,axiom,(
    ! [Z: produc1908205239877642774nteger,A4: set_Pr7604974323444597168e_term,F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger] :
      ( ( member4164122664394876845nteger @ Z @ ( collec6950949345391930603nteger @ ( produc6253627499356882019eger_o @ ( bNF_Gr4354642873724147921nteger @ A4 @ F ) ) ) )
     => ( member4242434998011752849e_term @ ( produc7822682618958472924nteger @ Z ) @ A4 ) ) )).

% Collect_case_prod_Grp_in
thf(fact_810_Collect__case__prod__Grp__in,axiom,(
    ! [Z: produc2285326912895808259nt_int,A4: set_Pr3642885161833720865e_term,F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ Z @ ( collec1790188477890212312nt_int @ ( produc1573362020775583542_int_o @ ( bNF_Gr5420098529801562302nt_int @ A4 @ F ) ) ) )
     => ( member3222579708246209666e_term @ ( produc6816164490631068361nt_int @ Z ) @ A4 ) ) )).

% Collect_case_prod_Grp_in
thf(fact_811_Collect__case__prod__Grp__in,axiom,(
    ! [Z: produc7773217078559923341nt_int,A4: set_in3461395444621081367e_term,F: ( int > option6357759511663192854e_term ) > product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ Z @ ( collec506566255779805410nt_int @ ( produc2558449545302689196_int_o @ ( bNF_Gr6359101500228959560nt_int @ A4 @ F ) ) ) )
     => ( member8845023287901829240e_term @ ( produc6230002227079971283nt_int @ Z ) @ A4 ) ) )).

% Collect_case_prod_Grp_in
thf(fact_812_prod__fun__imageE,axiom,(
    ! [C2: product_prod_nat_nat,F: nat > nat,G2: nat > nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G2 ) @ R2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C2
              = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G2 @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) )).

% prod_fun_imageE
thf(fact_813_prod__fun__imageE,axiom,(
    ! [C2: product_prod_nat_nat,F: product_prod_nat_nat > nat,G2: product_prod_nat_nat > nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( image_6706873702294942047at_nat @ ( produc5156754000680454454at_nat @ F @ G2 ) @ R2 ) )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( C2
              = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G2 @ Y3 ) ) )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 ) ) ) )).

% prod_fun_imageE
thf(fact_814_prod__fun__imageE,axiom,(
    ! [C2: produc859450856879609959at_nat,F: nat > product_prod_nat_nat,G2: nat > product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ ( image_1119892238107900587at_nat @ ( produc1153188318382123958at_nat @ F @ G2 ) @ R2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C2
              = ( produc6161850002892822231at_nat @ ( F @ X3 ) @ ( G2 @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) )).

% prod_fun_imageE
thf(fact_815_prod__fun__imageE,axiom,(
    ! [C2: produc859450856879609959at_nat,F: product_prod_nat_nat > product_prod_nat_nat,G2: product_prod_nat_nat > product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ ( image_6321802852832981957at_nat @ ( produc7147751436741673334at_nat @ F @ G2 ) @ R2 ) )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( C2
              = ( produc6161850002892822231at_nat @ ( F @ X3 ) @ ( G2 @ Y3 ) ) )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 ) ) ) )).

% prod_fun_imageE
thf(fact_816_prod__fun__imageE,axiom,(
    ! [C2: product_prod_nat_nat,F: ( code_integer > option6357759511663192854e_term ) > nat,G2: produc8923325533196201883nteger > nat,R2: set_Pr8056137968301705908nteger] :
      ( ( member8440522571783428010at_nat @ C2 @ ( image_9094096100532249404at_nat @ ( produc6033578056857175441er_nat @ F @ G2 ) @ R2 ) )
     => ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ( ( C2
              = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G2 @ Y3 ) ) )
           => ~ ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ R2 ) ) ) )).

% prod_fun_imageE
thf(fact_817_prod__fun__imageE,axiom,(
    ! [C2: product_prod_nat_nat,F: ( int > option6357759511663192854e_term ) > nat,G2: product_prod_int_int > nat,R2: set_Pr1872883991513573699nt_int] :
      ( ( member8440522571783428010at_nat @ C2 @ ( image_6286574459949954669at_nat @ ( produc5394751608776890016nt_nat @ F @ G2 ) @ R2 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( C2
              = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G2 @ Y3 ) ) )
           => ~ ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R2 ) ) ) )).

% prod_fun_imageE
thf(fact_818_prod__fun__imageE,axiom,(
    ! [C2: produc8763457246119570046nteger,F: nat > code_integer > option6357759511663192854e_term,G2: nat > produc8923325533196201883nteger,R2: set_Pr1261947904930325089at_nat] :
      ( ( member3068662437193594005nteger @ C2 @ ( image_7586372328468041018nteger @ ( produc6729254856488394607nteger @ F @ G2 ) @ R2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C2
              = ( produc6137756002093451184nteger @ ( F @ X3 ) @ ( G2 @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) )).

% prod_fun_imageE
thf(fact_819_prod__fun__imageE,axiom,(
    ! [C2: produc7773217078559923341nt_int,F: nat > int > option6357759511663192854e_term,G2: nat > product_prod_int_int,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7034335876925520548nt_int @ C2 @ ( image_1715401223925157705nt_int @ ( produc3178275182335418494nt_int @ F @ G2 ) @ R2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C2
              = ( produc4305682042979456191nt_int @ ( F @ X3 ) @ ( G2 @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) )).

% prod_fun_imageE
thf(fact_820_prod__fun__imageE,axiom,(
    ! [C2: produc859450856879609959at_nat,F: ( code_integer > option6357759511663192854e_term ) > product_prod_nat_nat,G2: produc8923325533196201883nteger > product_prod_nat_nat,R2: set_Pr8056137968301705908nteger] :
      ( ( member8206827879206165904at_nat @ C2 @ ( image_3866311809354581538at_nat @ ( produc8178386163773442717at_nat @ F @ G2 ) @ R2 ) )
     => ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ( ( C2
              = ( produc6161850002892822231at_nat @ ( F @ X3 ) @ ( G2 @ Y3 ) ) )
           => ~ ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ R2 ) ) ) )).

% prod_fun_imageE
thf(fact_821_prod__fun__imageE,axiom,(
    ! [C2: product_prod_nat_nat,F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > nat,G2: produc7822875418678951345atural > nat,R2: set_Pr4244404650639074861atural] :
      ( ( member8440522571783428010at_nat @ C2 @ ( image_3500444250886205059at_nat @ ( produc3022448295462886410al_nat @ F @ G2 ) @ R2 ) )
     => ~ ! [X3: produc4972180933644002618e_term > option6357759511663192854e_term,Y3: produc7822875418678951345atural] :
            ( ( C2
              = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G2 @ Y3 ) ) )
           => ~ ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X3 @ Y3 ) @ R2 ) ) ) )).

% prod_fun_imageE
thf(fact_822_fst__diag__snd,axiom,
    ( ( comp_P230220204662055659atural @ produc1537665297809786051atural
      @ ( comp_P4943825737035310295atural
        @ ^ [X4: produc7822875418678951345atural] :
            ( produc4480994950612372183atural @ X4 @ X4 )
        @ produc6591912806276919810atural ) )
    = produc6591912806276919810atural )).

% fst_diag_snd
thf(fact_823_fst__diag__snd,axiom,
    ( ( comp_P2690847479783168146list_f @ produc8633839776129222975list_f
      @ ( comp_P8324358310098339598list_f
        @ ^ [X4: produc1811169103916964701list_f] :
            ( produc7497196274192653651list_f @ X4 @ X4 )
        @ produc8710481132536983823list_f ) )
    = produc8710481132536983823list_f )).

% fst_diag_snd
thf(fact_824_snd__diag__fst,axiom,
    ( ( comp_P662548419035233652nteger @ produc4386931168992957931e_term
      @ ( comp_C5262483154866357838nteger
        @ ^ [X4: code_integer > option6357759511663192854e_term] :
            ( produc2189037350439096765e_term @ X4 @ X4 )
        @ produc2702340305111172292nteger ) )
    = produc2702340305111172292nteger )).

% snd_diag_fst
thf(fact_825_snd__diag__fst,axiom,
    ( ( comp_P5776439315172467588atural @ produc3958670729589399461e_term
      @ ( comp_P1227397885194180452atural
        @ ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term] :
            ( produc734250632046660983e_term @ X4 @ X4 )
        @ produc3051390257598303421atural ) )
    = produc3051390257598303421atural )).

% snd_diag_fst
thf(fact_826_snd__diag__fst,axiom,
    ( ( comp_P9158442376665023684nteger @ produc916109439378433339e_term
      @ ( comp_P1612206844032044750nteger
        @ ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term] :
            ( produc6915061378690470669e_term @ X4 @ X4 )
        @ produc7822682618958472924nteger ) )
    = produc7822682618958472924nteger )).

% snd_diag_fst
thf(fact_827_snd__diag__fst,axiom,
    ( ( comp_P1627152983954642052nt_int @ produc8590683033446154141e_term
      @ ( comp_P4947044428546866028nt_int
        @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term] :
            ( produc8124735558134971759e_term @ X4 @ X4 )
        @ produc6816164490631068361nt_int ) )
    = produc6816164490631068361nt_int )).

% snd_diag_fst
thf(fact_828_snd__diag__fst,axiom,
    ( ( comp_P607695144065014128nt_int @ produc2484719519985517193e_term
      @ ( comp_i2761858172409570284nt_int
        @ ^ [X4: int > option6357759511663192854e_term] :
            ( produc4510790414113558107e_term @ X4 @ X4 )
        @ produc6230002227079971283nt_int ) )
    = produc6230002227079971283nt_int )).

% snd_diag_fst
thf(fact_829_fst__def,axiom,
    ( produc2702340305111172292nteger
    = ( produc4834724593918014831e_term
      @ ^ [X12: code_integer > option6357759511663192854e_term,X23: produc8923325533196201883nteger] : X12 ) )).

% fst_def
thf(fact_830_fst__def,axiom,
    ( produc3051390257598303421atural
    = ( produc2323481435091179589e_term
      @ ^ [X12: produc4972180933644002618e_term > option6357759511663192854e_term,X23: produc7822875418678951345atural] : X12 ) )).

% fst_def
thf(fact_831_fst__def,axiom,
    ( produc7822682618958472924nteger
    = ( produc501307824270020591e_term
      @ ^ [X12: produc6241069584506657477e_term > option6357759511663192854e_term,X23: produc8923325533196201883nteger] : X12 ) )).

% fst_def
thf(fact_832_fst__def,axiom,
    ( produc6816164490631068361nt_int
    = ( produc8953926646514273613e_term
      @ ^ [X12: produc8551481072490612790e_term > option6357759511663192854e_term,X23: product_prod_int_int] : X12 ) )).

% fst_def
thf(fact_833_fst__def,axiom,
    ( produc6230002227079971283nt_int
    = ( produc4926158676823748301e_term
      @ ^ [X12: int > option6357759511663192854e_term,X23: product_prod_int_int] : X12 ) )).

% fst_def
thf(fact_834_snd__def,axiom,
    ( produc6591912806276919810atural
    = ( produc6302656368764539796atural
      @ ^ [X12: code_natural,X23: produc7822875418678951345atural] : X23 ) )).

% snd_def
thf(fact_835_snd__def,axiom,
    ( produc8710481132536983823list_f
    = ( produc6078222734141437267list_f
      @ ^ [X12: finfun1145668972854894571action,X23: produc1811169103916964701list_f] : X23 ) )).

% snd_def
thf(fact_836_sndOp__def,axiom,
    ( bNF_sn1036602493604259329nteger
    = ( ^ [P4: ( code_integer > option6357759511663192854e_term ) > ( code_integer > option6357759511663192854e_term ) > $o,Q3: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,Ac2: produc8763457246119570046nteger] :
          ( produc6137756002093451184nteger @ ( bNF_pi603945831373623569nteger @ P4 @ Q3 @ ( produc2702340305111172292nteger @ Ac2 ) @ ( produc6514156286822040066nteger @ Ac2 ) ) @ ( produc6514156286822040066nteger @ Ac2 ) ) ) )).

% sndOp_def
thf(fact_837_sndOp__def,axiom,
    ( bNF_sn2092289520945553049nteger
    = ( ^ [P4: ( code_integer > option6357759511663192854e_term ) > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,Q3: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,Ac2: produc8763457246119570046nteger] :
          ( produc8603105652947943368nteger @ ( bNF_pi3430033144021242281nteger @ P4 @ Q3 @ ( produc2702340305111172292nteger @ Ac2 ) @ ( produc6514156286822040066nteger @ Ac2 ) ) @ ( produc6514156286822040066nteger @ Ac2 ) ) ) )).

% sndOp_def
thf(fact_838_sndOp__def,axiom,
    ( bNF_sn7306468111849055197atural
    = ( ^ [P4: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > ( produc4972180933644002618e_term > option6357759511663192854e_term ) > $o,Q3: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o,Ac2: produc989692990947075319atural] :
          ( produc3831813291587773865atural @ ( bNF_pi5693743034599399149atural @ P4 @ Q3 @ ( produc3051390257598303421atural @ Ac2 ) @ ( produc3085575039554877691atural @ Ac2 ) ) @ ( produc3085575039554877691atural @ Ac2 ) ) ) )).

% sndOp_def
thf(fact_839_sndOp__def,axiom,
    ( bNF_sn1643940504687329305nteger
    = ( ^ [P4: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > ( code_integer > option6357759511663192854e_term ) > $o,Q3: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,Ac2: produc1908205239877642774nteger] :
          ( produc6137756002093451184nteger @ ( bNF_pi2981684127763018537nteger @ P4 @ Q3 @ ( produc7822682618958472924nteger @ Ac2 ) @ ( produc7856867400915047194nteger @ Ac2 ) ) @ ( produc7856867400915047194nteger @ Ac2 ) ) ) )).

% sndOp_def
thf(fact_840_sndOp__def,axiom,
    ( bNF_sn2594531154364564657nteger
    = ( ^ [P4: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,Q3: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,Ac2: produc1908205239877642774nteger] :
          ( produc8603105652947943368nteger @ ( bNF_pi981806077114908609nteger @ P4 @ Q3 @ ( produc7822682618958472924nteger @ Ac2 ) @ ( produc7856867400915047194nteger @ Ac2 ) ) @ ( produc7856867400915047194nteger @ Ac2 ) ) ) )).

% sndOp_def
thf(fact_841_sndOp__def,axiom,
    ( bNF_sn7816398935535452141nt_int
    = ( ^ [P4: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,Q3: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,Ac2: produc2285326912895808259nt_int] :
          ( produc5700946648718959541nt_int @ ( bNF_pi4681045200935680765nt_int @ P4 @ Q3 @ ( produc6816164490631068361nt_int @ Ac2 ) @ ( produc7328097813583171335nt_int @ Ac2 ) ) @ ( produc7328097813583171335nt_int @ Ac2 ) ) ) )).

% sndOp_def
thf(fact_842_sndOp__def,axiom,
    ( bNF_sn1641008659695686903nt_int
    = ( ^ [P4: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > ( int > option6357759511663192854e_term ) > $o,Q3: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,Ac2: produc2285326912895808259nt_int] :
          ( produc4305682042979456191nt_int @ ( bNF_pi1230435913411708935nt_int @ P4 @ Q3 @ ( produc6816164490631068361nt_int @ Ac2 ) @ ( produc7328097813583171335nt_int @ Ac2 ) ) @ ( produc7328097813583171335nt_int @ Ac2 ) ) ) )).

% sndOp_def
thf(fact_843_sndOp__def,axiom,
    ( bNF_sn789635790944691191nt_int
    = ( ^ [P4: ( int > option6357759511663192854e_term ) > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,Q3: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,Ac2: produc7773217078559923341nt_int] :
          ( produc5700946648718959541nt_int @ ( bNF_pi379063044660713223nt_int @ P4 @ Q3 @ ( produc6230002227079971283nt_int @ Ac2 ) @ ( produc3162348030201620241nt_int @ Ac2 ) ) @ ( produc3162348030201620241nt_int @ Ac2 ) ) ) )).

% sndOp_def
thf(fact_844_sndOp__def,axiom,
    ( bNF_sn5909627676663890945nt_int
    = ( ^ [P4: ( int > option6357759511663192854e_term ) > ( int > option6357759511663192854e_term ) > $o,Q3: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,Ac2: produc7773217078559923341nt_int] :
          ( produc4305682042979456191nt_int @ ( bNF_pi4273723537236733713nt_int @ P4 @ Q3 @ ( produc6230002227079971283nt_int @ Ac2 ) @ ( produc3162348030201620241nt_int @ Ac2 ) ) @ ( produc3162348030201620241nt_int @ Ac2 ) ) ) )).

% sndOp_def
thf(fact_845_sndOp__def,axiom,
    ( bNF_sn7364221859090291556atural
    = ( ^ [P4: code_natural > ( produc4972180933644002618e_term > option6357759511663192854e_term ) > $o,Q3: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o,Ac2: produc5835291356934675326atural] :
          ( produc3831813291587773865atural @ ( bNF_pi2877385506940030580atural @ P4 @ Q3 @ ( produc497848011034438852atural @ Ac2 ) @ ( produc6591912806276919810atural @ Ac2 ) ) @ ( produc6591912806276919810atural @ Ac2 ) ) ) )).

% sndOp_def
thf(fact_846_prod_Omap__comp,axiom,(
    ! [G1: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G22: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,F1: produc8923597787767437993list_f > produc1811169103916964701list_f,F22: produc8923597787767437993list_f > produc1811169103916964701list_f,V: produc6644936158991452263list_f] :
      ( ( produc4841148139688540918_a_b_c @ G1 @ G22 @ ( produc5505274914705194486list_f @ F1 @ F22 @ V ) )
      = ( produc8198191332872049718_a_b_c @ ( comp_P4767578765025054880list_f @ G1 @ F1 ) @ ( comp_P4767578765025054880list_f @ G22 @ F22 ) @ V ) ) )).

% prod.map_comp
thf(fact_847_map__prod__def,axiom,
    ( produc4718505701431157174atural
    = ( ^ [F2: code_natural > produc7822875418678951345atural,G: code_natural > produc7822875418678951345atural] :
          ( produc8282080750456430313atural
          @ ^ [X4: code_natural,Y6: code_natural] :
              ( produc4480994950612372183atural @ ( F2 @ X4 ) @ ( G @ Y6 ) ) ) ) )).

% map_prod_def
thf(fact_848_relChain__def,axiom,
    ( bNF_Ca968750328013420230at_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,As: nat > nat] :
        ! [I3: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J ) @ R3 )
         => ( ord_less_eq_nat @ ( As @ I3 ) @ ( As @ J ) ) ) ) )).

% relChain_def
thf(fact_849_relChain__def,axiom,
    ( bNF_Ca8308629720386654381at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat,As: product_prod_nat_nat > nat] :
        ! [I3: product_prod_nat_nat,J: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I3 @ J ) @ R3 )
         => ( ord_less_eq_nat @ ( As @ I3 ) @ ( As @ J ) ) ) ) )).

% relChain_def
thf(fact_850_pred__subset__eq2,axiom,(
    ! [R2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X4: nat,Y6: nat] :
            ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: nat,Y6: nat] :
            ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y6 ) @ S4 ) )
      = ( ord_le3146513528884898305at_nat @ R2 @ S4 ) ) )).

% pred_subset_eq2
thf(fact_851_pred__subset__eq2,axiom,(
    ! [R2: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X4: product_prod_nat_nat,Y6: product_prod_nat_nat] :
            ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: product_prod_nat_nat,Y6: product_prod_nat_nat] :
            ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y6 ) @ S4 ) )
      = ( ord_le3000389064537975527at_nat @ R2 @ S4 ) ) )).

% pred_subset_eq2
thf(fact_852_pred__subset__eq2,axiom,(
    ! [R2: set_Pr8056137968301705908nteger,S4: set_Pr8056137968301705908nteger] :
      ( ( ord_le3636971675376928563eger_o
        @ ^ [X4: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
            ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
            ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y6 ) @ S4 ) )
      = ( ord_le3216752416896350996nteger @ R2 @ S4 ) ) )).

% pred_subset_eq2
thf(fact_853_pred__subset__eq2,axiom,(
    ! [R2: set_Pr4244404650639074861atural,S4: set_Pr4244404650639074861atural] :
      ( ( ord_le7397293433748542458ural_o
        @ ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term,Y6: produc7822875418678951345atural] :
            ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term,Y6: produc7822875418678951345atural] :
            ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X4 @ Y6 ) @ S4 ) )
      = ( ord_le3616440322383431309atural @ R2 @ S4 ) ) )).

% pred_subset_eq2
thf(fact_854_pred__subset__eq2,axiom,(
    ! [R2: set_Pr1281608226676607948nteger,S4: set_Pr1281608226676607948nteger] :
      ( ( ord_le4340812435750786203eger_o
        @ ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
            ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
            ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y6 ) @ S4 ) )
      = ( ord_le653643898420964396nteger @ R2 @ S4 ) ) )).

% pred_subset_eq2
thf(fact_855_pred__subset__eq2,axiom,(
    ! [R2: set_Pr9222295170931077689nt_int,S4: set_Pr9222295170931077689nt_int] :
      ( ( ord_le5643404153117327598_int_o
        @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
            ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
            ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y6 ) @ S4 ) )
      = ( ord_le8725513860283290265nt_int @ R2 @ S4 ) ) )).

% pred_subset_eq2
thf(fact_856_pred__subset__eq2,axiom,(
    ! [R2: set_Pr1872883991513573699nt_int,S4: set_Pr1872883991513573699nt_int] :
      ( ( ord_le2124322318746777828_int_o
        @ ^ [X4: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
            ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
            ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y6 ) @ S4 ) )
      = ( ord_le135402666524580259nt_int @ R2 @ S4 ) ) )).

% pred_subset_eq2
thf(fact_857_conditional__action_Oset__cases,axiom,(
    ! [E: product_prod_nat_nat,A: condit1452497342716194761at_nat] :
      ( ( member8440522571783428010at_nat @ E @ ( condit6429450166605921294at_nat @ A ) )
     => ( A
        = ( condit7601234554295360766at_nat @ E ) ) ) )).

% conditional_action.set_cases
thf(fact_858_conditional__action_Oset__cases,axiom,(
    ! [E: produc859450856879609959at_nat,A: condit3449342828199732399at_nat] :
      ( ( member8206827879206165904at_nat @ E @ ( condit5307000522679409652at_nat @ A ) )
     => ( A
        = ( condit965456907942171876at_nat @ E ) ) ) )).

% conditional_action.set_cases
thf(fact_859_fun_Omap__transfer,axiom,(
    ! [Rb2: produc1811169103916964701list_f > produc1811169103916964701list_f > $o,Sd: list_n1973683603564934665_a_b_c > list_n1973683603564934665_a_b_c > $o] :
      ( bNF_re1616170189525647965_a_b_c @ ( bNF_re2373559492235539289_a_b_c @ Rb2 @ Sd )
      @ ( bNF_re2955035557799908905_a_b_c
        @ ( bNF_re5276636924349840729list_f
          @ ^ [Y5: produc8923597787767437993list_f,Z2: produc8923597787767437993list_f] : ( Y5 = Z2 )
          @ Rb2 )
        @ ( bNF_re7641334924310632669_a_b_c
          @ ^ [Y5: produc8923597787767437993list_f,Z2: produc8923597787767437993list_f] : ( Y5 = Z2 )
          @ Sd ) )
      @ comp_P4767578765025054880list_f
      @ comp_P4767578765025054880list_f ) )).

% fun.map_transfer
thf(fact_860_case__prod__beta_H,axiom,
    ( produc8282080750456430313atural
    = ( ^ [F2: code_natural > code_natural > produc6756470014200349799atural,X4: produc7822875418678951345atural] :
          ( F2 @ ( produc1773691148792264405atural @ X4 ) @ ( produc8662200840462671639atural @ X4 ) ) ) )).

% case_prod_beta'
thf(fact_861_conditional__action_Oset__intros,axiom,(
    ! [X1: product_prod_nat_nat] :
      ( member8440522571783428010at_nat @ X1 @ ( condit6429450166605921294at_nat @ ( condit7601234554295360766at_nat @ X1 ) ) ) )).

% conditional_action.set_intros
thf(fact_862_conditional__action_Oset__intros,axiom,(
    ! [X1: produc859450856879609959at_nat] :
      ( member8206827879206165904at_nat @ X1 @ ( condit5307000522679409652at_nat @ ( condit965456907942171876at_nat @ X1 ) ) ) )).

% conditional_action.set_intros
thf(fact_863_total__on__def,axiom,
    ( total_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A6 )
         => ! [Y6: nat] :
              ( ( member_nat @ Y6 @ A6 )
             => ( ( X4 != Y6 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y6 ) @ R3 )
                  | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ X4 ) @ R3 ) ) ) ) ) ) )).

% total_on_def
thf(fact_864_total__on__def,axiom,
    ( total_3592101749530773125at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
        ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A6 )
         => ! [Y6: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y6 @ A6 )
             => ( ( X4 != Y6 )
               => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y6 ) @ R3 )
                  | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y6 @ X4 ) @ R3 ) ) ) ) ) ) )).

% total_on_def
thf(fact_865_fst__convol,axiom,(
    ! [F: produc8923597787767437993list_f > list_n1973683603564934665_a_b_c,G2: produc8923597787767437993list_f > produc8853065311482726181list_f] :
      ( ( comp_P4767578765025054880list_f @ produc3535155421234910955list_f @ ( bNF_co7330362693151061550list_f @ F @ G2 ) )
      = F ) )).

% fst_convol
thf(fact_866_fun_Omap__comp,axiom,(
    ! [G2: list_n1973683603564934665_a_b_c > list_n1973683603564934665_a_b_c,F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,V: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( comp_l8432902307485339590list_f @ G2 @ ( comp_P4767578765025054880list_f @ F @ V ) )
      = ( comp_P4767578765025054880list_f @ ( comp_l6492537726383095040list_f @ G2 @ F ) @ V ) ) )).

% fun.map_comp
thf(fact_867_fun_Omap__comp,axiom,(
    ! [G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,F: produc8923597787767437993list_f > produc1811169103916964701list_f,V: produc8923597787767437993list_f > produc8923597787767437993list_f] :
      ( ( comp_P4767578765025054880list_f @ G2 @ ( comp_P4267156721827287172list_f @ F @ V ) )
      = ( comp_P1644921750638218854list_f @ ( comp_P4767578765025054880list_f @ G2 @ F ) @ V ) ) )).

% fun.map_comp
thf(fact_868_fun_Omap__comp,axiom,(
    ! [G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,F: produc1811169103916964701list_f > produc1811169103916964701list_f,V: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( comp_P4767578765025054880list_f @ G2 @ ( comp_P2673361230361438154list_f @ F @ V ) )
      = ( comp_P4767578765025054880list_f @ ( comp_P5271726855292207526list_f @ G2 @ F ) @ V ) ) )).

% fun.map_comp
thf(fact_869_map__prod__o__convol,axiom,(
    ! [H1: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,H2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,F: produc8923597787767437993list_f > produc1811169103916964701list_f,G2: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( comp_P2679607744079208442list_f @ ( produc4841148139688540918_a_b_c @ H1 @ H2 ) @ ( bNF_co5807002252975134268list_f @ F @ G2 ) )
      = ( bNF_co7315529798985115712_a_b_c @ ( comp_P4767578765025054880list_f @ H1 @ F ) @ ( comp_P4767578765025054880list_f @ H2 @ G2 ) ) ) )).

% map_prod_o_convol
thf(fact_870_convol__expand__snd_H,axiom,(
    ! [F: produc8923597787767437993list_f > produc1811169103916964701list_f,G2: produc8923597787767437993list_f > list_n1973683603564934665_a_b_c,H: produc8923597787767437993list_f > produc8853065311482726181list_f] :
      ( ( ( comp_P4767578765025054880list_f @ produc3535155421234910955list_f @ F )
        = G2 )
     => ( ( H
          = ( comp_P3270879886602560258list_f @ produc7287154213363835177list_f @ F ) )
        = ( ( bNF_co7330362693151061550list_f @ G2 @ H )
          = F ) ) ) )).

% convol_expand_snd'
thf(fact_871_convol__expand__snd,axiom,(
    ! [F: produc8923597787767437993list_f > produc1811169103916964701list_f,G2: produc8923597787767437993list_f > list_n1973683603564934665_a_b_c] :
      ( ( ( comp_P4767578765025054880list_f @ produc3535155421234910955list_f @ F )
        = G2 )
     => ( ( bNF_co7330362693151061550list_f @ G2 @ ( comp_P3270879886602560258list_f @ produc7287154213363835177list_f @ F ) )
        = F ) ) )).

% convol_expand_snd
thf(fact_872_convol__o,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,H: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( comp_P3349191292975441602list_f @ ( bNF_co3276643461748138758_a_b_c @ F @ G2 ) @ H )
      = ( bNF_co7315529798985115712_a_b_c @ ( comp_P4767578765025054880list_f @ F @ H ) @ ( comp_P4767578765025054880list_f @ G2 @ H ) ) ) )).

% convol_o
thf(fact_873_pair__in__Id__conv,axiom,(
    ! [A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ id_nat2 )
      = ( A = B ) ) )).

% pair_in_Id_conv
thf(fact_874_pair__in__Id__conv,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ id_Pro2258643101195443293at_nat )
      = ( A = B ) ) )).

% pair_in_Id_conv
thf(fact_875_conditional__action_Omap__comp,axiom,(
    ! [G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,F: produc8923597787767437993list_f > produc1811169103916964701list_f,V: condit573090327609778161list_f] :
      ( ( condit3995630822656669714_a_b_c @ G2 @ ( condit4069896782405135150list_f @ F @ V ) )
      = ( condit1468111593096988280_a_b_c @ ( comp_P4767578765025054880list_f @ G2 @ F ) @ V ) ) )).

% conditional_action.map_comp
thf(fact_876_rewriteL__comp__comp,axiom,(
    ! [F: list_n1973683603564934665_a_b_c > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,L: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,H: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( ( comp_l6492537726383095040list_f @ F @ G2 )
        = L )
     => ( ( comp_l8432902307485339590list_f @ F @ ( comp_P4767578765025054880list_f @ G2 @ H ) )
        = ( comp_P4767578765025054880list_f @ L @ H ) ) ) )).

% rewriteL_comp_comp
thf(fact_877_rewriteL__comp__comp,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > produc1811169103916964701list_f,L: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,H: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( ( comp_P5271726855292207526list_f @ F @ G2 )
        = L )
     => ( ( comp_P4767578765025054880list_f @ F @ ( comp_P2673361230361438154list_f @ G2 @ H ) )
        = ( comp_P4767578765025054880list_f @ L @ H ) ) ) )).

% rewriteL_comp_comp
thf(fact_878_rewriteL__comp__comp,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc8923597787767437993list_f > produc1811169103916964701list_f,L: produc8923597787767437993list_f > list_n1973683603564934665_a_b_c,H: produc8923597787767437993list_f > produc8923597787767437993list_f] :
      ( ( ( comp_P4767578765025054880list_f @ F @ G2 )
        = L )
     => ( ( comp_P4767578765025054880list_f @ F @ ( comp_P4267156721827287172list_f @ G2 @ H ) )
        = ( comp_P1644921750638218854list_f @ L @ H ) ) ) )).

% rewriteL_comp_comp
thf(fact_879_rewriteR__comp__comp,axiom,(
    ! [G2: produc8923597787767437993list_f > produc1811169103916964701list_f,H: produc8923597787767437993list_f > produc8923597787767437993list_f,R: produc8923597787767437993list_f > produc1811169103916964701list_f,F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ( ( comp_P4267156721827287172list_f @ G2 @ H )
        = R )
     => ( ( comp_P1644921750638218854list_f @ ( comp_P4767578765025054880list_f @ F @ G2 ) @ H )
        = ( comp_P4767578765025054880list_f @ F @ R ) ) ) )).

% rewriteR_comp_comp
thf(fact_880_rewriteR__comp__comp,axiom,(
    ! [G2: produc1811169103916964701list_f > produc1811169103916964701list_f,H: produc8923597787767437993list_f > produc1811169103916964701list_f,R: produc8923597787767437993list_f > produc1811169103916964701list_f,F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ( ( comp_P2673361230361438154list_f @ G2 @ H )
        = R )
     => ( ( comp_P4767578765025054880list_f @ ( comp_P5271726855292207526list_f @ F @ G2 ) @ H )
        = ( comp_P4767578765025054880list_f @ F @ R ) ) ) )).

% rewriteR_comp_comp
thf(fact_881_rewriteR__comp__comp,axiom,(
    ! [G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,H: produc8923597787767437993list_f > produc1811169103916964701list_f,R: produc8923597787767437993list_f > list_n1973683603564934665_a_b_c,F: list_n1973683603564934665_a_b_c > list_n1973683603564934665_a_b_c] :
      ( ( ( comp_P4767578765025054880list_f @ G2 @ H )
        = R )
     => ( ( comp_P4767578765025054880list_f @ ( comp_l6492537726383095040list_f @ F @ G2 ) @ H )
        = ( comp_l8432902307485339590list_f @ F @ R ) ) ) )).

% rewriteR_comp_comp
thf(fact_882_refl__on__def,axiom,
    ( refl_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
          ( ( ord_le3146513528884898305at_nat @ R3
            @ ( produc457027306803732586at_nat @ A6
              @ ^ [Uu: nat] : A6 ) )
          & ! [X4: nat] :
              ( ( member_nat @ X4 @ A6 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R3 ) ) ) ) )).

% refl_on_def
thf(fact_883_refl__on__def,axiom,
    ( refl_o7122697186039111552at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
          ( ( ord_le3000389064537975527at_nat @ R3
            @ ( produc2761391749766926216at_nat @ A6
              @ ^ [Uu: product_prod_nat_nat] : A6 ) )
          & ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A6 )
             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R3 ) ) ) ) )).

% refl_on_def
thf(fact_884_conditional__action_Opred__cong,axiom,(
    ! [X: condit1452497342716194761at_nat,Ya2: condit1452497342716194761at_nat,P: product_prod_nat_nat > $o,Pa: product_prod_nat_nat > $o] :
      ( ( X = Ya2 )
     => ( ! [Z4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ Z4 @ ( condit6429450166605921294at_nat @ Ya2 ) )
           => ( ( P @ Z4 )
              = ( Pa @ Z4 ) ) )
       => ( ( condit2957664943879695485at_nat @ P @ X )
          = ( condit2957664943879695485at_nat @ Pa @ Ya2 ) ) ) ) )).

% conditional_action.pred_cong
thf(fact_885_conditional__action_Opred__cong,axiom,(
    ! [X: condit3449342828199732399at_nat,Ya2: condit3449342828199732399at_nat,P: produc859450856879609959at_nat > $o,Pa: produc859450856879609959at_nat > $o] :
      ( ( X = Ya2 )
     => ( ! [Z4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ Z4 @ ( condit5307000522679409652at_nat @ Ya2 ) )
           => ( ( P @ Z4 )
              = ( Pa @ Z4 ) ) )
       => ( ( condit2125367562630040035at_nat @ P @ X )
          = ( condit2125367562630040035at_nat @ Pa @ Ya2 ) ) ) ) )).

% conditional_action.pred_cong
thf(fact_886_rewriteL__comp__comp2,axiom,(
    ! [F: list_n1973683603564934665_a_b_c > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,L1: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,L2: produc1811169103916964701list_f > produc1811169103916964701list_f,H: produc8923597787767437993list_f > produc1811169103916964701list_f,R: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( ( comp_l6492537726383095040list_f @ F @ G2 )
        = ( comp_P5271726855292207526list_f @ L1 @ L2 ) )
     => ( ( ( comp_P2673361230361438154list_f @ L2 @ H )
          = R )
       => ( ( comp_l8432902307485339590list_f @ F @ ( comp_P4767578765025054880list_f @ G2 @ H ) )
          = ( comp_P4767578765025054880list_f @ L1 @ R ) ) ) ) )).

% rewriteL_comp_comp2
thf(fact_887_rewriteL__comp__comp2,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > produc1811169103916964701list_f,L1: list_n1973683603564934665_a_b_c > list_n1973683603564934665_a_b_c,L2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,H: produc8923597787767437993list_f > produc1811169103916964701list_f,R: produc8923597787767437993list_f > list_n1973683603564934665_a_b_c] :
      ( ( ( comp_P5271726855292207526list_f @ F @ G2 )
        = ( comp_l6492537726383095040list_f @ L1 @ L2 ) )
     => ( ( ( comp_P4767578765025054880list_f @ L2 @ H )
          = R )
       => ( ( comp_P4767578765025054880list_f @ F @ ( comp_P2673361230361438154list_f @ G2 @ H ) )
          = ( comp_l8432902307485339590list_f @ L1 @ R ) ) ) ) )).

% rewriteL_comp_comp2
thf(fact_888_rewriteL__comp__comp2,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc8923597787767437993list_f > produc1811169103916964701list_f,L1: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,L2: produc8923597787767437993list_f > produc1811169103916964701list_f,H: produc8923597787767437993list_f > produc8923597787767437993list_f,R: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( ( comp_P4767578765025054880list_f @ F @ G2 )
        = ( comp_P4767578765025054880list_f @ L1 @ L2 ) )
     => ( ( ( comp_P4267156721827287172list_f @ L2 @ H )
          = R )
       => ( ( comp_P4767578765025054880list_f @ F @ ( comp_P4267156721827287172list_f @ G2 @ H ) )
          = ( comp_P4767578765025054880list_f @ L1 @ R ) ) ) ) )).

% rewriteL_comp_comp2
thf(fact_889_SigmaE,axiom,(
    ! [C2: produc39072733474941005at_nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ C2 @ ( produc7465995907834253422at_nat @ A4 @ B5 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A4 )
           => ! [Y3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc5496787636967163197at_nat @ X3 @ Y3 ) ) ) ) ) )).

% SigmaE
thf(fact_890_SigmaE,axiom,(
    ! [C2: produc7148259590854449153at_nat,A4: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ C2 @ ( produc3829605335166519074at_nat @ A4 @ B5 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A4 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc1860397064299428849at_nat @ X3 @ Y3 ) ) ) ) ) )).

% SigmaE
thf(fact_891_SigmaE,axiom,(
    ! [C2: produc349518998152878311at_nat,A4: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ C2 @ ( produc18557253517588744at_nat @ A4 @ B5 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A4 )
           => ! [Y3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc4662710985925991255at_nat @ X3 @ Y3 ) ) ) ) ) )).

% SigmaE
thf(fact_892_SigmaE,axiom,(
    ! [C2: product_prod_nat_nat,A4: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ) ) )).

% SigmaE
thf(fact_893_SigmaE,axiom,(
    ! [C2: produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ ( produc2761391749766926216at_nat @ A4 @ B5 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A4 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc6161850002892822231at_nat @ X3 @ Y3 ) ) ) ) ) )).

% SigmaE
thf(fact_894_SigmaE,axiom,(
    ! [C2: produc8763457246119570046nteger,A4: set_Co9149898834107579976e_term,B5: ( code_integer > option6357759511663192854e_term ) > set_Pr4811707699266497531nteger] :
      ( ( member3068662437193594005nteger @ C2 @ ( produc4741912807555180607nteger @ A4 @ B5 ) )
     => ~ ! [X3: code_integer > option6357759511663192854e_term] :
            ( ( member1535805642427569193e_term @ X3 @ A4 )
           => ! [Y3: produc8923325533196201883nteger] :
                ( ( member157494554546826820nteger @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc6137756002093451184nteger @ X3 @ Y3 ) ) ) ) ) )).

% SigmaE
thf(fact_895_SigmaE,axiom,(
    ! [C2: produc989692990947075319atural,A4: set_Pr2791530082921190693e_term,B5: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > set_Pr7564037697178849041atural] :
      ( ( member3245610415464309390atural @ C2 @ ( produc2485531694476998072atural @ A4 @ B5 ) )
     => ~ ! [X3: produc4972180933644002618e_term > option6357759511663192854e_term] :
            ( ( member2820825060907577222e_term @ X3 @ A4 )
           => ! [Y3: produc7822875418678951345atural] :
                ( ( member8280416476884352090atural @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc3831813291587773865atural @ X3 @ Y3 ) ) ) ) ) )).

% SigmaE
thf(fact_896_SigmaE,axiom,(
    ! [C2: produc1908205239877642774nteger,A4: set_Pr7604974323444597168e_term,B5: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr4811707699266497531nteger] :
      ( ( member4164122664394876845nteger @ C2 @ ( produc7256824055837167575nteger @ A4 @ B5 ) )
     => ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term] :
            ( ( member4242434998011752849e_term @ X3 @ A4 )
           => ! [Y3: produc8923325533196201883nteger] :
                ( ( member157494554546826820nteger @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc8603105652947943368nteger @ X3 @ Y3 ) ) ) ) ) )).

% SigmaE
thf(fact_897_SigmaE,axiom,(
    ! [C2: produc2285326912895808259nt_int,A4: set_Pr3642885161833720865e_term,B5: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7618704894036264090nt_int @ C2 @ ( produc5808185664642557892nt_int @ A4 @ B5 ) )
     => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term] :
            ( ( member3222579708246209666e_term @ X3 @ A4 )
           => ! [Y3: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ) ) ) )).

% SigmaE
thf(fact_898_SigmaE,axiom,(
    ! [C2: produc7773217078559923341nt_int,A4: set_in3461395444621081367e_term,B5: ( int > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7034335876925520548nt_int @ C2 @ ( produc1318478031590598990nt_int @ A4 @ B5 ) )
     => ~ ! [X3: int > option6357759511663192854e_term] :
            ( ( member8845023287901829240e_term @ X3 @ A4 )
           => ! [Y3: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ) ) ) )).

% SigmaE
thf(fact_899_rewriteR__comp__comp2,axiom,(
    ! [G2: produc8923597787767437993list_f > produc1811169103916964701list_f,H: produc8923597787767437993list_f > produc8923597787767437993list_f,R1: produc1811169103916964701list_f > produc1811169103916964701list_f,R22: produc8923597787767437993list_f > produc1811169103916964701list_f,F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,L: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ( ( comp_P4267156721827287172list_f @ G2 @ H )
        = ( comp_P2673361230361438154list_f @ R1 @ R22 ) )
     => ( ( ( comp_P5271726855292207526list_f @ F @ R1 )
          = L )
       => ( ( comp_P1644921750638218854list_f @ ( comp_P4767578765025054880list_f @ F @ G2 ) @ H )
          = ( comp_P4767578765025054880list_f @ L @ R22 ) ) ) ) )).

% rewriteR_comp_comp2
thf(fact_900_rewriteR__comp__comp2,axiom,(
    ! [G2: produc1811169103916964701list_f > produc1811169103916964701list_f,H: produc8923597787767437993list_f > produc1811169103916964701list_f,R1: produc8923597787767437993list_f > produc1811169103916964701list_f,R22: produc8923597787767437993list_f > produc8923597787767437993list_f,F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,L: produc8923597787767437993list_f > list_n1973683603564934665_a_b_c] :
      ( ( ( comp_P2673361230361438154list_f @ G2 @ H )
        = ( comp_P4267156721827287172list_f @ R1 @ R22 ) )
     => ( ( ( comp_P4767578765025054880list_f @ F @ R1 )
          = L )
       => ( ( comp_P4767578765025054880list_f @ ( comp_P5271726855292207526list_f @ F @ G2 ) @ H )
          = ( comp_P1644921750638218854list_f @ L @ R22 ) ) ) ) )).

% rewriteR_comp_comp2
thf(fact_901_rewriteR__comp__comp2,axiom,(
    ! [G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,H: produc8923597787767437993list_f > produc1811169103916964701list_f,R1: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,R22: produc8923597787767437993list_f > produc1811169103916964701list_f,F: list_n1973683603564934665_a_b_c > list_n1973683603564934665_a_b_c,L: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ( ( comp_P4767578765025054880list_f @ G2 @ H )
        = ( comp_P4767578765025054880list_f @ R1 @ R22 ) )
     => ( ( ( comp_l6492537726383095040list_f @ F @ R1 )
          = L )
       => ( ( comp_P4767578765025054880list_f @ ( comp_l6492537726383095040list_f @ F @ G2 ) @ H )
          = ( comp_P4767578765025054880list_f @ L @ R22 ) ) ) ) )).

% rewriteR_comp_comp2
thf(fact_902_typerep__conditional__action__def,axiom,
    ( typere4123026268246276339action
    = ( ^ [T: itself242674819721554163action] :
          ( typerep2 @ ( literal2 @ $false @ $true @ $true @ $false @ $false @ $false @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $false @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $false @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( cons_typerep @ ( typere7107042215916526635action @ type_lock_action ) @ nil_typerep ) ) ) )).

% typerep_conditional_action_def
thf(fact_903_comp__transfer,axiom,(
    ! [B5: produc1811169103916964701list_f > produc1811169103916964701list_f > $o,C4: list_n1973683603564934665_a_b_c > list_n1973683603564934665_a_b_c > $o,A4: produc8923597787767437993list_f > produc8923597787767437993list_f > $o] :
      ( bNF_re1616170189525647965_a_b_c @ ( bNF_re2373559492235539289_a_b_c @ B5 @ C4 ) @ ( bNF_re2955035557799908905_a_b_c @ ( bNF_re5276636924349840729list_f @ A4 @ B5 ) @ ( bNF_re7641334924310632669_a_b_c @ A4 @ C4 ) ) @ comp_P4767578765025054880list_f @ comp_P4767578765025054880list_f ) )).

% comp_transfer
thf(fact_904_o__rsp_I1_J,axiom,(
    ! [R23: produc1811169103916964701list_f > produc1811169103916964701list_f > $o,R32: list_n1973683603564934665_a_b_c > list_n1973683603564934665_a_b_c > $o,R12: produc8923597787767437993list_f > produc8923597787767437993list_f > $o] :
      ( bNF_re1616170189525647965_a_b_c @ ( bNF_re2373559492235539289_a_b_c @ R23 @ R32 ) @ ( bNF_re2955035557799908905_a_b_c @ ( bNF_re5276636924349840729list_f @ R12 @ R23 ) @ ( bNF_re7641334924310632669_a_b_c @ R12 @ R32 ) ) @ comp_P4767578765025054880list_f @ comp_P4767578765025054880list_f ) )).

% o_rsp(1)
thf(fact_905_conditional__action_Opred__mono__strong,axiom,(
    ! [P: product_prod_nat_nat > $o,X: condit1452497342716194761at_nat,Pa: product_prod_nat_nat > $o] :
      ( ( condit2957664943879695485at_nat @ P @ X )
     => ( ! [Z4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ Z4 @ ( condit6429450166605921294at_nat @ X ) )
           => ( ( P @ Z4 )
             => ( Pa @ Z4 ) ) )
       => ( condit2957664943879695485at_nat @ Pa @ X ) ) ) )).

% conditional_action.pred_mono_strong
thf(fact_906_conditional__action_Opred__mono__strong,axiom,(
    ! [P: produc859450856879609959at_nat > $o,X: condit3449342828199732399at_nat,Pa: produc859450856879609959at_nat > $o] :
      ( ( condit2125367562630040035at_nat @ P @ X )
     => ( ! [Z4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ Z4 @ ( condit5307000522679409652at_nat @ X ) )
           => ( ( P @ Z4 )
             => ( Pa @ Z4 ) ) )
       => ( condit2125367562630040035at_nat @ Pa @ X ) ) ) )).

% conditional_action.pred_mono_strong
thf(fact_907_o__rsp_I2_J,axiom,(
    ! [R12: produc8923597787767437993list_f > produc8923597787767437993list_f > $o] :
      ( bNF_re1616170189525647965_a_b_c
      @ ^ [Y5: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,Z2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] : ( Y5 = Z2 )
      @ ( bNF_re2955035557799908905_a_b_c
        @ ( bNF_re5276636924349840729list_f @ R12
          @ ^ [Y5: produc1811169103916964701list_f,Z2: produc1811169103916964701list_f] : ( Y5 = Z2 ) )
        @ ( bNF_re7641334924310632669_a_b_c @ R12
          @ ^ [Y5: list_n1973683603564934665_a_b_c,Z2: list_n1973683603564934665_a_b_c] : ( Y5 = Z2 ) ) )
      @ comp_P4767578765025054880list_f
      @ comp_P4767578765025054880list_f ) )).

% o_rsp(2)
thf(fact_908_next_Osimps,axiom,(
    ! [V: code_natural,W2: code_natural] :
      ( ( next @ ( produc3574140220909816553atural @ V @ W2 ) )
      = ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ one_one_Code_natural ) ) @ one_one_Code_natural ) @ ( produc3574140220909816553atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

% next.simps
thf(fact_909_IdI,axiom,(
    ! [A: nat] :
      ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ id_nat2 ) )).

% IdI
thf(fact_910_IdI,axiom,(
    ! [A: product_prod_nat_nat] :
      ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ id_Pro2258643101195443293at_nat ) )).

% IdI
thf(fact_911_IdE,axiom,(
    ! [P2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ id_nat2 )
     => ~ ! [X3: nat] :
            ( P2
           != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) )).

% IdE
thf(fact_912_IdE,axiom,(
    ! [P2: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ P2 @ id_Pro2258643101195443293at_nat )
     => ~ ! [X3: product_prod_nat_nat] :
            ( P2
           != ( produc6161850002892822231at_nat @ X3 @ X3 ) ) ) )).

% IdE
thf(fact_913_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X4: nat,Y6: nat] :
          ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y6 ) @ bot_bo2099793752762293965at_nat ) ) )).

% bot_empty_eq2
thf(fact_914_bot__empty__eq2,axiom,
    ( bot_bo4898103413517107610_nat_o
    = ( ^ [X4: product_prod_nat_nat,Y6: product_prod_nat_nat] :
          ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y6 ) @ bot_bo5327735625951526323at_nat ) ) )).

% bot_empty_eq2
thf(fact_915_bot__empty__eq2,axiom,
    ( bot_bo5358457235160185703eger_o
    = ( ^ [X4: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
          ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y6 ) @ bot_bo3145834390647256904nteger ) ) )).

% bot_empty_eq2
thf(fact_916_bot__empty__eq2,axiom,
    ( bot_bo6056521241689113134ural_o
    = ( ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term,Y6: produc7822875418678951345atural] :
          ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X4 @ Y6 ) @ bot_bo8406019360097795265atural ) ) )).

% bot_empty_eq2
thf(fact_917_bot__empty__eq2,axiom,
    ( bot_bo3000040243691356879eger_o
    = ( ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
          ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y6 ) @ bot_bo5443222936135328352nteger ) ) )).

% bot_empty_eq2
thf(fact_918_bot__empty__eq2,axiom,
    ( bot_bo8662317086119403298_int_o
    = ( ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
          ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y6 ) @ bot_bo572930865798478029nt_int ) ) )).

% bot_empty_eq2
thf(fact_919_bot__empty__eq2,axiom,
    ( bot_bo1403522918969695512_int_o
    = ( ^ [X4: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
          ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y6 ) @ bot_bo4508923176915781079nt_int ) ) )).

% bot_empty_eq2
thf(fact_920_comp__eq__id__dest,axiom,(
    ! [A: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,B: produc8923597787767437993list_f > produc1811169103916964701list_f,C2: produc8923597787767437993list_f > list_n1973683603564934665_a_b_c,V: produc8923597787767437993list_f] :
      ( ( ( comp_P4767578765025054880list_f @ A @ B )
        = ( comp_l8432902307485339590list_f @ id_lis6780652146352037754_a_b_c @ C2 ) )
     => ( ( A @ ( B @ V ) )
        = ( C2 @ V ) ) ) )).

% comp_eq_id_dest
thf(fact_921_GrD1,axiom,(
    ! [X: nat,Fx: nat,A4: set_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Fx ) @ ( bNF_Gr_nat_nat @ A4 @ F ) )
     => ( member_nat @ X @ A4 ) ) )).

% GrD1
thf(fact_922_GrD1,axiom,(
    ! [X: product_prod_nat_nat,Fx: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Fx ) @ ( bNF_Gr4088865321246080482at_nat @ A4 @ F ) )
     => ( member8440522571783428010at_nat @ X @ A4 ) ) )).

% GrD1
thf(fact_923_GrD1,axiom,(
    ! [X: code_integer > option6357759511663192854e_term,Fx: produc8923325533196201883nteger,A4: set_Co9149898834107579976e_term,F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Fx ) @ ( bNF_Gr5573520379198718565nteger @ A4 @ F ) )
     => ( member1535805642427569193e_term @ X @ A4 ) ) )).

% GrD1
thf(fact_924_GrD1,axiom,(
    ! [X: produc4972180933644002618e_term > option6357759511663192854e_term,Fx: produc7822875418678951345atural,A4: set_Pr2791530082921190693e_term,F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural] :
      ( ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X @ Fx ) @ ( bNF_Gr6432593794076504798atural @ A4 @ F ) )
     => ( member2820825060907577222e_term @ X @ A4 ) ) )).

% GrD1
thf(fact_925_GrD1,axiom,(
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Fx: produc8923325533196201883nteger,A4: set_Pr7604974323444597168e_term,F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Fx ) @ ( bNF_Gr1980514118581898493nteger @ A4 @ F ) )
     => ( member4242434998011752849e_term @ X @ A4 ) ) )).

% GrD1
thf(fact_926_GrD1,axiom,(
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Fx: product_prod_int_int,A4: set_Pr3642885161833720865e_term,F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Fx ) @ ( bNF_Gr3160678035669724394nt_int @ A4 @ F ) )
     => ( member3222579708246209666e_term @ X @ A4 ) ) )).

% GrD1
thf(fact_927_GrD1,axiom,(
    ! [X: int > option6357759511663192854e_term,Fx: product_prod_int_int,A4: set_in3461395444621081367e_term,F: ( int > option6357759511663192854e_term ) > product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Fx ) @ ( bNF_Gr5793145711989525364nt_int @ A4 @ F ) )
     => ( member8845023287901829240e_term @ X @ A4 ) ) )).

% GrD1
thf(fact_928_GrD2,axiom,(
    ! [X: nat,Fx: nat,A4: set_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Fx ) @ ( bNF_Gr_nat_nat @ A4 @ F ) )
     => ( ( F @ X )
        = Fx ) ) )).

% GrD2
thf(fact_929_GrD2,axiom,(
    ! [X: product_prod_nat_nat,Fx: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Fx ) @ ( bNF_Gr4088865321246080482at_nat @ A4 @ F ) )
     => ( ( F @ X )
        = Fx ) ) )).

% GrD2
thf(fact_930_GrD2,axiom,(
    ! [X: code_integer > option6357759511663192854e_term,Fx: produc8923325533196201883nteger,A4: set_Co9149898834107579976e_term,F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Fx ) @ ( bNF_Gr5573520379198718565nteger @ A4 @ F ) )
     => ( ( F @ X )
        = Fx ) ) )).

% GrD2
thf(fact_931_GrD2,axiom,(
    ! [X: produc4972180933644002618e_term > option6357759511663192854e_term,Fx: produc7822875418678951345atural,A4: set_Pr2791530082921190693e_term,F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural] :
      ( ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X @ Fx ) @ ( bNF_Gr6432593794076504798atural @ A4 @ F ) )
     => ( ( F @ X )
        = Fx ) ) )).

% GrD2
thf(fact_932_GrD2,axiom,(
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Fx: produc8923325533196201883nteger,A4: set_Pr7604974323444597168e_term,F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Fx ) @ ( bNF_Gr1980514118581898493nteger @ A4 @ F ) )
     => ( ( F @ X )
        = Fx ) ) )).

% GrD2
thf(fact_933_GrD2,axiom,(
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Fx: product_prod_int_int,A4: set_Pr3642885161833720865e_term,F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Fx ) @ ( bNF_Gr3160678035669724394nt_int @ A4 @ F ) )
     => ( ( F @ X )
        = Fx ) ) )).

% GrD2
thf(fact_934_GrD2,axiom,(
    ! [X: int > option6357759511663192854e_term,Fx: product_prod_int_int,A4: set_in3461395444621081367e_term,F: ( int > option6357759511663192854e_term ) > product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Fx ) @ ( bNF_Gr5793145711989525364nt_int @ A4 @ F ) )
     => ( ( F @ X )
        = Fx ) ) )).

% GrD2
thf(fact_935_fstOp__in,axiom,(
    ! [Ac3: product_prod_nat_nat,P: nat > nat > $o,Q: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ Ac3 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( relcompp_nat_nat_nat @ P @ Q ) ) ) )
     => ( member8440522571783428010at_nat @ ( bNF_fs6799373675401686564at_nat @ P @ Q @ Ac3 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P ) ) ) ) )).

% fstOp_in
thf(fact_936_fstOp__in,axiom,(
    ! [Ac3: produc859450856879609959at_nat,P: product_prod_nat_nat > product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ Ac3 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ ( relcom282571730187242960at_nat @ P @ Q ) ) ) )
     => ( member8206827879206165904at_nat @ ( bNF_fs8257191039976258191at_nat @ P @ Q @ Ac3 ) @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ P ) ) ) ) )).

% fstOp_in
thf(fact_937_sndOp__in,axiom,(
    ! [Ac3: product_prod_nat_nat,P: nat > nat > $o,Q: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ Ac3 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( relcompp_nat_nat_nat @ P @ Q ) ) ) )
     => ( member8440522571783428010at_nat @ ( bNF_sn8916957246138178918at_nat @ P @ Q @ Ac3 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ Q ) ) ) ) )).

% sndOp_in
thf(fact_938_sndOp__in,axiom,(
    ! [Ac3: produc859450856879609959at_nat,P: product_prod_nat_nat > product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ Ac3 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ ( relcom282571730187242960at_nat @ P @ Q ) ) ) )
     => ( member8206827879206165904at_nat @ ( bNF_sn5923954573255187405at_nat @ P @ Q @ Ac3 ) @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ Q ) ) ) ) )).

% sndOp_in
thf(fact_939_bsqr__max2,axiom,(
    ! [R: set_Pr8693737435421807431at_nat,A1: product_prod_nat_nat,A22: product_prod_nat_nat,B1: product_prod_nat_nat,B22: product_prod_nat_nat] :
      ( ( order_8331247224601271455at_nat @ ( field_4706345496283612622at_nat @ R ) @ R )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ ( produc6161850002892822231at_nat @ B1 @ B22 ) ) @ ( bNF_We3318688506120914846at_nat @ R ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( bNF_We6793920340776068943at_nat @ R @ A1 @ A22 ) @ ( bNF_We6793920340776068943at_nat @ R @ B1 @ B22 ) ) @ R ) ) ) )).

% bsqr_max2
thf(fact_940_bsqr__max2,axiom,(
    ! [R: set_Pr1261947904930325089at_nat,A1: nat,A22: nat,B1: nat,B22: nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( product_Pair_nat_nat @ B1 @ B22 ) ) @ ( bNF_We1732890022510290291qr_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We2380939065524664066x2_nat @ R @ A1 @ A22 ) @ ( bNF_We2380939065524664066x2_nat @ R @ B1 @ B22 ) ) @ R ) ) ) )).

% bsqr_max2
thf(fact_941_convol__mem__GrpI,axiom,(
    ! [X: nat,A4: set_nat,G2: nat > nat] :
      ( ( member_nat @ X @ A4 )
     => ( member8440522571783428010at_nat @ ( bNF_co805650143699787099at_nat @ id_nat @ G2 @ X ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A4 @ G2 ) ) ) ) ) )).

% convol_mem_GrpI
thf(fact_942_convol__mem__GrpI,axiom,(
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X @ A4 )
     => ( member8206827879206165904at_nat @ ( bNF_co3897469944465340696at_nat @ id_Pro7559728917991913458at_nat @ G2 @ X ) @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ ( bNF_Gr7275288873855648526at_nat @ A4 @ G2 ) ) ) ) ) )).

% convol_mem_GrpI
thf(fact_943_conditional__action_Orel__cong,axiom,(
    ! [X: condit1452497342716194761at_nat,Ya2: condit1452497342716194761at_nat,Y: condit1452497342716194761at_nat,Xa: condit1452497342716194761at_nat,R2: product_prod_nat_nat > product_prod_nat_nat > $o,Ra2: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( X = Ya2 )
     => ( ( Y = Xa )
       => ( ! [Z4: product_prod_nat_nat,Yb: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Z4 @ ( condit6429450166605921294at_nat @ Ya2 ) )
             => ( ( member8440522571783428010at_nat @ Yb @ ( condit6429450166605921294at_nat @ Xa ) )
               => ( ( R2 @ Z4 @ Yb )
                  = ( Ra2 @ Z4 @ Yb ) ) ) )
         => ( ( condit3216346422009540699at_nat @ R2 @ X @ Y )
            = ( condit3216346422009540699at_nat @ Ra2 @ Ya2 @ Xa ) ) ) ) ) )).

% conditional_action.rel_cong
thf(fact_944_conditional__action_Orel__cong,axiom,(
    ! [X: condit1452497342716194761at_nat,Ya2: condit1452497342716194761at_nat,Y: condit3449342828199732399at_nat,Xa: condit3449342828199732399at_nat,R2: product_prod_nat_nat > produc859450856879609959at_nat > $o,Ra2: product_prod_nat_nat > produc859450856879609959at_nat > $o] :
      ( ( X = Ya2 )
     => ( ( Y = Xa )
       => ( ! [Z4: product_prod_nat_nat,Yb: produc859450856879609959at_nat] :
              ( ( member8440522571783428010at_nat @ Z4 @ ( condit6429450166605921294at_nat @ Ya2 ) )
             => ( ( member8206827879206165904at_nat @ Yb @ ( condit5307000522679409652at_nat @ Xa ) )
               => ( ( R2 @ Z4 @ Yb )
                  = ( Ra2 @ Z4 @ Yb ) ) ) )
         => ( ( condit8505942925542983873at_nat @ R2 @ X @ Y )
            = ( condit8505942925542983873at_nat @ Ra2 @ Ya2 @ Xa ) ) ) ) ) )).

% conditional_action.rel_cong
thf(fact_945_conditional__action_Orel__cong,axiom,(
    ! [X: condit3449342828199732399at_nat,Ya2: condit3449342828199732399at_nat,Y: condit1452497342716194761at_nat,Xa: condit1452497342716194761at_nat,R2: produc859450856879609959at_nat > product_prod_nat_nat > $o,Ra2: produc859450856879609959at_nat > product_prod_nat_nat > $o] :
      ( ( X = Ya2 )
     => ( ( Y = Xa )
       => ( ! [Z4: produc859450856879609959at_nat,Yb: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ Z4 @ ( condit5307000522679409652at_nat @ Ya2 ) )
             => ( ( member8440522571783428010at_nat @ Yb @ ( condit6429450166605921294at_nat @ Xa ) )
               => ( ( R2 @ Z4 @ Yb )
                  = ( Ra2 @ Z4 @ Yb ) ) ) )
         => ( ( condit4869552352875249525at_nat @ R2 @ X @ Y )
            = ( condit4869552352875249525at_nat @ Ra2 @ Ya2 @ Xa ) ) ) ) ) )).

% conditional_action.rel_cong
thf(fact_946_conditional__action_Orel__cong,axiom,(
    ! [X: condit3449342828199732399at_nat,Ya2: condit3449342828199732399at_nat,Y: condit3449342828199732399at_nat,Xa: condit3449342828199732399at_nat,R2: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,Ra2: produc859450856879609959at_nat > produc859450856879609959at_nat > $o] :
      ( ( X = Ya2 )
     => ( ( Y = Xa )
       => ( ! [Z4: produc859450856879609959at_nat,Yb: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ Z4 @ ( condit5307000522679409652at_nat @ Ya2 ) )
             => ( ( member8206827879206165904at_nat @ Yb @ ( condit5307000522679409652at_nat @ Xa ) )
               => ( ( R2 @ Z4 @ Yb )
                  = ( Ra2 @ Z4 @ Yb ) ) ) )
         => ( ( condit7775146776940579035at_nat @ R2 @ X @ Y )
            = ( condit7775146776940579035at_nat @ Ra2 @ Ya2 @ Xa ) ) ) ) ) )).

% conditional_action.rel_cong
thf(fact_947_image2__eqI,axiom,(
    ! [B: nat,F: product_prod_nat_nat > nat,X: product_prod_nat_nat,C2: nat,G2: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G2 @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A4 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( bNF_Gr8954129895687480649at_nat @ A4 @ F @ G2 ) ) ) ) ) )).

% image2_eqI
thf(fact_948_image2__eqI,axiom,(
    ! [B: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,C2: product_prod_nat_nat,G2: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G2 @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A4 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C2 ) @ ( bNF_Gr8808630244184418791at_nat @ A4 @ F @ G2 ) ) ) ) ) )).

% image2_eqI
thf(fact_949_image2__eqI,axiom,(
    ! [B: nat,F: produc859450856879609959at_nat > nat,X: produc859450856879609959at_nat,C2: nat,G2: produc859450856879609959at_nat > nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G2 @ X ) )
       => ( ( member8206827879206165904at_nat @ X @ A4 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( bNF_Gr97278709799963567at_nat @ A4 @ F @ G2 ) ) ) ) ) )).

% image2_eqI
thf(fact_950_image2__eqI,axiom,(
    ! [B: product_prod_nat_nat,F: produc859450856879609959at_nat > product_prod_nat_nat,X: produc859450856879609959at_nat,C2: product_prod_nat_nat,G2: produc859450856879609959at_nat > product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G2 @ X ) )
       => ( ( member8206827879206165904at_nat @ X @ A4 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C2 ) @ ( bNF_Gr978357984662150989at_nat @ A4 @ F @ G2 ) ) ) ) ) )).

% image2_eqI
thf(fact_951_image2__eqI,axiom,(
    ! [B: code_integer > option6357759511663192854e_term,F: product_prod_nat_nat > code_integer > option6357759511663192854e_term,X: product_prod_nat_nat,C2: produc8923325533196201883nteger,G2: product_prod_nat_nat > produc8923325533196201883nteger,A4: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G2 @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A4 )
         => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ B @ C2 ) @ ( bNF_Gr7498812625829202080nteger @ A4 @ F @ G2 ) ) ) ) ) )).

% image2_eqI
thf(fact_952_image2__eqI,axiom,(
    ! [B: int > option6357759511663192854e_term,F: product_prod_nat_nat > int > option6357759511663192854e_term,X: product_prod_nat_nat,C2: product_prod_int_int,G2: product_prod_nat_nat > product_prod_int_int,A4: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G2 @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A4 )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ B @ C2 ) @ ( bNF_Gr236874688385193391nt_int @ A4 @ F @ G2 ) ) ) ) ) )).

% image2_eqI
thf(fact_953_image2__eqI,axiom,(
    ! [B: code_integer > option6357759511663192854e_term,F: produc859450856879609959at_nat > code_integer > option6357759511663192854e_term,X: produc859450856879609959at_nat,C2: produc8923325533196201883nteger,G2: produc859450856879609959at_nat > produc8923325533196201883nteger,A4: set_Pr8693737435421807431at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G2 @ X ) )
       => ( ( member8206827879206165904at_nat @ X @ A4 )
         => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ B @ C2 ) @ ( bNF_Gr6656827289399605946nteger @ A4 @ F @ G2 ) ) ) ) ) )).

% image2_eqI
thf(fact_954_image2__eqI,axiom,(
    ! [B: produc4972180933644002618e_term > option6357759511663192854e_term,F: product_prod_nat_nat > produc4972180933644002618e_term > option6357759511663192854e_term,X: product_prod_nat_nat,C2: produc7822875418678951345atural,G2: product_prod_nat_nat > produc7822875418678951345atural,A4: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G2 @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A4 )
         => ( member3245610415464309390atural @ ( produc3831813291587773865atural @ B @ C2 ) @ ( bNF_Gr2085308689215053465atural @ A4 @ F @ G2 ) ) ) ) ) )).

% image2_eqI
thf(fact_955_image2__eqI,axiom,(
    ! [B: produc6241069584506657477e_term > option6357759511663192854e_term,F: product_prod_nat_nat > produc6241069584506657477e_term > option6357759511663192854e_term,X: product_prod_nat_nat,C2: produc8923325533196201883nteger,G2: product_prod_nat_nat > produc8923325533196201883nteger,A4: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G2 @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A4 )
         => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ B @ C2 ) @ ( bNF_Gr6856601050575222968nteger @ A4 @ F @ G2 ) ) ) ) ) )).

% image2_eqI
thf(fact_956_image2__eqI,axiom,(
    ! [B: produc8551481072490612790e_term > option6357759511663192854e_term,F: product_prod_nat_nat > produc8551481072490612790e_term > option6357759511663192854e_term,X: product_prod_nat_nat,C2: product_prod_int_int,G2: product_prod_nat_nat > product_prod_int_int,A4: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G2 @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A4 )
         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ B @ C2 ) @ ( bNF_Gr1652745948325567653nt_int @ A4 @ F @ G2 ) ) ) ) ) )).

% image2_eqI
thf(fact_957_FieldI2,axiom,(
    ! [I2: produc859450856879609959at_nat,J2: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I2 @ J2 ) @ R2 )
     => ( member8206827879206165904at_nat @ J2 @ ( field_3044959742847878580at_nat @ R2 ) ) ) )).

% FieldI2
thf(fact_958_FieldI2,axiom,(
    ! [I2: nat,J2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R2 )
     => ( member_nat @ J2 @ ( field_nat @ R2 ) ) ) )).

% FieldI2
thf(fact_959_FieldI2,axiom,(
    ! [I2: product_prod_nat_nat,J2: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R2 )
     => ( member8440522571783428010at_nat @ J2 @ ( field_4706345496283612622at_nat @ R2 ) ) ) )).

% FieldI2
thf(fact_960_range__snd,axiom,
    ( ( image_3837677093542026988atural @ produc6591912806276919810atural @ top_to8002910594584184420atural )
    = top_to5116171525255943265atural )).

% range_snd
thf(fact_961_range__snd,axiom,
    ( ( image_4621308865571145409list_f @ produc8710481132536983823list_f @ top_to1815313666553408473list_f )
    = top_to3090985321854968643list_f )).

% range_snd
thf(fact_962_range__fst,axiom,
    ( ( image_4933254100216391447e_term @ produc2702340305111172292nteger @ top_to2348597233010745700nteger )
    = top_to6545799489128733944e_term )).

% range_fst
thf(fact_963_range__fst,axiom,
    ( ( image_5638616612462892909e_term @ produc3051390257598303421atural @ top_to1252183740382221533atural )
    = top_to7040477076443024341e_term )).

% range_fst
thf(fact_964_range__fst,axiom,
    ( ( image_4436781500471753239e_term @ produc7822682618958472924nteger @ top_to7512759353274530428nteger )
    = top_to2630549280111655008e_term )).

% range_fst
thf(fact_965_range__fst,axiom,
    ( ( image_142517975987106933e_term @ produc6816164490631068361nt_int @ top_to3181862456159035625nt_int )
    = top_to58670533896299217e_term )).

% range_fst
thf(fact_966_range__fst,axiom,
    ( ( image_4217645046469565813e_term @ produc6230002227079971283nt_int @ top_to2069137843433766899nt_int )
    = top_to1459954387912989127e_term )).

% range_fst
thf(fact_967_FieldI1,axiom,(
    ! [I2: produc859450856879609959at_nat,J2: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I2 @ J2 ) @ R2 )
     => ( member8206827879206165904at_nat @ I2 @ ( field_3044959742847878580at_nat @ R2 ) ) ) )).

% FieldI1
thf(fact_968_FieldI1,axiom,(
    ! [I2: nat,J2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R2 )
     => ( member_nat @ I2 @ ( field_nat @ R2 ) ) ) )).

% FieldI1
thf(fact_969_FieldI1,axiom,(
    ! [I2: product_prod_nat_nat,J2: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R2 )
     => ( member8440522571783428010at_nat @ I2 @ ( field_4706345496283612622at_nat @ R2 ) ) ) )).

% FieldI1
thf(fact_970_conditional__action_Orel__mono__strong,axiom,(
    ! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,X: condit1452497342716194761at_nat,Y: condit1452497342716194761at_nat,Ra2: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( condit3216346422009540699at_nat @ R2 @ X @ Y )
     => ( ! [Z4: product_prod_nat_nat,Yb: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ Z4 @ ( condit6429450166605921294at_nat @ X ) )
           => ( ( member8440522571783428010at_nat @ Yb @ ( condit6429450166605921294at_nat @ Y ) )
             => ( ( R2 @ Z4 @ Yb )
               => ( Ra2 @ Z4 @ Yb ) ) ) )
       => ( condit3216346422009540699at_nat @ Ra2 @ X @ Y ) ) ) )).

% conditional_action.rel_mono_strong
thf(fact_971_conditional__action_Orel__mono__strong,axiom,(
    ! [R2: product_prod_nat_nat > produc859450856879609959at_nat > $o,X: condit1452497342716194761at_nat,Y: condit3449342828199732399at_nat,Ra2: product_prod_nat_nat > produc859450856879609959at_nat > $o] :
      ( ( condit8505942925542983873at_nat @ R2 @ X @ Y )
     => ( ! [Z4: product_prod_nat_nat,Yb: produc859450856879609959at_nat] :
            ( ( member8440522571783428010at_nat @ Z4 @ ( condit6429450166605921294at_nat @ X ) )
           => ( ( member8206827879206165904at_nat @ Yb @ ( condit5307000522679409652at_nat @ Y ) )
             => ( ( R2 @ Z4 @ Yb )
               => ( Ra2 @ Z4 @ Yb ) ) ) )
       => ( condit8505942925542983873at_nat @ Ra2 @ X @ Y ) ) ) )).

% conditional_action.rel_mono_strong
thf(fact_972_conditional__action_Orel__mono__strong,axiom,(
    ! [R2: produc859450856879609959at_nat > product_prod_nat_nat > $o,X: condit3449342828199732399at_nat,Y: condit1452497342716194761at_nat,Ra2: produc859450856879609959at_nat > product_prod_nat_nat > $o] :
      ( ( condit4869552352875249525at_nat @ R2 @ X @ Y )
     => ( ! [Z4: produc859450856879609959at_nat,Yb: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ Z4 @ ( condit5307000522679409652at_nat @ X ) )
           => ( ( member8440522571783428010at_nat @ Yb @ ( condit6429450166605921294at_nat @ Y ) )
             => ( ( R2 @ Z4 @ Yb )
               => ( Ra2 @ Z4 @ Yb ) ) ) )
       => ( condit4869552352875249525at_nat @ Ra2 @ X @ Y ) ) ) )).

% conditional_action.rel_mono_strong
thf(fact_973_conditional__action_Orel__mono__strong,axiom,(
    ! [R2: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,X: condit3449342828199732399at_nat,Y: condit3449342828199732399at_nat,Ra2: produc859450856879609959at_nat > produc859450856879609959at_nat > $o] :
      ( ( condit7775146776940579035at_nat @ R2 @ X @ Y )
     => ( ! [Z4: produc859450856879609959at_nat,Yb: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ Z4 @ ( condit5307000522679409652at_nat @ X ) )
           => ( ( member8206827879206165904at_nat @ Yb @ ( condit5307000522679409652at_nat @ Y ) )
             => ( ( R2 @ Z4 @ Yb )
               => ( Ra2 @ Z4 @ Yb ) ) ) )
       => ( condit7775146776940579035at_nat @ Ra2 @ X @ Y ) ) ) )).

% conditional_action.rel_mono_strong
thf(fact_974_conditional__action_Orel__refl__strong,axiom,(
    ! [X: condit1452497342716194761at_nat,Ra2: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [Z4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ Z4 @ ( condit6429450166605921294at_nat @ X ) )
         => ( Ra2 @ Z4 @ Z4 ) )
     => ( condit3216346422009540699at_nat @ Ra2 @ X @ X ) ) )).

% conditional_action.rel_refl_strong
thf(fact_975_conditional__action_Orel__refl__strong,axiom,(
    ! [X: condit3449342828199732399at_nat,Ra2: produc859450856879609959at_nat > produc859450856879609959at_nat > $o] :
      ( ! [Z4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ Z4 @ ( condit5307000522679409652at_nat @ X ) )
         => ( Ra2 @ Z4 @ Z4 ) )
     => ( condit7775146776940579035at_nat @ Ra2 @ X @ X ) ) )).

% conditional_action.rel_refl_strong
thf(fact_976_converse__iff,axiom,(
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( converse_nat_nat @ R ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R ) ) )).

% converse_iff
thf(fact_977_converse__iff,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( conver5251360641855610363at_nat @ R ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ A ) @ R ) ) )).

% converse_iff
thf(fact_978_converse__iff,axiom,(
    ! [A: produc8923325533196201883nteger,B: code_integer > option6357759511663192854e_term,R: set_Pr8056137968301705908nteger] :
      ( ( member4545690668652318835e_term @ ( produc2242505987757549526e_term @ A @ B ) @ ( conver2866147139040501132nteger @ R ) )
      = ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ B @ A ) @ R ) ) )).

% converse_iff
thf(fact_979_converse__iff,axiom,(
    ! [A: product_prod_int_int,B: int > option6357759511663192854e_term,R: set_Pr1872883991513573699nt_int] :
      ( ( member3093737243930835296e_term @ ( produc7516973260301520963e_term @ A @ B ) @ ( conver6302444324914690715nt_int @ R ) )
      = ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ B @ A ) @ R ) ) )).

% converse_iff
thf(fact_980_converse__iff,axiom,(
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,R: set_Pr5868096553338476434e_term] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ A @ B ) @ ( conver8194269161559375282e_term @ R ) )
      = ( member4545690668652318835e_term @ ( produc2242505987757549526e_term @ B @ A ) @ R ) ) )).

% converse_iff
thf(fact_981_converse__iff,axiom,(
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,R: set_Pr9020418513037773311e_term] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ ( conver290363505381979679e_term @ R ) )
      = ( member3093737243930835296e_term @ ( produc7516973260301520963e_term @ B @ A ) @ R ) ) )).

% converse_iff
thf(fact_982_converse__iff,axiom,(
    ! [A: produc7822875418678951345atural,B: produc4972180933644002618e_term > option6357759511663192854e_term,R: set_Pr4244404650639074861atural] :
      ( ( member1425327671947302374e_term @ ( produc3460696219379557065e_term @ A @ B ) @ ( conver4625362202874268037atural @ R ) )
      = ( member3245610415464309390atural @ ( produc3831813291587773865atural @ B @ A ) @ R ) ) )).

% converse_iff
thf(fact_983_converse__iff,axiom,(
    ! [A: produc8923325533196201883nteger,B: produc6241069584506657477e_term > option6357759511663192854e_term,R: set_Pr1281608226676607948nteger] :
      ( ( member3254845646535258459e_term @ ( produc1898379890548282302e_term @ A @ B ) @ ( conver173282527379661732nteger @ R ) )
      = ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ B @ A ) @ R ) ) )).

% converse_iff
thf(fact_984_converse__iff,axiom,(
    ! [A: product_prod_int_int,B: produc8551481072490612790e_term > option6357759511663192854e_term,R: set_Pr9222295170931077689nt_int] :
      ( ( member7154775377335690986e_term @ ( produc5219906170222623181e_term @ A @ B ) @ ( conver8152939771091638161nt_int @ R ) )
      = ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ B @ A ) @ R ) ) )).

% converse_iff
thf(fact_985_converse__iff,axiom,(
    ! [A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural,R: set_Pr7460367489947184517e_term] :
      ( ( member3245610415464309390atural @ ( produc3831813291587773865atural @ A @ B ) @ ( conver4254245130666051237e_term @ R ) )
      = ( member1425327671947302374e_term @ ( produc3460696219379557065e_term @ B @ A ) @ R ) ) )).

% converse_iff
thf(fact_986_Id__on__iff,axiom,(
    ! [X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ ( id_on_7917584626360399108at_nat @ A4 ) )
      = ( ( X = Y )
        & ( member8206827879206165904at_nat @ X @ A4 ) ) ) )).

% Id_on_iff
thf(fact_987_Id__on__iff,axiom,(
    ! [X: nat,Y: nat,A4: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( id_on_nat @ A4 ) )
      = ( ( X = Y )
        & ( member_nat @ X @ A4 ) ) ) )).

% Id_on_iff
thf(fact_988_Id__on__iff,axiom,(
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( id_on_2554058798563519774at_nat @ A4 ) )
      = ( ( X = Y )
        & ( member8440522571783428010at_nat @ X @ A4 ) ) ) )).

% Id_on_iff
thf(fact_989_Id__on__eqI,axiom,(
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( A = B )
     => ( ( member_nat @ A @ A4 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A4 ) ) ) ) )).

% Id_on_eqI
thf(fact_990_Id__on__eqI,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( A = B )
     => ( ( member8440522571783428010at_nat @ A @ A4 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( id_on_2554058798563519774at_nat @ A4 ) ) ) ) )).

% Id_on_eqI
thf(fact_991_Id__on__eqI,axiom,(
    ! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( A = B )
     => ( ( member8206827879206165904at_nat @ A @ A4 )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ ( id_on_7917584626360399108at_nat @ A4 ) ) ) ) )).

% Id_on_eqI
thf(fact_992_Id__onE,axiom,(
    ! [C2: produc349518998152878311at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ C2 @ ( id_on_7917584626360399108at_nat @ A4 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A4 )
           => ( C2
             != ( produc4662710985925991255at_nat @ X3 @ X3 ) ) ) ) )).

% Id_onE
thf(fact_993_Id__onE,axiom,(
    ! [C2: product_prod_nat_nat,A4: set_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( id_on_nat @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( C2
             != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) )).

% Id_onE
thf(fact_994_Id__onE,axiom,(
    ! [C2: produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ ( id_on_2554058798563519774at_nat @ A4 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A4 )
           => ( C2
             != ( produc6161850002892822231at_nat @ X3 @ X3 ) ) ) ) )).

% Id_onE
thf(fact_995_surj__fun__eq,axiom,(
    ! [F: produc8923597787767437993list_f > produc1811169103916964701list_f,X7: set_Pr8454954044441558665list_f,G1: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G22: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ( ( image_4621308865571145409list_f @ F @ X7 )
        = top_to3090985321854968643list_f )
     => ( ! [X3: produc8923597787767437993list_f] :
            ( ( member1992181931100238290list_f @ X3 @ X7 )
           => ( ( comp_P4767578765025054880list_f @ G1 @ F @ X3 )
              = ( comp_P4767578765025054880list_f @ G22 @ F @ X3 ) ) )
       => ( G1 = G22 ) ) ) )).

% surj_fun_eq
thf(fact_996_comp__surj,axiom,(
    ! [F: produc8923597787767437993list_f > produc1811169103916964701list_f,G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ( ( image_4621308865571145409list_f @ F @ top_to1815313666553408473list_f )
        = top_to3090985321854968643list_f )
     => ( ( ( image_6684545758539136421_a_b_c @ G2 @ top_to3090985321854968643list_f )
          = top_to66987640601337145_a_b_c )
       => ( ( image_6608088440629302181_a_b_c @ ( comp_P4767578765025054880list_f @ G2 @ F ) @ top_to1815313666553408473list_f )
          = top_to66987640601337145_a_b_c ) ) ) )).

% comp_surj
thf(fact_997_fun_Oset__map,axiom,(
    ! [F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,V: produc8923597787767437993list_f > produc1811169103916964701list_f] :
      ( ( image_6608088440629302181_a_b_c @ ( comp_P4767578765025054880list_f @ F @ V ) @ top_to1815313666553408473list_f )
      = ( image_6684545758539136421_a_b_c @ F @ ( image_4621308865571145409list_f @ V @ top_to1815313666553408473list_f ) ) ) )).

% fun.set_map
thf(fact_998_IdD,axiom,(
    ! [A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ id_nat2 )
     => ( A = B ) ) )).

% IdD
thf(fact_999_IdD,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ id_Pro2258643101195443293at_nat )
     => ( A = B ) ) )).

% IdD
thf(fact_1000_prod__set__simps_I2_J,axiom,(
    ! [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( basic_1149804855828023554nteger @ ( produc6137756002093451184nteger @ X @ Y ) )
      = ( insert4913895101485356395nteger @ Y @ bot_bo4276436098303576167nteger ) ) )).

% prod_set_simps(2)
thf(fact_1001_prod__set__simps_I2_J,axiom,(
    ! [X: produc4972180933644002618e_term > option6357759511663192854e_term,Y: produc7822875418678951345atural] :
      ( ( basic_3858280913897308155atural @ ( produc3831813291587773865atural @ X @ Y ) )
      = ( insert3813444986968105857atural @ Y @ bot_bo7028766096215927677atural ) ) )).

% prod_set_simps(2)
thf(fact_1002_prod__set__simps_I2_J,axiom,(
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( basic_8629573275257477658nteger @ ( produc8603105652947943368nteger @ X @ Y ) )
      = ( insert4913895101485356395nteger @ Y @ bot_bo4276436098303576167nteger ) ) )).

% prod_set_simps(2)
thf(fact_1003_prod__set__simps_I2_J,axiom,(
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( basic_7234646000605779463nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
      = ( insert5033312907999012233nt_int @ Y @ bot_bo1796632182523588997nt_int ) ) )).

% prod_set_simps(2)
thf(fact_1004_prod__set__simps_I2_J,axiom,(
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( basic_3943112340473720337nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
      = ( insert5033312907999012233nt_int @ Y @ bot_bo1796632182523588997nt_int ) ) )).

% prod_set_simps(2)
thf(fact_1005_fun_Omap__cong,axiom,(
    ! [X: produc8923597787767437993list_f > produc1811169103916964701list_f,Ya2: produc8923597787767437993list_f > produc1811169103916964701list_f,F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ( X = Ya2 )
     => ( ! [Z4: produc1811169103916964701list_f] :
            ( ( member6304641671555014644list_f @ Z4 @ ( image_4621308865571145409list_f @ Ya2 @ top_to1815313666553408473list_f ) )
           => ( ( F @ Z4 )
              = ( G2 @ Z4 ) ) )
       => ( ( comp_P4767578765025054880list_f @ F @ X )
          = ( comp_P4767578765025054880list_f @ G2 @ Ya2 ) ) ) ) )).

% fun.map_cong
thf(fact_1006_fun_Omap__cong0,axiom,(
    ! [X: produc8923597787767437993list_f > produc1811169103916964701list_f,F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,G2: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ! [Z4: produc1811169103916964701list_f] :
          ( ( member6304641671555014644list_f @ Z4 @ ( image_4621308865571145409list_f @ X @ top_to1815313666553408473list_f ) )
         => ( ( F @ Z4 )
            = ( G2 @ Z4 ) ) )
     => ( ( comp_P4767578765025054880list_f @ F @ X )
        = ( comp_P4767578765025054880list_f @ G2 @ X ) ) ) )).

% fun.map_cong0
thf(fact_1007_fun_Oinj__map__strong,axiom,(
    ! [X: produc8923597787767437993list_f > produc1811169103916964701list_f,Xa: produc8923597787767437993list_f > produc1811169103916964701list_f,F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c,Fa: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ! [Z4: produc1811169103916964701list_f,Za2: produc1811169103916964701list_f] :
          ( ( member6304641671555014644list_f @ Z4 @ ( image_4621308865571145409list_f @ X @ top_to1815313666553408473list_f ) )
         => ( ( member6304641671555014644list_f @ Za2 @ ( image_4621308865571145409list_f @ Xa @ top_to1815313666553408473list_f ) )
           => ( ( ( F @ Z4 )
                = ( Fa @ Za2 ) )
             => ( Z4 = Za2 ) ) ) )
     => ( ( ( comp_P4767578765025054880list_f @ F @ X )
          = ( comp_P4767578765025054880list_f @ Fa @ Xa ) )
       => ( X = Xa ) ) ) )).

% fun.inj_map_strong
thf(fact_1008_prod__set__simps_I1_J,axiom,(
    ! [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( basic_4672511774439847616nteger @ ( produc6137756002093451184nteger @ X @ Y ) )
      = ( insert4303373725994308034e_term @ X @ bot_bo1530185625139755228e_term ) ) )).

% prod_set_simps(1)
thf(fact_1009_prod__set__simps_I1_J,axiom,(
    ! [X: produc4972180933644002618e_term > option6357759511663192854e_term,Y: produc7822875418678951345atural] :
      ( ( basic_2650174300292291769atural @ ( produc3831813291587773865atural @ X @ Y ) )
      = ( insert4479822593746981279e_term @ X @ bot_bo4014960707591977401e_term ) ) )).

% prod_set_simps(1)
thf(fact_1010_prod__set__simps_I1_J,axiom,(
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( basic_7421466661652461272nteger @ ( produc8603105652947943368nteger @ X @ Y ) )
      = ( insert5901432530851156906e_term @ X @ bot_bo8828404948115383876e_term ) ) )).

% prod_set_simps(1)
thf(fact_1011_prod__set__simps_I1_J,axiom,(
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( basic_7218179853841504965nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
      = ( insert2341259790341569179e_term @ X @ bot_bo974140798557118645e_term ) ) )).

% prod_set_simps(1)
thf(fact_1012_prod__set__simps_I1_J,axiom,(
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( basic_3459673265925145551nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
      = ( insert5768611539565238801e_term @ X @ bot_bo4665192896540886443e_term ) ) )).

% prod_set_simps(1)
thf(fact_1013_fun_Oin__rel,axiom,(
    ! [R2: list_n1973683603564934665_a_b_c > produc8853065311482726181list_f > $o,A: produc8923597787767437993list_f > list_n1973683603564934665_a_b_c,B: produc8923597787767437993list_f > produc8853065311482726181list_f] :
      ( ( bNF_re6046208945862828881list_f
        @ ^ [Y5: produc8923597787767437993list_f,Z2: produc8923597787767437993list_f] : ( Y5 = Z2 )
        @ R2
        @ A
        @ B )
      = ( ? [Z3: produc8923597787767437993list_f > produc1811169103916964701list_f] :
            ( ( member7024593645017551904list_f @ Z3
              @ ( collec6052127551248529762list_f
                @ ^ [X4: produc8923597787767437993list_f > produc1811169103916964701list_f] :
                    ( ord_le2653721700351411443list_f @ ( image_4621308865571145409list_f @ X4 @ top_to1815313666553408473list_f ) @ ( collec2591973510557893426list_f @ ( produc2111220630807053972st_f_o @ R2 ) ) ) ) )
            & ( ( comp_P4767578765025054880list_f @ produc3535155421234910955list_f @ Z3 )
              = A )
            & ( ( comp_P3270879886602560258list_f @ produc7287154213363835177list_f @ Z3 )
              = B ) ) ) ) )).

% fun.in_rel
thf(fact_1014_type__copy__map__cong0,axiom,(
    ! [M: produc8923597787767437993list_f > produc1811169103916964701list_f,G2: produc8923597787767437993list_f > produc8923597787767437993list_f,X: produc8923597787767437993list_f,N: produc1811169103916964701list_f > produc1811169103916964701list_f,H: produc8923597787767437993list_f > produc1811169103916964701list_f,F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ( ( M @ ( G2 @ X ) )
        = ( N @ ( H @ X ) ) )
     => ( ( comp_P1644921750638218854list_f @ ( comp_P4767578765025054880list_f @ F @ M ) @ G2 @ X )
        = ( comp_P4767578765025054880list_f @ ( comp_P5271726855292207526list_f @ F @ N ) @ H @ X ) ) ) )).

% type_copy_map_cong0
thf(fact_1015_type__copy__map__cong0,axiom,(
    ! [M: produc1811169103916964701list_f > produc1811169103916964701list_f,G2: produc8923597787767437993list_f > produc1811169103916964701list_f,X: produc8923597787767437993list_f,N: produc8923597787767437993list_f > produc1811169103916964701list_f,H: produc8923597787767437993list_f > produc8923597787767437993list_f,F: produc1811169103916964701list_f > list_n1973683603564934665_a_b_c] :
      ( ( ( M @ ( G2 @ X ) )
        = ( N @ ( H @ X ) ) )
     => ( ( comp_P4767578765025054880list_f @ ( comp_P5271726855292207526list_f @ F @ M ) @ G2 @ X )
        = ( comp_P1644921750638218854list_f @ ( comp_P4767578765025054880list_f @ F @ N ) @ H @ X ) ) ) )).

% type_copy_map_cong0
thf(fact_1016_DiffI,axiom,(
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ A4 )
     => ( ~ ( member8440522571783428010at_nat @ C2 @ B5 )
       => ( member8440522571783428010at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ) ) )).

% DiffI
thf(fact_1017_DiffI,axiom,(
    ! [C2: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ A4 )
     => ( ~ ( member8206827879206165904at_nat @ C2 @ B5 )
       => ( member8206827879206165904at_nat @ C2 @ ( minus_8321449233255521966at_nat @ A4 @ B5 ) ) ) ) )).

% DiffI
thf(fact_1018_DiffE,axiom,(
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
     => ~ ( ( member8440522571783428010at_nat @ C2 @ A4 )
         => ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) )).

% DiffE
thf(fact_1019_DiffE,axiom,(
    ! [C2: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ ( minus_8321449233255521966at_nat @ A4 @ B5 ) )
     => ~ ( ( member8206827879206165904at_nat @ C2 @ A4 )
         => ( member8206827879206165904at_nat @ C2 @ B5 ) ) ) )).

% DiffE
thf(fact_1020_relcompE,axiom,(
    ! [Xz: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Xz @ ( relcomp_nat_nat_nat @ R @ S ) )
     => ~ ! [X3: nat,Y3: nat,Z4: nat] :
            ( ( Xz
              = ( product_Pair_nat_nat @ X3 @ Z4 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z4 ) @ S ) ) ) ) )).

% relcompE
thf(fact_1021_relcompE,axiom,(
    ! [Xz: produc859450856879609959at_nat,R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ Xz @ ( relcom7295539661566034944at_nat @ R @ S ) )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z4: product_prod_nat_nat] :
            ( ( Xz
              = ( produc6161850002892822231at_nat @ X3 @ Z4 ) )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R )
             => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z4 ) @ S ) ) ) ) )).

% relcompE
thf(fact_1022_relcompE,axiom,(
    ! [Xz: produc8763457246119570046nteger,R: set_Pr8056137968301705908nteger,S: set_Pr5052581263052047815nteger] :
      ( ( member3068662437193594005nteger @ Xz @ ( relcom3229198627874052691nteger @ R @ S ) )
     => ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Z4: produc8923325533196201883nteger] :
            ( ( Xz
              = ( produc6137756002093451184nteger @ X3 @ Z4 ) )
           => ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ R )
             => ~ ( member335077259223437328nteger @ ( produc4911887526399440471nteger @ Y3 @ Z4 ) @ S ) ) ) ) )).

% relcompE
thf(fact_1023_relcompE,axiom,(
    ! [Xz: produc7773217078559923341nt_int,R: set_Pr1872883991513573699nt_int,S: set_Pr2560585780119916871nt_int] :
      ( ( member7034335876925520548nt_int @ Xz @ ( relcom825500880559691426nt_int @ R @ S ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int,Z4: product_prod_int_int] :
            ( ( Xz
              = ( produc4305682042979456191nt_int @ X3 @ Z4 ) )
           => ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R )
             => ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y3 @ Z4 ) @ S ) ) ) ) )).

% relcompE
thf(fact_1024_relcompE,axiom,(
    ! [Xz: produc8763457246119570046nteger,R: set_Pr6140516264755836645e_term,S: set_Pr8056137968301705908nteger] :
      ( ( member3068662437193594005nteger @ Xz @ ( relcom3624616710273232564nteger @ R @ S ) )
     => ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: code_integer > option6357759511663192854e_term,Z4: produc8923325533196201883nteger] :
            ( ( Xz
              = ( produc6137756002093451184nteger @ X3 @ Z4 ) )
           => ( ( member1138991337916034606e_term @ ( produc2189037350439096765e_term @ X3 @ Y3 ) @ R )
             => ~ ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ Y3 @ Z4 ) @ S ) ) ) ) )).

% relcompE
thf(fact_1025_relcompE,axiom,(
    ! [Xz: produc989692990947075319atural,R: set_Pr4244404650639074861atural,S: set_Pr250711821884878663atural] :
      ( ( member3245610415464309390atural @ Xz @ ( relcom3245834559762200496atural @ R @ S ) )
     => ~ ! [X3: produc4972180933644002618e_term > option6357759511663192854e_term,Y3: produc7822875418678951345atural,Z4: produc7822875418678951345atural] :
            ( ( Xz
              = ( produc3831813291587773865atural @ X3 @ Z4 ) )
           => ( ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X3 @ Y3 ) @ R )
             => ~ ( member3756964795865971088atural @ ( produc4480994950612372183atural @ Y3 @ Z4 ) @ S ) ) ) ) )).

% relcompE
thf(fact_1026_relcompE,axiom,(
    ! [Xz: produc1908205239877642774nteger,R: set_Pr1281608226676607948nteger,S: set_Pr5052581263052047815nteger] :
      ( ( member4164122664394876845nteger @ Xz @ ( relcom5385145687114743995nteger @ R @ S ) )
     => ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Z4: produc8923325533196201883nteger] :
            ( ( Xz
              = ( produc8603105652947943368nteger @ X3 @ Z4 ) )
           => ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R )
             => ~ ( member335077259223437328nteger @ ( produc4911887526399440471nteger @ Y3 @ Z4 ) @ S ) ) ) ) )).

% relcompE
thf(fact_1027_relcompE,axiom,(
    ! [Xz: produc2285326912895808259nt_int,R: set_Pr9222295170931077689nt_int,S: set_Pr2560585780119916871nt_int] :
      ( ( member7618704894036264090nt_int @ Xz @ ( relcom3053196381443395756nt_int @ R @ S ) )
     => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int,Z4: product_prod_int_int] :
            ( ( Xz
              = ( produc5700946648718959541nt_int @ X3 @ Z4 ) )
           => ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R )
             => ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y3 @ Z4 ) @ S ) ) ) ) )).

% relcompE
thf(fact_1028_relcompE,axiom,(
    ! [Xz: produc7773217078559923341nt_int,R: set_Pr7140365069852192003e_term,S: set_Pr1872883991513573699nt_int] :
      ( ( member7034335876925520548nt_int @ Xz @ ( relcom393131893966929076nt_int @ R @ S ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: int > option6357759511663192854e_term,Z4: product_prod_int_int] :
            ( ( Xz
              = ( produc4305682042979456191nt_int @ X3 @ Z4 ) )
           => ( ( member6538663773838357324e_term @ ( produc4510790414113558107e_term @ X3 @ Y3 ) @ R )
             => ~ ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ Y3 @ Z4 ) @ S ) ) ) ) )).

% relcompE
thf(fact_1029_relcompE,axiom,(
    ! [Xz: produc8763457246119570046nteger,R: set_Pr7612807928715620173e_term,S: set_Pr1281608226676607948nteger] :
      ( ( member3068662437193594005nteger @ Xz @ ( relcom4524264481439647948nteger @ R @ S ) )
     => ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc6241069584506657477e_term > option6357759511663192854e_term,Z4: produc8923325533196201883nteger] :
            ( ( Xz
              = ( produc6137756002093451184nteger @ X3 @ Z4 ) )
           => ( ( member6074385462936892566e_term @ ( produc4288197420388063525e_term @ X3 @ Y3 ) @ R )
             => ~ ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ Y3 @ Z4 ) @ S ) ) ) ) )).

% relcompE
thf(fact_1030_relcomp_OrelcompI,axiom,(
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C2: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( relcomp_nat_nat_nat @ R @ S ) ) ) ) )).

% relcomp.relcompI
thf(fact_1031_relcomp_OrelcompI,axiom,(
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,C2: product_prod_nat_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C2 ) @ S )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C2 ) @ ( relcom7295539661566034944at_nat @ R @ S ) ) ) ) )).

% relcomp.relcompI
thf(fact_1032_relcomp_OrelcompI,axiom,(
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,R: set_Pr8056137968301705908nteger,C2: produc8923325533196201883nteger,S: set_Pr5052581263052047815nteger] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ A @ B ) @ R )
     => ( ( member335077259223437328nteger @ ( produc4911887526399440471nteger @ B @ C2 ) @ S )
       => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ A @ C2 ) @ ( relcom3229198627874052691nteger @ R @ S ) ) ) ) )).

% relcomp.relcompI
thf(fact_1033_relcomp_OrelcompI,axiom,(
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,R: set_Pr1872883991513573699nt_int,C2: product_prod_int_int,S: set_Pr2560585780119916871nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ R )
     => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B @ C2 ) @ S )
       => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ C2 ) @ ( relcom825500880559691426nt_int @ R @ S ) ) ) ) )).

% relcomp.relcompI
thf(fact_1034_relcomp_OrelcompI,axiom,(
    ! [A: code_integer > option6357759511663192854e_term,B: code_integer > option6357759511663192854e_term,R: set_Pr6140516264755836645e_term,C2: produc8923325533196201883nteger,S: set_Pr8056137968301705908nteger] :
      ( ( member1138991337916034606e_term @ ( produc2189037350439096765e_term @ A @ B ) @ R )
     => ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ B @ C2 ) @ S )
       => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ A @ C2 ) @ ( relcom3624616710273232564nteger @ R @ S ) ) ) ) )).

% relcomp.relcompI
thf(fact_1035_relcomp_OrelcompI,axiom,(
    ! [A: int > option6357759511663192854e_term,B: int > option6357759511663192854e_term,R: set_Pr7140365069852192003e_term,C2: product_prod_int_int,S: set_Pr1872883991513573699nt_int] :
      ( ( member6538663773838357324e_term @ ( produc4510790414113558107e_term @ A @ B ) @ R )
     => ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ B @ C2 ) @ S )
       => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ C2 ) @ ( relcom393131893966929076nt_int @ R @ S ) ) ) ) )).

% relcomp.relcompI
thf(fact_1036_relcomp_OrelcompI,axiom,(
    ! [A: produc4972180933644002618e_term > option6357759511663192854e_term,B: produc7822875418678951345atural,R: set_Pr4244404650639074861atural,C2: produc7822875418678951345atural,S: set_Pr250711821884878663atural] :
      ( ( member3245610415464309390atural @ ( produc3831813291587773865atural @ A @ B ) @ R )
     => ( ( member3756964795865971088atural @ ( produc4480994950612372183atural @ B @ C2 ) @ S )
       => ( member3245610415464309390atural @ ( produc3831813291587773865atural @ A @ C2 ) @ ( relcom3245834559762200496atural @ R @ S ) ) ) ) )).

% relcomp.relcompI
thf(fact_1037_relcomp_OrelcompI,axiom,(
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,R: set_Pr1281608226676607948nteger,C2: produc8923325533196201883nteger,S: set_Pr5052581263052047815nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ R )
     => ( ( member335077259223437328nteger @ ( produc4911887526399440471nteger @ B @ C2 ) @ S )
       => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ C2 ) @ ( relcom5385145687114743995nteger @ R @ S ) ) ) ) )).

% relcomp.relcompI
thf(fact_1038_relcomp_OrelcompI,axiom,(
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,R: set_Pr9222295170931077689nt_int,C2: product_prod_int_int,S: set_Pr2560585780119916871nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ R )
     => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B @ C2 ) @ S )
       => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ C2 ) @ ( relcom3053196381443395756nt_int @ R @ S ) ) ) ) )).

% relcomp.relcompI
thf(fact_1039_relcomp_OrelcompI,axiom,(
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: code_integer > option6357759511663192854e_term,R: set_Pr2125308852348728269e_term,C2: produc8923325533196201883nteger,S: set_Pr8056137968301705908nteger] :
      ( ( member5754429765941377302e_term @ ( produc4391412222137964453e_term @ A @ B ) @ R )
     => ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ B @ C2 ) @ S )
       => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ C2 ) @ ( relcom4075915465181424204nteger @ R @ S ) ) ) ) )).

% relcomp.relcompI
thf(fact_1040_case__swap,axiom,(
    ! [F: code_natural > code_natural > produc6756470014200349799atural,P2: produc7822875418678951345atural] :
      ( ( produc8282080750456430313atural
        @ ^ [Y6: code_natural,X4: code_natural] :
            ( F @ X4 @ Y6 )
        @ ( produc5279793246624815241atural @ P2 ) )
      = ( produc8282080750456430313atural @ F @ P2 ) ) )).

% case_swap
thf(fact_1041_case__prod__map__prod,axiom,(
    ! [H: code_natural > code_natural > produc6756470014200349799atural,F: code_natural > code_natural,G2: code_natural > code_natural,X: produc7822875418678951345atural] :
      ( ( produc8282080750456430313atural @ H @ ( produc8950045413935083162atural @ F @ G2 @ X ) )
      = ( produc8282080750456430313atural
        @ ^ [L3: code_natural,R3: code_natural] :
            ( H @ ( F @ L3 ) @ ( G2 @ R3 ) )
        @ X ) ) )).

% case_prod_map_prod
thf(fact_1042_list_Oset_I1_J,axiom,
    ( ( set_typerep2 @ nil_typerep )
    = bot_bot_set_typerep )).

% list.set(1)
thf(fact_1043_inj__on__convol__ident,axiom,(
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger,X7: set_Co9149898834107579976e_term] :
      ( inj_on8181760918054529539nteger
      @ ^ [X4: code_integer > option6357759511663192854e_term] :
          ( produc6137756002093451184nteger @ X4 @ ( F @ X4 ) )
      @ X7 ) )).

% inj_on_convol_ident
thf(fact_1044_inj__on__convol__ident,axiom,(
    ! [F: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural,X7: set_Pr2791530082921190693e_term] :
      ( inj_on8316333876640852697atural
      @ ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term] :
          ( produc3831813291587773865atural @ X4 @ ( F @ X4 ) )
      @ X7 ) )).

% inj_on_convol_ident
thf(fact_1045_inj__on__convol__ident,axiom,(
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger,X7: set_Pr7604974323444597168e_term] :
      ( inj_on1754559267590340355nteger
      @ ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term] :
          ( produc8603105652947943368nteger @ X4 @ ( F @ X4 ) )
      @ X7 ) )).

% inj_on_convol_ident
thf(fact_1046_inj__on__convol__ident,axiom,(
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int,X7: set_Pr3642885161833720865e_term] :
      ( inj_on4793825870093415905nt_int
      @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term] :
          ( produc5700946648718959541nt_int @ X4 @ ( F @ X4 ) )
      @ X7 ) )).

% inj_on_convol_ident
thf(fact_1047_inj__on__convol__ident,axiom,(
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int,X7: set_in3461395444621081367e_term] :
      ( inj_on5552665323793282529nt_int
      @ ^ [X4: int > option6357759511663192854e_term] :
          ( produc4305682042979456191nt_int @ X4 @ ( F @ X4 ) )
      @ X7 ) )).

% inj_on_convol_ident
thf(fact_1048_inf_Ocobounded1,axiom,(
    ! [A: nat,B: nat] :
      ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) )).

% inf.cobounded1
thf(fact_1049_prod_Ocase__distrib,axiom,(
    ! [H: produc6756470014200349799atural > produc6756470014200349799atural,F: code_natural > code_natural > produc6756470014200349799atural,Prod: produc7822875418678951345atural] :
      ( ( H @ ( produc8282080750456430313atural @ F @ Prod ) )
      = ( produc8282080750456430313atural
        @ ^ [X12: code_natural,X23: code_natural] :
            ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) )).

% prod.case_distrib
thf(fact_1050_Compr__image__eq,axiom,(
    ! [F: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( image_5168914502847457605at_nat @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_5168914502847457605at_nat @ F
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) )).

% Compr_image_eq
thf(fact_1051_Compr__image__eq,axiom,(
    ! [F: produc859450856879609959at_nat > product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( image_6706873702294942047at_nat @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_6706873702294942047at_nat @ F
        @ ( collec7088162979684241874at_nat
          @ ^ [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) )).

% Compr_image_eq
thf(fact_1052_Compr__image__eq,axiom,(
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ ( image_1119892238107900587at_nat @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_1119892238107900587at_nat @ F
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) )).

% Compr_image_eq
thf(fact_1053_Compr__image__eq,axiom,(
    ! [F: produc859450856879609959at_nat > produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ ( image_6321802852832981957at_nat @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_6321802852832981957at_nat @ F
        @ ( collec7088162979684241874at_nat
          @ ^ [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) )).

% Compr_image_eq
thf(fact_1054_interrupt__action_Oset__intros_I2_J,axiom,(
    ! [X2: product_prod_nat_nat] :
      ( member8440522571783428010at_nat @ X2 @ ( interr3917558629499790676at_nat @ ( interr6945818126708391058at_nat @ X2 ) ) ) )).

% interrupt_action.set_intros(2)
thf(fact_1055_interrupt__action_Oset__intros_I2_J,axiom,(
    ! [X2: produc859450856879609959at_nat] :
      ( member8206827879206165904at_nat @ X2 @ ( interr4604075546494911546at_nat @ ( interr7700225897186592376at_nat @ X2 ) ) ) )).

% interrupt_action.set_intros(2)
thf(fact_1056_interrupt__action_Oset__intros_I3_J,axiom,(
    ! [X32: product_prod_nat_nat] :
      ( member8440522571783428010at_nat @ X32 @ ( interr3917558629499790676at_nat @ ( interr4077302776001012533at_nat @ X32 ) ) ) )).

% interrupt_action.set_intros(3)
thf(fact_1057_interrupt__action_Oset__intros_I3_J,axiom,(
    ! [X32: produc859450856879609959at_nat] :
      ( member8206827879206165904at_nat @ X32 @ ( interr4604075546494911546at_nat @ ( interr1275353696496618651at_nat @ X32 ) ) ) )).

% interrupt_action.set_intros(3)
thf(fact_1058_transfer__start,axiom,(
    ! [P: $o] :
      ( P
     => ~ ( rel_o_o
          @ ^ [Y5: $o,Z2: $o] : ( Y5 = Z2 )
          @ P
          @ $false ) ) )).

% transfer_start
thf(fact_1059_add_Oright__neutral,axiom,(
    ! [A: code_natural] :
      ( ( plus_p4538020629002901425atural @ A @ zero_z2226904508553997617atural )
      = A ) )).

% add.right_neutral
thf(fact_1060_add_Oright__neutral,axiom,(
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) )).

% add.right_neutral
thf(fact_1061_add_Oright__neutral,axiom,(
    ! [A: literal] :
      ( ( plus_plus_literal @ A @ zero_zero_literal )
      = A ) )).

% add.right_neutral
thf(fact_1062_subset__CollectI,axiom,(
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Q: product_prod_nat_nat > $o,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ B5 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le3146513528884898305at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ B5 )
                & ( Q @ X4 ) ) )
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) ) ) ) )).

% subset_CollectI
thf(fact_1063_subset__CollectI,axiom,(
    ! [B5: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,Q: produc859450856879609959at_nat > $o,P: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B5 @ A4 )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ B5 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le3000389064537975527at_nat
          @ ( collec7088162979684241874at_nat
            @ ^ [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ B5 )
                & ( Q @ X4 ) ) )
          @ ( collec7088162979684241874at_nat
            @ ^ [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) ) ) ) )).

% subset_CollectI
thf(fact_1064_neq0__conv,axiom,(
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) )).

% neq0_conv
thf(fact_1065_subset__Collect__iff,axiom,(
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ B5
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ B5 )
             => ( P @ X4 ) ) ) ) ) )).

% subset_Collect_iff
thf(fact_1066_subset__Collect__iff,axiom,(
    ! [B5: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B5 @ A4 )
     => ( ( ord_le3000389064537975527at_nat @ B5
          @ ( collec7088162979684241874at_nat
            @ ^ [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ B5 )
             => ( P @ X4 ) ) ) ) ) )).

% subset_Collect_iff
thf(fact_1067_prop__restrict,axiom,(
    ! [X: product_prod_nat_nat,Z6: set_Pr1261947904930325089at_nat,X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ Z6 )
     => ( ( ord_le3146513528884898305at_nat @ Z6
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ X7 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) )).

% prop_restrict
thf(fact_1068_prop__restrict,axiom,(
    ! [X: produc859450856879609959at_nat,Z6: set_Pr8693737435421807431at_nat,X7: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X @ Z6 )
     => ( ( ord_le3000389064537975527at_nat @ Z6
          @ ( collec7088162979684241874at_nat
            @ ^ [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ X7 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) )).

% prop_restrict
thf(fact_1069_Collect__restrict,axiom,(
    ! [X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ X7 )
            & ( P @ X4 ) ) )
      @ X7 ) )).

% Collect_restrict
thf(fact_1070_Collect__restrict,axiom,(
    ! [X7: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ X7 )
            & ( P @ X4 ) ) )
      @ X7 ) )).

% Collect_restrict
thf(fact_1071_sup__set__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( sup_su798857527126471912_nat_o
            @ ^ [X4: product_prod_nat_nat] :
                ( member8440522571783428010at_nat @ X4 @ A6 )
            @ ^ [X4: product_prod_nat_nat] :
                ( member8440522571783428010at_nat @ X4 @ B7 ) ) ) ) )).

% sup_set_def
thf(fact_1072_sup__set__def,axiom,
    ( sup_su718114333110466843at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B7: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ( sup_su8986005896011022210_nat_o
            @ ^ [X4: produc859450856879609959at_nat] :
                ( member8206827879206165904at_nat @ X4 @ A6 )
            @ ^ [X4: produc859450856879609959at_nat] :
                ( member8206827879206165904at_nat @ X4 @ B7 ) ) ) ) )).

% sup_set_def
thf(fact_1073_Un__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A6 )
              | ( member8440522571783428010at_nat @ X4 @ B7 ) ) ) ) )).

% Un_def
thf(fact_1074_Un__def,axiom,
    ( sup_su718114333110466843at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B7: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ^ [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ A6 )
              | ( member8206827879206165904at_nat @ X4 @ B7 ) ) ) ) )).

% Un_def
thf(fact_1075_split__seed__def,axiom,
    ( split_seed
    = ( ^ [S3: produc7822875418678951345atural] :
          ( produc8282080750456430313atural
          @ ^ [V2: code_natural,W: code_natural] :
              ( produc8282080750456430313atural
              @ ^ [V4: code_natural,W3: code_natural] :
                  ( produc4480994950612372183atural @ ( produc3574140220909816553atural @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V2 ) @ W3 ) @ ( produc3574140220909816553atural @ V4 @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W ) ) )
              @ ( produc6591912806276919810atural @ ( next @ S3 ) ) )
          @ S3 ) ) )).

% split_seed_def
thf(fact_1076_sup__Un__eq,axiom,(
    ! [R2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su798857527126471912_nat_o
        @ ^ [X4: product_prod_nat_nat] :
            ( member8440522571783428010at_nat @ X4 @ R2 )
        @ ^ [X4: product_prod_nat_nat] :
            ( member8440522571783428010at_nat @ X4 @ S4 ) )
      = ( ^ [X4: product_prod_nat_nat] :
            ( member8440522571783428010at_nat @ X4 @ ( sup_su6327502436637775413at_nat @ R2 @ S4 ) ) ) ) )).

% sup_Un_eq
thf(fact_1077_sup__Un__eq,axiom,(
    ! [R2: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( sup_su8986005896011022210_nat_o
        @ ^ [X4: produc859450856879609959at_nat] :
            ( member8206827879206165904at_nat @ X4 @ R2 )
        @ ^ [X4: produc859450856879609959at_nat] :
            ( member8206827879206165904at_nat @ X4 @ S4 ) )
      = ( ^ [X4: produc859450856879609959at_nat] :
            ( member8206827879206165904at_nat @ X4 @ ( sup_su718114333110466843at_nat @ R2 @ S4 ) ) ) ) )).

% sup_Un_eq
thf(fact_1078_sup__Un__eq2,axiom,(
    ! [R2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X4: nat,Y6: nat] :
            ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: nat,Y6: nat] :
            ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y6 ) @ S4 ) )
      = ( ^ [X4: nat,Y6: nat] :
            ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y6 ) @ ( sup_su6327502436637775413at_nat @ R2 @ S4 ) ) ) ) )).

% sup_Un_eq2
thf(fact_1079_sup__Un__eq2,axiom,(
    ! [R2: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( sup_su362511073950362882_nat_o
        @ ^ [X4: product_prod_nat_nat,Y6: product_prod_nat_nat] :
            ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: product_prod_nat_nat,Y6: product_prod_nat_nat] :
            ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y6 ) @ S4 ) )
      = ( ^ [X4: product_prod_nat_nat,Y6: product_prod_nat_nat] :
            ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y6 ) @ ( sup_su718114333110466843at_nat @ R2 @ S4 ) ) ) ) )).

% sup_Un_eq2
thf(fact_1080_sup__Un__eq2,axiom,(
    ! [R2: set_Pr8056137968301705908nteger,S4: set_Pr8056137968301705908nteger] :
      ( ( sup_su249214085273793023eger_o
        @ ^ [X4: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
            ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
            ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y6 ) @ S4 ) )
      = ( ^ [X4: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
            ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y6 ) @ ( sup_su4607367565216292320nteger @ R2 @ S4 ) ) ) ) )).

% sup_Un_eq2
thf(fact_1081_sup__Un__eq2,axiom,(
    ! [R2: set_Pr4244404650639074861atural,S4: set_Pr4244404650639074861atural] :
      ( ( sup_su3291028051651642566ural_o
        @ ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term,Y6: produc7822875418678951345atural] :
            ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term,Y6: produc7822875418678951345atural] :
            ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X4 @ Y6 ) @ S4 ) )
      = ( ^ [X4: produc4972180933644002618e_term > option6357759511663192854e_term,Y6: produc7822875418678951345atural] :
            ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X4 @ Y6 ) @ ( sup_su7554080439416909657atural @ R2 @ S4 ) ) ) ) )).

% sup_Un_eq2
thf(fact_1082_sup__Un__eq2,axiom,(
    ! [R2: set_Pr1281608226676607948nteger,S4: set_Pr1281608226676607948nteger] :
      ( ( sup_su234547053653886311eger_o
        @ ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
            ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
            ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y6 ) @ S4 ) )
      = ( ^ [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
            ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y6 ) @ ( sup_su4591284015454442744nteger @ R2 @ S4 ) ) ) ) )).

% sup_Un_eq2
thf(fact_1083_sup__Un__eq2,axiom,(
    ! [R2: set_Pr9222295170931077689nt_int,S4: set_Pr9222295170931077689nt_int] :
      ( ( sup_su4182031696650224058_int_o
        @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
            ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
            ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y6 ) @ S4 ) )
      = ( ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
            ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y6 ) @ ( sup_su3382966977382714213nt_int @ R2 @ S4 ) ) ) ) )).

% sup_Un_eq2
thf(fact_1084_sup__Un__eq2,axiom,(
    ! [R2: set_Pr1872883991513573699nt_int,S4: set_Pr1872883991513573699nt_int] :
      ( ( sup_su600977994968626096_int_o
        @ ^ [X4: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
            ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
            ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y6 ) @ S4 ) )
      = ( ^ [X4: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
            ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y6 ) @ ( sup_su3298353300217089135nt_int @ R2 @ S4 ) ) ) ) )).

% sup_Un_eq2
thf(fact_1085_ImageE,axiom,(
    ! [B: nat,R: set_Pr1261947904930325089at_nat,A4: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ R @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ R )
           => ~ ( member_nat @ X3 @ A4 ) ) ) )).

% ImageE
thf(fact_1086_ImageE,axiom,(
    ! [B: product_prod_nat_nat,R: set_Pr846279711151727201at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_4802155899172601389at_nat @ R @ A4 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ X3 @ B ) @ R )
           => ~ ( member8206827879206165904at_nat @ X3 @ A4 ) ) ) )).

% ImageE
thf(fact_1087_ImageE,axiom,(
    ! [B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_2484338115069272851at_nat @ R @ A4 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ B ) @ R )
           => ~ ( member8440522571783428010at_nat @ X3 @ A4 ) ) ) )).

% ImageE
thf(fact_1088_ImageE,axiom,(
    ! [B: produc859450856879609959at_nat,R: set_Pr9129816769188007085at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ B @ ( image_8438546471840335737at_nat @ R @ A4 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ X3 @ B ) @ R )
           => ~ ( member8440522571783428010at_nat @ X3 @ A4 ) ) ) )).

% ImageE
thf(fact_1089_ImageE,axiom,(
    ! [B: produc859450856879609959at_nat,R: set_Pr553994874890374343at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ B @ ( image_2092125385209402259at_nat @ R @ A4 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ B ) @ R )
           => ~ ( member8206827879206165904at_nat @ X3 @ A4 ) ) ) )).

% ImageE
thf(fact_1090_ImageE,axiom,(
    ! [B: produc8923325533196201883nteger,R: set_Pr8056137968301705908nteger,A4: set_Co9149898834107579976e_term] :
      ( ( member157494554546826820nteger @ B @ ( image_3609208789005719156nteger @ R @ A4 ) )
     => ~ ! [X3: code_integer > option6357759511663192854e_term] :
            ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ B ) @ R )
           => ~ ( member1535805642427569193e_term @ X3 @ A4 ) ) ) )).

% ImageE
thf(fact_1091_ImageE,axiom,(
    ! [B: produc7822875418678951345atural,R: set_Pr4244404650639074861atural,A4: set_Pr2791530082921190693e_term] :
      ( ( member8280416476884352090atural @ B @ ( image_3613525429136488045atural @ R @ A4 ) )
     => ~ ! [X3: produc4972180933644002618e_term > option6357759511663192854e_term] :
            ( ( member3245610415464309390atural @ ( produc3831813291587773865atural @ X3 @ B ) @ R )
           => ~ ( member2820825060907577222e_term @ X3 @ A4 ) ) ) )).

% ImageE
thf(fact_1092_ImageE,axiom,(
    ! [B: produc8923325533196201883nteger,R: set_Pr1281608226676607948nteger,A4: set_Pr7604974323444597168e_term] :
      ( ( member157494554546826820nteger @ B @ ( image_8384817790496657548nteger @ R @ A4 ) )
     => ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term] :
            ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ B ) @ R )
           => ~ ( member4242434998011752849e_term @ X3 @ A4 ) ) ) )).

% ImageE
thf(fact_1093_ImageE,axiom,(
    ! [B: product_prod_int_int,R: set_Pr9222295170931077689nt_int,A4: set_Pr3642885161833720865e_term] :
      ( ( member5262025264175285858nt_int @ B @ ( image_7245241130373604473nt_int @ R @ A4 ) )
     => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term] :
            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ B ) @ R )
           => ~ ( member3222579708246209666e_term @ X3 @ A4 ) ) ) )).

% ImageE
thf(fact_1094_ImageE,axiom,(
    ! [B: product_prod_int_int,R: set_Pr1872883991513573699nt_int,A4: set_in3461395444621081367e_term] :
      ( ( member5262025264175285858nt_int @ B @ ( image_5859975413754942339nt_int @ R @ A4 ) )
     => ~ ! [X3: int > option6357759511663192854e_term] :
            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ B ) @ R )
           => ~ ( member8845023287901829240e_term @ X3 @ A4 ) ) ) )).

% ImageE
thf(fact_1095_conditional__action_Oin__rel,axiom,
    ( condit637652166054216492nteger
    = ( ^ [R4: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A5: condit970840934589502176e_term,B4: condit2712864380631382371nteger] :
        ? [Z3: condit3546104913763457868nteger] :
          ( ( member4692730287278693091nteger @ Z3
            @ ( collec7894718386245043233nteger
              @ ^ [X4: condit3546104913763457868nteger] :
                  ( ord_le3216752416896350996nteger @ ( condit1485266954482049841nteger @ X4 ) @ ( collec6350289969311926995nteger @ ( produc127349428274296955eger_o @ R4 ) ) ) ) )
          & ( ( condit2191420391077584106e_term @ produc2702340305111172292nteger @ Z3 )
            = A5 )
          & ( ( condit3247640175922477123nteger @ produc6514156286822040066nteger @ Z3 )
            = B4 ) ) ) )).

% conditional_action.in_rel
thf(fact_1096_conditional__action_Oin__rel,axiom,
    ( condit8576614724009541413atural
    = ( ^ [R4: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o,A5: condit2742226398119032253e_term,B4: condit5465194378543733881atural] :
        ? [Z3: condit630041368442733253atural] :
          ( ( member6732191204198495708atural @ Z3
            @ ( collec4385107202880966170atural
              @ ^ [X4: condit630041368442733253atural] :
                  ( ord_le3616440322383431309atural @ ( condit5031902942079090474atural @ X4 ) @ ( collec6032437096461363148atural @ ( produc2278936416606902338ural_o @ R4 ) ) ) ) )
          & ( ( condit6832744779408388160e_term @ produc3051390257598303421atural @ Z3 )
            = A5 )
          & ( ( condit3057704127283464224atural @ produc3085575039554877691atural @ Z3 )
            = B4 ) ) ) )).

% conditional_action.in_rel
thf(fact_1097_conditional__action_Oin__rel,axiom,
    ( condit4124535048514935108nteger
    = ( ^ [R4: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A5: condit7555670638642438728e_term,B4: condit2712864380631382371nteger] :
        ? [Z3: condit6890616981335042148nteger] :
          ( ( member3769394780236028795nteger @ Z3
            @ ( collec1422310778918499257nteger
              @ ^ [X4: condit6890616981335042148nteger] :
                  ( ord_le653643898420964396nteger @ ( condit5950415191009657929nteger @ X4 ) @ ( collec6950949345391930603nteger @ ( produc6253627499356882019eger_o @ R4 ) ) ) ) )
          & ( ( condit5630909667417248490e_term @ produc7822682618958472924nteger @ Z3 )
            = A5 )
          & ( ( condit473727461810382379nteger @ produc7856867400915047194nteger @ Z3 )
            = B4 ) ) ) )).

% conditional_action.in_rel
thf(fact_1098_conditional__action_Oin__rel,axiom,
    ( condit8107536958121069873nt_int
    = ( ^ [R4: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A5: condit5657465404729207993e_term,B4: condit1149335772477489793nt_int] :
        ? [Z3: condit7703456510367084241nt_int] :
          ( ( member7338111355302933992nt_int @ Z3
            @ ( collec358744292499306022nt_int
              @ ^ [X4: condit7703456510367084241nt_int] :
                  ( ord_le8725513860283290265nt_int @ ( condit7641442619515482422nt_int @ X4 ) @ ( collec1790188477890212312nt_int @ ( produc1573362020775583542_int_o @ R4 ) ) ) ) )
          & ( ( condit6665956976869506376e_term @ produc6816164490631068361nt_int @ Z3 )
            = A5 )
          & ( ( condit2143695903752878364nt_int @ produc7328097813583171335nt_int @ Z3 )
            = B4 ) ) ) )).

% conditional_action.in_rel
thf(fact_1099_conditional__action_Oin__rel,axiom,
    ( condit5124865079037370427nt_int
    = ( ^ [R4: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A5: condit4685264965002145711e_term,B4: condit1149335772477489793nt_int] :
        ? [Z3: condit2639935803418221019nt_int] :
          ( ( member6933286338431872370nt_int @ Z3
            @ ( collec4281315781365710512nt_int
              @ ^ [X4: condit2639935803418221019nt_int] :
                  ( ord_le135402666524580259nt_int @ ( condit3959696847226599744nt_int @ X4 ) @ ( collec506566255779805410nt_int @ ( produc2558449545302689196_int_o @ R4 ) ) ) ) )
          & ( ( condit3702801411652404040e_term @ produc6230002227079971283nt_int @ Z3 )
            = A5 )
          & ( ( condit748182627455203218nt_int @ produc3162348030201620241nt_int @ Z3 )
            = B4 ) ) ) )).

% conditional_action.in_rel
thf(fact_1100_conditional__action_Oin__rel,axiom,
    ( condit1127689873879419692atural
    = ( ^ [R4: code_natural > produc7822875418678951345atural > $o,A5: condit6493346598134228598atural,B4: condit5465194378543733881atural] :
        ? [Z3: condit1810779906380472396atural] :
          ( ( member5077444122367321059atural @ Z3
            @ ( collec3264392531641636129atural
              @ ^ [X4: condit1810779906380472396atural] :
                  ( ord_le2202040463148982292atural @ ( condit6773436539961436209atural @ X4 ) @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ R4 ) ) ) ) )
          & ( ( condit7067702762122744960atural @ produc497848011034438852atural @ Z3 )
            = A5 )
          & ( ( condit4604360551831230809atural @ produc6591912806276919810atural @ Z3 )
            = B4 ) ) ) )).

% conditional_action.in_rel
thf(fact_1101_conditional__action_Oin__rel,axiom,
    ( condit1543564991105445733list_f
    = ( ^ [R4: finfun1145668972854894571action > produc1811169103916964701list_f > $o,A5: condit3482957019168584761action,B4: condit8002672114479986987list_f] :
        ? [Z3: condit573090327609778161list_f] :
          ( ( member3716888906625679770list_f @ Z3
            @ ( collec7820802666344048348list_f
              @ ^ [X4: condit573090327609778161list_f] :
                  ( ord_le3852096572277380649list_f @ ( condit6827580605863688246list_f @ X4 ) @ ( collec2941370454899091988list_f @ ( produc7065790152434098210st_f_o @ R4 ) ) ) ) )
          & ( ( condit7155426171764300860action @ produc3089225533250117581list_f @ Z3 )
            = A5 )
          & ( ( condit4069896782405135150list_f @ produc8710481132536983823list_f @ Z3 )
            = B4 ) ) ) )).

% conditional_action.in_rel
thf(fact_1102_conditional__action_Orel__compp__Grp,axiom,
    ( condit637652166054216492nteger
    = ( ^ [R4: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
          ( relcom8952203456716109636nteger
          @ ( conver3963888511192105379e_term
            @ ( bNF_Gr6558335477642860668e_term
              @ ( collec7894718386245043233nteger
                @ ^ [X4: condit3546104913763457868nteger] :
                    ( ord_le3216752416896350996nteger @ ( condit1485266954482049841nteger @ X4 ) @ ( collec6350289969311926995nteger @ ( produc127349428274296955eger_o @ R4 ) ) ) )
              @ ( condit2191420391077584106e_term @ produc2702340305111172292nteger ) ) )
          @ ( bNF_Gr6778840809880969415nteger
            @ ( collec7894718386245043233nteger
              @ ^ [X4: condit3546104913763457868nteger] :
                  ( ord_le3216752416896350996nteger @ ( condit1485266954482049841nteger @ X4 ) @ ( collec6350289969311926995nteger @ ( produc127349428274296955eger_o @ R4 ) ) ) )
            @ ( condit3247640175922477123nteger @ produc6514156286822040066nteger ) ) ) ) )).

% conditional_action.rel_compp_Grp
thf(fact_1103_conditional__action_Orel__compp__Grp,axiom,
    ( condit8576614724009541413atural
    = ( ^ [R4: ( produc4972180933644002618e_term > option6357759511663192854e_term ) > produc7822875418678951345atural > $o] :
          ( relcom5764181667456148612atural
          @ ( conver779324577884746617e_term
            @ ( bNF_Gr681077883941426002e_term
              @ ( collec4385107202880966170atural
                @ ^ [X4: condit630041368442733253atural] :
                    ( ord_le3616440322383431309atural @ ( condit5031902942079090474atural @ X4 ) @ ( collec6032437096461363148atural @ ( produc2278936416606902338ural_o @ R4 ) ) ) )
              @ ( condit6832744779408388160e_term @ produc3051390257598303421atural ) ) )
          @ ( bNF_Gr6818132734868028196atural
            @ ( collec4385107202880966170atural
              @ ^ [X4: condit630041368442733253atural] :
                  ( ord_le3616440322383431309atural @ ( condit5031902942079090474atural @ X4 ) @ ( collec6032437096461363148atural @ ( produc2278936416606902338ural_o @ R4 ) ) ) )
            @ ( condit3057704127283464224atural @ produc3085575039554877691atural ) ) ) ) )).

% conditional_action.rel_compp_Grp
thf(fact_1104_conditional__action_Orel__compp__Grp,axiom,
    ( condit4124535048514935108nteger
    = ( ^ [R4: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
          ( relcom8070249692364219716nteger
          @ ( conver3477549252740975267e_term
            @ ( bNF_Gr3379302558797654652e_term
              @ ( collec1422310778918499257nteger
                @ ^ [X4: condit6890616981335042148nteger] :
                    ( ord_le653643898420964396nteger @ ( condit5950415191009657929nteger @ X4 ) @ ( collec6950949345391930603nteger @ ( produc6253627499356882019eger_o @ R4 ) ) ) )
              @ ( condit5630909667417248490e_term @ produc7822682618958472924nteger ) ) )
          @ ( bNF_Gr5930519763323370031nteger
            @ ( collec1422310778918499257nteger
              @ ^ [X4: condit6890616981335042148nteger] :
                  ( ord_le653643898420964396nteger @ ( condit5950415191009657929nteger @ X4 ) @ ( collec6950949345391930603nteger @ ( produc6253627499356882019eger_o @ R4 ) ) ) )
            @ ( condit473727461810382379nteger @ produc7856867400915047194nteger ) ) ) ) )).

% conditional_action.rel_compp_Grp
thf(fact_1105_conditional__action_Orel__compp__Grp,axiom,
    ( condit8107536958121069873nt_int
    = ( ^ [R4: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
          ( relcom605141853646172036nt_int
          @ ( conver6238527978116907649e_term
            @ ( bNF_Gr5342119887459643994e_term
              @ ( collec358744292499306022nt_int
                @ ^ [X4: condit7703456510367084241nt_int] :
                    ( ord_le8725513860283290265nt_int @ ( condit7641442619515482422nt_int @ X4 ) @ ( collec1790188477890212312nt_int @ ( produc1573362020775583542_int_o @ R4 ) ) ) )
              @ ( condit6665956976869506376e_term @ produc6816164490631068361nt_int ) ) )
          @ ( bNF_Gr4032692272425925664nt_int
            @ ( collec358744292499306022nt_int
              @ ^ [X4: condit7703456510367084241nt_int] :
                  ( ord_le8725513860283290265nt_int @ ( condit7641442619515482422nt_int @ X4 ) @ ( collec1790188477890212312nt_int @ ( produc1573362020775583542_int_o @ R4 ) ) ) )
            @ ( condit2143695903752878364nt_int @ produc7328097813583171335nt_int ) ) ) ) )).

% conditional_action.rel_compp_Grp
thf(fact_1106_conditional__action_Orel__compp__Grp,axiom,
    ( condit5124865079037370427nt_int
    = ( ^ [R4: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
          ( relcom478388072180681092nt_int
          @ ( conver2776829400750641793e_term
            @ ( bNF_Gr5588081758082759002e_term
              @ ( collec4281315781365710512nt_int
                @ ^ [X4: condit2639935803418221019nt_int] :
                    ( ord_le135402666524580259nt_int @ ( condit3959696847226599744nt_int @ X4 ) @ ( collec506566255779805410nt_int @ ( produc2558449545302689196_int_o @ R4 ) ) ) )
              @ ( condit3702801411652404040e_term @ produc6230002227079971283nt_int ) ) )
          @ ( bNF_Gr4086170905571730198nt_int
            @ ( collec4281315781365710512nt_int
              @ ^ [X4: condit2639935803418221019nt_int] :
                  ( ord_le135402666524580259nt_int @ ( condit3959696847226599744nt_int @ X4 ) @ ( collec506566255779805410nt_int @ ( produc2558449545302689196_int_o @ R4 ) ) ) )
            @ ( condit748182627455203218nt_int @ produc3162348030201620241nt_int ) ) ) ) )).

% conditional_action.rel_compp_Grp
thf(fact_1107_conditional__action_Orel__compp__Grp,axiom,
    ( condit1127689873879419692atural
    = ( ^ [R4: code_natural > produc7822875418678951345atural > $o] :
          ( relcom7477549861947874628atural
          @ ( conver9184724556494050873atural
            @ ( bNF_Gr659660228831487250atural
              @ ( collec3264392531641636129atural
                @ ^ [X4: condit1810779906380472396atural] :
                    ( ord_le2202040463148982292atural @ ( condit6773436539961436209atural @ X4 ) @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ R4 ) ) ) )
              @ ( condit7067702762122744960atural @ produc497848011034438852atural ) ) )
          @ ( bNF_Gr7139796620294019805atural
            @ ( collec3264392531641636129atural
              @ ^ [X4: condit1810779906380472396atural] :
                  ( ord_le2202040463148982292atural @ ( condit6773436539961436209atural @ X4 ) @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ R4 ) ) ) )
            @ ( condit4604360551831230809atural @ produc6591912806276919810atural ) ) ) ) )).

% conditional_action.rel_compp_Grp
thf(fact_1108_conditional__action_Orel__compp__Grp,axiom,
    ( condit1543564991105445733list_f
    = ( ^ [R4: finfun1145668972854894571action > produc1811169103916964701list_f > $o] :
          ( relcom7515417596351238834list_f
          @ ( conver7720072225776492021action
            @ ( bNF_Gr6437747679974042972action
              @ ( collec7820802666344048348list_f
                @ ^ [X4: condit573090327609778161list_f] :
                    ( ord_le3852096572277380649list_f @ ( condit6827580605863688246list_f @ X4 ) @ ( collec2941370454899091988list_f @ ( produc7065790152434098210st_f_o @ R4 ) ) ) )
              @ ( condit7155426171764300860action @ produc3089225533250117581list_f ) ) )
          @ ( bNF_Gr9208479453086946510list_f
            @ ( collec7820802666344048348list_f
              @ ^ [X4: condit573090327609778161list_f] :
                  ( ord_le3852096572277380649list_f @ ( condit6827580605863688246list_f @ X4 ) @ ( collec2941370454899091988list_f @ ( produc7065790152434098210st_f_o @ R4 ) ) ) )
            @ ( condit4069896782405135150list_f @ produc8710481132536983823list_f ) ) ) ) )).

% conditional_action.rel_compp_Grp
thf(fact_1109_Image__singleton__iff,axiom,(
    ! [B: nat,R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) )).

% Image_singleton_iff
thf(fact_1110_Image__singleton__iff,axiom,(
    ! [B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_2484338115069272851at_nat @ R @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R ) ) )).

% Image_singleton_iff
thf(fact_1111_Image__singleton__iff,axiom,(
    ! [B: produc8923325533196201883nteger,R: set_Pr8056137968301705908nteger,A: code_integer > option6357759511663192854e_term] :
      ( ( member157494554546826820nteger @ B @ ( image_3609208789005719156nteger @ R @ ( insert4303373725994308034e_term @ A @ bot_bo1530185625139755228e_term ) ) )
      = ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ A @ B ) @ R ) ) )).

% Image_singleton_iff
thf(fact_1112_Image__singleton__iff,axiom,(
    ! [B: produc7822875418678951345atural,R: set_Pr4244404650639074861atural,A: produc4972180933644002618e_term > option6357759511663192854e_term] :
      ( ( member8280416476884352090atural @ B @ ( image_3613525429136488045atural @ R @ ( insert4479822593746981279e_term @ A @ bot_bo4014960707591977401e_term ) ) )
      = ( member3245610415464309390atural @ ( produc3831813291587773865atural @ A @ B ) @ R ) ) )).

% Image_singleton_iff
thf(fact_1113_Image__singleton__iff,axiom,(
    ! [B: produc8923325533196201883nteger,R: set_Pr1281608226676607948nteger,A: produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( member157494554546826820nteger @ B @ ( image_8384817790496657548nteger @ R @ ( insert5901432530851156906e_term @ A @ bot_bo8828404948115383876e_term ) ) )
      = ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ R ) ) )).

% Image_singleton_iff
thf(fact_1114_Image__singleton__iff,axiom,(
    ! [B: product_prod_int_int,R: set_Pr9222295170931077689nt_int,A: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( member5262025264175285858nt_int @ B @ ( image_7245241130373604473nt_int @ R @ ( insert2341259790341569179e_term @ A @ bot_bo974140798557118645e_term ) ) )
      = ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ R ) ) )).

% Image_singleton_iff
thf(fact_1115_Image__singleton__iff,axiom,(
    ! [B: product_prod_int_int,R: set_Pr1872883991513573699nt_int,A: int > option6357759511663192854e_term] :
      ( ( member5262025264175285858nt_int @ B @ ( image_5859975413754942339nt_int @ R @ ( insert5768611539565238801e_term @ A @ bot_bo4665192896540886443e_term ) ) )
      = ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ R ) ) )).

% Image_singleton_iff
thf(fact_1116_add__Suc__right,axiom,(
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
      = ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) )).

% add_Suc_right

% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,(
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) )).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,(
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) )).

thf(help_If_3_1_If_001t__Predicate__Opred_It__Product____Type__Ounit_J_T,axiom,(
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) )).

thf(help_If_2_1_If_001t__Predicate__Opred_It__Product____Type__Ounit_J_T,axiom,(
    ! [X: pred_Product_unit,Y: pred_Product_unit] :
      ( ( if_pred_Product_unit @ $false @ X @ Y )
      = Y ) )).

thf(help_If_1_1_If_001t__Predicate__Opred_It__Product____Type__Ounit_J_T,axiom,(
    ! [X: pred_Product_unit,Y: pred_Product_unit] :
      ( ( if_pred_Product_unit @ $true @ X @ Y )
      = X ) )).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( thr_a_d_a_b_c_e_f @ ( produc8331056147679645153list_f @ ls @ ( produc8608434180808776663list_f @ nts @ jswss ) ) )
    = nts )).
