TPTP Problem File: SLH0474^1.p

%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Rewrite_Properties_Reduction/0021_Ground_Reduction_on_GTRS/prob_00038_001523__14265870_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1973 ( 659 unt; 690 typ;   0 def)
%            Number of atoms       : 3704 (1122 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 10925 ( 292   ~;  25   |; 261   &;8931   @)
%                                         (   0 <=>;1416  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   6 avg)
%            Number of types       :  103 ( 102 usr)
%            Number of type conns  : 2743 (2743   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  591 ( 588 usr;  35 con; 0-4 aty)
%            Number of variables   : 4128 ( 884   ^;3148   !;  96   ?;4128   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:19:35.354
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_I_062_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    set_te1997870499059199101rm_a_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_J,type,
    set_te1821851435260963965rm_c_d: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    set_Pr2693567550035855805_nat_o: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_Pr5768764684362454787et_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Nat__Onat_J,type,
    produc7378404338329476677_d_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    produc2720409071189015237_b_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
    produc105651233775526341rm_c_d: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc1234881154892807749rm_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_Pr1261947904930325089at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc7491599851749785783at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc2400336064389900727et_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_Sum_sum_nat_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J,type,
    set_Pr6903500605879609269_c_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    set_Pr4934435412358123699_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
    set_Pr3149072824959771635_nat_o: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
    set_Pr2101469702781467981_o_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    produc6739137655528088327_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc573827679308723021et_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    product_prod_nat_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    set_Product_prod_o_o: $tType ).

thf(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
    sum_sum_nat_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
    set_option_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Term__Oterm_Itf__c_Mtf__d_J_J,type,
    set_term_c_d: $tType ).

thf(ty_n_t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    set_term_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    product_prod_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    product_prod_o_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
    set_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    product_prod_o_o: $tType ).

thf(ty_n_t__Term__Oterm_Itf__c_Mtf__d_J,type,
    term_c_d: $tType ).

thf(ty_n_t__Term__Oterm_Itf__a_Mtf__b_J,type,
    term_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

% Explicit typings (588)
thf(sy_c_Abstract__Rewriting_Orestrict_001_Eo,type,
    abstract_restrict_o: set_Product_prod_o_o > set_o > set_Product_prod_o_o ).

thf(sy_c_Abstract__Rewriting_Orestrict_001t__Nat__Onat,type,
    abstra1219450721576609224ct_nat: set_Pr1261947904930325089at_nat > set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Abstract__Rewriting_Orestrict_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    abstra4478131260412406025at_nat: set_Pr8693737435421807431at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_Abstract__Rewriting_Orestrict_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    abstra2354209648719431185rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b ).

thf(sy_c_Abstract__Rewriting_Orestrict_001t__Set__Oset_It__Nat__Onat_J,type,
    abstra6515489984838218110et_nat: set_Pr5488025237498180813et_nat > set_set_nat > set_Pr5488025237498180813et_nat ).

thf(sy_c_Abstract__Rewriting_Orestrict_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    abstra6662565036008789004rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Abstract__Rewriting_Orestrict_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    abstra1087653880669894540rm_c_d: set_Pr5080540287643815273rm_c_d > set_term_c_d > set_Pr5080540287643815273rm_c_d ).

thf(sy_c_BNF__Def_OGr_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    bNF_Gr3285426624312675404rm_c_d: set_term_a_b > ( term_a_b > term_c_d ) > set_Pr4645549399426460137rm_c_d ).

thf(sy_c_BNF__Def_OGr_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bNF_Gr4940740347456048204rm_a_b: set_term_c_d > ( term_c_d > term_a_b ) > set_Pr4821568463224695273rm_a_b ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    bNF_Gr6034662968427871246rm_c_d: set_Pr4386577575007340137rm_a_b > ( term_a_b > term_c_d ) > set_Pr5080540287643815273rm_c_d ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bNF_Gr7689976691571244046rm_a_b: set_Pr5080540287643815273rm_c_d > ( term_c_d > term_a_b ) > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    bNF_Gr7274730409055677163rm_c_d: set_term_a_b > set_Pr5080540287643815273rm_c_d > ( term_a_b > term_c_d ) > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bNF_Gr8930044132199049963rm_a_b: set_term_c_d > set_Pr4386577575007340137rm_a_b > ( term_c_d > term_a_b ) > set_Pr5080540287643815273rm_c_d ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_M_Eo_J,type,
    comple6214475593288795910_nat_o: set_nat_o > nat > $o ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_M_062_It__Term__Oterm_Itf__c_Mtf__d_J_M_Eo_J_J,type,
    comple7831901308807748069_c_d_o: set_te8875761578035853502_c_d_o > term_a_b > term_c_d > $o ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Term__Oterm_Itf__c_Mtf__d_J_M_062_It__Term__Oterm_Itf__a_Mtf__b_J_M_Eo_J_J,type,
    comple6699983181919050213_a_b_o: set_te7068155017529424062_a_b_o > term_c_d > term_a_b > $o ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
    complete_Inf_Inf_o: set_o > $o ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7806235888213564991et_nat: set_set_nat > set_nat ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_J,type,
    comple146610104219517168rm_c_d: set_se244340041498719817rm_c_d > set_Pr4645549399426460137rm_c_d ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    comple322629168017752304rm_a_b: set_se5852867905277097545rm_a_b > set_Pr4821568463224695273rm_a_b ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
    comple8317665133742190828_nat_o: set_nat_o > nat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_M_062_It__Term__Oterm_Itf__c_Mtf__d_J_M_Eo_J_J,type,
    comple4776141392677706955_c_d_o: set_te8875761578035853502_c_d_o > term_a_b > term_c_d > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Term__Oterm_Itf__c_Mtf__d_J_M_062_It__Term__Oterm_Itf__a_Mtf__b_J_M_Eo_J_J,type,
    comple3644223265789009099_a_b_o: set_te7068155017529424062_a_b_o > term_c_d > term_a_b > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
    complete_Sup_Sup_o: set_o > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7399068483239264473et_nat: set_set_nat > set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    comple5685304695842803022at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_J,type,
    comple5229724214311509974rm_c_d: set_se244340041498719817rm_c_d > set_Pr4645549399426460137rm_c_d ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    comple5405743278109745110rm_a_b: set_se5852867905277097545rm_a_b > set_Pr4821568463224695273rm_a_b ).

thf(sy_c_Equiv__Relations_Oequiv_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
    equiv_3005225879699351953rm_c_d: set_te1821851435260963965rm_c_d > set_Pr6132369733443093455rm_c_d > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001_062_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    equiv_214858368621111697rm_a_b: set_te1997870499059199101rm_a_b > set_Pr8788715879341982671rm_a_b > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001t__Nat__Onat,type,
    equiv_equiv_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    equiv_5407799809235660215at_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001t__Set__Oset_It__Nat__Onat_J,type,
    equiv_equiv_set_nat: set_set_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
    equiv_7517305544406289097at_nat: set_Sum_sum_nat_nat > set_Pr1076074158304066111at_nat > $o ).

thf(sy_c_Equiv__Relations_Oequivp_001t__Nat__Onat,type,
    equiv_equivp_nat: ( nat > nat > $o ) > $o ).

thf(sy_c_Equiv__Relations_Oquotient_001t__Nat__Onat,type,
    equiv_quotient_nat: set_nat > set_Pr1261947904930325089at_nat > set_set_nat ).

thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
    finite_Fpow_nat: set_nat > set_set_nat ).

thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
    finite_card_nat: set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
    finite4943504547041397169rm_c_d: set_Pr4645549399426460137rm_c_d > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    finite2153137035963156913rm_a_b: set_Pr4821568463224695273rm_a_b > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
    finite_card_set_nat: set_set_nat > nat ).

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Nat__Onat_J,type,
    finite5523153139673422903on_nat: set_option_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
    finite6534129571051747890rm_c_d: set_Pr4645549399426460137rm_c_d > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    finite3743762059973507634rm_a_b: set_Pr4821568463224695273rm_a_b > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
    finite1152437895449049373et_nat: set_set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite6187706683773761046at_nat: set_Sum_sum_nat_nat > $o ).

thf(sy_c_Fun_Oinj__on_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_J,type,
    inj_on8823995452884580655rm_c_d: ( ( term_a_b > term_c_d ) > set_Pr4645549399426460137rm_c_d ) > set_te1821851435260963965rm_c_d > $o ).

thf(sy_c_Fun_Oinj__on_001_062_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    inj_on2179106752557545775rm_a_b: ( ( term_c_d > term_a_b ) > set_Pr4821568463224695273rm_a_b ) > set_te1997870499059199101rm_a_b > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    inj_on8105003582846801791et_nat: ( nat > set_set_nat ) > set_nat > $o ).

thf(sy_c_Fun__Def_Oin__rel_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    fun_in8391593592550394988rm_c_d: set_Pr4645549399426460137rm_c_d > term_a_b > term_c_d > $o ).

thf(sy_c_Fun__Def_Oin__rel_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fun_in823535278838991980rm_a_b: set_Pr4821568463224695273rm_a_b > term_c_d > term_a_b > $o ).

thf(sy_c_Fun__Def_Orp__inv__image_001t__Nat__Onat_001t__Nat__Onat,type,
    fun_rp3004530424201022588at_nat: produc3843707927480180839at_nat > ( nat > nat ) > produc3843707927480180839at_nat ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_Pr7557018466319803784at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    the_Pr3853923214622141328rm_a_b: ( produc357393685978478089rm_a_b > $o ) > produc357393685978478089rm_a_b ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
    the_Pr2724693293504859920rm_c_d: ( produc8451535801715972489rm_c_d > $o ) > produc8451535801715972489rm_c_d ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    the_Pr9157697819281395472rm_a_b: ( produc5661168290637732233rm_a_b > $o ) > produc5661168290637732233rm_a_b ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Term__Oterm_Itf__c_Mtf__d_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001_062_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Nat__Onat_J_M_Eo_J,type,
    produc14787907101276426_nat_o: ( term_c_d > nat > produc7378404338329476677_d_nat > $o ) > produc7378404338329476677_d_nat > produc7378404338329476677_d_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Nat__Onat_001_Eo,type,
    produc8272582700005955334_nat_o: ( term_c_d > nat > $o ) > produc7378404338329476677_d_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001_062_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_M_Eo_J,type,
    produc3271877877324961290_c_d_o: ( term_c_d > term_a_b > produc8451535801715972489rm_c_d > $o ) > produc5661168290637732233rm_a_b > produc8451535801715972489rm_c_d > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001_062_It__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_M_Eo_J,type,
    produc5873602711790147082_a_b_o: ( term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o ) > produc5661168290637732233rm_a_b > produc5661168290637732233rm_a_b > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001_Eo,type,
    produc327966649874835522_a_b_o: ( term_c_d > term_a_b > $o ) > produc5661168290637732233rm_a_b > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    produc3367825333614200230_b_nat: ( term_c_d > term_a_b > nat ) > produc5661168290637732233rm_a_b > nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
    produc5370438780080785971rm_c_d: ( term_c_d > term_a_b > produc8451535801715972489rm_c_d ) > produc5661168290637732233rm_a_b > produc8451535801715972489rm_c_d ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc2580071269002545715rm_a_b: ( term_c_d > term_a_b > produc5661168290637732233rm_a_b ) > produc5661168290637732233rm_a_b > produc5661168290637732233rm_a_b ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Set__Oset_I_Eo_J,type,
    produc2530863894259861282_set_o: ( term_c_d > term_a_b > set_o ) > produc5661168290637732233rm_a_b > set_o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc7500204723417323868et_nat: ( term_c_d > term_a_b > set_nat ) > produc5661168290637732233rm_a_b > set_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__c_Mtf__d_J_001_062_It__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_M_Eo_J,type,
    produc8575123938295100938_c_d_o: ( term_c_d > term_c_d > produc4531938369520450825rm_c_d > $o ) > produc4531938369520450825rm_c_d > produc4531938369520450825rm_c_d > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__c_Mtf__d_J_001_Eo,type,
    produc6447316180697048258_c_d_o: ( term_c_d > term_c_d > $o ) > produc4531938369520450825rm_c_d > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Nat__Onat,type,
    produc8025820600754661670_d_nat: ( term_c_d > term_c_d > nat ) > produc4531938369520450825rm_c_d > nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Set__Oset_I_Eo_J,type,
    produc1641171031246963618_set_o: ( term_c_d > term_c_d > set_o ) > produc4531938369520450825rm_c_d > set_o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc6663801938352323804et_nat: ( term_c_d > term_c_d > set_nat ) > produc4531938369520450825rm_c_d > set_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Nat__Onat,type,
    product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    produc2541846788875228641rm_a_b: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    produc6190307670391109985rm_c_d: produc8451535801715972489rm_c_d > produc5661168290637732233rm_a_b ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    produc7845621393534482785rm_a_b: produc5661168290637732233rm_a_b > produc8451535801715972489rm_c_d ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    produc2270710238195588321rm_c_d: produc4531938369520450825rm_c_d > produc4531938369520450825rm_c_d ).

thf(sy_c_Relation_OId__on_001t__Nat__Onat,type,
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thf(sy_c_Relation_OImage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: set_Pr1261947904930325089at_nat > set_nat > set_nat ).

thf(sy_c_Relation_OImage_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
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thf(sy_c_Relation_OImage_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
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thf(sy_c_Relation_Oantisym__on_001t__Nat__Onat,type,
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thf(sy_c_Relation_Oconverse_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Relation_Oconverse_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_Relation_Oconverse_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
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thf(sy_c_Relation_Oconverse_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    conver2033648821598950629rm_c_d: set_Pr4645549399426460137rm_c_d > set_Pr4821568463224695273rm_a_b ).

thf(sy_c_Relation_Oconverse_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    conver3688962544742323429rm_a_b: set_Pr4821568463224695273rm_a_b > set_Pr4645549399426460137rm_c_d ).

thf(sy_c_Relation_Oconverse_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    conver7337423426258204773rm_c_d: set_Pr5080540287643815273rm_c_d > set_Pr5080540287643815273rm_c_d ).

thf(sy_c_Relation_Oconversep_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Relation_Oconversep_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
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thf(sy_c_Relation_Oconversep_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    conver8884510217699480863rm_c_d: ( term_a_b > term_c_d > $o ) > term_c_d > term_a_b > $o ).

thf(sy_c_Relation_Oconversep_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    conver1316451903988077855rm_a_b: ( term_c_d > term_a_b > $o ) > term_a_b > term_c_d > $o ).

thf(sy_c_Relation_Oconversep_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    conver4964912785503959199rm_c_d: ( term_c_d > term_c_d > $o ) > term_c_d > term_c_d > $o ).

thf(sy_c_Relation_Oinv__image_001t__Nat__Onat_001_062_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    inv_im908594706308016235rm_a_b: set_Pr1261947904930325089at_nat > ( ( term_c_d > term_a_b ) > nat ) > set_Pr8788715879341982671rm_a_b ).

thf(sy_c_Relation_Oinv__image_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Relation_Oinv__image_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Relation_Oinv__image_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Relation_Oinv__image_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Relation_Oinv__image_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
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thf(sy_c_Relation_Oinv__image_001t__Nat__Onat_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
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thf(sy_c_Relation_Oinv__image_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Relation_Oinv__image_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Relation_Oinv__image_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Relation_Oinv__image_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Relation_Oinv__image_001t__Set__Oset_It__Nat__Onat_J_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Relation_Oinv__image_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Relation_Oinv__image_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Relation_Oinv__image_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Relation_Oinv__image_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    inv_im2609263929585225528_b_nat: set_Pr4386577575007340137rm_a_b > ( nat > term_a_b ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Relation_Oinv__image_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    inv_im3004224898869642620rm_a_b: set_Pr4386577575007340137rm_a_b > ( term_a_b > term_a_b ) > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Relation_Oinv__image_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    inv_im6652685780385523964rm_c_d: set_Pr4386577575007340137rm_a_b > ( term_c_d > term_a_b ) > set_Pr5080540287643815273rm_c_d ).

thf(sy_c_Relation_Oinv__image_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Nat__Onat,type,
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thf(sy_c_Relation_Oinv__image_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
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thf(sy_c_Relation_Oinv__image_001t__Term__Oterm_Itf__c_Mtf__d_J_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    inv_im2733088348190002300rm_c_d: set_Pr5080540287643815273rm_c_d > ( term_c_d > term_c_d ) > set_Pr5080540287643815273rm_c_d ).

thf(sy_c_Relation_Oirrefl__on_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
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thf(sy_c_Relation_Oirrefl__on_001_062_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    irrefl7486889657381885777rm_a_b: set_te1997870499059199101rm_a_b > set_Pr8788715879341982671rm_a_b > $o ).

thf(sy_c_Relation_Oirrefl__on_001_Eo,type,
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thf(sy_c_Relation_Oirrefl__on_001t__Nat__Onat,type,
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    member7278092155855023563_c_d_o: ( produc8451535801715972489rm_c_d > $o ) > set_Pr2504828417160952554_c_d_o > $o ).

thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_M_Eo_J,type,
    member656444953465433547_a_b_o: ( produc5661168290637732233rm_a_b > $o ) > set_Pr6472531492093999850_a_b_o > $o ).

thf(sy_c_member_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_M_062_It__Term__Oterm_Itf__c_Mtf__d_J_M_Eo_J_J,type,
    member8627074640717963399_c_d_o: ( term_a_b > term_c_d > $o ) > set_te8875761578035853502_c_d_o > $o ).

thf(sy_c_member_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
    member5930539741918748382rm_c_d: ( term_a_b > term_c_d ) > set_te1821851435260963965rm_c_d > $o ).

thf(sy_c_member_001_062_It__Term__Oterm_Itf__c_Mtf__d_J_M_062_It__Term__Oterm_Itf__a_Mtf__b_J_M_Eo_J_J,type,
    member7495156513829265543_a_b_o: ( term_c_d > term_a_b > $o ) > set_te7068155017529424062_a_b_o > $o ).

thf(sy_c_member_001_062_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member3140172230840508126rm_a_b: ( term_c_d > term_a_b ) > set_te1997870499059199101rm_a_b > $o ).

thf(sy_c_member_001_Eo,type,
    member_o: $o > set_o > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_M_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_J,type,
    member7058256936922213400rm_c_d: produc4272247648830171759rm_c_d > set_Pr6132369733443093455rm_c_d > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_M_062_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member4454048570373794840rm_a_b: produc1668039282281753199rm_a_b > set_Pr8788715879341982671rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member4905127681128465508et_nat: produc573827679308723021et_nat > set_Pr5768764684362454787et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    member6310962623043647828_nat_o: product_prod_nat_o > set_Pr3149072824959771635_nat_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member3782324328723991648et_nat: produc2400336064389900727et_nat > set_Pr400265656397884439et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    member2200362271989506215_nat_o: produc652263959640188542_nat_o > set_Pr6558897299634722526_nat_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member3348759134392003351at_nat: produc8373899037510109440at_nat > set_Pr2539167527615954998at_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__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__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member8066425057025219984rm_a_b: produc7765306094179330407rm_a_b > set_Pr5038301440468608839rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_J,type,
    member5070412009747381648rm_c_d: produc4769293046901492071rm_c_d > set_Pr7649946878678271815rm_c_d > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J_J,type,
    member2666392743443611024_b_nat: produc2365273780597721447_b_nat > set_Pr4200871855241742151_b_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member8417600551952982416rm_a_b: produc4523971788519308903rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_J,type,
    member3585784829053206928rm_c_d: produc8915528102474309223rm_c_d > set_Pr8929823917597237063rm_c_d > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Nat__Onat_J_J,type,
    member8400340274391571856_d_nat: produc8099221311545682279_d_nat > set_Pr991062860374239047_d_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member4039383142722929040rm_a_b: produc145754379289255527rm_a_b > set_Pr626007440406323015rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_J,type,
    member8430939456677929360rm_c_d: produc4537310693244255847rm_c_d > set_Pr6583054764951797575rm_c_d > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    member1847065620493055006_nat_o: produc6739137655528088327_nat_o > set_Pr2693567550035855805_nat_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member8873588116083876704at_nat: produc7491599851749785783at_nat > set_Pr3601174868274201367at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member4902808451920011912at_nat: produc3819362883047884511at_nat > set_Pr1076074158304066111at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member5869715511025134514rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
    member4740485589907853106rm_c_d: produc8451535801715972489rm_c_d > set_Pr4645549399426460137rm_c_d > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member1950118078829612850rm_a_b: produc5661168290637732233rm_a_b > set_Pr4821568463224695273rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J,type,
    member820888157712331442rm_c_d: produc4531938369520450825rm_c_d > set_Pr5080540287643815273rm_c_d > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_J,type,
    member9082614701894843282rm_c_d: set_Pr4645549399426460137rm_c_d > set_se244340041498719817rm_c_d > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member35261728838302610rm_a_b: set_Pr4821568463224695273rm_a_b > set_se5852867905277097545rm_a_b > $o ).

thf(sy_c_member_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8583185029347631382at_nat: sum_sum_nat_nat > set_Sum_sum_nat_nat > $o ).

thf(sy_c_member_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    member_term_a_b: term_a_b > set_term_a_b > $o ).

thf(sy_c_member_001t__Term__Oterm_Itf__c_Mtf__d_J,type,
    member_term_c_d: term_c_d > set_term_c_d > $o ).

thf(sy_v__092_060R_062,type,
    r: set_Pr4645549399426460137rm_c_d ).

% Relevant facts (1276)
thf(fact_0_split__part,axiom,
    ! [P: $o,Q: term_a_b > term_a_b > $o] :
      ( ( produc6251846659952913346_a_b_o
        @ ^ [A: term_a_b,B: term_a_b] :
            ( P
            & ( Q @ A @ B ) ) )
      = ( ^ [Ab: produc357393685978478089rm_a_b] :
            ( P
            & ( produc6251846659952913346_a_b_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_1_split__part,axiom,
    ! [P: $o,Q: nat > nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [A: nat,B: nat] :
            ( P
            & ( Q @ A @ B ) ) )
      = ( ^ [Ab: product_prod_nat_nat] :
            ( P
            & ( produc6081775807080527818_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_2_split__part,axiom,
    ! [P: $o,Q: term_a_b > term_c_d > $o] :
      ( ( produc3147824153920350274_c_d_o
        @ ^ [A: term_a_b,B: term_c_d] :
            ( P
            & ( Q @ A @ B ) ) )
      = ( ^ [Ab: produc8451535801715972489rm_c_d] :
            ( P
            & ( produc3147824153920350274_c_d_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_3_split__part,axiom,
    ! [P: $o,Q: term_c_d > term_a_b > $o] :
      ( ( produc327966649874835522_a_b_o
        @ ^ [A: term_c_d,B: term_a_b] :
            ( P
            & ( Q @ A @ B ) ) )
      = ( ^ [Ab: produc5661168290637732233rm_a_b] :
            ( P
            & ( produc327966649874835522_a_b_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_4_converse__inject,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d,S: set_Pr5080540287643815273rm_c_d] :
      ( ( ( conver7337423426258204773rm_c_d @ R )
        = ( conver7337423426258204773rm_c_d @ S ) )
      = ( R = S ) ) ).

% converse_inject
thf(fact_5_converse__inject,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ( conver7608559976937845093rm_a_b @ R )
        = ( conver7608559976937845093rm_a_b @ S ) )
      = ( R = S ) ) ).

% converse_inject
thf(fact_6_converse__inject,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( converse_nat_nat @ R )
        = ( converse_nat_nat @ S ) )
      = ( R = S ) ) ).

% converse_inject
thf(fact_7_converse__inject,axiom,
    ! [R: set_Pr4821568463224695273rm_a_b,S: set_Pr4821568463224695273rm_a_b] :
      ( ( ( conver3688962544742323429rm_a_b @ R )
        = ( conver3688962544742323429rm_a_b @ S ) )
      = ( R = S ) ) ).

% converse_inject
thf(fact_8_converse__inject,axiom,
    ! [R: set_Pr4645549399426460137rm_c_d,S: set_Pr4645549399426460137rm_c_d] :
      ( ( ( conver2033648821598950629rm_c_d @ R )
        = ( conver2033648821598950629rm_c_d @ S ) )
      = ( R = S ) ) ).

% converse_inject
thf(fact_9_converse__converse,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d] :
      ( ( conver7337423426258204773rm_c_d @ ( conver7337423426258204773rm_c_d @ R ) )
      = R ) ).

% converse_converse
thf(fact_10_converse__converse,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( conver7608559976937845093rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) )
      = R ) ).

% converse_converse
thf(fact_11_converse__converse,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( converse_nat_nat @ ( converse_nat_nat @ R ) )
      = R ) ).

% converse_converse
thf(fact_12_converse__converse,axiom,
    ! [R: set_Pr4645549399426460137rm_c_d] :
      ( ( conver3688962544742323429rm_a_b @ ( conver2033648821598950629rm_c_d @ R ) )
      = R ) ).

% converse_converse
thf(fact_13_converse__converse,axiom,
    ! [R: set_Pr4821568463224695273rm_a_b] :
      ( ( conver2033648821598950629rm_c_d @ ( conver3688962544742323429rm_a_b @ R ) )
      = R ) ).

% converse_converse
thf(fact_14_prod_Odisc__eq__case,axiom,
    ! [Prod: produc357393685978478089rm_a_b] :
      ( produc6251846659952913346_a_b_o
      @ ^ [Uu: term_a_b,Uv: term_a_b] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_15_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( produc6081775807080527818_nat_o
      @ ^ [Uu: nat,Uv: nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_16_prod_Odisc__eq__case,axiom,
    ! [Prod: produc8451535801715972489rm_c_d] :
      ( produc3147824153920350274_c_d_o
      @ ^ [Uu: term_a_b,Uv: term_c_d] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_17_prod_Odisc__eq__case,axiom,
    ! [Prod: produc5661168290637732233rm_a_b] :
      ( produc327966649874835522_a_b_o
      @ ^ [Uu: term_c_d,Uv: term_a_b] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_18_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: term_a_b > term_c_d > $o,Prod: produc8451535801715972489rm_c_d] :
      ( ( H @ ( produc3147824153920350274_c_d_o @ F @ Prod ) )
      = ( produc3147824153920350274_c_d_o
        @ ^ [X1: term_a_b,X2: term_c_d] : ( H @ ( F @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_19_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: term_c_d > term_a_b > $o,Prod: produc5661168290637732233rm_a_b] :
      ( ( H @ ( produc327966649874835522_a_b_o @ F @ Prod ) )
      = ( produc327966649874835522_a_b_o
        @ ^ [X1: term_c_d,X2: term_a_b] : ( H @ ( F @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_20_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X1: nat,X2: nat] : ( H @ ( F @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_21_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: term_a_b > term_a_b > $o,Prod: produc357393685978478089rm_a_b] :
      ( ( H @ ( produc6251846659952913346_a_b_o @ F @ Prod ) )
      = ( produc6251846659952913346_a_b_o
        @ ^ [X1: term_a_b,X2: term_a_b] : ( H @ ( F @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_22_prod_Ocase__distrib,axiom,
    ! [H: $o > produc5661168290637732233rm_a_b,F: term_a_b > term_c_d > $o,Prod: produc8451535801715972489rm_c_d] :
      ( ( H @ ( produc3147824153920350274_c_d_o @ F @ Prod ) )
      = ( produc2919470507846847027rm_a_b
        @ ^ [X1: term_a_b,X2: term_c_d] : ( H @ ( F @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_23_prod_Ocase__distrib,axiom,
    ! [H: produc5661168290637732233rm_a_b > $o,F: term_a_b > term_c_d > produc5661168290637732233rm_a_b,Prod: produc8451535801715972489rm_c_d] :
      ( ( H @ ( produc2919470507846847027rm_a_b @ F @ Prod ) )
      = ( produc3147824153920350274_c_d_o
        @ ^ [X1: term_a_b,X2: term_c_d] : ( H @ ( F @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_24_prod_Ocase__distrib,axiom,
    ! [H: $o > produc8451535801715972489rm_c_d > $o,F: term_a_b > term_c_d > $o,Prod: produc8451535801715972489rm_c_d] :
      ( ( H @ ( produc3147824153920350274_c_d_o @ F @ Prod ) )
      = ( produc7053737044884821514_c_d_o
        @ ^ [X1: term_a_b,X2: term_c_d] : ( H @ ( F @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_25_prod_Ocase__distrib,axiom,
    ! [H: $o > produc5661168290637732233rm_a_b > $o,F: term_c_d > term_a_b > $o,Prod: produc5661168290637732233rm_a_b] :
      ( ( H @ ( produc327966649874835522_a_b_o @ F @ Prod ) )
      = ( produc5873602711790147082_a_b_o
        @ ^ [X1: term_c_d,X2: term_a_b] : ( H @ ( F @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_26_prod_Ocase__distrib,axiom,
    ! [H: ( produc5661168290637732233rm_a_b > $o ) > $o,F: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o,Prod: produc5661168290637732233rm_a_b] :
      ( ( H @ ( produc5873602711790147082_a_b_o @ F @ Prod ) )
      = ( produc327966649874835522_a_b_o
        @ ^ [X1: term_c_d,X2: term_a_b] : ( H @ ( F @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_27_prod_Ocase__distrib,axiom,
    ! [H: ( produc8451535801715972489rm_c_d > $o ) > $o,F: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o,Prod: produc8451535801715972489rm_c_d] :
      ( ( H @ ( produc7053737044884821514_c_d_o @ F @ Prod ) )
      = ( produc3147824153920350274_c_d_o
        @ ^ [X1: term_a_b,X2: term_c_d] : ( H @ ( F @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_28_case__prod__app,axiom,
    ( produc5873602711790147082_a_b_o
    = ( ^ [F2: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o,X: produc5661168290637732233rm_a_b,Y: produc5661168290637732233rm_a_b] :
          ( produc327966649874835522_a_b_o
          @ ^ [L: term_c_d,R2: term_a_b] : ( F2 @ L @ R2 @ Y )
          @ X ) ) ) ).

% case_prod_app
thf(fact_29_case__prod__app,axiom,
    ( produc7053737044884821514_c_d_o
    = ( ^ [F2: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o,X: produc8451535801715972489rm_c_d,Y: produc8451535801715972489rm_c_d] :
          ( produc3147824153920350274_c_d_o
          @ ^ [L: term_a_b,R2: term_c_d] : ( F2 @ L @ R2 @ Y )
          @ X ) ) ) ).

% case_prod_app
thf(fact_30_Ball__def,axiom,
    ( ball_P7180823220487229619rm_a_b
    = ( ^ [A2: set_Pr4386577575007340137rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
        ! [X: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ A2 )
         => ( P2 @ X ) ) ) ) ).

% Ball_def
thf(fact_31_Ball__def,axiom,
    ( ball_P8096063237992195499at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
        ! [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ A2 )
         => ( P2 @ X ) ) ) ) ).

% Ball_def
thf(fact_32_Ball__def,axiom,
    ( ball_o
    = ( ^ [A2: set_o,P2: $o > $o] :
        ! [X: $o] :
          ( ( member_o @ X @ A2 )
         => ( P2 @ X ) ) ) ) ).

% Ball_def
thf(fact_33_Ball__def,axiom,
    ( ball_set_nat
    = ( ^ [A2: set_set_nat,P2: set_nat > $o] :
        ! [X: set_nat] :
          ( ( member_set_nat @ X @ A2 )
         => ( P2 @ X ) ) ) ) ).

% Ball_def
thf(fact_34_Ball__def,axiom,
    ( ball_nat
    = ( ^ [A2: set_nat,P2: nat > $o] :
        ! [X: nat] :
          ( ( member_nat @ X @ A2 )
         => ( P2 @ X ) ) ) ) ).

% Ball_def
thf(fact_35_ball__reg,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ R3 )
         => ( ( P @ X3 )
           => ( Q @ X3 ) ) )
     => ( ! [X3: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X3 @ R3 )
           => ( P @ X3 ) )
       => ! [X4: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X4 @ R3 )
           => ( Q @ X4 ) ) ) ) ).

% ball_reg
thf(fact_36_ball__reg,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ R3 )
         => ( ( P @ X3 )
           => ( Q @ X3 ) ) )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ R3 )
           => ( P @ X3 ) )
       => ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ R3 )
           => ( Q @ X4 ) ) ) ) ).

% ball_reg
thf(fact_37_ball__reg,axiom,
    ! [R3: set_o,P: $o > $o,Q: $o > $o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ R3 )
         => ( ( P @ X3 )
           => ( Q @ X3 ) ) )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ R3 )
           => ( P @ X3 ) )
       => ! [X4: $o] :
            ( ( member_o @ X4 @ R3 )
           => ( Q @ X4 ) ) ) ) ).

% ball_reg
thf(fact_38_ball__reg,axiom,
    ! [R3: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ R3 )
         => ( ( P @ X3 )
           => ( Q @ X3 ) ) )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ R3 )
           => ( P @ X3 ) )
       => ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ R3 )
           => ( Q @ X4 ) ) ) ) ).

% ball_reg
thf(fact_39_ball__reg,axiom,
    ! [R3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ R3 )
         => ( ( P @ X3 )
           => ( Q @ X3 ) ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ R3 )
           => ( P @ X3 ) )
       => ! [X4: nat] :
            ( ( member_nat @ X4 @ R3 )
           => ( Q @ X4 ) ) ) ) ).

% ball_reg
thf(fact_40_irrefl__on__distinct,axiom,
    ( irrefl6620156600162922239rm_a_b
    = ( ^ [A2: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
        ! [X: produc4523971788519308903rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ X @ R2 )
         => ( produc161078424057476524_a_b_o
            @ ^ [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
                ( ( member5869715511025134514rm_a_b @ A @ A2 )
               => ( ( member5869715511025134514rm_a_b @ B @ A2 )
                 => ( A != B ) ) )
            @ X ) ) ) ) ).

% irrefl_on_distinct
thf(fact_41_irrefl__on__distinct,axiom,
    ( irrefl_on_o
    = ( ^ [A2: set_o,R2: set_Product_prod_o_o] :
        ! [X: product_prod_o_o] :
          ( ( member7466972457876170832od_o_o @ X @ R2 )
         => ( produc6197397395684419436_o_o_o
            @ ^ [A: $o,B: $o] :
                ( ( member_o @ A @ A2 )
               => ( ( member_o @ B @ A2 )
                 => ( A = ~ B ) ) )
            @ X ) ) ) ) ).

% irrefl_on_distinct
thf(fact_42_irrefl__on__distinct,axiom,
    ( irrefl_on_set_nat
    = ( ^ [A2: set_set_nat,R2: set_Pr5488025237498180813et_nat] :
        ! [X: produc7819656566062154093et_nat] :
          ( ( member8277197624267554838et_nat @ X @ R2 )
         => ( produc6247414631856714078_nat_o
            @ ^ [A: set_nat,B: set_nat] :
                ( ( member_set_nat @ A @ A2 )
               => ( ( member_set_nat @ B @ A2 )
                 => ( A != B ) ) )
            @ X ) ) ) ) ).

% irrefl_on_distinct
thf(fact_43_irrefl__on__distinct,axiom,
    ( irrefl7408258699751485431at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
        ! [X: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X @ R2 )
         => ( produc6590410687421337004_nat_o
            @ ^ [A: product_prod_nat_nat,B: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ A @ A2 )
               => ( ( member8440522571783428010at_nat @ B @ A2 )
                 => ( A != B ) ) )
            @ X ) ) ) ) ).

% irrefl_on_distinct
thf(fact_44_irrefl__on__distinct,axiom,
    ( irrefl_on_term_a_b
    = ( ^ [A2: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
        ! [X: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ R2 )
         => ( produc6251846659952913346_a_b_o
            @ ^ [A: term_a_b,B: term_a_b] :
                ( ( member_term_a_b @ A @ A2 )
               => ( ( member_term_a_b @ B @ A2 )
                 => ( A != B ) ) )
            @ X ) ) ) ) ).

% irrefl_on_distinct
thf(fact_45_irrefl__on__distinct,axiom,
    ( irrefl_on_nat
    = ( ^ [A2: set_nat,R2: set_Pr1261947904930325089at_nat] :
        ! [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ R2 )
         => ( produc6081775807080527818_nat_o
            @ ^ [A: nat,B: nat] :
                ( ( member_nat @ A @ A2 )
               => ( ( member_nat @ B @ A2 )
                 => ( A != B ) ) )
            @ X ) ) ) ) ).

% irrefl_on_distinct
thf(fact_46_llrg__def,axiom,
    ( rewrit4370356227420379230_d_c_d
    = ( ^ [R4: set_Pr5080540287643815273rm_c_d] :
        ! [X: produc4531938369520450825rm_c_d] :
          ( ( member820888157712331442rm_c_d @ X @ R4 )
         => ( produc6447316180697048258_c_d_o
            @ ^ [L: term_c_d,R2: term_c_d] :
                ( ( terms_9172279080238252641rm_c_d @ L )
                & ( term_ground_c_d @ R2 ) )
            @ X ) ) ) ) ).

% llrg_def
thf(fact_47_llrg__def,axiom,
    ( rewrit5000278123254648670_b_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
        ! [X: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ R4 )
         => ( produc6251846659952913346_a_b_o
            @ ^ [L: term_a_b,R2: term_a_b] :
                ( ( terms_5523818207328828897rm_a_b @ L )
                & ( term_ground_a_b @ R2 ) )
            @ X ) ) ) ) ).

% llrg_def
thf(fact_48_llrg__def,axiom,
    ( rewrit8648738996164072414_b_c_d
    = ( ^ [R4: set_Pr4645549399426460137rm_c_d] :
        ! [X: produc8451535801715972489rm_c_d] :
          ( ( member4740485589907853106rm_c_d @ X @ R4 )
         => ( produc3147824153920350274_c_d_o
            @ ^ [L: term_a_b,R2: term_c_d] :
                ( ( terms_5523818207328828897rm_a_b @ L )
                & ( term_ground_c_d @ R2 ) )
            @ X ) ) ) ) ).

% llrg_def
thf(fact_49_llrg__def,axiom,
    ( rewrit721895354510955486_d_a_b
    = ( ^ [R4: set_Pr4821568463224695273rm_a_b] :
        ! [X: produc5661168290637732233rm_a_b] :
          ( ( member1950118078829612850rm_a_b @ X @ R4 )
         => ( produc327966649874835522_a_b_o
            @ ^ [L: term_c_d,R2: term_a_b] :
                ( ( terms_9172279080238252641rm_c_d @ L )
                & ( term_ground_a_b @ R2 ) )
            @ X ) ) ) ) ).

% llrg_def
thf(fact_50_irrefl__on__converse,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ( irrefl_on_term_c_d @ A3 @ ( conver7337423426258204773rm_c_d @ R ) )
      = ( irrefl_on_term_c_d @ A3 @ R ) ) ).

% irrefl_on_converse
thf(fact_51_irrefl__on__converse,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( irrefl_on_term_a_b @ A3 @ ( conver7608559976937845093rm_a_b @ R ) )
      = ( irrefl_on_term_a_b @ A3 @ R ) ) ).

% irrefl_on_converse
thf(fact_52_irrefl__on__converse,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ A3 @ ( converse_nat_nat @ R ) )
      = ( irrefl_on_nat @ A3 @ R ) ) ).

% irrefl_on_converse
thf(fact_53_irrefl__on__converse,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( irrefl7408258699751485431at_nat @ A3 @ ( conver5251360641855610363at_nat @ R ) )
      = ( irrefl7408258699751485431at_nat @ A3 @ R ) ) ).

% irrefl_on_converse
thf(fact_54_gound__linear,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( terms_5523818207328828897rm_a_b @ T ) ) ).

% gound_linear
thf(fact_55_gound__linear,axiom,
    ! [T: term_c_d] :
      ( ( term_ground_c_d @ T )
     => ( terms_9172279080238252641rm_c_d @ T ) ) ).

% gound_linear
thf(fact_56_internal__case__prod__def,axiom,
    produc7818728302573061600_a_b_o = produc5873602711790147082_a_b_o ).

% internal_case_prod_def
thf(fact_57_internal__case__prod__def,axiom,
    produc6084579033702138589rm_a_b = produc2919470507846847027rm_a_b ).

% internal_case_prod_def
thf(fact_58_internal__case__prod__def,axiom,
    produc8998862635667736032_c_d_o = produc7053737044884821514_c_d_o ).

% internal_case_prod_def
thf(fact_59_internal__case__prod__def,axiom,
    produc3341718396117254252_a_b_o = produc6251846659952913346_a_b_o ).

% internal_case_prod_def
thf(fact_60_internal__case__prod__def,axiom,
    produc9139807813499881076_nat_o = produc6081775807080527818_nat_o ).

% internal_case_prod_def
thf(fact_61_internal__case__prod__def,axiom,
    produc237695890084691180_c_d_o = produc3147824153920350274_c_d_o ).

% internal_case_prod_def
thf(fact_62_internal__case__prod__def,axiom,
    produc6641210422893952236_a_b_o = produc327966649874835522_a_b_o ).

% internal_case_prod_def
thf(fact_63_curry__case__prod,axiom,
    ! [F: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o] :
      ( ( produc1963403694558327721_a_b_o @ ( produc5873602711790147082_a_b_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_64_curry__case__prod,axiom,
    ! [F: term_a_b > term_c_d > produc5661168290637732233rm_a_b] :
      ( ( produc1511760821894535124rm_a_b @ ( produc2919470507846847027rm_a_b @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_65_curry__case__prod,axiom,
    ! [F: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o] :
      ( ( produc3143538027653002153_c_d_o @ ( produc7053737044884821514_c_d_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_66_curry__case__prod,axiom,
    ! [F: term_a_b > term_a_b > $o] :
      ( ( produc6008550485034061667_a_b_o @ ( produc6251846659952913346_a_b_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_67_curry__case__prod,axiom,
    ! [F: nat > nat > $o] :
      ( ( produc1310100445399344235_nat_o @ ( produc6081775807080527818_nat_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_68_curry__case__prod,axiom,
    ! [F: term_a_b > term_c_d > $o] :
      ( ( produc2904527979001498595_c_d_o @ ( produc3147824153920350274_c_d_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_69_curry__case__prod,axiom,
    ! [F: term_c_d > term_a_b > $o] :
      ( ( produc84670474955983843_a_b_o @ ( produc327966649874835522_a_b_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_70_case__prod__curry,axiom,
    ! [F: produc5661168290637732233rm_a_b > produc5661168290637732233rm_a_b > $o] :
      ( ( produc5873602711790147082_a_b_o @ ( produc1963403694558327721_a_b_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_71_case__prod__curry,axiom,
    ! [F: produc8451535801715972489rm_c_d > produc5661168290637732233rm_a_b] :
      ( ( produc2919470507846847027rm_a_b @ ( produc1511760821894535124rm_a_b @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_72_case__prod__curry,axiom,
    ! [F: produc8451535801715972489rm_c_d > produc8451535801715972489rm_c_d > $o] :
      ( ( produc7053737044884821514_c_d_o @ ( produc3143538027653002153_c_d_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_73_case__prod__curry,axiom,
    ! [F: produc357393685978478089rm_a_b > $o] :
      ( ( produc6251846659952913346_a_b_o @ ( produc6008550485034061667_a_b_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_74_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o @ ( produc1310100445399344235_nat_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_75_case__prod__curry,axiom,
    ! [F: produc8451535801715972489rm_c_d > $o] :
      ( ( produc3147824153920350274_c_d_o @ ( produc2904527979001498595_c_d_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_76_case__prod__curry,axiom,
    ! [F: produc5661168290637732233rm_a_b > $o] :
      ( ( produc327966649874835522_a_b_o @ ( produc84670474955983843_a_b_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_77_case__swap,axiom,
    ! [F: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o,P3: produc5661168290637732233rm_a_b] :
      ( ( produc432089842495231498_a_b_o
        @ ^ [Y: term_a_b,X: term_c_d] : ( F @ X @ Y )
        @ ( produc7845621393534482785rm_a_b @ P3 ) )
      = ( produc5873602711790147082_a_b_o @ F @ P3 ) ) ).

% case_swap
thf(fact_78_case__swap,axiom,
    ! [F: term_a_b > term_c_d > produc5661168290637732233rm_a_b,P3: produc8451535801715972489rm_c_d] :
      ( ( produc2580071269002545715rm_a_b
        @ ^ [Y: term_c_d,X: term_a_b] : ( F @ X @ Y )
        @ ( produc6190307670391109985rm_c_d @ P3 ) )
      = ( produc2919470507846847027rm_a_b @ F @ P3 ) ) ).

% case_swap
thf(fact_79_case__swap,axiom,
    ! [F: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o,P3: produc8451535801715972489rm_c_d] :
      ( ( produc3271877877324961290_c_d_o
        @ ^ [Y: term_c_d,X: term_a_b] : ( F @ X @ Y )
        @ ( produc6190307670391109985rm_c_d @ P3 ) )
      = ( produc7053737044884821514_c_d_o @ F @ P3 ) ) ).

% case_swap
thf(fact_80_case__swap,axiom,
    ! [F: term_a_b > term_c_d > produc5661168290637732233rm_a_b > $o,P3: produc8451535801715972489rm_c_d] :
      ( ( produc5873602711790147082_a_b_o
        @ ^ [Y: term_c_d,X: term_a_b] : ( F @ X @ Y )
        @ ( produc6190307670391109985rm_c_d @ P3 ) )
      = ( produc432089842495231498_a_b_o @ F @ P3 ) ) ).

% case_swap
thf(fact_81_case__swap,axiom,
    ! [F: term_c_d > term_a_b > produc5661168290637732233rm_a_b,P3: produc5661168290637732233rm_a_b] :
      ( ( produc2919470507846847027rm_a_b
        @ ^ [Y: term_a_b,X: term_c_d] : ( F @ X @ Y )
        @ ( produc7845621393534482785rm_a_b @ P3 ) )
      = ( produc2580071269002545715rm_a_b @ F @ P3 ) ) ).

% case_swap
thf(fact_82_case__swap,axiom,
    ! [F: term_c_d > term_a_b > produc8451535801715972489rm_c_d > $o,P3: produc5661168290637732233rm_a_b] :
      ( ( produc7053737044884821514_c_d_o
        @ ^ [Y: term_a_b,X: term_c_d] : ( F @ X @ Y )
        @ ( produc7845621393534482785rm_a_b @ P3 ) )
      = ( produc3271877877324961290_c_d_o @ F @ P3 ) ) ).

% case_swap
thf(fact_83_case__swap,axiom,
    ! [F: term_a_b > term_a_b > $o,P3: produc357393685978478089rm_a_b] :
      ( ( produc6251846659952913346_a_b_o
        @ ^ [Y: term_a_b,X: term_a_b] : ( F @ X @ Y )
        @ ( produc2541846788875228641rm_a_b @ P3 ) )
      = ( produc6251846659952913346_a_b_o @ F @ P3 ) ) ).

% case_swap
thf(fact_84_case__swap,axiom,
    ! [F: nat > nat > $o,P3: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [Y: nat,X: nat] : ( F @ X @ Y )
        @ ( product_swap_nat_nat @ P3 ) )
      = ( produc6081775807080527818_nat_o @ F @ P3 ) ) ).

% case_swap
thf(fact_85_case__swap,axiom,
    ! [F: term_c_d > term_a_b > $o,P3: produc5661168290637732233rm_a_b] :
      ( ( produc3147824153920350274_c_d_o
        @ ^ [Y: term_a_b,X: term_c_d] : ( F @ X @ Y )
        @ ( produc7845621393534482785rm_a_b @ P3 ) )
      = ( produc327966649874835522_a_b_o @ F @ P3 ) ) ).

% case_swap
thf(fact_86_case__swap,axiom,
    ! [F: term_a_b > term_c_d > $o,P3: produc8451535801715972489rm_c_d] :
      ( ( produc327966649874835522_a_b_o
        @ ^ [Y: term_c_d,X: term_a_b] : ( F @ X @ Y )
        @ ( produc6190307670391109985rm_c_d @ P3 ) )
      = ( produc3147824153920350274_c_d_o @ F @ P3 ) ) ).

% case_swap
thf(fact_87_llrg__ground__rhs,axiom,
    ! [R5: set_Pr4821568463224695273rm_a_b,L2: term_c_d,R: term_a_b] :
      ( ( rewrit721895354510955486_d_a_b @ R5 )
     => ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ L2 @ R ) @ R5 )
       => ( term_ground_a_b @ R ) ) ) ).

% llrg_ground_rhs
thf(fact_88_llrg__ground__rhs,axiom,
    ! [R5: set_Pr4645549399426460137rm_c_d,L2: term_a_b,R: term_c_d] :
      ( ( rewrit8648738996164072414_b_c_d @ R5 )
     => ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ L2 @ R ) @ R5 )
       => ( term_ground_c_d @ R ) ) ) ).

% llrg_ground_rhs
thf(fact_89_llrg__ground__rhs,axiom,
    ! [R5: set_Pr5080540287643815273rm_c_d,L2: term_c_d,R: term_c_d] :
      ( ( rewrit4370356227420379230_d_c_d @ R5 )
     => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ L2 @ R ) @ R5 )
       => ( term_ground_c_d @ R ) ) ) ).

% llrg_ground_rhs
thf(fact_90_llrg__ground__rhs,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,L2: term_a_b,R: term_a_b] :
      ( ( rewrit5000278123254648670_b_a_b @ R5 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R ) @ R5 )
       => ( term_ground_a_b @ R ) ) ) ).

% llrg_ground_rhs
thf(fact_91_converse__unfold,axiom,
    ( conver7337423426258204773rm_c_d
    = ( ^ [R2: set_Pr5080540287643815273rm_c_d] :
          ( collec4273722079467896820rm_c_d
          @ ( produc6447316180697048258_c_d_o
            @ ^ [Y: term_c_d,X: term_c_d] : ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X @ Y ) @ R2 ) ) ) ) ) ).

% converse_unfold
thf(fact_92_converse__unfold,axiom,
    ( conver7608559976937845093rm_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b] :
          ( collec99177395925924084rm_a_b
          @ ( produc6251846659952913346_a_b_o
            @ ^ [Y: term_a_b,X: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 ) ) ) ) ) ).

% converse_unfold
thf(fact_93_converse__unfold,axiom,
    ( converse_nat_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [Y: nat,X: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% converse_unfold
thf(fact_94_converse__unfold,axiom,
    ( conver3688962544742323429rm_a_b
    = ( ^ [R2: set_Pr4821568463224695273rm_a_b] :
          ( collec8193319511663418484rm_c_d
          @ ( produc3147824153920350274_c_d_o
            @ ^ [Y: term_a_b,X: term_c_d] : ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ X @ Y ) @ R2 ) ) ) ) ) ).

% converse_unfold
thf(fact_95_converse__unfold,axiom,
    ( conver2033648821598950629rm_c_d
    = ( ^ [R2: set_Pr4645549399426460137rm_c_d] :
          ( collec5402952000585178228rm_a_b
          @ ( produc327966649874835522_a_b_o
            @ ^ [Y: term_c_d,X: term_a_b] : ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ X @ Y ) @ R2 ) ) ) ) ) ).

% converse_unfold
thf(fact_96_converse__inv__image,axiom,
    ! [R3: set_Pr5080540287643815273rm_c_d,F: term_c_d > term_c_d] :
      ( ( conver7337423426258204773rm_c_d @ ( inv_im2733088348190002300rm_c_d @ R3 @ F ) )
      = ( inv_im2733088348190002300rm_c_d @ ( conver7337423426258204773rm_c_d @ R3 ) @ F ) ) ).

% converse_inv_image
thf(fact_97_converse__inv__image,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,F: term_c_d > term_a_b] :
      ( ( conver7337423426258204773rm_c_d @ ( inv_im6652685780385523964rm_c_d @ R3 @ F ) )
      = ( inv_im6652685780385523964rm_c_d @ ( conver7608559976937845093rm_a_b @ R3 ) @ F ) ) ).

% converse_inv_image
thf(fact_98_converse__inv__image,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F: term_c_d > nat] :
      ( ( conver7337423426258204773rm_c_d @ ( inv_im5296270683936134712rm_c_d @ R3 @ F ) )
      = ( inv_im5296270683936134712rm_c_d @ ( converse_nat_nat @ R3 ) @ F ) ) ).

% converse_inv_image
thf(fact_99_converse__inv__image,axiom,
    ! [R3: set_Pr5080540287643815273rm_c_d,F: term_a_b > term_c_d] :
      ( ( conver7608559976937845093rm_a_b @ ( inv_im8307999503528896764rm_a_b @ R3 @ F ) )
      = ( inv_im8307999503528896764rm_a_b @ ( conver7337423426258204773rm_c_d @ R3 ) @ F ) ) ).

% converse_inv_image
thf(fact_100_converse__inv__image,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,F: term_a_b > term_a_b] :
      ( ( conver7608559976937845093rm_a_b @ ( inv_im3004224898869642620rm_a_b @ R3 @ F ) )
      = ( inv_im3004224898869642620rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ F ) ) ).

% converse_inv_image
thf(fact_101_converse__inv__image,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F: term_a_b > nat] :
      ( ( conver7608559976937845093rm_a_b @ ( inv_im1647809802420253368rm_a_b @ R3 @ F ) )
      = ( inv_im1647809802420253368rm_a_b @ ( converse_nat_nat @ R3 ) @ F ) ) ).

% converse_inv_image
thf(fact_102_converse__inv__image,axiom,
    ! [R3: set_Pr5080540287643815273rm_c_d,F: nat > term_c_d] :
      ( ( converse_nat_nat @ ( inv_im7267259196725686968_d_nat @ R3 @ F ) )
      = ( inv_im7267259196725686968_d_nat @ ( conver7337423426258204773rm_c_d @ R3 ) @ F ) ) ).

% converse_inv_image
thf(fact_103_converse__inv__image,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,F: nat > term_a_b] :
      ( ( converse_nat_nat @ ( inv_im2609263929585225528_b_nat @ R3 @ F ) )
      = ( inv_im2609263929585225528_b_nat @ ( conver7608559976937845093rm_a_b @ R3 ) @ F ) ) ).

% converse_inv_image
thf(fact_104_converse__inv__image,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( converse_nat_nat @ ( inv_image_nat_nat @ R3 @ F ) )
      = ( inv_image_nat_nat @ ( converse_nat_nat @ R3 ) @ F ) ) ).

% converse_inv_image
thf(fact_105_prod_Oinject,axiom,
    ! [X12: term_a_b,X22: term_c_d,Y1: term_a_b,Y2: term_c_d] :
      ( ( ( produc1445286645097778113rm_c_d @ X12 @ X22 )
        = ( produc1445286645097778113rm_c_d @ Y1 @ Y2 ) )
      = ( ( X12 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_106_prod_Oinject,axiom,
    ! [X12: term_c_d,X22: term_a_b,Y1: term_c_d,Y2: term_a_b] :
      ( ( ( produc3100600368241150913rm_a_b @ X12 @ X22 )
        = ( produc3100600368241150913rm_a_b @ Y1 @ Y2 ) )
      = ( ( X12 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_107_prod_Oinject,axiom,
    ! [X12: term_c_d,X22: term_c_d,Y1: term_c_d,Y2: term_c_d] :
      ( ( ( produc6749061249757032257rm_c_d @ X12 @ X22 )
        = ( produc6749061249757032257rm_c_d @ Y1 @ Y2 ) )
      = ( ( X12 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_108_prod_Oinject,axiom,
    ! [X12: term_a_b,X22: term_a_b,Y1: term_a_b,Y2: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ X12 @ X22 )
        = ( produc7020197800436672577rm_a_b @ Y1 @ Y2 ) )
      = ( ( X12 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_109_prod_Oinject,axiom,
    ! [X12: nat,X22: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X12 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X12 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_110_old_Oprod_Oinject,axiom,
    ! [A4: term_a_b,B2: term_c_d,A5: term_a_b,B3: term_c_d] :
      ( ( ( produc1445286645097778113rm_c_d @ A4 @ B2 )
        = ( produc1445286645097778113rm_c_d @ A5 @ B3 ) )
      = ( ( A4 = A5 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_111_old_Oprod_Oinject,axiom,
    ! [A4: term_c_d,B2: term_a_b,A5: term_c_d,B3: term_a_b] :
      ( ( ( produc3100600368241150913rm_a_b @ A4 @ B2 )
        = ( produc3100600368241150913rm_a_b @ A5 @ B3 ) )
      = ( ( A4 = A5 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_112_old_Oprod_Oinject,axiom,
    ! [A4: term_c_d,B2: term_c_d,A5: term_c_d,B3: term_c_d] :
      ( ( ( produc6749061249757032257rm_c_d @ A4 @ B2 )
        = ( produc6749061249757032257rm_c_d @ A5 @ B3 ) )
      = ( ( A4 = A5 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_113_old_Oprod_Oinject,axiom,
    ! [A4: term_a_b,B2: term_a_b,A5: term_a_b,B3: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ A4 @ B2 )
        = ( produc7020197800436672577rm_a_b @ A5 @ B3 ) )
      = ( ( A4 = A5 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_114_old_Oprod_Oinject,axiom,
    ! [A4: nat,B2: nat,A5: nat,B3: nat] :
      ( ( ( product_Pair_nat_nat @ A4 @ B2 )
        = ( product_Pair_nat_nat @ A5 @ B3 ) )
      = ( ( A4 = A5 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_115_curryI,axiom,
    ! [F: produc4531938369520450825rm_c_d > $o,A4: term_c_d,B2: term_c_d] :
      ( ( F @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) )
     => ( produc6204020005778196579_c_d_o @ F @ A4 @ B2 ) ) ).

% curryI
thf(fact_116_curryI,axiom,
    ! [F: produc357393685978478089rm_a_b > $o,A4: term_a_b,B2: term_a_b] :
      ( ( F @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) )
     => ( produc6008550485034061667_a_b_o @ F @ A4 @ B2 ) ) ).

% curryI
thf(fact_117_curryI,axiom,
    ! [F: product_prod_nat_nat > $o,A4: nat,B2: nat] :
      ( ( F @ ( product_Pair_nat_nat @ A4 @ B2 ) )
     => ( produc1310100445399344235_nat_o @ F @ A4 @ B2 ) ) ).

% curryI
thf(fact_118_curryI,axiom,
    ! [F: produc5661168290637732233rm_a_b > $o,A4: term_c_d,B2: term_a_b] :
      ( ( F @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) )
     => ( produc84670474955983843_a_b_o @ F @ A4 @ B2 ) ) ).

% curryI
thf(fact_119_curryI,axiom,
    ! [F: produc8451535801715972489rm_c_d > $o,A4: term_a_b,B2: term_c_d] :
      ( ( F @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) )
     => ( produc2904527979001498595_c_d_o @ F @ A4 @ B2 ) ) ).

% curryI
thf(fact_120_swap__swap,axiom,
    ! [P3: produc5661168290637732233rm_a_b] :
      ( ( produc6190307670391109985rm_c_d @ ( produc7845621393534482785rm_a_b @ P3 ) )
      = P3 ) ).

% swap_swap
thf(fact_121_swap__swap,axiom,
    ! [P3: produc8451535801715972489rm_c_d] :
      ( ( produc7845621393534482785rm_a_b @ ( produc6190307670391109985rm_c_d @ P3 ) )
      = P3 ) ).

% swap_swap
thf(fact_122_mem__case__prodI2,axiom,
    ! [P3: product_prod_nat_nat,Z: $o,C: nat > nat > set_o] :
      ( ! [A6: nat,B4: nat] :
          ( ( P3
            = ( product_Pair_nat_nat @ A6 @ B4 ) )
         => ( member_o @ Z @ ( C @ A6 @ B4 ) ) )
     => ( member_o @ Z @ ( produc59986286002894506_set_o @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_123_mem__case__prodI2,axiom,
    ! [P3: product_prod_nat_nat,Z: nat,C: nat > nat > set_nat] :
      ( ! [A6: nat,B4: nat] :
          ( ( P3
            = ( product_Pair_nat_nat @ A6 @ B4 ) )
         => ( member_nat @ Z @ ( C @ A6 @ B4 ) ) )
     => ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_124_mem__case__prodI2,axiom,
    ! [P3: product_prod_nat_nat,Z: set_nat,C: nat > nat > set_set_nat] :
      ( ! [A6: nat,B4: nat] :
          ( ( P3
            = ( product_Pair_nat_nat @ A6 @ B4 ) )
         => ( member_set_nat @ Z @ ( C @ A6 @ B4 ) ) )
     => ( member_set_nat @ Z @ ( produc8404753619367356554et_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_125_mem__case__prodI2,axiom,
    ! [P3: product_prod_nat_nat,Z: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat] :
      ( ! [A6: nat,B4: nat] :
          ( ( P3
            = ( product_Pair_nat_nat @ A6 @ B4 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C @ A6 @ B4 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_126_mem__case__prodI2,axiom,
    ! [P3: produc8451535801715972489rm_c_d,Z: $o,C: term_a_b > term_c_d > set_o] :
      ( ! [A6: term_a_b,B4: term_c_d] :
          ( ( P3
            = ( produc1445286645097778113rm_c_d @ A6 @ B4 ) )
         => ( member_o @ Z @ ( C @ A6 @ B4 ) ) )
     => ( member_o @ Z @ ( produc7074531164778841890_set_o @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_127_mem__case__prodI2,axiom,
    ! [P3: produc8451535801715972489rm_c_d,Z: nat,C: term_a_b > term_c_d > set_nat] :
      ( ! [A6: term_a_b,B4: term_c_d] :
          ( ( P3
            = ( produc1445286645097778113rm_c_d @ A6 @ B4 ) )
         => ( member_nat @ Z @ ( C @ A6 @ B4 ) ) )
     => ( member_nat @ Z @ ( produc5232004564063148380et_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_128_mem__case__prodI2,axiom,
    ! [P3: produc5661168290637732233rm_a_b,Z: $o,C: term_c_d > term_a_b > set_o] :
      ( ! [A6: term_c_d,B4: term_a_b] :
          ( ( P3
            = ( produc3100600368241150913rm_a_b @ A6 @ B4 ) )
         => ( member_o @ Z @ ( C @ A6 @ B4 ) ) )
     => ( member_o @ Z @ ( produc2530863894259861282_set_o @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_129_mem__case__prodI2,axiom,
    ! [P3: produc5661168290637732233rm_a_b,Z: nat,C: term_c_d > term_a_b > set_nat] :
      ( ! [A6: term_c_d,B4: term_a_b] :
          ( ( P3
            = ( produc3100600368241150913rm_a_b @ A6 @ B4 ) )
         => ( member_nat @ Z @ ( C @ A6 @ B4 ) ) )
     => ( member_nat @ Z @ ( produc7500204723417323868et_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_130_mem__case__prodI2,axiom,
    ! [P3: produc4531938369520450825rm_c_d,Z: $o,C: term_c_d > term_c_d > set_o] :
      ( ! [A6: term_c_d,B4: term_c_d] :
          ( ( P3
            = ( produc6749061249757032257rm_c_d @ A6 @ B4 ) )
         => ( member_o @ Z @ ( C @ A6 @ B4 ) ) )
     => ( member_o @ Z @ ( produc1641171031246963618_set_o @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_131_mem__case__prodI2,axiom,
    ! [P3: produc4531938369520450825rm_c_d,Z: nat,C: term_c_d > term_c_d > set_nat] :
      ( ! [A6: term_c_d,B4: term_c_d] :
          ( ( P3
            = ( produc6749061249757032257rm_c_d @ A6 @ B4 ) )
         => ( member_nat @ Z @ ( C @ A6 @ B4 ) ) )
     => ( member_nat @ Z @ ( produc6663801938352323804et_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_132_mem__case__prodI,axiom,
    ! [Z: $o,C: nat > nat > set_o,A4: nat,B2: nat] :
      ( ( member_o @ Z @ ( C @ A4 @ B2 ) )
     => ( member_o @ Z @ ( produc59986286002894506_set_o @ C @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_133_mem__case__prodI,axiom,
    ! [Z: nat,C: nat > nat > set_nat,A4: nat,B2: nat] :
      ( ( member_nat @ Z @ ( C @ A4 @ B2 ) )
     => ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_134_mem__case__prodI,axiom,
    ! [Z: set_nat,C: nat > nat > set_set_nat,A4: nat,B2: nat] :
      ( ( member_set_nat @ Z @ ( C @ A4 @ B2 ) )
     => ( member_set_nat @ Z @ ( produc8404753619367356554et_nat @ C @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_135_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,A4: nat,B2: nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( C @ A4 @ B2 ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_136_mem__case__prodI,axiom,
    ! [Z: $o,C: term_a_b > term_c_d > set_o,A4: term_a_b,B2: term_c_d] :
      ( ( member_o @ Z @ ( C @ A4 @ B2 ) )
     => ( member_o @ Z @ ( produc7074531164778841890_set_o @ C @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_137_mem__case__prodI,axiom,
    ! [Z: nat,C: term_a_b > term_c_d > set_nat,A4: term_a_b,B2: term_c_d] :
      ( ( member_nat @ Z @ ( C @ A4 @ B2 ) )
     => ( member_nat @ Z @ ( produc5232004564063148380et_nat @ C @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_138_mem__case__prodI,axiom,
    ! [Z: $o,C: term_c_d > term_a_b > set_o,A4: term_c_d,B2: term_a_b] :
      ( ( member_o @ Z @ ( C @ A4 @ B2 ) )
     => ( member_o @ Z @ ( produc2530863894259861282_set_o @ C @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_139_mem__case__prodI,axiom,
    ! [Z: nat,C: term_c_d > term_a_b > set_nat,A4: term_c_d,B2: term_a_b] :
      ( ( member_nat @ Z @ ( C @ A4 @ B2 ) )
     => ( member_nat @ Z @ ( produc7500204723417323868et_nat @ C @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_140_mem__case__prodI,axiom,
    ! [Z: $o,C: term_c_d > term_c_d > set_o,A4: term_c_d,B2: term_c_d] :
      ( ( member_o @ Z @ ( C @ A4 @ B2 ) )
     => ( member_o @ Z @ ( produc1641171031246963618_set_o @ C @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_141_mem__case__prodI,axiom,
    ! [Z: nat,C: term_c_d > term_c_d > set_nat,A4: term_c_d,B2: term_c_d] :
      ( ( member_nat @ Z @ ( C @ A4 @ B2 ) )
     => ( member_nat @ Z @ ( produc6663801938352323804et_nat @ C @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_142_case__prodI2_H,axiom,
    ! [P3: produc5661168290637732233rm_a_b,C: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o,X5: produc5661168290637732233rm_a_b] :
      ( ! [A6: term_c_d,B4: term_a_b] :
          ( ( ( produc3100600368241150913rm_a_b @ A6 @ B4 )
            = P3 )
         => ( C @ A6 @ B4 @ X5 ) )
     => ( produc5873602711790147082_a_b_o @ C @ P3 @ X5 ) ) ).

% case_prodI2'
thf(fact_143_case__prodI2_H,axiom,
    ! [P3: produc8451535801715972489rm_c_d,C: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o,X5: produc8451535801715972489rm_c_d] :
      ( ! [A6: term_a_b,B4: term_c_d] :
          ( ( ( produc1445286645097778113rm_c_d @ A6 @ B4 )
            = P3 )
         => ( C @ A6 @ B4 @ X5 ) )
     => ( produc7053737044884821514_c_d_o @ C @ P3 @ X5 ) ) ).

% case_prodI2'
thf(fact_144_converse__iff,axiom,
    ! [A4: term_c_d,B2: term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) @ ( conver7337423426258204773rm_c_d @ R ) )
      = ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ B2 @ A4 ) @ R ) ) ).

% converse_iff
thf(fact_145_converse__iff,axiom,
    ! [A4: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) @ ( conver7608559976937845093rm_a_b @ R ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A4 ) @ R ) ) ).

% converse_iff
thf(fact_146_converse__iff,axiom,
    ! [A4: nat,B2: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( converse_nat_nat @ R ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ R ) ) ).

% converse_iff
thf(fact_147_converse__iff,axiom,
    ! [A4: term_c_d,B2: term_a_b,R: set_Pr4645549399426460137rm_c_d] :
      ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) @ ( conver2033648821598950629rm_c_d @ R ) )
      = ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ B2 @ A4 ) @ R ) ) ).

% converse_iff
thf(fact_148_converse__iff,axiom,
    ! [A4: term_a_b,B2: term_c_d,R: set_Pr4821568463224695273rm_a_b] :
      ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) @ ( conver3688962544742323429rm_a_b @ R ) )
      = ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ B2 @ A4 ) @ R ) ) ).

% converse_iff
thf(fact_149_in__inv__image,axiom,
    ! [X5: term_c_d,Y3: term_c_d,R: set_Pr5080540287643815273rm_c_d,F: term_c_d > term_c_d] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Y3 ) @ ( inv_im2733088348190002300rm_c_d @ R @ F ) )
      = ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ ( F @ X5 ) @ ( F @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_150_in__inv__image,axiom,
    ! [X5: term_c_d,Y3: term_c_d,R: set_Pr4386577575007340137rm_a_b,F: term_c_d > term_a_b] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Y3 ) @ ( inv_im6652685780385523964rm_c_d @ R @ F ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ X5 ) @ ( F @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_151_in__inv__image,axiom,
    ! [X5: term_c_d,Y3: term_c_d,R: set_Pr1261947904930325089at_nat,F: term_c_d > nat] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Y3 ) @ ( inv_im5296270683936134712rm_c_d @ R @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_152_in__inv__image,axiom,
    ! [X5: term_a_b,Y3: term_a_b,R: set_Pr5080540287643815273rm_c_d,F: term_a_b > term_c_d] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y3 ) @ ( inv_im8307999503528896764rm_a_b @ R @ F ) )
      = ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ ( F @ X5 ) @ ( F @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_153_in__inv__image,axiom,
    ! [X5: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,F: term_a_b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y3 ) @ ( inv_im3004224898869642620rm_a_b @ R @ F ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ X5 ) @ ( F @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_154_in__inv__image,axiom,
    ! [X5: term_a_b,Y3: term_a_b,R: set_Pr1261947904930325089at_nat,F: term_a_b > nat] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y3 ) @ ( inv_im1647809802420253368rm_a_b @ R @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_155_in__inv__image,axiom,
    ! [X5: nat,Y3: nat,R: set_Pr5080540287643815273rm_c_d,F: nat > term_c_d] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ ( inv_im7267259196725686968_d_nat @ R @ F ) )
      = ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ ( F @ X5 ) @ ( F @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_156_in__inv__image,axiom,
    ! [X5: nat,Y3: nat,R: set_Pr4386577575007340137rm_a_b,F: nat > term_a_b] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ ( inv_im2609263929585225528_b_nat @ R @ F ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ X5 ) @ ( F @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_157_in__inv__image,axiom,
    ! [X5: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ ( inv_image_nat_nat @ R @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_158_swap__simp,axiom,
    ! [X5: term_c_d,Y3: term_c_d] :
      ( ( produc2270710238195588321rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Y3 ) )
      = ( produc6749061249757032257rm_c_d @ Y3 @ X5 ) ) ).

% swap_simp
thf(fact_159_swap__simp,axiom,
    ! [X5: term_a_b,Y3: term_a_b] :
      ( ( produc2541846788875228641rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y3 ) )
      = ( produc7020197800436672577rm_a_b @ Y3 @ X5 ) ) ).

% swap_simp
thf(fact_160_swap__simp,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) )
      = ( product_Pair_nat_nat @ Y3 @ X5 ) ) ).

% swap_simp
thf(fact_161_swap__simp,axiom,
    ! [X5: term_a_b,Y3: term_c_d] :
      ( ( produc6190307670391109985rm_c_d @ ( produc1445286645097778113rm_c_d @ X5 @ Y3 ) )
      = ( produc3100600368241150913rm_a_b @ Y3 @ X5 ) ) ).

% swap_simp
thf(fact_162_swap__simp,axiom,
    ! [X5: term_c_d,Y3: term_a_b] :
      ( ( produc7845621393534482785rm_a_b @ ( produc3100600368241150913rm_a_b @ X5 @ Y3 ) )
      = ( produc1445286645097778113rm_c_d @ Y3 @ X5 ) ) ).

% swap_simp
thf(fact_163_curry__conv,axiom,
    ( produc84670474955983843_a_b_o
    = ( ^ [F2: produc5661168290637732233rm_a_b > $o,A: term_c_d,B: term_a_b] : ( F2 @ ( produc3100600368241150913rm_a_b @ A @ B ) ) ) ) ).

% curry_conv
thf(fact_164_curry__conv,axiom,
    ( produc2904527979001498595_c_d_o
    = ( ^ [F2: produc8451535801715972489rm_c_d > $o,A: term_a_b,B: term_c_d] : ( F2 @ ( produc1445286645097778113rm_c_d @ A @ B ) ) ) ) ).

% curry_conv
thf(fact_165_case__prod__conv,axiom,
    ! [F: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o,A4: term_c_d,B2: term_a_b] :
      ( ( produc5873602711790147082_a_b_o @ F @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) )
      = ( F @ A4 @ B2 ) ) ).

% case_prod_conv
thf(fact_166_case__prod__conv,axiom,
    ! [F: term_a_b > term_c_d > produc5661168290637732233rm_a_b,A4: term_a_b,B2: term_c_d] :
      ( ( produc2919470507846847027rm_a_b @ F @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) )
      = ( F @ A4 @ B2 ) ) ).

% case_prod_conv
thf(fact_167_case__prod__conv,axiom,
    ! [F: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o,A4: term_a_b,B2: term_c_d] :
      ( ( produc7053737044884821514_c_d_o @ F @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) )
      = ( F @ A4 @ B2 ) ) ).

% case_prod_conv
thf(fact_168_case__prod__conv,axiom,
    ! [F: term_a_b > term_a_b > $o,A4: term_a_b,B2: term_a_b] :
      ( ( produc6251846659952913346_a_b_o @ F @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) )
      = ( F @ A4 @ B2 ) ) ).

% case_prod_conv
thf(fact_169_case__prod__conv,axiom,
    ! [F: nat > nat > $o,A4: nat,B2: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A4 @ B2 ) )
      = ( F @ A4 @ B2 ) ) ).

% case_prod_conv
thf(fact_170_case__prod__conv,axiom,
    ! [F: term_a_b > term_c_d > $o,A4: term_a_b,B2: term_c_d] :
      ( ( produc3147824153920350274_c_d_o @ F @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) )
      = ( F @ A4 @ B2 ) ) ).

% case_prod_conv
thf(fact_171_case__prod__conv,axiom,
    ! [F: term_c_d > term_a_b > $o,A4: term_c_d,B2: term_a_b] :
      ( ( produc327966649874835522_a_b_o @ F @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) )
      = ( F @ A4 @ B2 ) ) ).

% case_prod_conv
thf(fact_172_case__prodI,axiom,
    ! [F: term_c_d > term_c_d > $o,A4: term_c_d,B2: term_c_d] :
      ( ( F @ A4 @ B2 )
     => ( produc6447316180697048258_c_d_o @ F @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) ) ) ).

% case_prodI
thf(fact_173_case__prodI,axiom,
    ! [F: term_a_b > term_a_b > $o,A4: term_a_b,B2: term_a_b] :
      ( ( F @ A4 @ B2 )
     => ( produc6251846659952913346_a_b_o @ F @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) ) ) ).

% case_prodI
thf(fact_174_case__prodI,axiom,
    ! [F: nat > nat > $o,A4: nat,B2: nat] :
      ( ( F @ A4 @ B2 )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ).

% case_prodI
thf(fact_175_case__prodI,axiom,
    ! [F: term_a_b > term_c_d > $o,A4: term_a_b,B2: term_c_d] :
      ( ( F @ A4 @ B2 )
     => ( produc3147824153920350274_c_d_o @ F @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) ) ) ).

% case_prodI
thf(fact_176_case__prodI,axiom,
    ! [F: term_c_d > term_a_b > $o,A4: term_c_d,B2: term_a_b] :
      ( ( F @ A4 @ B2 )
     => ( produc327966649874835522_a_b_o @ F @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) ) ) ).

% case_prodI
thf(fact_177_case__prodI2,axiom,
    ! [P3: produc4531938369520450825rm_c_d,C: term_c_d > term_c_d > $o] :
      ( ! [A6: term_c_d,B4: term_c_d] :
          ( ( P3
            = ( produc6749061249757032257rm_c_d @ A6 @ B4 ) )
         => ( C @ A6 @ B4 ) )
     => ( produc6447316180697048258_c_d_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_178_case__prodI2,axiom,
    ! [P3: produc357393685978478089rm_a_b,C: term_a_b > term_a_b > $o] :
      ( ! [A6: term_a_b,B4: term_a_b] :
          ( ( P3
            = ( produc7020197800436672577rm_a_b @ A6 @ B4 ) )
         => ( C @ A6 @ B4 ) )
     => ( produc6251846659952913346_a_b_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_179_case__prodI2,axiom,
    ! [P3: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A6: nat,B4: nat] :
          ( ( P3
            = ( product_Pair_nat_nat @ A6 @ B4 ) )
         => ( C @ A6 @ B4 ) )
     => ( produc6081775807080527818_nat_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_180_case__prodI2,axiom,
    ! [P3: produc8451535801715972489rm_c_d,C: term_a_b > term_c_d > $o] :
      ( ! [A6: term_a_b,B4: term_c_d] :
          ( ( P3
            = ( produc1445286645097778113rm_c_d @ A6 @ B4 ) )
         => ( C @ A6 @ B4 ) )
     => ( produc3147824153920350274_c_d_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_181_case__prodI2,axiom,
    ! [P3: produc5661168290637732233rm_a_b,C: term_c_d > term_a_b > $o] :
      ( ! [A6: term_c_d,B4: term_a_b] :
          ( ( P3
            = ( produc3100600368241150913rm_a_b @ A6 @ B4 ) )
         => ( C @ A6 @ B4 ) )
     => ( produc327966649874835522_a_b_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_182_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc8451535801715972489rm_c_d] :
      ~ ! [A6: term_a_b,B4: term_c_d] :
          ( Y3
         != ( produc1445286645097778113rm_c_d @ A6 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_183_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc5661168290637732233rm_a_b] :
      ~ ! [A6: term_c_d,B4: term_a_b] :
          ( Y3
         != ( produc3100600368241150913rm_a_b @ A6 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_184_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc4531938369520450825rm_c_d] :
      ~ ! [A6: term_c_d,B4: term_c_d] :
          ( Y3
         != ( produc6749061249757032257rm_c_d @ A6 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_185_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc357393685978478089rm_a_b] :
      ~ ! [A6: term_a_b,B4: term_a_b] :
          ( Y3
         != ( produc7020197800436672577rm_a_b @ A6 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_186_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_nat_nat] :
      ~ ! [A6: nat,B4: nat] :
          ( Y3
         != ( product_Pair_nat_nat @ A6 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_187_curryD,axiom,
    ! [F: produc4531938369520450825rm_c_d > $o,A4: term_c_d,B2: term_c_d] :
      ( ( produc6204020005778196579_c_d_o @ F @ A4 @ B2 )
     => ( F @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) ) ) ).

% curryD
thf(fact_188_curryD,axiom,
    ! [F: produc357393685978478089rm_a_b > $o,A4: term_a_b,B2: term_a_b] :
      ( ( produc6008550485034061667_a_b_o @ F @ A4 @ B2 )
     => ( F @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) ) ) ).

% curryD
thf(fact_189_curryD,axiom,
    ! [F: product_prod_nat_nat > $o,A4: nat,B2: nat] :
      ( ( produc1310100445399344235_nat_o @ F @ A4 @ B2 )
     => ( F @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ).

% curryD
thf(fact_190_curryD,axiom,
    ! [F: produc5661168290637732233rm_a_b > $o,A4: term_c_d,B2: term_a_b] :
      ( ( produc84670474955983843_a_b_o @ F @ A4 @ B2 )
     => ( F @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) ) ) ).

% curryD
thf(fact_191_curryD,axiom,
    ! [F: produc8451535801715972489rm_c_d > $o,A4: term_a_b,B2: term_c_d] :
      ( ( produc2904527979001498595_c_d_o @ F @ A4 @ B2 )
     => ( F @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) ) ) ).

% curryD
thf(fact_192_curryE,axiom,
    ! [F: produc4531938369520450825rm_c_d > $o,A4: term_c_d,B2: term_c_d] :
      ( ( produc6204020005778196579_c_d_o @ F @ A4 @ B2 )
     => ( F @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) ) ) ).

% curryE
thf(fact_193_curryE,axiom,
    ! [F: produc357393685978478089rm_a_b > $o,A4: term_a_b,B2: term_a_b] :
      ( ( produc6008550485034061667_a_b_o @ F @ A4 @ B2 )
     => ( F @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) ) ) ).

% curryE
thf(fact_194_curryE,axiom,
    ! [F: product_prod_nat_nat > $o,A4: nat,B2: nat] :
      ( ( produc1310100445399344235_nat_o @ F @ A4 @ B2 )
     => ( F @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ).

% curryE
thf(fact_195_curryE,axiom,
    ! [F: produc5661168290637732233rm_a_b > $o,A4: term_c_d,B2: term_a_b] :
      ( ( produc84670474955983843_a_b_o @ F @ A4 @ B2 )
     => ( F @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) ) ) ).

% curryE
thf(fact_196_curryE,axiom,
    ! [F: produc8451535801715972489rm_c_d > $o,A4: term_a_b,B2: term_c_d] :
      ( ( produc2904527979001498595_c_d_o @ F @ A4 @ B2 )
     => ( F @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) ) ) ).

% curryE
thf(fact_197_curry__def,axiom,
    ( produc84670474955983843_a_b_o
    = ( ^ [C2: produc5661168290637732233rm_a_b > $o,X: term_c_d,Y: term_a_b] : ( C2 @ ( produc3100600368241150913rm_a_b @ X @ Y ) ) ) ) ).

% curry_def
thf(fact_198_curry__def,axiom,
    ( produc2904527979001498595_c_d_o
    = ( ^ [C2: produc8451535801715972489rm_c_d > $o,X: term_a_b,Y: term_c_d] : ( C2 @ ( produc1445286645097778113rm_c_d @ X @ Y ) ) ) ) ).

% curry_def
thf(fact_199_surj__pair,axiom,
    ! [P3: produc8451535801715972489rm_c_d] :
    ? [X3: term_a_b,Y4: term_c_d] :
      ( P3
      = ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_200_surj__pair,axiom,
    ! [P3: produc5661168290637732233rm_a_b] :
    ? [X3: term_c_d,Y4: term_a_b] :
      ( P3
      = ( produc3100600368241150913rm_a_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_201_surj__pair,axiom,
    ! [P3: produc4531938369520450825rm_c_d] :
    ? [X3: term_c_d,Y4: term_c_d] :
      ( P3
      = ( produc6749061249757032257rm_c_d @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_202_surj__pair,axiom,
    ! [P3: produc357393685978478089rm_a_b] :
    ? [X3: term_a_b,Y4: term_a_b] :
      ( P3
      = ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_203_surj__pair,axiom,
    ! [P3: product_prod_nat_nat] :
    ? [X3: nat,Y4: nat] :
      ( P3
      = ( product_Pair_nat_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_204_mem__Collect__eq,axiom,
    ! [A4: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ A4 @ ( collec99177395925924084rm_a_b @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_205_mem__Collect__eq,axiom,
    ! [A4: $o,P: $o > $o] :
      ( ( member_o @ A4 @ ( collect_o @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_206_mem__Collect__eq,axiom,
    ! [A4: produc8451535801715972489rm_c_d,P: produc8451535801715972489rm_c_d > $o] :
      ( ( member4740485589907853106rm_c_d @ A4 @ ( collec8193319511663418484rm_c_d @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_207_mem__Collect__eq,axiom,
    ! [A4: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A4 @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_208_mem__Collect__eq,axiom,
    ! [A4: term_c_d,P: term_c_d > $o] :
      ( ( member_term_c_d @ A4 @ ( collect_term_c_d @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_209_mem__Collect__eq,axiom,
    ! [A4: term_a_b,P: term_a_b > $o] :
      ( ( member_term_a_b @ A4 @ ( collect_term_a_b @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_210_mem__Collect__eq,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A4 @ ( collect_set_nat @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_211_mem__Collect__eq,axiom,
    ! [A4: nat,P: nat > $o] :
      ( ( member_nat @ A4 @ ( collect_nat @ P ) )
      = ( P @ A4 ) ) ).

% mem_Collect_eq
thf(fact_212_Collect__mem__eq,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b] :
      ( ( collec99177395925924084rm_a_b
        @ ^ [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_213_Collect__mem__eq,axiom,
    ! [A3: set_o] :
      ( ( collect_o
        @ ^ [X: $o] : ( member_o @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_214_Collect__mem__eq,axiom,
    ! [A3: set_Pr4645549399426460137rm_c_d] :
      ( ( collec8193319511663418484rm_c_d
        @ ^ [X: produc8451535801715972489rm_c_d] : ( member4740485589907853106rm_c_d @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_215_Collect__mem__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_216_Collect__mem__eq,axiom,
    ! [A3: set_term_c_d] :
      ( ( collect_term_c_d
        @ ^ [X: term_c_d] : ( member_term_c_d @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_217_Collect__mem__eq,axiom,
    ! [A3: set_term_a_b] :
      ( ( collect_term_a_b
        @ ^ [X: term_a_b] : ( member_term_a_b @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_218_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] : ( member_set_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_219_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_220_Collect__cong,axiom,
    ! [P: produc8451535801715972489rm_c_d > $o,Q: produc8451535801715972489rm_c_d > $o] :
      ( ! [X3: produc8451535801715972489rm_c_d] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec8193319511663418484rm_c_d @ P )
        = ( collec8193319511663418484rm_c_d @ Q ) ) ) ).

% Collect_cong
thf(fact_221_Collect__cong,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec3392354462482085612at_nat @ P )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_222_Collect__cong,axiom,
    ! [P: term_c_d > $o,Q: term_c_d > $o] :
      ( ! [X3: term_c_d] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_term_c_d @ P )
        = ( collect_term_c_d @ Q ) ) ) ).

% Collect_cong
thf(fact_223_Collect__cong,axiom,
    ! [P: term_a_b > $o,Q: term_a_b > $o] :
      ( ! [X3: term_a_b] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_term_a_b @ P )
        = ( collect_term_a_b @ Q ) ) ) ).

% Collect_cong
thf(fact_224_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_225_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_226_prod__cases,axiom,
    ! [P: produc8451535801715972489rm_c_d > $o,P3: produc8451535801715972489rm_c_d] :
      ( ! [A6: term_a_b,B4: term_c_d] : ( P @ ( produc1445286645097778113rm_c_d @ A6 @ B4 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_227_prod__cases,axiom,
    ! [P: produc5661168290637732233rm_a_b > $o,P3: produc5661168290637732233rm_a_b] :
      ( ! [A6: term_c_d,B4: term_a_b] : ( P @ ( produc3100600368241150913rm_a_b @ A6 @ B4 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_228_prod__cases,axiom,
    ! [P: produc4531938369520450825rm_c_d > $o,P3: produc4531938369520450825rm_c_d] :
      ( ! [A6: term_c_d,B4: term_c_d] : ( P @ ( produc6749061249757032257rm_c_d @ A6 @ B4 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_229_prod__cases,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,P3: produc357393685978478089rm_a_b] :
      ( ! [A6: term_a_b,B4: term_a_b] : ( P @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_230_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
      ( ! [A6: nat,B4: nat] : ( P @ ( product_Pair_nat_nat @ A6 @ B4 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_231_Pair__inject,axiom,
    ! [A4: term_a_b,B2: term_c_d,A5: term_a_b,B3: term_c_d] :
      ( ( ( produc1445286645097778113rm_c_d @ A4 @ B2 )
        = ( produc1445286645097778113rm_c_d @ A5 @ B3 ) )
     => ~ ( ( A4 = A5 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_232_Pair__inject,axiom,
    ! [A4: term_c_d,B2: term_a_b,A5: term_c_d,B3: term_a_b] :
      ( ( ( produc3100600368241150913rm_a_b @ A4 @ B2 )
        = ( produc3100600368241150913rm_a_b @ A5 @ B3 ) )
     => ~ ( ( A4 = A5 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_233_Pair__inject,axiom,
    ! [A4: term_c_d,B2: term_c_d,A5: term_c_d,B3: term_c_d] :
      ( ( ( produc6749061249757032257rm_c_d @ A4 @ B2 )
        = ( produc6749061249757032257rm_c_d @ A5 @ B3 ) )
     => ~ ( ( A4 = A5 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_234_Pair__inject,axiom,
    ! [A4: term_a_b,B2: term_a_b,A5: term_a_b,B3: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ A4 @ B2 )
        = ( produc7020197800436672577rm_a_b @ A5 @ B3 ) )
     => ~ ( ( A4 = A5 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_235_Pair__inject,axiom,
    ! [A4: nat,B2: nat,A5: nat,B3: nat] :
      ( ( ( product_Pair_nat_nat @ A4 @ B2 )
        = ( product_Pair_nat_nat @ A5 @ B3 ) )
     => ~ ( ( A4 = A5 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_236_mem__case__prodE,axiom,
    ! [Z: $o,C: nat > nat > set_o,P3: product_prod_nat_nat] :
      ( ( member_o @ Z @ ( produc59986286002894506_set_o @ C @ P3 ) )
     => ~ ! [X3: nat,Y4: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X3 @ Y4 ) )
           => ~ ( member_o @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_237_mem__case__prodE,axiom,
    ! [Z: nat,C: nat > nat > set_nat,P3: product_prod_nat_nat] :
      ( ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P3 ) )
     => ~ ! [X3: nat,Y4: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X3 @ Y4 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_238_mem__case__prodE,axiom,
    ! [Z: set_nat,C: nat > nat > set_set_nat,P3: product_prod_nat_nat] :
      ( ( member_set_nat @ Z @ ( produc8404753619367356554et_nat @ C @ P3 ) )
     => ~ ! [X3: nat,Y4: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X3 @ Y4 ) )
           => ~ ( member_set_nat @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_239_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,P3: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C @ P3 ) )
     => ~ ! [X3: nat,Y4: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X3 @ Y4 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_240_mem__case__prodE,axiom,
    ! [Z: $o,C: term_a_b > term_c_d > set_o,P3: produc8451535801715972489rm_c_d] :
      ( ( member_o @ Z @ ( produc7074531164778841890_set_o @ C @ P3 ) )
     => ~ ! [X3: term_a_b,Y4: term_c_d] :
            ( ( P3
              = ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) )
           => ~ ( member_o @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_241_mem__case__prodE,axiom,
    ! [Z: nat,C: term_a_b > term_c_d > set_nat,P3: produc8451535801715972489rm_c_d] :
      ( ( member_nat @ Z @ ( produc5232004564063148380et_nat @ C @ P3 ) )
     => ~ ! [X3: term_a_b,Y4: term_c_d] :
            ( ( P3
              = ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_242_mem__case__prodE,axiom,
    ! [Z: $o,C: term_c_d > term_a_b > set_o,P3: produc5661168290637732233rm_a_b] :
      ( ( member_o @ Z @ ( produc2530863894259861282_set_o @ C @ P3 ) )
     => ~ ! [X3: term_c_d,Y4: term_a_b] :
            ( ( P3
              = ( produc3100600368241150913rm_a_b @ X3 @ Y4 ) )
           => ~ ( member_o @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_243_mem__case__prodE,axiom,
    ! [Z: nat,C: term_c_d > term_a_b > set_nat,P3: produc5661168290637732233rm_a_b] :
      ( ( member_nat @ Z @ ( produc7500204723417323868et_nat @ C @ P3 ) )
     => ~ ! [X3: term_c_d,Y4: term_a_b] :
            ( ( P3
              = ( produc3100600368241150913rm_a_b @ X3 @ Y4 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_244_mem__case__prodE,axiom,
    ! [Z: $o,C: term_c_d > term_c_d > set_o,P3: produc4531938369520450825rm_c_d] :
      ( ( member_o @ Z @ ( produc1641171031246963618_set_o @ C @ P3 ) )
     => ~ ! [X3: term_c_d,Y4: term_c_d] :
            ( ( P3
              = ( produc6749061249757032257rm_c_d @ X3 @ Y4 ) )
           => ~ ( member_o @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_245_mem__case__prodE,axiom,
    ! [Z: nat,C: term_c_d > term_c_d > set_nat,P3: produc4531938369520450825rm_c_d] :
      ( ( member_nat @ Z @ ( produc6663801938352323804et_nat @ C @ P3 ) )
     => ~ ! [X3: term_c_d,Y4: term_c_d] :
            ( ( P3
              = ( produc6749061249757032257rm_c_d @ X3 @ Y4 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_246_internal__case__prod__conv,axiom,
    ! [C: term_c_d > term_a_b > $o,A4: term_c_d,B2: term_a_b] :
      ( ( produc6641210422893952236_a_b_o @ C @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) )
      = ( C @ A4 @ B2 ) ) ).

% internal_case_prod_conv
thf(fact_247_internal__case__prod__conv,axiom,
    ! [C: term_a_b > term_c_d > $o,A4: term_a_b,B2: term_c_d] :
      ( ( produc237695890084691180_c_d_o @ C @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) )
      = ( C @ A4 @ B2 ) ) ).

% internal_case_prod_conv
thf(fact_248_case__prod__Pair__iden,axiom,
    ! [P3: produc8451535801715972489rm_c_d] :
      ( ( produc5709838018925087283rm_c_d @ produc1445286645097778113rm_c_d @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_249_case__prod__Pair__iden,axiom,
    ! [P3: produc5661168290637732233rm_a_b] :
      ( ( produc2580071269002545715rm_a_b @ produc3100600368241150913rm_a_b @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_250_case__prod__Pair__iden,axiom,
    ! [P3: produc4531938369520450825rm_c_d] :
      ( ( produc6200404326136932915rm_c_d @ produc6749061249757032257rm_c_d @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_251_case__prod__Pair__iden,axiom,
    ! [P3: produc357393685978478089rm_a_b] :
      ( ( produc2089504961790700083rm_a_b @ produc7020197800436672577rm_a_b @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_252_case__prod__Pair__iden,axiom,
    ! [P3: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_253_pred__equals__eq2,axiom,
    ! [R3: set_Pr4645549399426460137rm_c_d,S2: set_Pr4645549399426460137rm_c_d] :
      ( ( ( ^ [X: term_a_b,Y: term_c_d] : ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ X @ Y ) @ R3 ) )
        = ( ^ [X: term_a_b,Y: term_c_d] : ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ X @ Y ) @ S2 ) ) )
      = ( R3 = S2 ) ) ).

% pred_equals_eq2
thf(fact_254_pred__equals__eq2,axiom,
    ! [R3: set_Pr4821568463224695273rm_a_b,S2: set_Pr4821568463224695273rm_a_b] :
      ( ( ( ^ [X: term_c_d,Y: term_a_b] : ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ X @ Y ) @ R3 ) )
        = ( ^ [X: term_c_d,Y: term_a_b] : ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ X @ Y ) @ S2 ) ) )
      = ( R3 = S2 ) ) ).

% pred_equals_eq2
thf(fact_255_pred__equals__eq2,axiom,
    ! [R3: set_Pr5080540287643815273rm_c_d,S2: set_Pr5080540287643815273rm_c_d] :
      ( ( ( ^ [X: term_c_d,Y: term_c_d] : ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X @ Y ) @ R3 ) )
        = ( ^ [X: term_c_d,Y: term_c_d] : ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X @ Y ) @ S2 ) ) )
      = ( R3 = S2 ) ) ).

% pred_equals_eq2
thf(fact_256_pred__equals__eq2,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
      ( ( ( ^ [X: term_a_b,Y: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 ) )
        = ( ^ [X: term_a_b,Y: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ S2 ) ) )
      = ( R3 = S2 ) ) ).

% pred_equals_eq2
thf(fact_257_pred__equals__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 ) )
        = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S2 ) ) )
      = ( R3 = S2 ) ) ).

% pred_equals_eq2
thf(fact_258_case__prodE_H,axiom,
    ! [C: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o,P3: produc5661168290637732233rm_a_b,Z: produc5661168290637732233rm_a_b] :
      ( ( produc5873602711790147082_a_b_o @ C @ P3 @ Z )
     => ~ ! [X3: term_c_d,Y4: term_a_b] :
            ( ( P3
              = ( produc3100600368241150913rm_a_b @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 @ Z ) ) ) ).

% case_prodE'
thf(fact_259_case__prodE_H,axiom,
    ! [C: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o,P3: produc8451535801715972489rm_c_d,Z: produc8451535801715972489rm_c_d] :
      ( ( produc7053737044884821514_c_d_o @ C @ P3 @ Z )
     => ~ ! [X3: term_a_b,Y4: term_c_d] :
            ( ( P3
              = ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 @ Z ) ) ) ).

% case_prodE'
thf(fact_260_case__prodD_H,axiom,
    ! [R3: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o,A4: term_c_d,B2: term_a_b,C: produc5661168290637732233rm_a_b] :
      ( ( produc5873602711790147082_a_b_o @ R3 @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) @ C )
     => ( R3 @ A4 @ B2 @ C ) ) ).

% case_prodD'
thf(fact_261_case__prodD_H,axiom,
    ! [R3: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o,A4: term_a_b,B2: term_c_d,C: produc8451535801715972489rm_c_d] :
      ( ( produc7053737044884821514_c_d_o @ R3 @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) @ C )
     => ( R3 @ A4 @ B2 @ C ) ) ).

% case_prodD'
thf(fact_262_inv__image__def,axiom,
    ( inv_im8307999503528896764rm_a_b
    = ( ^ [R2: set_Pr5080540287643815273rm_c_d,F2: term_a_b > term_c_d] :
          ( collec99177395925924084rm_a_b
          @ ( produc6251846659952913346_a_b_o
            @ ^ [X: term_a_b,Y: term_a_b] : ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_263_inv__image__def,axiom,
    ( inv_im3004224898869642620rm_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b] :
          ( collec99177395925924084rm_a_b
          @ ( produc6251846659952913346_a_b_o
            @ ^ [X: term_a_b,Y: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_264_inv__image__def,axiom,
    ( inv_im1647809802420253368rm_a_b
    = ( ^ [R2: set_Pr1261947904930325089at_nat,F2: term_a_b > nat] :
          ( collec99177395925924084rm_a_b
          @ ( produc6251846659952913346_a_b_o
            @ ^ [X: term_a_b,Y: term_a_b] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_265_inv__image__def,axiom,
    ( inv_im7267259196725686968_d_nat
    = ( ^ [R2: set_Pr5080540287643815273rm_c_d,F2: nat > term_c_d] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y: nat] : ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_266_inv__image__def,axiom,
    ( inv_im2609263929585225528_b_nat
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b,F2: nat > term_a_b] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_267_inv__image__def,axiom,
    ( inv_image_nat_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat,F2: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ) ) ) ).

% inv_image_def
thf(fact_268_curry__K,axiom,
    ! [C: $o] :
      ( ( produc84670474955983843_a_b_o
        @ ^ [X: produc5661168290637732233rm_a_b] : C )
      = ( ^ [X: term_c_d,Y: term_a_b] : C ) ) ).

% curry_K
thf(fact_269_curry__K,axiom,
    ! [C: $o] :
      ( ( produc2904527979001498595_c_d_o
        @ ^ [X: produc8451535801715972489rm_c_d] : C )
      = ( ^ [X: term_a_b,Y: term_c_d] : C ) ) ).

% curry_K
thf(fact_270_old_Oprod_Ocase,axiom,
    ! [F: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o,X12: term_c_d,X22: term_a_b] :
      ( ( produc5873602711790147082_a_b_o @ F @ ( produc3100600368241150913rm_a_b @ X12 @ X22 ) )
      = ( F @ X12 @ X22 ) ) ).

% old.prod.case
thf(fact_271_old_Oprod_Ocase,axiom,
    ! [F: term_a_b > term_c_d > produc5661168290637732233rm_a_b,X12: term_a_b,X22: term_c_d] :
      ( ( produc2919470507846847027rm_a_b @ F @ ( produc1445286645097778113rm_c_d @ X12 @ X22 ) )
      = ( F @ X12 @ X22 ) ) ).

% old.prod.case
thf(fact_272_old_Oprod_Ocase,axiom,
    ! [F: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o,X12: term_a_b,X22: term_c_d] :
      ( ( produc7053737044884821514_c_d_o @ F @ ( produc1445286645097778113rm_c_d @ X12 @ X22 ) )
      = ( F @ X12 @ X22 ) ) ).

% old.prod.case
thf(fact_273_old_Oprod_Ocase,axiom,
    ! [F: term_a_b > term_a_b > $o,X12: term_a_b,X22: term_a_b] :
      ( ( produc6251846659952913346_a_b_o @ F @ ( produc7020197800436672577rm_a_b @ X12 @ X22 ) )
      = ( F @ X12 @ X22 ) ) ).

% old.prod.case
thf(fact_274_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > $o,X12: nat,X22: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X12 @ X22 ) )
      = ( F @ X12 @ X22 ) ) ).

% old.prod.case
thf(fact_275_old_Oprod_Ocase,axiom,
    ! [F: term_a_b > term_c_d > $o,X12: term_a_b,X22: term_c_d] :
      ( ( produc3147824153920350274_c_d_o @ F @ ( produc1445286645097778113rm_c_d @ X12 @ X22 ) )
      = ( F @ X12 @ X22 ) ) ).

% old.prod.case
thf(fact_276_old_Oprod_Ocase,axiom,
    ! [F: term_c_d > term_a_b > $o,X12: term_c_d,X22: term_a_b] :
      ( ( produc327966649874835522_a_b_o @ F @ ( produc3100600368241150913rm_a_b @ X12 @ X22 ) )
      = ( F @ X12 @ X22 ) ) ).

% old.prod.case
thf(fact_277_converse_Ocases,axiom,
    ! [A1: term_c_d,A22: term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A1 @ A22 ) @ ( conver7337423426258204773rm_c_d @ R ) )
     => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A22 @ A1 ) @ R ) ) ).

% converse.cases
thf(fact_278_converse_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A22 @ A1 ) @ R ) ) ).

% converse.cases
thf(fact_279_converse_Ocases,axiom,
    ! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( converse_nat_nat @ R ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A22 @ A1 ) @ R ) ) ).

% converse.cases
thf(fact_280_converse_Ocases,axiom,
    ! [A1: term_c_d,A22: term_a_b,R: set_Pr4645549399426460137rm_c_d] :
      ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ A1 @ A22 ) @ ( conver2033648821598950629rm_c_d @ R ) )
     => ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ A22 @ A1 ) @ R ) ) ).

% converse.cases
thf(fact_281_converse_Ocases,axiom,
    ! [A1: term_a_b,A22: term_c_d,R: set_Pr4821568463224695273rm_a_b] :
      ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ A1 @ A22 ) @ ( conver3688962544742323429rm_a_b @ R ) )
     => ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ A22 @ A1 ) @ R ) ) ).

% converse.cases
thf(fact_282_converse_Osimps,axiom,
    ! [A1: term_c_d,A22: term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A1 @ A22 ) @ ( conver7337423426258204773rm_c_d @ R ) )
      = ( ? [A: term_c_d,B: term_c_d] :
            ( ( A1 = B )
            & ( A22 = A )
            & ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A @ B ) @ R ) ) ) ) ).

% converse.simps
thf(fact_283_converse_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R ) )
      = ( ? [A: term_a_b,B: term_a_b] :
            ( ( A1 = B )
            & ( A22 = A )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R ) ) ) ) ).

% converse.simps
thf(fact_284_converse_Osimps,axiom,
    ! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( converse_nat_nat @ R ) )
      = ( ? [A: nat,B: nat] :
            ( ( A1 = B )
            & ( A22 = A )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ).

% converse.simps
thf(fact_285_converse_Osimps,axiom,
    ! [A1: term_c_d,A22: term_a_b,R: set_Pr4645549399426460137rm_c_d] :
      ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ A1 @ A22 ) @ ( conver2033648821598950629rm_c_d @ R ) )
      = ( ? [A: term_a_b,B: term_c_d] :
            ( ( A1 = B )
            & ( A22 = A )
            & ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ A @ B ) @ R ) ) ) ) ).

% converse.simps
thf(fact_286_converse_Osimps,axiom,
    ! [A1: term_a_b,A22: term_c_d,R: set_Pr4821568463224695273rm_a_b] :
      ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ A1 @ A22 ) @ ( conver3688962544742323429rm_a_b @ R ) )
      = ( ? [A: term_c_d,B: term_a_b] :
            ( ( A1 = B )
            & ( A22 = A )
            & ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ A @ B ) @ R ) ) ) ) ).

% converse.simps
thf(fact_287_converseD,axiom,
    ! [A4: term_c_d,B2: term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) @ ( conver7337423426258204773rm_c_d @ R ) )
     => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ B2 @ A4 ) @ R ) ) ).

% converseD
thf(fact_288_converseD,axiom,
    ! [A4: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) @ ( conver7608559976937845093rm_a_b @ R ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A4 ) @ R ) ) ).

% converseD
thf(fact_289_converseD,axiom,
    ! [A4: nat,B2: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( converse_nat_nat @ R ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ R ) ) ).

% converseD
thf(fact_290_converseD,axiom,
    ! [A4: term_c_d,B2: term_a_b,R: set_Pr4645549399426460137rm_c_d] :
      ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) @ ( conver2033648821598950629rm_c_d @ R ) )
     => ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ B2 @ A4 ) @ R ) ) ).

% converseD
thf(fact_291_converseD,axiom,
    ! [A4: term_a_b,B2: term_c_d,R: set_Pr4821568463224695273rm_a_b] :
      ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) @ ( conver3688962544742323429rm_a_b @ R ) )
     => ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ B2 @ A4 ) @ R ) ) ).

% converseD
thf(fact_292_converseE,axiom,
    ! [Yx: produc4531938369520450825rm_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ( member820888157712331442rm_c_d @ Yx @ ( conver7337423426258204773rm_c_d @ R ) )
     => ~ ! [X3: term_c_d,Y4: term_c_d] :
            ( ( Yx
              = ( produc6749061249757032257rm_c_d @ Y4 @ X3 ) )
           => ~ ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X3 @ Y4 ) @ R ) ) ) ).

% converseE
thf(fact_293_converseE,axiom,
    ! [Yx: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Yx @ ( conver7608559976937845093rm_a_b @ R ) )
     => ~ ! [X3: term_a_b,Y4: term_a_b] :
            ( ( Yx
              = ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R ) ) ) ).

% converseE
thf(fact_294_converseE,axiom,
    ! [Yx: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Yx @ ( converse_nat_nat @ R ) )
     => ~ ! [X3: nat,Y4: nat] :
            ( ( Yx
              = ( product_Pair_nat_nat @ Y4 @ X3 ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R ) ) ) ).

% converseE
thf(fact_295_converseE,axiom,
    ! [Yx: produc5661168290637732233rm_a_b,R: set_Pr4645549399426460137rm_c_d] :
      ( ( member1950118078829612850rm_a_b @ Yx @ ( conver2033648821598950629rm_c_d @ R ) )
     => ~ ! [X3: term_a_b,Y4: term_c_d] :
            ( ( Yx
              = ( produc3100600368241150913rm_a_b @ Y4 @ X3 ) )
           => ~ ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) @ R ) ) ) ).

% converseE
thf(fact_296_converseE,axiom,
    ! [Yx: produc8451535801715972489rm_c_d,R: set_Pr4821568463224695273rm_a_b] :
      ( ( member4740485589907853106rm_c_d @ Yx @ ( conver3688962544742323429rm_a_b @ R ) )
     => ~ ! [X3: term_c_d,Y4: term_a_b] :
            ( ( Yx
              = ( produc1445286645097778113rm_c_d @ Y4 @ X3 ) )
           => ~ ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ X3 @ Y4 ) @ R ) ) ) ).

% converseE
thf(fact_297_converseI,axiom,
    ! [A4: term_c_d,B2: term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) @ R )
     => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ B2 @ A4 ) @ ( conver7337423426258204773rm_c_d @ R ) ) ) ).

% converseI
thf(fact_298_converseI,axiom,
    ! [A4: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) @ R )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A4 ) @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).

% converseI
thf(fact_299_converseI,axiom,
    ! [A4: nat,B2: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ ( converse_nat_nat @ R ) ) ) ).

% converseI
thf(fact_300_converseI,axiom,
    ! [A4: term_a_b,B2: term_c_d,R: set_Pr4645549399426460137rm_c_d] :
      ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) @ R )
     => ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ B2 @ A4 ) @ ( conver2033648821598950629rm_c_d @ R ) ) ) ).

% converseI
thf(fact_301_converseI,axiom,
    ! [A4: term_c_d,B2: term_a_b,R: set_Pr4821568463224695273rm_a_b] :
      ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) @ R )
     => ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ B2 @ A4 ) @ ( conver3688962544742323429rm_a_b @ R ) ) ) ).

% converseI
thf(fact_302_irrefl__on__def,axiom,
    ( irrefl_on_term_c_d
    = ( ^ [A2: set_term_c_d,R2: set_Pr5080540287643815273rm_c_d] :
        ! [X: term_c_d] :
          ( ( member_term_c_d @ X @ A2 )
         => ~ ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X @ X ) @ R2 ) ) ) ) ).

% irrefl_on_def
thf(fact_303_irrefl__on__def,axiom,
    ( irrefl_on_term_a_b
    = ( ^ [A2: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
        ! [X: term_a_b] :
          ( ( member_term_a_b @ X @ A2 )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R2 ) ) ) ) ).

% irrefl_on_def
thf(fact_304_irrefl__on__def,axiom,
    ( irrefl_on_nat
    = ( ^ [A2: set_nat,R2: set_Pr1261947904930325089at_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A2 )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R2 ) ) ) ) ).

% irrefl_on_def
thf(fact_305_irrefl__on__def,axiom,
    ( irrefl7408258699751485431at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
        ! [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ A2 )
         => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ R2 ) ) ) ) ).

% irrefl_on_def
thf(fact_306_irrefl__onI,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ! [A6: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A6 @ A3 )
         => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ A6 ) @ R ) )
     => ( irrefl6620156600162922239rm_a_b @ A3 @ R ) ) ).

% irrefl_onI
thf(fact_307_irrefl__onI,axiom,
    ! [A3: set_o,R: set_Product_prod_o_o] :
      ( ! [A6: $o] :
          ( ( member_o @ A6 @ A3 )
         => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A6 @ A6 ) @ R ) )
     => ( irrefl_on_o @ A3 @ R ) ) ).

% irrefl_onI
thf(fact_308_irrefl__onI,axiom,
    ! [A3: set_set_nat,R: set_Pr5488025237498180813et_nat] :
      ( ! [A6: set_nat] :
          ( ( member_set_nat @ A6 @ A3 )
         => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A6 @ A6 ) @ R ) )
     => ( irrefl_on_set_nat @ A3 @ R ) ) ).

% irrefl_onI
thf(fact_309_irrefl__onI,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ! [A6: term_c_d] :
          ( ( member_term_c_d @ A6 @ A3 )
         => ~ ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A6 @ A6 ) @ R ) )
     => ( irrefl_on_term_c_d @ A3 @ R ) ) ).

% irrefl_onI
thf(fact_310_irrefl__onI,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [A6: term_a_b] :
          ( ( member_term_a_b @ A6 @ A3 )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ A6 ) @ R ) )
     => ( irrefl_on_term_a_b @ A3 @ R ) ) ).

% irrefl_onI
thf(fact_311_irrefl__onI,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [A6: nat] :
          ( ( member_nat @ A6 @ A3 )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ A6 ) @ R ) )
     => ( irrefl_on_nat @ A3 @ R ) ) ).

% irrefl_onI
thf(fact_312_irrefl__onI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ! [A6: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ A6 @ A3 )
         => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A6 @ A6 ) @ R ) )
     => ( irrefl7408258699751485431at_nat @ A3 @ R ) ) ).

% irrefl_onI
thf(fact_313_irrefl__onD,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A4: produc357393685978478089rm_a_b] :
      ( ( irrefl6620156600162922239rm_a_b @ A3 @ R )
     => ( ( member5869715511025134514rm_a_b @ A4 @ A3 )
       => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ A4 ) @ R ) ) ) ).

% irrefl_onD
thf(fact_314_irrefl__onD,axiom,
    ! [A3: set_o,R: set_Product_prod_o_o,A4: $o] :
      ( ( irrefl_on_o @ A3 @ R )
     => ( ( member_o @ A4 @ A3 )
       => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A4 @ A4 ) @ R ) ) ) ).

% irrefl_onD
thf(fact_315_irrefl__onD,axiom,
    ! [A3: set_set_nat,R: set_Pr5488025237498180813et_nat,A4: set_nat] :
      ( ( irrefl_on_set_nat @ A3 @ R )
     => ( ( member_set_nat @ A4 @ A3 )
       => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A4 @ A4 ) @ R ) ) ) ).

% irrefl_onD
thf(fact_316_irrefl__onD,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d,A4: term_c_d] :
      ( ( irrefl_on_term_c_d @ A3 @ R )
     => ( ( member_term_c_d @ A4 @ A3 )
       => ~ ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ A4 ) @ R ) ) ) ).

% irrefl_onD
thf(fact_317_irrefl__onD,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A4: term_a_b] :
      ( ( irrefl_on_term_a_b @ A3 @ R )
     => ( ( member_term_a_b @ A4 @ A3 )
       => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R ) ) ) ).

% irrefl_onD
thf(fact_318_irrefl__onD,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat,A4: nat] :
      ( ( irrefl_on_nat @ A3 @ R )
     => ( ( member_nat @ A4 @ A3 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R ) ) ) ).

% irrefl_onD
thf(fact_319_irrefl__onD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A4: product_prod_nat_nat] :
      ( ( irrefl7408258699751485431at_nat @ A3 @ R )
     => ( ( member8440522571783428010at_nat @ A4 @ A3 )
       => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ A4 ) @ R ) ) ) ).

% irrefl_onD
thf(fact_320_case__prodE2,axiom,
    ! [Q: ( produc5661168290637732233rm_a_b > $o ) > $o,P: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o,Z: produc5661168290637732233rm_a_b] :
      ( ( Q @ ( produc5873602711790147082_a_b_o @ P @ Z ) )
     => ~ ! [X3: term_c_d,Y4: term_a_b] :
            ( ( Z
              = ( produc3100600368241150913rm_a_b @ X3 @ Y4 ) )
           => ~ ( Q @ ( P @ X3 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_321_case__prodE2,axiom,
    ! [Q: produc5661168290637732233rm_a_b > $o,P: term_a_b > term_c_d > produc5661168290637732233rm_a_b,Z: produc8451535801715972489rm_c_d] :
      ( ( Q @ ( produc2919470507846847027rm_a_b @ P @ Z ) )
     => ~ ! [X3: term_a_b,Y4: term_c_d] :
            ( ( Z
              = ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) )
           => ~ ( Q @ ( P @ X3 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_322_case__prodE2,axiom,
    ! [Q: ( produc8451535801715972489rm_c_d > $o ) > $o,P: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o,Z: produc8451535801715972489rm_c_d] :
      ( ( Q @ ( produc7053737044884821514_c_d_o @ P @ Z ) )
     => ~ ! [X3: term_a_b,Y4: term_c_d] :
            ( ( Z
              = ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) )
           => ~ ( Q @ ( P @ X3 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_323_case__prodE2,axiom,
    ! [Q: $o > $o,P: term_a_b > term_a_b > $o,Z: produc357393685978478089rm_a_b] :
      ( ( Q @ ( produc6251846659952913346_a_b_o @ P @ Z ) )
     => ~ ! [X3: term_a_b,Y4: term_a_b] :
            ( ( Z
              = ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) )
           => ~ ( Q @ ( P @ X3 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_324_case__prodE2,axiom,
    ! [Q: $o > $o,P: nat > nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P @ Z ) )
     => ~ ! [X3: nat,Y4: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y4 ) )
           => ~ ( Q @ ( P @ X3 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_325_case__prodE2,axiom,
    ! [Q: $o > $o,P: term_a_b > term_c_d > $o,Z: produc8451535801715972489rm_c_d] :
      ( ( Q @ ( produc3147824153920350274_c_d_o @ P @ Z ) )
     => ~ ! [X3: term_a_b,Y4: term_c_d] :
            ( ( Z
              = ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) )
           => ~ ( Q @ ( P @ X3 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_326_case__prodE2,axiom,
    ! [Q: $o > $o,P: term_c_d > term_a_b > $o,Z: produc5661168290637732233rm_a_b] :
      ( ( Q @ ( produc327966649874835522_a_b_o @ P @ Z ) )
     => ~ ! [X3: term_c_d,Y4: term_a_b] :
            ( ( Z
              = ( produc3100600368241150913rm_a_b @ X3 @ Y4 ) )
           => ~ ( Q @ ( P @ X3 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_327_case__prod__eta,axiom,
    ! [F: produc5661168290637732233rm_a_b > produc5661168290637732233rm_a_b > $o] :
      ( ( produc5873602711790147082_a_b_o
        @ ^ [X: term_c_d,Y: term_a_b] : ( F @ ( produc3100600368241150913rm_a_b @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_328_case__prod__eta,axiom,
    ! [F: produc8451535801715972489rm_c_d > produc5661168290637732233rm_a_b] :
      ( ( produc2919470507846847027rm_a_b
        @ ^ [X: term_a_b,Y: term_c_d] : ( F @ ( produc1445286645097778113rm_c_d @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_329_case__prod__eta,axiom,
    ! [F: produc8451535801715972489rm_c_d > produc8451535801715972489rm_c_d > $o] :
      ( ( produc7053737044884821514_c_d_o
        @ ^ [X: term_a_b,Y: term_c_d] : ( F @ ( produc1445286645097778113rm_c_d @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_330_case__prod__eta,axiom,
    ! [F: produc357393685978478089rm_a_b > $o] :
      ( ( produc6251846659952913346_a_b_o
        @ ^ [X: term_a_b,Y: term_a_b] : ( F @ ( produc7020197800436672577rm_a_b @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_331_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_332_case__prod__eta,axiom,
    ! [F: produc8451535801715972489rm_c_d > $o] :
      ( ( produc3147824153920350274_c_d_o
        @ ^ [X: term_a_b,Y: term_c_d] : ( F @ ( produc1445286645097778113rm_c_d @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_333_case__prod__eta,axiom,
    ! [F: produc5661168290637732233rm_a_b > $o] :
      ( ( produc327966649874835522_a_b_o
        @ ^ [X: term_c_d,Y: term_a_b] : ( F @ ( produc3100600368241150913rm_a_b @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_334_cond__case__prod__eta,axiom,
    ! [F: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o,G: produc5661168290637732233rm_a_b > produc5661168290637732233rm_a_b > $o] :
      ( ! [X3: term_c_d,Y4: term_a_b] :
          ( ( F @ X3 @ Y4 )
          = ( G @ ( produc3100600368241150913rm_a_b @ X3 @ Y4 ) ) )
     => ( ( produc5873602711790147082_a_b_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_335_cond__case__prod__eta,axiom,
    ! [F: term_a_b > term_c_d > produc5661168290637732233rm_a_b,G: produc8451535801715972489rm_c_d > produc5661168290637732233rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_c_d] :
          ( ( F @ X3 @ Y4 )
          = ( G @ ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) ) )
     => ( ( produc2919470507846847027rm_a_b @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_336_cond__case__prod__eta,axiom,
    ! [F: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o,G: produc8451535801715972489rm_c_d > produc8451535801715972489rm_c_d > $o] :
      ( ! [X3: term_a_b,Y4: term_c_d] :
          ( ( F @ X3 @ Y4 )
          = ( G @ ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) ) )
     => ( ( produc7053737044884821514_c_d_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_337_cond__case__prod__eta,axiom,
    ! [F: term_a_b > term_a_b > $o,G: produc357393685978478089rm_a_b > $o] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( F @ X3 @ Y4 )
          = ( G @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) ) )
     => ( ( produc6251846659952913346_a_b_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_338_cond__case__prod__eta,axiom,
    ! [F: nat > nat > $o,G: product_prod_nat_nat > $o] :
      ( ! [X3: nat,Y4: nat] :
          ( ( F @ X3 @ Y4 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y4 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_339_cond__case__prod__eta,axiom,
    ! [F: term_a_b > term_c_d > $o,G: produc8451535801715972489rm_c_d > $o] :
      ( ! [X3: term_a_b,Y4: term_c_d] :
          ( ( F @ X3 @ Y4 )
          = ( G @ ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) ) )
     => ( ( produc3147824153920350274_c_d_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_340_cond__case__prod__eta,axiom,
    ! [F: term_c_d > term_a_b > $o,G: produc5661168290637732233rm_a_b > $o] :
      ( ! [X3: term_c_d,Y4: term_a_b] :
          ( ( F @ X3 @ Y4 )
          = ( G @ ( produc3100600368241150913rm_a_b @ X3 @ Y4 ) ) )
     => ( ( produc327966649874835522_a_b_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_341_case__prodD,axiom,
    ! [F: term_c_d > term_c_d > $o,A4: term_c_d,B2: term_c_d] :
      ( ( produc6447316180697048258_c_d_o @ F @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) )
     => ( F @ A4 @ B2 ) ) ).

% case_prodD
thf(fact_342_case__prodD,axiom,
    ! [F: term_a_b > term_a_b > $o,A4: term_a_b,B2: term_a_b] :
      ( ( produc6251846659952913346_a_b_o @ F @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) )
     => ( F @ A4 @ B2 ) ) ).

% case_prodD
thf(fact_343_case__prodD,axiom,
    ! [F: nat > nat > $o,A4: nat,B2: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A4 @ B2 ) )
     => ( F @ A4 @ B2 ) ) ).

% case_prodD
thf(fact_344_case__prodD,axiom,
    ! [F: term_a_b > term_c_d > $o,A4: term_a_b,B2: term_c_d] :
      ( ( produc3147824153920350274_c_d_o @ F @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) )
     => ( F @ A4 @ B2 ) ) ).

% case_prodD
thf(fact_345_case__prodD,axiom,
    ! [F: term_c_d > term_a_b > $o,A4: term_c_d,B2: term_a_b] :
      ( ( produc327966649874835522_a_b_o @ F @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) )
     => ( F @ A4 @ B2 ) ) ).

% case_prodD
thf(fact_346_case__prodE,axiom,
    ! [C: term_c_d > term_c_d > $o,P3: produc4531938369520450825rm_c_d] :
      ( ( produc6447316180697048258_c_d_o @ C @ P3 )
     => ~ ! [X3: term_c_d,Y4: term_c_d] :
            ( ( P3
              = ( produc6749061249757032257rm_c_d @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_347_case__prodE,axiom,
    ! [C: term_a_b > term_a_b > $o,P3: produc357393685978478089rm_a_b] :
      ( ( produc6251846659952913346_a_b_o @ C @ P3 )
     => ~ ! [X3: term_a_b,Y4: term_a_b] :
            ( ( P3
              = ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_348_case__prodE,axiom,
    ! [C: nat > nat > $o,P3: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P3 )
     => ~ ! [X3: nat,Y4: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_349_case__prodE,axiom,
    ! [C: term_a_b > term_c_d > $o,P3: produc8451535801715972489rm_c_d] :
      ( ( produc3147824153920350274_c_d_o @ C @ P3 )
     => ~ ! [X3: term_a_b,Y4: term_c_d] :
            ( ( P3
              = ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_350_case__prodE,axiom,
    ! [C: term_c_d > term_a_b > $o,P3: produc5661168290637732233rm_a_b] :
      ( ( produc327966649874835522_a_b_o @ C @ P3 )
     => ~ ! [X3: term_c_d,Y4: term_a_b] :
            ( ( P3
              = ( produc3100600368241150913rm_a_b @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_351_ballI2,axiom,
    ! [A3: set_Pr5080540287643815273rm_c_d,P: term_c_d > term_c_d > $o] :
      ( ! [X3: term_c_d,Y4: term_c_d] :
          ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X3 @ Y4 ) @ A3 )
         => ( P @ X3 @ Y4 ) )
     => ! [X4: produc4531938369520450825rm_c_d] :
          ( ( member820888157712331442rm_c_d @ X4 @ A3 )
         => ( produc6447316180697048258_c_d_o @ P @ X4 ) ) ) ).

% ballI2
thf(fact_352_ballI2,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b,P: term_a_b > term_a_b > $o] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ A3 )
         => ( P @ X3 @ Y4 ) )
     => ! [X4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X4 @ A3 )
         => ( produc6251846659952913346_a_b_o @ P @ X4 ) ) ) ).

% ballI2
thf(fact_353_ballI2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ A3 )
         => ( P @ X3 @ Y4 ) )
     => ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A3 )
         => ( produc6081775807080527818_nat_o @ P @ X4 ) ) ) ).

% ballI2
thf(fact_354_ballI2,axiom,
    ! [A3: set_Pr4645549399426460137rm_c_d,P: term_a_b > term_c_d > $o] :
      ( ! [X3: term_a_b,Y4: term_c_d] :
          ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ X3 @ Y4 ) @ A3 )
         => ( P @ X3 @ Y4 ) )
     => ! [X4: produc8451535801715972489rm_c_d] :
          ( ( member4740485589907853106rm_c_d @ X4 @ A3 )
         => ( produc3147824153920350274_c_d_o @ P @ X4 ) ) ) ).

% ballI2
thf(fact_355_ballI2,axiom,
    ! [A3: set_Pr4821568463224695273rm_a_b,P: term_c_d > term_a_b > $o] :
      ( ! [X3: term_c_d,Y4: term_a_b] :
          ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ X3 @ Y4 ) @ A3 )
         => ( P @ X3 @ Y4 ) )
     => ! [X4: produc5661168290637732233rm_a_b] :
          ( ( member1950118078829612850rm_a_b @ X4 @ A3 )
         => ( produc327966649874835522_a_b_o @ P @ X4 ) ) ) ).

% ballI2
thf(fact_356_split__cong,axiom,
    ! [Q2: produc5661168290637732233rm_a_b,F: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o,G: term_c_d > term_a_b > produc5661168290637732233rm_a_b > $o,P3: produc5661168290637732233rm_a_b] :
      ( ! [X3: term_c_d,Y4: term_a_b] :
          ( ( ( produc3100600368241150913rm_a_b @ X3 @ Y4 )
            = Q2 )
         => ( ( F @ X3 @ Y4 )
            = ( G @ X3 @ Y4 ) ) )
     => ( ( P3 = Q2 )
       => ( ( produc5873602711790147082_a_b_o @ F @ P3 )
          = ( produc5873602711790147082_a_b_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_357_split__cong,axiom,
    ! [Q2: produc8451535801715972489rm_c_d,F: term_a_b > term_c_d > produc5661168290637732233rm_a_b,G: term_a_b > term_c_d > produc5661168290637732233rm_a_b,P3: produc8451535801715972489rm_c_d] :
      ( ! [X3: term_a_b,Y4: term_c_d] :
          ( ( ( produc1445286645097778113rm_c_d @ X3 @ Y4 )
            = Q2 )
         => ( ( F @ X3 @ Y4 )
            = ( G @ X3 @ Y4 ) ) )
     => ( ( P3 = Q2 )
       => ( ( produc2919470507846847027rm_a_b @ F @ P3 )
          = ( produc2919470507846847027rm_a_b @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_358_split__cong,axiom,
    ! [Q2: produc8451535801715972489rm_c_d,F: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o,G: term_a_b > term_c_d > produc8451535801715972489rm_c_d > $o,P3: produc8451535801715972489rm_c_d] :
      ( ! [X3: term_a_b,Y4: term_c_d] :
          ( ( ( produc1445286645097778113rm_c_d @ X3 @ Y4 )
            = Q2 )
         => ( ( F @ X3 @ Y4 )
            = ( G @ X3 @ Y4 ) ) )
     => ( ( P3 = Q2 )
       => ( ( produc7053737044884821514_c_d_o @ F @ P3 )
          = ( produc7053737044884821514_c_d_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_359_split__cong,axiom,
    ! [Q2: produc357393685978478089rm_a_b,F: term_a_b > term_a_b > $o,G: term_a_b > term_a_b > $o,P3: produc357393685978478089rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b] :
          ( ( ( produc7020197800436672577rm_a_b @ X3 @ Y4 )
            = Q2 )
         => ( ( F @ X3 @ Y4 )
            = ( G @ X3 @ Y4 ) ) )
     => ( ( P3 = Q2 )
       => ( ( produc6251846659952913346_a_b_o @ F @ P3 )
          = ( produc6251846659952913346_a_b_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_360_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: nat > nat > $o,G: nat > nat > $o,P3: product_prod_nat_nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( ( product_Pair_nat_nat @ X3 @ Y4 )
            = Q2 )
         => ( ( F @ X3 @ Y4 )
            = ( G @ X3 @ Y4 ) ) )
     => ( ( P3 = Q2 )
       => ( ( produc6081775807080527818_nat_o @ F @ P3 )
          = ( produc6081775807080527818_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_361_split__cong,axiom,
    ! [Q2: produc8451535801715972489rm_c_d,F: term_a_b > term_c_d > $o,G: term_a_b > term_c_d > $o,P3: produc8451535801715972489rm_c_d] :
      ( ! [X3: term_a_b,Y4: term_c_d] :
          ( ( ( produc1445286645097778113rm_c_d @ X3 @ Y4 )
            = Q2 )
         => ( ( F @ X3 @ Y4 )
            = ( G @ X3 @ Y4 ) ) )
     => ( ( P3 = Q2 )
       => ( ( produc3147824153920350274_c_d_o @ F @ P3 )
          = ( produc3147824153920350274_c_d_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_362_split__cong,axiom,
    ! [Q2: produc5661168290637732233rm_a_b,F: term_c_d > term_a_b > $o,G: term_c_d > term_a_b > $o,P3: produc5661168290637732233rm_a_b] :
      ( ! [X3: term_c_d,Y4: term_a_b] :
          ( ( ( produc3100600368241150913rm_a_b @ X3 @ Y4 )
            = Q2 )
         => ( ( F @ X3 @ Y4 )
            = ( G @ X3 @ Y4 ) ) )
     => ( ( P3 = Q2 )
       => ( ( produc327966649874835522_a_b_o @ F @ P3 )
          = ( produc327966649874835522_a_b_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_363_llrg__srsteps__with__root__step__inv__ground,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,F3: set_Pr4934435412358123699_a_nat] :
      ( ( rewrit5000278123254648670_b_a_b @ R5 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F3 @ ( conver7608559976937845093rm_a_b @ R5 ) ) )
       => ( term_ground_a_b @ S ) ) ) ).

% llrg_srsteps_with_root_step_inv_ground
thf(fact_364_llrg__srsteps__with__root__step__inv__ground,axiom,
    ! [R5: set_Pr5080540287643815273rm_c_d,S: term_c_d,T: term_c_d,F3: set_Pr6903500605879609269_c_nat] :
      ( ( rewrit4370356227420379230_d_c_d @ R5 )
     => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ S @ T ) @ ( srstep2269559354477410592ep_c_d @ F3 @ ( conver7337423426258204773rm_c_d @ R5 ) ) )
       => ( term_ground_c_d @ S ) ) ) ).

% llrg_srsteps_with_root_step_inv_ground
thf(fact_365_llrg__srsteps__with__root__step__ground,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,F3: set_Pr4934435412358123699_a_nat] :
      ( ( rewrit5000278123254648670_b_a_b @ R5 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F3 @ R5 ) )
       => ( term_ground_a_b @ T ) ) ) ).

% llrg_srsteps_with_root_step_ground
thf(fact_366_llrg__srsteps__with__root__step__ground,axiom,
    ! [R5: set_Pr5080540287643815273rm_c_d,S: term_c_d,T: term_c_d,F3: set_Pr6903500605879609269_c_nat] :
      ( ( rewrit4370356227420379230_d_c_d @ R5 )
     => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ S @ T ) @ ( srstep2269559354477410592ep_c_d @ F3 @ R5 ) )
       => ( term_ground_c_d @ T ) ) ) ).

% llrg_srsteps_with_root_step_ground
thf(fact_367_irreflp__on__irrefl__on__eq,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ( irreflp_on_term_c_d @ A3
        @ ^ [A: term_c_d,B: term_c_d] : ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A @ B ) @ R ) )
      = ( irrefl_on_term_c_d @ A3 @ R ) ) ).

% irreflp_on_irrefl_on_eq
thf(fact_368_irreflp__on__irrefl__on__eq,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( irreflp_on_term_a_b @ A3
        @ ^ [A: term_a_b,B: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R ) )
      = ( irrefl_on_term_a_b @ A3 @ R ) ) ).

% irreflp_on_irrefl_on_eq
thf(fact_369_irreflp__on__irrefl__on__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( irrefl2576046763318297307at_nat @ A3
        @ ^ [A: product_prod_nat_nat,B: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R ) )
      = ( irrefl7408258699751485431at_nat @ A3 @ R ) ) ).

% irreflp_on_irrefl_on_eq
thf(fact_370_irreflp__on__irrefl__on__eq,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( irreflp_on_nat @ A3
        @ ^ [A: nat,B: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
      = ( irrefl_on_nat @ A3 @ R ) ) ).

% irreflp_on_irrefl_on_eq
thf(fact_371_refl__on__def_H,axiom,
    ( refl_o5344404488616963464rm_a_b
    = ( ^ [A2: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
          ( ! [X: produc4523971788519308903rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ X @ R2 )
             => ( produc161078424057476524_a_b_o
                @ ^ [Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
                    ( ( member5869715511025134514rm_a_b @ Y @ A2 )
                    & ( member5869715511025134514rm_a_b @ Z2 @ A2 ) )
                @ X ) )
          & ! [X: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X @ A2 )
             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ X ) @ R2 ) ) ) ) ) ).

% refl_on_def'
thf(fact_372_refl__on__def_H,axiom,
    ( refl_o7122697186039111552at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
          ( ! [X: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X @ R2 )
             => ( produc6590410687421337004_nat_o
                @ ^ [Y: product_prod_nat_nat,Z2: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y @ A2 )
                    & ( member8440522571783428010at_nat @ Z2 @ A2 ) )
                @ X ) )
          & ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A2 )
             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ R2 ) ) ) ) ) ).

% refl_on_def'
thf(fact_373_refl__on__def_H,axiom,
    ( refl_on_o
    = ( ^ [A2: set_o,R2: set_Product_prod_o_o] :
          ( ! [X: product_prod_o_o] :
              ( ( member7466972457876170832od_o_o @ X @ R2 )
             => ( produc6197397395684419436_o_o_o
                @ ^ [Y: $o,Z2: $o] :
                    ( ( member_o @ Y @ A2 )
                    & ( member_o @ Z2 @ A2 ) )
                @ X ) )
          & ! [X: $o] :
              ( ( member_o @ X @ A2 )
             => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ X ) @ R2 ) ) ) ) ) ).

% refl_on_def'
thf(fact_374_refl__on__def_H,axiom,
    ( refl_on_set_nat
    = ( ^ [A2: set_set_nat,R2: set_Pr5488025237498180813et_nat] :
          ( ! [X: produc7819656566062154093et_nat] :
              ( ( member8277197624267554838et_nat @ X @ R2 )
             => ( produc6247414631856714078_nat_o
                @ ^ [Y: set_nat,Z2: set_nat] :
                    ( ( member_set_nat @ Y @ A2 )
                    & ( member_set_nat @ Z2 @ A2 ) )
                @ X ) )
          & ! [X: set_nat] :
              ( ( member_set_nat @ X @ A2 )
             => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ X ) @ R2 ) ) ) ) ) ).

% refl_on_def'
thf(fact_375_refl__on__def_H,axiom,
    ( refl_on_term_c_d
    = ( ^ [A2: set_term_c_d,R2: set_Pr5080540287643815273rm_c_d] :
          ( ! [X: produc4531938369520450825rm_c_d] :
              ( ( member820888157712331442rm_c_d @ X @ R2 )
             => ( produc6447316180697048258_c_d_o
                @ ^ [Y: term_c_d,Z2: term_c_d] :
                    ( ( member_term_c_d @ Y @ A2 )
                    & ( member_term_c_d @ Z2 @ A2 ) )
                @ X ) )
          & ! [X: term_c_d] :
              ( ( member_term_c_d @ X @ A2 )
             => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X @ X ) @ R2 ) ) ) ) ) ).

% refl_on_def'
thf(fact_376_refl__on__def_H,axiom,
    ( refl_on_term_a_b
    = ( ^ [A2: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
          ( ! [X: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X @ R2 )
             => ( produc6251846659952913346_a_b_o
                @ ^ [Y: term_a_b,Z2: term_a_b] :
                    ( ( member_term_a_b @ Y @ A2 )
                    & ( member_term_a_b @ Z2 @ A2 ) )
                @ X ) )
          & ! [X: term_a_b] :
              ( ( member_term_a_b @ X @ A2 )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R2 ) ) ) ) ) ).

% refl_on_def'
thf(fact_377_refl__on__def_H,axiom,
    ( refl_on_nat
    = ( ^ [A2: set_nat,R2: set_Pr1261947904930325089at_nat] :
          ( ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ R2 )
             => ( produc6081775807080527818_nat_o
                @ ^ [Y: nat,Z2: nat] :
                    ( ( member_nat @ Y @ A2 )
                    & ( member_nat @ Z2 @ A2 ) )
                @ X ) )
          & ! [X: nat] :
              ( ( member_nat @ X @ A2 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R2 ) ) ) ) ) ).

% refl_on_def'
thf(fact_378_srsteps__with__root__step__ground_I2_J,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,F3: set_Pr4934435412358123699_a_nat] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ R5 )
         => ( produc6251846659952913346_a_b_o
            @ ^ [S3: term_a_b,T2: term_a_b] :
                ( ( term_ground_a_b @ S3 )
                & ( term_ground_a_b @ T2 ) )
            @ X3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F3 @ R5 ) )
       => ( term_ground_a_b @ T ) ) ) ).

% srsteps_with_root_step_ground(2)
thf(fact_379_srsteps__with__root__step__ground_I2_J,axiom,
    ! [R5: set_Pr5080540287643815273rm_c_d,S: term_c_d,T: term_c_d,F3: set_Pr6903500605879609269_c_nat] :
      ( ! [X3: produc4531938369520450825rm_c_d] :
          ( ( member820888157712331442rm_c_d @ X3 @ R5 )
         => ( produc6447316180697048258_c_d_o
            @ ^ [S3: term_c_d,T2: term_c_d] :
                ( ( term_ground_c_d @ S3 )
                & ( term_ground_c_d @ T2 ) )
            @ X3 ) )
     => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ S @ T ) @ ( srstep2269559354477410592ep_c_d @ F3 @ R5 ) )
       => ( term_ground_c_d @ T ) ) ) ).

% srsteps_with_root_step_ground(2)
thf(fact_380_srsteps__with__root__step__ground_I1_J,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,F3: set_Pr4934435412358123699_a_nat] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ R5 )
         => ( produc6251846659952913346_a_b_o
            @ ^ [S3: term_a_b,T2: term_a_b] :
                ( ( term_ground_a_b @ S3 )
                & ( term_ground_a_b @ T2 ) )
            @ X3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F3 @ R5 ) )
       => ( term_ground_a_b @ S ) ) ) ).

% srsteps_with_root_step_ground(1)
thf(fact_381_srsteps__with__root__step__ground_I1_J,axiom,
    ! [R5: set_Pr5080540287643815273rm_c_d,S: term_c_d,T: term_c_d,F3: set_Pr6903500605879609269_c_nat] :
      ( ! [X3: produc4531938369520450825rm_c_d] :
          ( ( member820888157712331442rm_c_d @ X3 @ R5 )
         => ( produc6447316180697048258_c_d_o
            @ ^ [S3: term_c_d,T2: term_c_d] :
                ( ( term_ground_c_d @ S3 )
                & ( term_ground_c_d @ T2 ) )
            @ X3 ) )
     => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ S @ T ) @ ( srstep2269559354477410592ep_c_d @ F3 @ R5 ) )
       => ( term_ground_c_d @ S ) ) ) ).

% srsteps_with_root_step_ground(1)
thf(fact_382_restrict__def,axiom,
    ( abstra2354209648719431185rm_a_b
    = ( ^ [R2: set_Pr2972776593051762503rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
          ( collec1344989105561874898rm_a_b
          @ ( produc161078424057476524_a_b_o
            @ ^ [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
                ( ( member5869715511025134514rm_a_b @ X @ S4 )
                & ( member5869715511025134514rm_a_b @ Y @ S4 )
                & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_383_restrict__def,axiom,
    ( abstra4478131260412406025at_nat
    = ( ^ [R2: set_Pr8693737435421807431at_nat,S4: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ S4 )
                & ( member8440522571783428010at_nat @ Y @ S4 )
                & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_384_restrict__def,axiom,
    ( abstract_restrict_o
    = ( ^ [R2: set_Product_prod_o_o,S4: set_o] :
          ( collec3167064739498627218od_o_o
          @ ( produc6197397395684419436_o_o_o
            @ ^ [X: $o,Y: $o] :
                ( ( member_o @ X @ S4 )
                & ( member_o @ Y @ S4 )
                & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_385_restrict__def,axiom,
    ( abstra6515489984838218110et_nat
    = ( ^ [R2: set_Pr5488025237498180813et_nat,S4: set_set_nat] :
          ( collec6662362479098859352et_nat
          @ ( produc6247414631856714078_nat_o
            @ ^ [X: set_nat,Y: set_nat] :
                ( ( member_set_nat @ X @ S4 )
                & ( member_set_nat @ Y @ S4 )
                & ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_386_restrict__def,axiom,
    ( abstra1087653880669894540rm_c_d
    = ( ^ [R2: set_Pr5080540287643815273rm_c_d,S4: set_term_c_d] :
          ( collec4273722079467896820rm_c_d
          @ ( produc6447316180697048258_c_d_o
            @ ^ [X: term_c_d,Y: term_c_d] :
                ( ( member_term_c_d @ X @ S4 )
                & ( member_term_c_d @ Y @ S4 )
                & ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X @ Y ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_387_restrict__def,axiom,
    ( abstra6662565036008789004rm_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b,S4: set_term_a_b] :
          ( collec99177395925924084rm_a_b
          @ ( produc6251846659952913346_a_b_o
            @ ^ [X: term_a_b,Y: term_a_b] :
                ( ( member_term_a_b @ X @ S4 )
                & ( member_term_a_b @ Y @ S4 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_388_restrict__def,axiom,
    ( abstra1219450721576609224ct_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat,S4: set_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y: nat] :
                ( ( member_nat @ X @ S4 )
                & ( member_nat @ Y @ S4 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ).

% restrict_def
thf(fact_389_same__fst__def,axiom,
    ( same_f843731131676418198rm_a_b
    = ( ^ [P2: term_a_b > $o,R6: term_a_b > set_Pr4386577575007340137rm_a_b] :
          ( collec1344989105561874898rm_a_b
          @ ( produc161078424057476524_a_b_o
            @ ( produc4352215818379867658_a_b_o
              @ ^ [X6: term_a_b,Y5: term_a_b] :
                  ( produc6251846659952913346_a_b_o
                  @ ^ [X: term_a_b,Y: term_a_b] :
                      ( ( X6 = X )
                      & ( P2 @ X )
                      & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Y ) @ ( R6 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_390_same__fst__def,axiom,
    ( same_fst_nat_nat
    = ( ^ [P2: nat > $o,R6: nat > set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [X6: nat,Y5: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [X: nat,Y: nat] :
                      ( ( X6 = X )
                      & ( P2 @ X )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Y ) @ ( R6 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_391_same__fst__def,axiom,
    ( same_f4492192013192299542rm_c_d
    = ( ^ [P2: term_a_b > $o,R6: term_a_b > set_Pr5080540287643815273rm_c_d] :
          ( collec5736545419516875218rm_c_d
          @ ( produc8669550388439012780_c_d_o
            @ ( produc7053737044884821514_c_d_o
              @ ^ [X6: term_a_b,Y5: term_c_d] :
                  ( produc3147824153920350274_c_d_o
                  @ ^ [X: term_a_b,Y: term_c_d] :
                      ( ( X6 = X )
                      & ( P2 @ X )
                      & ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ Y5 @ Y ) @ ( R6 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_392_same__fst__def,axiom,
    ( same_f6147505736335672342rm_a_b
    = ( ^ [P2: term_c_d > $o,R6: term_c_d > set_Pr4386577575007340137rm_a_b] :
          ( collec6190143733186597330rm_a_b
          @ ( produc1056133890898515372_a_b_o
            @ ( produc5873602711790147082_a_b_o
              @ ^ [X6: term_c_d,Y5: term_a_b] :
                  ( produc327966649874835522_a_b_o
                  @ ^ [X: term_c_d,Y: term_a_b] :
                      ( ( X6 = X )
                      & ( P2 @ X )
                      & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Y ) @ ( R6 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_393_refl__on__converse,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ( refl_on_term_c_d @ A3 @ ( conver7337423426258204773rm_c_d @ R ) )
      = ( refl_on_term_c_d @ A3 @ R ) ) ).

% refl_on_converse
thf(fact_394_refl__on__converse,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( refl_on_term_a_b @ A3 @ ( conver7608559976937845093rm_a_b @ R ) )
      = ( refl_on_term_a_b @ A3 @ R ) ) ).

% refl_on_converse
thf(fact_395_refl__on__converse,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( refl_on_nat @ A3 @ ( converse_nat_nat @ R ) )
      = ( refl_on_nat @ A3 @ R ) ) ).

% refl_on_converse
thf(fact_396_same__fstI,axiom,
    ! [P: term_a_b > $o,X5: term_a_b,Y6: term_c_d,Y3: term_c_d,R3: term_a_b > set_Pr5080540287643815273rm_c_d] :
      ( ( P @ X5 )
     => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ Y6 @ Y3 ) @ ( R3 @ X5 ) )
       => ( member3585784829053206928rm_c_d @ ( produc4766828187215050455rm_c_d @ ( produc1445286645097778113rm_c_d @ X5 @ Y6 ) @ ( produc1445286645097778113rm_c_d @ X5 @ Y3 ) ) @ ( same_f4492192013192299542rm_c_d @ P @ R3 ) ) ) ) ).

% same_fstI
thf(fact_397_same__fstI,axiom,
    ! [P: term_c_d > $o,X5: term_c_d,Y6: term_c_d,Y3: term_c_d,R3: term_c_d > set_Pr5080540287643815273rm_c_d] :
      ( ( P @ X5 )
     => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ Y6 @ Y3 ) @ ( R3 @ X5 ) )
       => ( member8430939456677929360rm_c_d @ ( produc5257394494426896087rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Y6 ) @ ( produc6749061249757032257rm_c_d @ X5 @ Y3 ) ) @ ( same_f572594580996777878rm_c_d @ P @ R3 ) ) ) ) ).

% same_fstI
thf(fact_398_same__fstI,axiom,
    ! [P: term_c_d > $o,X5: term_c_d,Y6: term_a_b,Y3: term_a_b,R3: term_c_d > set_Pr4386577575007340137rm_a_b] :
      ( ( P @ X5 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Y3 ) @ ( R3 @ X5 ) )
       => ( member4039383142722929040rm_a_b @ ( produc2254039926008297175rm_a_b @ ( produc3100600368241150913rm_a_b @ X5 @ Y6 ) @ ( produc3100600368241150913rm_a_b @ X5 @ Y3 ) ) @ ( same_f6147505736335672342rm_a_b @ P @ R3 ) ) ) ) ).

% same_fstI
thf(fact_399_same__fstI,axiom,
    ! [P: term_a_b > $o,X5: term_a_b,Y6: term_a_b,Y3: term_a_b,R3: term_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( P @ X5 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Y3 ) @ ( R3 @ X5 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y6 ) @ ( produc7020197800436672577rm_a_b @ X5 @ Y3 ) ) @ ( same_f843731131676418198rm_a_b @ P @ R3 ) ) ) ) ).

% same_fstI
thf(fact_400_same__fstI,axiom,
    ! [P: nat > $o,X5: nat,Y6: nat,Y3: nat,R3: nat > set_Pr1261947904930325089at_nat] :
      ( ( P @ X5 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y3 ) @ ( R3 @ X5 ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X5 @ Y6 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) @ ( same_fst_nat_nat @ P @ R3 ) ) ) ) ).

% same_fstI
thf(fact_401_irreflp__on__def,axiom,
    ( irreflp_on_nat
    = ( ^ [A2: set_nat,R6: nat > nat > $o] :
        ! [X: nat] :
          ( ( member_nat @ X @ A2 )
         => ~ ( R6 @ X @ X ) ) ) ) ).

% irreflp_on_def
thf(fact_402_irreflp__onI,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b,R3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
      ( ! [A6: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A6 @ A3 )
         => ~ ( R3 @ A6 @ A6 ) )
     => ( irrefl1118632579264806371rm_a_b @ A3 @ R3 ) ) ).

% irreflp_onI
thf(fact_403_irreflp__onI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R3: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [A6: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ A6 @ A3 )
         => ~ ( R3 @ A6 @ A6 ) )
     => ( irrefl2576046763318297307at_nat @ A3 @ R3 ) ) ).

% irreflp_onI
thf(fact_404_irreflp__onI,axiom,
    ! [A3: set_o,R3: $o > $o > $o] :
      ( ! [A6: $o] :
          ( ( member_o @ A6 @ A3 )
         => ~ ( R3 @ A6 @ A6 ) )
     => ( irreflp_on_o @ A3 @ R3 ) ) ).

% irreflp_onI
thf(fact_405_irreflp__onI,axiom,
    ! [A3: set_set_nat,R3: set_nat > set_nat > $o] :
      ( ! [A6: set_nat] :
          ( ( member_set_nat @ A6 @ A3 )
         => ~ ( R3 @ A6 @ A6 ) )
     => ( irreflp_on_set_nat @ A3 @ R3 ) ) ).

% irreflp_onI
thf(fact_406_irreflp__onI,axiom,
    ! [A3: set_nat,R3: nat > nat > $o] :
      ( ! [A6: nat] :
          ( ( member_nat @ A6 @ A3 )
         => ~ ( R3 @ A6 @ A6 ) )
     => ( irreflp_on_nat @ A3 @ R3 ) ) ).

% irreflp_onI
thf(fact_407_irreflp__onD,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b,R3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o,A4: produc357393685978478089rm_a_b] :
      ( ( irrefl1118632579264806371rm_a_b @ A3 @ R3 )
     => ( ( member5869715511025134514rm_a_b @ A4 @ A3 )
       => ~ ( R3 @ A4 @ A4 ) ) ) ).

% irreflp_onD
thf(fact_408_irreflp__onD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R3: product_prod_nat_nat > product_prod_nat_nat > $o,A4: product_prod_nat_nat] :
      ( ( irrefl2576046763318297307at_nat @ A3 @ R3 )
     => ( ( member8440522571783428010at_nat @ A4 @ A3 )
       => ~ ( R3 @ A4 @ A4 ) ) ) ).

% irreflp_onD
thf(fact_409_irreflp__onD,axiom,
    ! [A3: set_o,R3: $o > $o > $o,A4: $o] :
      ( ( irreflp_on_o @ A3 @ R3 )
     => ( ( member_o @ A4 @ A3 )
       => ~ ( R3 @ A4 @ A4 ) ) ) ).

% irreflp_onD
thf(fact_410_irreflp__onD,axiom,
    ! [A3: set_set_nat,R3: set_nat > set_nat > $o,A4: set_nat] :
      ( ( irreflp_on_set_nat @ A3 @ R3 )
     => ( ( member_set_nat @ A4 @ A3 )
       => ~ ( R3 @ A4 @ A4 ) ) ) ).

% irreflp_onD
thf(fact_411_irreflp__onD,axiom,
    ! [A3: set_nat,R3: nat > nat > $o,A4: nat] :
      ( ( irreflp_on_nat @ A3 @ R3 )
     => ( ( member_nat @ A4 @ A3 )
       => ~ ( R3 @ A4 @ A4 ) ) ) ).

% irreflp_onD
thf(fact_412_refl__onD2,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X5: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A3 @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X5 @ Y3 ) @ R )
       => ( member5869715511025134514rm_a_b @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_413_refl__onD2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X5: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) @ R )
       => ( member8440522571783428010at_nat @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_414_refl__onD2,axiom,
    ! [A3: set_o,R: set_Product_prod_o_o,X5: $o,Y3: $o] :
      ( ( refl_on_o @ A3 @ R )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y3 ) @ R )
       => ( member_o @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_415_refl__onD2,axiom,
    ! [A3: set_set_nat,R: set_Pr5488025237498180813et_nat,X5: set_nat,Y3: set_nat] :
      ( ( refl_on_set_nat @ A3 @ R )
     => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y3 ) @ R )
       => ( member_set_nat @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_416_refl__onD2,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d,X5: term_c_d,Y3: term_c_d] :
      ( ( refl_on_term_c_d @ A3 @ R )
     => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Y3 ) @ R )
       => ( member_term_c_d @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_417_refl__onD2,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X5: term_a_b,Y3: term_a_b] :
      ( ( refl_on_term_a_b @ A3 @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y3 ) @ R )
       => ( member_term_a_b @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_418_refl__onD2,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat,X5: nat,Y3: nat] :
      ( ( refl_on_nat @ A3 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R )
       => ( member_nat @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_419_refl__onD1,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X5: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A3 @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X5 @ Y3 ) @ R )
       => ( member5869715511025134514rm_a_b @ X5 @ A3 ) ) ) ).

% refl_onD1
thf(fact_420_refl__onD1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X5: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) @ R )
       => ( member8440522571783428010at_nat @ X5 @ A3 ) ) ) ).

% refl_onD1
thf(fact_421_refl__onD1,axiom,
    ! [A3: set_o,R: set_Product_prod_o_o,X5: $o,Y3: $o] :
      ( ( refl_on_o @ A3 @ R )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y3 ) @ R )
       => ( member_o @ X5 @ A3 ) ) ) ).

% refl_onD1
thf(fact_422_refl__onD1,axiom,
    ! [A3: set_set_nat,R: set_Pr5488025237498180813et_nat,X5: set_nat,Y3: set_nat] :
      ( ( refl_on_set_nat @ A3 @ R )
     => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y3 ) @ R )
       => ( member_set_nat @ X5 @ A3 ) ) ) ).

% refl_onD1
thf(fact_423_refl__onD1,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d,X5: term_c_d,Y3: term_c_d] :
      ( ( refl_on_term_c_d @ A3 @ R )
     => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Y3 ) @ R )
       => ( member_term_c_d @ X5 @ A3 ) ) ) ).

% refl_onD1
thf(fact_424_refl__onD1,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X5: term_a_b,Y3: term_a_b] :
      ( ( refl_on_term_a_b @ A3 @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y3 ) @ R )
       => ( member_term_a_b @ X5 @ A3 ) ) ) ).

% refl_onD1
thf(fact_425_refl__onD1,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat,X5: nat,Y3: nat] :
      ( ( refl_on_nat @ A3 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R )
       => ( member_nat @ X5 @ A3 ) ) ) ).

% refl_onD1
thf(fact_426_refl__onD,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A4: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A3 @ R )
     => ( ( member5869715511025134514rm_a_b @ A4 @ A3 )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ A4 ) @ R ) ) ) ).

% refl_onD
thf(fact_427_refl__onD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A4: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R )
     => ( ( member8440522571783428010at_nat @ A4 @ A3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ A4 ) @ R ) ) ) ).

% refl_onD
thf(fact_428_refl__onD,axiom,
    ! [A3: set_o,R: set_Product_prod_o_o,A4: $o] :
      ( ( refl_on_o @ A3 @ R )
     => ( ( member_o @ A4 @ A3 )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A4 @ A4 ) @ R ) ) ) ).

% refl_onD
thf(fact_429_refl__onD,axiom,
    ! [A3: set_set_nat,R: set_Pr5488025237498180813et_nat,A4: set_nat] :
      ( ( refl_on_set_nat @ A3 @ R )
     => ( ( member_set_nat @ A4 @ A3 )
       => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A4 @ A4 ) @ R ) ) ) ).

% refl_onD
thf(fact_430_refl__onD,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d,A4: term_c_d] :
      ( ( refl_on_term_c_d @ A3 @ R )
     => ( ( member_term_c_d @ A4 @ A3 )
       => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ A4 ) @ R ) ) ) ).

% refl_onD
thf(fact_431_refl__onD,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A4: term_a_b] :
      ( ( refl_on_term_a_b @ A3 @ R )
     => ( ( member_term_a_b @ A4 @ A3 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R ) ) ) ).

% refl_onD
thf(fact_432_refl__onD,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat,A4: nat] :
      ( ( refl_on_nat @ A3 @ R )
     => ( ( member_nat @ A4 @ A3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R ) ) ) ).

% refl_onD
thf(fact_433_rp__inv__image__def,axiom,
    ( fun_rp3004530424201022588at_nat
    = ( produc1827612380487965875at_nat
      @ ^ [R6: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat,F2: nat > nat] : ( produc2922128104949294807at_nat @ ( inv_image_nat_nat @ R6 @ F2 ) @ ( inv_image_nat_nat @ S4 @ F2 ) ) ) ) ).

% rp_inv_image_def
thf(fact_434_refl__on__domain,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A4: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A3 @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B2 ) @ R )
       => ( ( member5869715511025134514rm_a_b @ A4 @ A3 )
          & ( member5869715511025134514rm_a_b @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_435_refl__on__domain,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A4: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ B2 ) @ R )
       => ( ( member8440522571783428010at_nat @ A4 @ A3 )
          & ( member8440522571783428010at_nat @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_436_refl__on__domain,axiom,
    ! [A3: set_o,R: set_Product_prod_o_o,A4: $o,B2: $o] :
      ( ( refl_on_o @ A3 @ R )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A4 @ B2 ) @ R )
       => ( ( member_o @ A4 @ A3 )
          & ( member_o @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_437_refl__on__domain,axiom,
    ! [A3: set_set_nat,R: set_Pr5488025237498180813et_nat,A4: set_nat,B2: set_nat] :
      ( ( refl_on_set_nat @ A3 @ R )
     => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A4 @ B2 ) @ R )
       => ( ( member_set_nat @ A4 @ A3 )
          & ( member_set_nat @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_438_refl__on__domain,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d,A4: term_c_d,B2: term_c_d] :
      ( ( refl_on_term_c_d @ A3 @ R )
     => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) @ R )
       => ( ( member_term_c_d @ A4 @ A3 )
          & ( member_term_c_d @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_439_refl__on__domain,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A4: term_a_b,B2: term_a_b] :
      ( ( refl_on_term_a_b @ A3 @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) @ R )
       => ( ( member_term_a_b @ A4 @ A3 )
          & ( member_term_a_b @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_440_refl__on__domain,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat,A4: nat,B2: nat] :
      ( ( refl_on_nat @ A3 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R )
       => ( ( member_nat @ A4 @ A3 )
          & ( member_nat @ B2 @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_441_lex__prod__def,axiom,
    ( lex_pr5892420715117795161rm_c_d
    = ( ^ [Ra: set_Pr5080540287643815273rm_c_d,Rb: set_Pr5080540287643815273rm_c_d] :
          ( collec1358328010286821842rm_c_d
          @ ( produc341233818425275820_c_d_o
            @ ( produc8575123938295100938_c_d_o
              @ ^ [A: term_c_d,B: term_c_d] :
                  ( produc6447316180697048258_c_d_o
                  @ ^ [A7: term_c_d,B5: term_c_d] :
                      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A @ A7 ) @ Ra )
                      | ( ( A = A7 )
                        & ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ B @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_442_lex__prod__def,axiom,
    ( lex_pr8119212446852010005_d_nat
    = ( ^ [Ra: set_Pr5080540287643815273rm_c_d,Rb: set_Pr1261947904930325089at_nat] :
          ( collec3801099787273043922_d_nat
          @ ( produc4132314953574007724_nat_o
            @ ( produc14787907101276426_nat_o
              @ ^ [A: term_c_d,B: nat] :
                  ( produc8272582700005955334_nat_o
                  @ ^ [A7: term_c_d,B5: nat] :
                      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A @ A7 ) @ Ra )
                      | ( ( A = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_443_lex__prod__def,axiom,
    ( lex_pr3461217179711548565_b_nat
    = ( ^ [Ra: set_Pr4386577575007340137rm_a_b,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7290524293179858898_b_nat
          @ ( produc1396999915110114220_nat_o
            @ ( produc2189794613626067210_nat_o
              @ ^ [A: term_a_b,B: nat] :
                  ( produc9162275563018852998_nat_o
                  @ ^ [A7: term_a_b,B5: nat] :
                      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A7 ) @ Ra )
                      | ( ( A = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_444_lex__prod__def,axiom,
    ( lex_pr6148223934062457749rm_c_d
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr5080540287643815273rm_c_d] :
          ( collec471171522628853714rm_c_d
          @ ( produc7763675869049316268_c_d_o
            @ ( produc827177795400628490_c_d_o
              @ ^ [A: nat,B: term_c_d] :
                  ( produc4132180821779813254_c_d_o
                  @ ^ [A7: nat,B5: term_c_d] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A7 ) @ Ra )
                      | ( ( A = A7 )
                        & ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ B @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_445_lex__prod__def,axiom,
    ( lex_pr2499763052546576405rm_a_b
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr4386577575007340137rm_a_b] :
          ( collec3467184569906692050rm_a_b
          @ ( produc813038845109486508_a_b_o
            @ ( produc1326667778449004810_a_b_o
              @ ^ [A: nat,B: term_a_b] :
                  ( produc7236203327812376326_a_b_o
                  @ ^ [A7: nat,B5: term_a_b] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A7 ) @ Ra )
                      | ( ( A = A7 )
                        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_446_lex__prod__def,axiom,
    ( lex_pr6163557265797435481rm_a_b
    = ( ^ [Ra: set_Pr4386577575007340137rm_a_b,Rb: set_Pr4386577575007340137rm_a_b] :
          ( collec1344989105561874898rm_a_b
          @ ( produc161078424057476524_a_b_o
            @ ( produc4352215818379867658_a_b_o
              @ ^ [A: term_a_b,B: term_a_b] :
                  ( produc6251846659952913346_a_b_o
                  @ ^ [A7: term_a_b,B5: term_a_b] :
                      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A7 ) @ Ra )
                      | ( ( A = A7 )
                        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_447_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A: nat,B: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A7: nat,B5: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A7 ) @ Ra )
                      | ( ( A = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_448_lex__prod__def,axiom,
    ( lex_pr588646110458541017rm_c_d
    = ( ^ [Ra: set_Pr4386577575007340137rm_a_b,Rb: set_Pr5080540287643815273rm_c_d] :
          ( collec5736545419516875218rm_c_d
          @ ( produc8669550388439012780_c_d_o
            @ ( produc7053737044884821514_c_d_o
              @ ^ [A: term_a_b,B: term_c_d] :
                  ( produc3147824153920350274_c_d_o
                  @ ^ [A7: term_a_b,B5: term_c_d] :
                      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A7 ) @ Ra )
                      | ( ( A = A7 )
                        & ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ B @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_449_lex__prod__def,axiom,
    ( lex_pr2243959833601913817rm_a_b
    = ( ^ [Ra: set_Pr5080540287643815273rm_c_d,Rb: set_Pr4386577575007340137rm_a_b] :
          ( collec6190143733186597330rm_a_b
          @ ( produc1056133890898515372_a_b_o
            @ ( produc5873602711790147082_a_b_o
              @ ^ [A: term_c_d,B: term_a_b] :
                  ( produc327966649874835522_a_b_o
                  @ ^ [A7: term_c_d,B5: term_a_b] :
                      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A @ A7 ) @ Ra )
                      | ( ( A = A7 )
                        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_450_irreflp__irrefl__eq,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d] :
      ( ( irreflp_on_term_c_d @ top_top_set_term_c_d
        @ ^ [A: term_c_d,B: term_c_d] : ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A @ B ) @ R ) )
      = ( irrefl_on_term_c_d @ top_top_set_term_c_d @ R ) ) ).

% irreflp_irrefl_eq
thf(fact_451_irreflp__irrefl__eq,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( irreflp_on_term_a_b @ top_top_set_term_a_b
        @ ^ [A: term_a_b,B: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R ) )
      = ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).

% irreflp_irrefl_eq
thf(fact_452_irreflp__irrefl__eq,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( irrefl2576046763318297307at_nat @ top_to4669805908274784177at_nat
        @ ^ [A: product_prod_nat_nat,B: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R ) )
      = ( irrefl7408258699751485431at_nat @ top_to4669805908274784177at_nat @ R ) ) ).

% irreflp_irrefl_eq
thf(fact_453_irreflp__irrefl__eq,axiom,
    ! [R: set_Pr1076074158304066111at_nat] :
      ( ( irrefl3240489695051909285at_nat @ top_to6661820994512907621at_nat
        @ ^ [A: sum_sum_nat_nat,B: sum_sum_nat_nat] : ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ A @ B ) @ R ) )
      = ( irrefl688463893611329673at_nat @ top_to6661820994512907621at_nat @ R ) ) ).

% irreflp_irrefl_eq
thf(fact_454_irreflp__irrefl__eq,axiom,
    ! [R: set_Pr5488025237498180813et_nat] :
      ( ( irreflp_on_set_nat @ top_top_set_set_nat
        @ ^ [A: set_nat,B: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ R ) )
      = ( irrefl_on_set_nat @ top_top_set_set_nat @ R ) ) ).

% irreflp_irrefl_eq
thf(fact_455_irreflp__irrefl__eq,axiom,
    ! [R: set_Pr8788715879341982671rm_a_b] :
      ( ( irrefl4183777908429839213rm_a_b @ top_to220653634128270125rm_a_b
        @ ^ [A: term_c_d > term_a_b,B: term_c_d > term_a_b] : ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ A @ B ) @ R ) )
      = ( irrefl7486889657381885777rm_a_b @ top_to220653634128270125rm_a_b @ R ) ) ).

% irreflp_irrefl_eq
thf(fact_456_irreflp__irrefl__eq,axiom,
    ! [R: set_Pr6132369733443093455rm_c_d] :
      ( ( irrefl6974145419508079469rm_c_d @ top_to44634570330034989rm_c_d
        @ ^ [A: term_a_b > term_c_d,B: term_a_b > term_c_d] : ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ A @ B ) @ R ) )
      = ( irrefl1053885131605350225rm_c_d @ top_to44634570330034989rm_c_d @ R ) ) ).

% irreflp_irrefl_eq
thf(fact_457_irreflp__irrefl__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( irreflp_on_nat @ top_top_set_nat
        @ ^ [A: nat,B: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
      = ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).

% irreflp_irrefl_eq
thf(fact_458_trans__on__join,axiom,
    ( trans_5404704112166290345rm_a_b
    = ( ^ [A2: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
        ! [X: produc4523971788519308903rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ X @ R2 )
         => ( produc161078424057476524_a_b_o
            @ ^ [Y: produc357393685978478089rm_a_b,Y12: produc357393685978478089rm_a_b] :
                ( ( member5869715511025134514rm_a_b @ Y @ A2 )
               => ( ( member5869715511025134514rm_a_b @ Y12 @ A2 )
                 => ! [Z2: produc4523971788519308903rm_a_b] :
                      ( ( member8417600551952982416rm_a_b @ Z2 @ R2 )
                     => ( produc161078424057476524_a_b_o
                        @ ^ [Y22: produc357393685978478089rm_a_b,Aa: produc357393685978478089rm_a_b] :
                            ( ( Y12 = Y22 )
                           => ( ( member5869715511025134514rm_a_b @ Aa @ A2 )
                             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Aa ) @ R2 ) ) )
                        @ Z2 ) ) ) )
            @ X ) ) ) ) ).

% trans_on_join
thf(fact_459_trans__on__join,axiom,
    ( trans_on_o
    = ( ^ [A2: set_o,R2: set_Product_prod_o_o] :
        ! [X: product_prod_o_o] :
          ( ( member7466972457876170832od_o_o @ X @ R2 )
         => ( produc6197397395684419436_o_o_o
            @ ^ [Y: $o,Y12: $o] :
                ( ( member_o @ Y @ A2 )
               => ( ( member_o @ Y12 @ A2 )
                 => ! [Z2: product_prod_o_o] :
                      ( ( member7466972457876170832od_o_o @ Z2 @ R2 )
                     => ( produc6197397395684419436_o_o_o
                        @ ^ [Y22: $o,Aa: $o] :
                            ( ( Y12 = Y22 )
                           => ( ( member_o @ Aa @ A2 )
                             => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ Aa ) @ R2 ) ) )
                        @ Z2 ) ) ) )
            @ X ) ) ) ) ).

% trans_on_join
thf(fact_460_trans__on__join,axiom,
    ( trans_on_set_nat
    = ( ^ [A2: set_set_nat,R2: set_Pr5488025237498180813et_nat] :
        ! [X: produc7819656566062154093et_nat] :
          ( ( member8277197624267554838et_nat @ X @ R2 )
         => ( produc6247414631856714078_nat_o
            @ ^ [Y: set_nat,Y12: set_nat] :
                ( ( member_set_nat @ Y @ A2 )
               => ( ( member_set_nat @ Y12 @ A2 )
                 => ! [Z2: produc7819656566062154093et_nat] :
                      ( ( member8277197624267554838et_nat @ Z2 @ R2 )
                     => ( produc6247414631856714078_nat_o
                        @ ^ [Y22: set_nat,Aa: set_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member_set_nat @ Aa @ A2 )
                             => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y @ Aa ) @ R2 ) ) )
                        @ Z2 ) ) ) )
            @ X ) ) ) ) ).

% trans_on_join
thf(fact_461_trans__on__join,axiom,
    ( trans_on_term_c_d
    = ( ^ [A2: set_term_c_d,R2: set_Pr5080540287643815273rm_c_d] :
        ! [X: produc4531938369520450825rm_c_d] :
          ( ( member820888157712331442rm_c_d @ X @ R2 )
         => ( produc6447316180697048258_c_d_o
            @ ^ [Y: term_c_d,Y12: term_c_d] :
                ( ( member_term_c_d @ Y @ A2 )
               => ( ( member_term_c_d @ Y12 @ A2 )
                 => ! [Z2: produc4531938369520450825rm_c_d] :
                      ( ( member820888157712331442rm_c_d @ Z2 @ R2 )
                     => ( produc6447316180697048258_c_d_o
                        @ ^ [Y22: term_c_d,Aa: term_c_d] :
                            ( ( Y12 = Y22 )
                           => ( ( member_term_c_d @ Aa @ A2 )
                             => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ Y @ Aa ) @ R2 ) ) )
                        @ Z2 ) ) ) )
            @ X ) ) ) ) ).

% trans_on_join
thf(fact_462_trans__on__join,axiom,
    ( trans_3241703717017304737at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
        ! [X: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X @ R2 )
         => ( produc6590410687421337004_nat_o
            @ ^ [Y: product_prod_nat_nat,Y12: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y @ A2 )
               => ( ( member8440522571783428010at_nat @ Y12 @ A2 )
                 => ! [Z2: produc859450856879609959at_nat] :
                      ( ( member8206827879206165904at_nat @ Z2 @ R2 )
                     => ( produc6590410687421337004_nat_o
                        @ ^ [Y22: product_prod_nat_nat,Aa: product_prod_nat_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member8440522571783428010at_nat @ Aa @ A2 )
                             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Aa ) @ R2 ) ) )
                        @ Z2 ) ) ) )
            @ X ) ) ) ) ).

% trans_on_join
thf(fact_463_trans__on__join,axiom,
    ( trans_on_term_a_b
    = ( ^ [A2: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
        ! [X: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ R2 )
         => ( produc6251846659952913346_a_b_o
            @ ^ [Y: term_a_b,Y12: term_a_b] :
                ( ( member_term_a_b @ Y @ A2 )
               => ( ( member_term_a_b @ Y12 @ A2 )
                 => ! [Z2: produc357393685978478089rm_a_b] :
                      ( ( member5869715511025134514rm_a_b @ Z2 @ R2 )
                     => ( produc6251846659952913346_a_b_o
                        @ ^ [Y22: term_a_b,Aa: term_a_b] :
                            ( ( Y12 = Y22 )
                           => ( ( member_term_a_b @ Aa @ A2 )
                             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Aa ) @ R2 ) ) )
                        @ Z2 ) ) ) )
            @ X ) ) ) ) ).

% trans_on_join
thf(fact_464_trans__on__join,axiom,
    ( trans_on_nat
    = ( ^ [A2: set_nat,R2: set_Pr1261947904930325089at_nat] :
        ! [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ R2 )
         => ( produc6081775807080527818_nat_o
            @ ^ [Y: nat,Y12: nat] :
                ( ( member_nat @ Y @ A2 )
               => ( ( member_nat @ Y12 @ A2 )
                 => ! [Z2: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ Z2 @ R2 )
                     => ( produc6081775807080527818_nat_o
                        @ ^ [Y22: nat,Aa: nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member_nat @ Aa @ A2 )
                             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Aa ) @ R2 ) ) )
                        @ Z2 ) ) ) )
            @ X ) ) ) ) ).

% trans_on_join
thf(fact_465_The__split__eq,axiom,
    ! [X5: term_c_d,Y3: term_c_d] :
      ( ( the_Pr8028467898164114064rm_c_d
        @ ( produc6447316180697048258_c_d_o
          @ ^ [X6: term_c_d,Y5: term_c_d] :
              ( ( X5 = X6 )
              & ( Y3 = Y5 ) ) ) )
      = ( produc6749061249757032257rm_c_d @ X5 @ Y3 ) ) ).

% The_split_eq
thf(fact_466_The__split__eq,axiom,
    ! [X5: term_a_b,Y3: term_a_b] :
      ( ( the_Pr3853923214622141328rm_a_b
        @ ( produc6251846659952913346_a_b_o
          @ ^ [X6: term_a_b,Y5: term_a_b] :
              ( ( X5 = X6 )
              & ( Y3 = Y5 ) ) ) )
      = ( produc7020197800436672577rm_a_b @ X5 @ Y3 ) ) ).

% The_split_eq
thf(fact_467_The__split__eq,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( the_Pr7557018466319803784at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X6: nat,Y5: nat] :
              ( ( X5 = X6 )
              & ( Y3 = Y5 ) ) ) )
      = ( product_Pair_nat_nat @ X5 @ Y3 ) ) ).

% The_split_eq
thf(fact_468_The__split__eq,axiom,
    ! [X5: term_a_b,Y3: term_c_d] :
      ( ( the_Pr2724693293504859920rm_c_d
        @ ( produc3147824153920350274_c_d_o
          @ ^ [X6: term_a_b,Y5: term_c_d] :
              ( ( X5 = X6 )
              & ( Y3 = Y5 ) ) ) )
      = ( produc1445286645097778113rm_c_d @ X5 @ Y3 ) ) ).

% The_split_eq
thf(fact_469_The__split__eq,axiom,
    ! [X5: term_c_d,Y3: term_a_b] :
      ( ( the_Pr9157697819281395472rm_a_b
        @ ( produc327966649874835522_a_b_o
          @ ^ [X6: term_c_d,Y5: term_a_b] :
              ( ( X5 = X6 )
              & ( Y3 = Y5 ) ) ) )
      = ( produc3100600368241150913rm_a_b @ X5 @ Y3 ) ) ).

% The_split_eq
thf(fact_470_ssubst__Pair__rhs,axiom,
    ! [R: term_a_b,S: term_c_d,R3: set_Pr4645549399426460137rm_c_d,S5: term_c_d] :
      ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ R @ S ) @ R3 )
     => ( ( S5 = S )
       => ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ R @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_471_ssubst__Pair__rhs,axiom,
    ! [R: term_c_d,S: term_a_b,R3: set_Pr4821568463224695273rm_a_b,S5: term_a_b] :
      ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ R @ S ) @ R3 )
     => ( ( S5 = S )
       => ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ R @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_472_ssubst__Pair__rhs,axiom,
    ! [R: term_c_d,S: term_c_d,R3: set_Pr5080540287643815273rm_c_d,S5: term_c_d] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ R @ S ) @ R3 )
     => ( ( S5 = S )
       => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ R @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_473_ssubst__Pair__rhs,axiom,
    ! [R: term_a_b,S: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S5: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S ) @ R3 )
     => ( ( S5 = S )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_474_ssubst__Pair__rhs,axiom,
    ! [R: nat,S: nat,R3: set_Pr1261947904930325089at_nat,S5: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R3 )
     => ( ( S5 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_475_converse__def,axiom,
    ( conver7337423426258204773rm_c_d
    = ( ^ [R2: set_Pr5080540287643815273rm_c_d] :
          ( collec4273722079467896820rm_c_d
          @ ( produc6447316180697048258_c_d_o
            @ ( conver4964912785503959199rm_c_d
              @ ^ [X: term_c_d,Y: term_c_d] : ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X @ Y ) @ R2 ) ) ) ) ) ) ).

% converse_def
thf(fact_476_converse__def,axiom,
    ( conver7608559976937845093rm_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b] :
          ( collec99177395925924084rm_a_b
          @ ( produc6251846659952913346_a_b_o
            @ ( conver5236049336183599519rm_a_b
              @ ^ [X: term_a_b,Y: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 ) ) ) ) ) ) ).

% converse_def
thf(fact_477_converse__def,axiom,
    ( converse_nat_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ( conversep_nat_nat
              @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ).

% converse_def
thf(fact_478_converse__def,axiom,
    ( conver3688962544742323429rm_a_b
    = ( ^ [R2: set_Pr4821568463224695273rm_a_b] :
          ( collec8193319511663418484rm_c_d
          @ ( produc3147824153920350274_c_d_o
            @ ( conver1316451903988077855rm_a_b
              @ ^ [X: term_c_d,Y: term_a_b] : ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ X @ Y ) @ R2 ) ) ) ) ) ) ).

% converse_def
thf(fact_479_converse__def,axiom,
    ( conver2033648821598950629rm_c_d
    = ( ^ [R2: set_Pr4645549399426460137rm_c_d] :
          ( collec5402952000585178228rm_a_b
          @ ( produc327966649874835522_a_b_o
            @ ( conver8884510217699480863rm_c_d
              @ ^ [X: term_a_b,Y: term_c_d] : ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ X @ Y ) @ R2 ) ) ) ) ) ) ).

% converse_def
thf(fact_480_pair__imageI,axiom,
    ! [A4: term_a_b,B2: term_c_d,A3: set_Pr4645549399426460137rm_c_d,F: term_a_b > term_c_d > $o] :
      ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) @ A3 )
     => ( member_o @ ( F @ A4 @ B2 ) @ ( image_2891144671118473044_c_d_o @ ( produc3147824153920350274_c_d_o @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_481_pair__imageI,axiom,
    ! [A4: term_c_d,B2: term_a_b,A3: set_Pr4821568463224695273rm_a_b,F: term_c_d > term_a_b > $o] :
      ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) @ A3 )
     => ( member_o @ ( F @ A4 @ B2 ) @ ( image_5492869505583658836_a_b_o @ ( produc327966649874835522_a_b_o @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_482_pair__imageI,axiom,
    ! [A4: nat,B2: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ A3 )
     => ( member_nat @ ( F @ A4 @ B2 ) @ ( image_2486076414777270412at_nat @ ( produc6842872674320459806at_nat @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_483_pair__imageI,axiom,
    ! [A4: nat,B2: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ A3 )
     => ( member_o @ ( F @ A4 @ B2 ) @ ( image_3693632289388996572_nat_o @ ( produc6081775807080527818_nat_o @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_484_pair__imageI,axiom,
    ! [A4: nat,B2: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ A3 )
     => ( member_set_nat @ ( F @ A4 @ B2 ) @ ( image_15824709712370754et_nat @ ( produc6189476227299908564et_nat @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_485_pair__imageI,axiom,
    ! [A4: nat,B2: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F @ A4 @ B2 ) @ ( image_5168914502847457605at_nat @ ( produc2626176000494625587at_nat @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_486_pair__imageI,axiom,
    ! [A4: term_a_b,B2: term_c_d,A3: set_Pr4645549399426460137rm_c_d,F: term_a_b > term_c_d > nat] :
      ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) @ A3 )
     => ( member_nat @ ( F @ A4 @ B2 ) @ ( image_3202112734296366868_d_nat @ ( produc2342468372103988646_d_nat @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_487_pair__imageI,axiom,
    ! [A4: term_c_d,B2: term_a_b,A3: set_Pr4821568463224695273rm_a_b,F: term_c_d > term_a_b > nat] :
      ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) @ A3 )
     => ( member_nat @ ( F @ A4 @ B2 ) @ ( image_2243577467222712084_b_nat @ ( produc3367825333614200230_b_nat @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_488_pair__imageI,axiom,
    ! [A4: term_c_d,B2: term_c_d,A3: set_Pr5080540287643815273rm_c_d,F: term_c_d > term_c_d > $o] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) @ A3 )
     => ( member_o @ ( F @ A4 @ B2 ) @ ( image_6190636697895171028_c_d_o @ ( produc6447316180697048258_c_d_o @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_489_pair__imageI,axiom,
    ! [A4: term_c_d,B2: term_c_d,A3: set_Pr5080540287643815273rm_c_d,F: term_c_d > term_c_d > nat] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) @ A3 )
     => ( member_nat @ ( F @ A4 @ B2 ) @ ( image_8885464962947039892_d_nat @ ( produc8025820600754661670_d_nat @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_490_image__eqI,axiom,
    ! [B2: $o,F: $o > $o,X5: $o,A3: set_o] :
      ( ( B2
        = ( F @ X5 ) )
     => ( ( member_o @ X5 @ A3 )
       => ( member_o @ B2 @ ( image_o_o @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_491_image__eqI,axiom,
    ! [B2: nat,F: $o > nat,X5: $o,A3: set_o] :
      ( ( B2
        = ( F @ X5 ) )
     => ( ( member_o @ X5 @ A3 )
       => ( member_nat @ B2 @ ( image_o_nat @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_492_image__eqI,axiom,
    ! [B2: $o,F: nat > $o,X5: nat,A3: set_nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( ( member_nat @ X5 @ A3 )
       => ( member_o @ B2 @ ( image_nat_o @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_493_image__eqI,axiom,
    ! [B2: nat,F: nat > nat,X5: nat,A3: set_nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( ( member_nat @ X5 @ A3 )
       => ( member_nat @ B2 @ ( image_nat_nat2 @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_494_image__eqI,axiom,
    ! [B2: set_nat,F: $o > set_nat,X5: $o,A3: set_o] :
      ( ( B2
        = ( F @ X5 ) )
     => ( ( member_o @ X5 @ A3 )
       => ( member_set_nat @ B2 @ ( image_o_set_nat @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_495_image__eqI,axiom,
    ! [B2: $o,F: set_nat > $o,X5: set_nat,A3: set_set_nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( ( member_set_nat @ X5 @ A3 )
       => ( member_o @ B2 @ ( image_set_nat_o @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_496_image__eqI,axiom,
    ! [B2: nat,F: set_nat > nat,X5: set_nat,A3: set_set_nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( ( member_set_nat @ X5 @ A3 )
       => ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_497_image__eqI,axiom,
    ! [B2: set_nat,F: nat > set_nat,X5: nat,A3: set_nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( ( member_nat @ X5 @ A3 )
       => ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_498_image__eqI,axiom,
    ! [B2: $o,F: product_prod_nat_nat > $o,X5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( ( member8440522571783428010at_nat @ X5 @ A3 )
       => ( member_o @ B2 @ ( image_3693632289388996572_nat_o @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_499_image__eqI,axiom,
    ! [B2: nat,F: product_prod_nat_nat > nat,X5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( ( member8440522571783428010at_nat @ X5 @ A3 )
       => ( member_nat @ B2 @ ( image_2486076414777270412at_nat @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_500_UNIV__I,axiom,
    ! [X5: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X5 @ top_to1314267278846557113rm_a_b ) ).

% UNIV_I
thf(fact_501_UNIV__I,axiom,
    ! [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ top_to4669805908274784177at_nat ) ).

% UNIV_I
thf(fact_502_UNIV__I,axiom,
    ! [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ).

% UNIV_I
thf(fact_503_UNIV__I,axiom,
    ! [X5: sum_sum_nat_nat] : ( member8583185029347631382at_nat @ X5 @ top_to6661820994512907621at_nat ) ).

% UNIV_I
thf(fact_504_UNIV__I,axiom,
    ! [X5: set_nat] : ( member_set_nat @ X5 @ top_top_set_set_nat ) ).

% UNIV_I
thf(fact_505_UNIV__I,axiom,
    ! [X5: term_c_d > term_a_b] : ( member3140172230840508126rm_a_b @ X5 @ top_to220653634128270125rm_a_b ) ).

% UNIV_I
thf(fact_506_UNIV__I,axiom,
    ! [X5: term_a_b > term_c_d] : ( member5930539741918748382rm_c_d @ X5 @ top_to44634570330034989rm_c_d ) ).

% UNIV_I
thf(fact_507_UNIV__I,axiom,
    ! [X5: nat] : ( member_nat @ X5 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_508_converse__UNIV,axiom,
    ( ( conver7337423426258204773rm_c_d @ top_to2008229991483032249rm_c_d )
    = top_to2008229991483032249rm_c_d ) ).

% converse_UNIV
thf(fact_509_converse__UNIV,axiom,
    ( ( conver7608559976937845093rm_a_b @ top_to1314267278846557113rm_a_b )
    = top_to1314267278846557113rm_a_b ) ).

% converse_UNIV
thf(fact_510_converse__UNIV,axiom,
    ( ( converse_nat_nat @ top_to4669805908274784177at_nat )
    = top_to4669805908274784177at_nat ) ).

% converse_UNIV
thf(fact_511_converse__UNIV,axiom,
    ( ( conver2033648821598950629rm_c_d @ top_to1573239103265677113rm_c_d )
    = top_to1749258167063912249rm_a_b ) ).

% converse_UNIV
thf(fact_512_converse__UNIV,axiom,
    ( ( conver3688962544742323429rm_a_b @ top_to1749258167063912249rm_a_b )
    = top_to1573239103265677113rm_c_d ) ).

% converse_UNIV
thf(fact_513_conversep__iff,axiom,
    ( conver8884510217699480863rm_c_d
    = ( ^ [R2: term_a_b > term_c_d > $o,A: term_c_d,B: term_a_b] : ( R2 @ B @ A ) ) ) ).

% conversep_iff
thf(fact_514_conversep__iff,axiom,
    ( conver1316451903988077855rm_a_b
    = ( ^ [R2: term_c_d > term_a_b > $o,A: term_a_b,B: term_c_d] : ( R2 @ B @ A ) ) ) ).

% conversep_iff
thf(fact_515_conversep__inject,axiom,
    ! [R: term_a_b > term_c_d > $o,S: term_a_b > term_c_d > $o] :
      ( ( ( conver8884510217699480863rm_c_d @ R )
        = ( conver8884510217699480863rm_c_d @ S ) )
      = ( R = S ) ) ).

% conversep_inject
thf(fact_516_conversep__inject,axiom,
    ! [R: term_c_d > term_a_b > $o,S: term_c_d > term_a_b > $o] :
      ( ( ( conver1316451903988077855rm_a_b @ R )
        = ( conver1316451903988077855rm_a_b @ S ) )
      = ( R = S ) ) ).

% conversep_inject
thf(fact_517_conversep__conversep,axiom,
    ! [R: term_c_d > term_a_b > $o] :
      ( ( conver8884510217699480863rm_c_d @ ( conver1316451903988077855rm_a_b @ R ) )
      = R ) ).

% conversep_conversep
thf(fact_518_conversep__conversep,axiom,
    ! [R: term_a_b > term_c_d > $o] :
      ( ( conver1316451903988077855rm_a_b @ ( conver8884510217699480863rm_c_d @ R ) )
      = R ) ).

% conversep_conversep
thf(fact_519_surj__swap,axiom,
    ( ( image_3339315644087553349rm_a_b @ produc6190307670391109985rm_c_d @ top_to1573239103265677113rm_c_d )
    = top_to1749258167063912249rm_a_b ) ).

% surj_swap
thf(fact_520_surj__swap,axiom,
    ( ( image_6407262405037280581rm_c_d @ produc7845621393534482785rm_a_b @ top_to1749258167063912249rm_a_b )
    = top_to1573239103265677113rm_c_d ) ).

% surj_swap
thf(fact_521_trans__on__converse,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ( trans_on_term_c_d @ A3 @ ( conver7337423426258204773rm_c_d @ R ) )
      = ( trans_on_term_c_d @ A3 @ R ) ) ).

% trans_on_converse
thf(fact_522_trans__on__converse,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( trans_on_term_a_b @ A3 @ ( conver7608559976937845093rm_a_b @ R ) )
      = ( trans_on_term_a_b @ A3 @ R ) ) ).

% trans_on_converse
thf(fact_523_trans__on__converse,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ A3 @ ( converse_nat_nat @ R ) )
      = ( trans_on_nat @ A3 @ R ) ) ).

% trans_on_converse
thf(fact_524_trans__on__converse,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( trans_3241703717017304737at_nat @ A3 @ ( conver5251360641855610363at_nat @ R ) )
      = ( trans_3241703717017304737at_nat @ A3 @ R ) ) ).

% trans_on_converse
thf(fact_525_pair__in__swap__image,axiom,
    ! [Y3: term_c_d,X5: term_c_d,A3: set_Pr5080540287643815273rm_c_d] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ Y3 @ X5 ) @ ( image_6620249462377639237rm_c_d @ produc2270710238195588321rm_c_d @ A3 ) )
      = ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Y3 ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_526_pair__in__swap__image,axiom,
    ! [Y3: term_a_b,X5: term_a_b,A3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X5 ) @ ( image_3126328586747194693rm_a_b @ produc2541846788875228641rm_a_b @ A3 ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y3 ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_527_pair__in__swap__image,axiom,
    ! [Y3: nat,X5: nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X5 ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A3 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_528_pair__in__swap__image,axiom,
    ! [Y3: term_c_d,X5: term_a_b,A3: set_Pr4645549399426460137rm_c_d] :
      ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ Y3 @ X5 ) @ ( image_3339315644087553349rm_a_b @ produc6190307670391109985rm_c_d @ A3 ) )
      = ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ X5 @ Y3 ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_529_pair__in__swap__image,axiom,
    ! [Y3: term_a_b,X5: term_c_d,A3: set_Pr4821568463224695273rm_a_b] :
      ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ Y3 @ X5 ) @ ( image_6407262405037280581rm_c_d @ produc7845621393534482785rm_a_b @ A3 ) )
      = ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ X5 @ Y3 ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_530_irreflp__on__converse,axiom,
    ! [A3: set_nat,R: nat > nat > $o] :
      ( ( irreflp_on_nat @ A3 @ ( conversep_nat_nat @ R ) )
      = ( irreflp_on_nat @ A3 @ R ) ) ).

% irreflp_on_converse
thf(fact_531_trans__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R_A )
     => ( ( trans_on_nat @ top_top_set_nat @ R_B )
       => ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_532_trans__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr5488025237498180813et_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R_A )
     => ( ( trans_on_set_nat @ top_top_set_set_nat @ R_B )
       => ( trans_4870676645132490583et_nat @ top_to173688896754200935et_nat @ ( lex_prod_nat_set_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_533_trans__lex__prod,axiom,
    ! [R_A: set_Pr5488025237498180813et_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_set_nat @ top_top_set_set_nat @ R_A )
     => ( ( trans_on_nat @ top_top_set_nat @ R_B )
       => ( trans_738568395637599831at_nat @ top_to3374598108630517863at_nat @ ( lex_prod_set_nat_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_534_trans__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( trans_on_nat @ top_top_set_nat @ R_B )
       => ( trans_690021572951380832at_nat @ top_to2567003362141017830at_nat @ ( lex_pr4669217892513370978at_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_535_trans__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R_A )
     => ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R_B )
       => ( trans_8787906625838855558at_nat @ top_to7745748144976627212at_nat @ ( lex_pr8029265285556086080at_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_536_trans__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1076074158304066111at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R_A )
     => ( ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R_B )
       => ( trans_1779813735273096528at_nat @ top_to678699698364323680at_nat @ ( lex_pr6930960718592407552at_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_537_trans__lex__prod,axiom,
    ! [R_A: set_Pr1076074158304066111at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R_A )
     => ( ( trans_on_nat @ top_top_set_nat @ R_B )
       => ( trans_8252949245602683984at_nat @ top_to8411399450232720992at_nat @ ( lex_pr7572635211855035776at_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_538_trans__lex__prod,axiom,
    ! [R_A: set_Pr5488025237498180813et_nat,R_B: set_Pr5488025237498180813et_nat] :
      ( ( trans_on_set_nat @ top_top_set_set_nat @ R_A )
     => ( ( trans_on_set_nat @ top_top_set_set_nat @ R_B )
       => ( trans_1943384616820666125et_nat @ top_to3040159065575275037et_nat @ ( lex_pr6462853727283027389et_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_539_trans__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr5488025237498180813et_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( trans_on_set_nat @ top_top_set_set_nat @ R_B )
       => ( trans_185108081036603414et_nat @ top_to4194386802282351772et_nat @ ( lex_pr6326417796491097880et_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_540_trans__lex__prod,axiom,
    ! [R_A: set_Pr1076074158304066111at_nat,R_B: set_Pr5488025237498180813et_nat] :
      ( ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R_A )
     => ( ( trans_on_set_nat @ top_top_set_set_nat @ R_B )
       => ( trans_2415969763348723974et_nat @ top_to6784681116597541910et_nat @ ( lex_pr277683831282854198et_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_541_in__lex__prod,axiom,
    ! [A4: term_c_d,B2: nat,A5: term_c_d,B3: nat,R: set_Pr5080540287643815273rm_c_d,S: set_Pr1261947904930325089at_nat] :
      ( ( member8400340274391571856_d_nat @ ( produc1117254357280306391_d_nat @ ( produc7136022372351851517_d_nat @ A4 @ B2 ) @ ( produc7136022372351851517_d_nat @ A5 @ B3 ) ) @ ( lex_pr8119212446852010005_d_nat @ R @ S ) )
      = ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ A5 ) @ R )
        | ( ( A4 = A5 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_542_in__lex__prod,axiom,
    ! [A4: term_a_b,B2: nat,A5: term_a_b,B3: nat,R: set_Pr4386577575007340137rm_a_b,S: set_Pr1261947904930325089at_nat] :
      ( ( member2666392743443611024_b_nat @ ( produc6590571091770987735_b_nat @ ( produc2478027105211390077_b_nat @ A4 @ B2 ) @ ( produc2478027105211390077_b_nat @ A5 @ B3 ) ) @ ( lex_pr3461217179711548565_b_nat @ R @ S ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A5 ) @ R )
        | ( ( A4 = A5 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_543_in__lex__prod,axiom,
    ! [A4: nat,B2: term_c_d,A5: nat,B3: term_c_d,R: set_Pr1261947904930325089at_nat,S: set_Pr5080540287643815273rm_c_d] :
      ( ( member5070412009747381648rm_c_d @ ( produc8279382839690433751rm_c_d @ ( produc5165033859562299261rm_c_d @ A4 @ B2 ) @ ( produc5165033859562299261rm_c_d @ A5 @ B3 ) ) @ ( lex_pr6148223934062457749rm_c_d @ R @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A5 ) @ R )
        | ( ( A4 = A5 )
          & ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ B2 @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_544_in__lex__prod,axiom,
    ! [A4: nat,B2: term_a_b,A5: nat,B3: term_a_b,R: set_Pr1261947904930325089at_nat,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member8066425057025219984rm_a_b @ ( produc3440225595649897687rm_a_b @ ( produc1516572978046417917rm_a_b @ A4 @ B2 ) @ ( produc1516572978046417917rm_a_b @ A5 @ B3 ) ) @ ( lex_pr2499763052546576405rm_a_b @ R @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A5 ) @ R )
        | ( ( A4 = A5 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_545_in__lex__prod,axiom,
    ! [A4: term_c_d,B2: term_c_d,A5: term_c_d,B3: term_c_d,R: set_Pr5080540287643815273rm_c_d,S: set_Pr5080540287643815273rm_c_d] :
      ( ( member8430939456677929360rm_c_d @ ( produc5257394494426896087rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) @ ( produc6749061249757032257rm_c_d @ A5 @ B3 ) ) @ ( lex_pr5892420715117795161rm_c_d @ R @ S ) )
      = ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ A5 ) @ R )
        | ( ( A4 = A5 )
          & ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ B2 @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_546_in__lex__prod,axiom,
    ! [A4: term_a_b,B2: term_a_b,A5: term_a_b,B3: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) @ ( produc7020197800436672577rm_a_b @ A5 @ B3 ) ) @ ( lex_pr6163557265797435481rm_a_b @ R @ S ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A5 ) @ R )
        | ( ( A4 = A5 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_547_in__lex__prod,axiom,
    ! [A4: term_c_d,B2: term_a_b,A5: term_c_d,B3: term_a_b,R: set_Pr5080540287643815273rm_c_d,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member4039383142722929040rm_a_b @ ( produc2254039926008297175rm_a_b @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) @ ( produc3100600368241150913rm_a_b @ A5 @ B3 ) ) @ ( lex_pr2243959833601913817rm_a_b @ R @ S ) )
      = ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ A5 ) @ R )
        | ( ( A4 = A5 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_548_in__lex__prod,axiom,
    ! [A4: term_a_b,B2: term_c_d,A5: term_a_b,B3: term_c_d,R: set_Pr4386577575007340137rm_a_b,S: set_Pr5080540287643815273rm_c_d] :
      ( ( member3585784829053206928rm_c_d @ ( produc4766828187215050455rm_c_d @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) @ ( produc1445286645097778113rm_c_d @ A5 @ B3 ) ) @ ( lex_pr588646110458541017rm_c_d @ R @ S ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A5 ) @ R )
        | ( ( A4 = A5 )
          & ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ B2 @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_549_in__lex__prod,axiom,
    ! [A4: nat,B2: nat,A5: nat,B3: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ A5 @ B3 ) ) @ ( lex_prod_nat_nat @ R @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A5 ) @ R )
        | ( ( A4 = A5 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_550_theI__unique,axiom,
    ! [P: produc5661168290637732233rm_a_b > $o,X5: produc5661168290637732233rm_a_b] :
      ( ? [X4: produc5661168290637732233rm_a_b] :
          ( ( P @ X4 )
          & ! [Y4: produc5661168290637732233rm_a_b] :
              ( ( P @ Y4 )
             => ( Y4 = X4 ) ) )
     => ( ( P @ X5 )
        = ( X5
          = ( the_Pr9157697819281395472rm_a_b @ P ) ) ) ) ).

% theI_unique
thf(fact_551_theI__unique,axiom,
    ! [P: produc8451535801715972489rm_c_d > $o,X5: produc8451535801715972489rm_c_d] :
      ( ? [X4: produc8451535801715972489rm_c_d] :
          ( ( P @ X4 )
          & ! [Y4: produc8451535801715972489rm_c_d] :
              ( ( P @ Y4 )
             => ( Y4 = X4 ) ) )
     => ( ( P @ X5 )
        = ( X5
          = ( the_Pr2724693293504859920rm_c_d @ P ) ) ) ) ).

% theI_unique
thf(fact_552_refl__lex__prod,axiom,
    ! [R_B: set_Pr4386577575007340137rm_a_b,R_A: set_Pr5080540287643815273rm_c_d] :
      ( ( refl_on_term_a_b @ top_top_set_term_a_b @ R_B )
     => ( refl_o1424807056421441800rm_a_b @ top_to1749258167063912249rm_a_b @ ( lex_pr2243959833601913817rm_a_b @ R_A @ R_B ) ) ) ).

% refl_lex_prod
thf(fact_553_refl__lex__prod,axiom,
    ! [R_B: set_Pr5080540287643815273rm_c_d,R_A: set_Pr4386577575007340137rm_a_b] :
      ( ( refl_on_term_c_d @ top_top_set_term_c_d @ R_B )
     => ( refl_o4215174567499682056rm_c_d @ top_to1573239103265677113rm_c_d @ ( lex_pr588646110458541017rm_c_d @ R_A @ R_B ) ) ) ).

% refl_lex_prod
thf(fact_554_refl__lex__prod,axiom,
    ! [R_B: set_Pr1261947904930325089at_nat,R_A: set_Pr1261947904930325089at_nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R_B )
     => ( refl_o7122697186039111552at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ).

% refl_lex_prod
thf(fact_555_transI,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d] :
      ( ! [X3: term_c_d,Y4: term_c_d,Z3: term_c_d] :
          ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X3 @ Y4 ) @ R )
         => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ Y4 @ Z3 ) @ R )
           => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X3 @ Z3 ) @ R ) ) )
     => ( trans_on_term_c_d @ top_top_set_term_c_d @ R ) ) ).

% transI
thf(fact_556_transI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b,Z3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z3 ) @ R )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ R ) ) )
     => ( trans_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).

% transI
thf(fact_557_transI,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y4: product_prod_nat_nat,Z3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ Z3 ) @ R )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Z3 ) @ R ) ) )
     => ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R ) ) ).

% transI
thf(fact_558_transI,axiom,
    ! [R: set_Pr1076074158304066111at_nat] :
      ( ! [X3: sum_sum_nat_nat,Y4: sum_sum_nat_nat,Z3: sum_sum_nat_nat] :
          ( ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ X3 @ Y4 ) @ R )
         => ( ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ Y4 @ Z3 ) @ R )
           => ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ X3 @ Z3 ) @ R ) ) )
     => ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R ) ) ).

% transI
thf(fact_559_transI,axiom,
    ! [R: set_Pr5488025237498180813et_nat] :
      ( ! [X3: set_nat,Y4: set_nat,Z3: set_nat] :
          ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ Y4 ) @ R )
         => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y4 @ Z3 ) @ R )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ Z3 ) @ R ) ) )
     => ( trans_on_set_nat @ top_top_set_set_nat @ R ) ) ).

% transI
thf(fact_560_transI,axiom,
    ! [R: set_Pr8788715879341982671rm_a_b] :
      ( ! [X3: term_c_d > term_a_b,Y4: term_c_d > term_a_b,Z3: term_c_d > term_a_b] :
          ( ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ X3 @ Y4 ) @ R )
         => ( ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ Y4 @ Z3 ) @ R )
           => ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ X3 @ Z3 ) @ R ) ) )
     => ( trans_5020182533697283623rm_a_b @ top_to220653634128270125rm_a_b @ R ) ) ).

% transI
thf(fact_561_transI,axiom,
    ! [R: set_Pr6132369733443093455rm_c_d] :
      ( ! [X3: term_a_b > term_c_d,Y4: term_a_b > term_c_d,Z3: term_a_b > term_c_d] :
          ( ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ X3 @ Y4 ) @ R )
         => ( ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ Y4 @ Z3 ) @ R )
           => ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ X3 @ Z3 ) @ R ) ) )
     => ( trans_7810550044775523879rm_c_d @ top_to44634570330034989rm_c_d @ R ) ) ).

% transI
thf(fact_562_transI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y4: nat,Z3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z3 ) @ R )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ R ) ) )
     => ( trans_on_nat @ top_top_set_nat @ R ) ) ).

% transI
thf(fact_563_transE,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d,X5: term_c_d,Y3: term_c_d,Z: term_c_d] :
      ( ( trans_on_term_c_d @ top_top_set_term_c_d @ R )
     => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Y3 ) @ R )
       => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ Y3 @ Z ) @ R )
         => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Z ) @ R ) ) ) ) ).

% transE
thf(fact_564_transE,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X5: term_a_b,Y3: term_a_b,Z: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y3 ) @ R )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Z ) @ R ) ) ) ) ).

% transE
thf(fact_565_transE,axiom,
    ! [R: set_Pr8693737435421807431at_nat,X5: product_prod_nat_nat,Y3: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) @ R )
       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z ) @ R )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Z ) @ R ) ) ) ) ).

% transE
thf(fact_566_transE,axiom,
    ! [R: set_Pr1076074158304066111at_nat,X5: sum_sum_nat_nat,Y3: sum_sum_nat_nat,Z: sum_sum_nat_nat] :
      ( ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R )
     => ( ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ X5 @ Y3 ) @ R )
       => ( ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ Y3 @ Z ) @ R )
         => ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ X5 @ Z ) @ R ) ) ) ) ).

% transE
thf(fact_567_transE,axiom,
    ! [R: set_Pr5488025237498180813et_nat,X5: set_nat,Y3: set_nat,Z: set_nat] :
      ( ( trans_on_set_nat @ top_top_set_set_nat @ R )
     => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y3 ) @ R )
       => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y3 @ Z ) @ R )
         => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Z ) @ R ) ) ) ) ).

% transE
thf(fact_568_transE,axiom,
    ! [R: set_Pr8788715879341982671rm_a_b,X5: term_c_d > term_a_b,Y3: term_c_d > term_a_b,Z: term_c_d > term_a_b] :
      ( ( trans_5020182533697283623rm_a_b @ top_to220653634128270125rm_a_b @ R )
     => ( ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ X5 @ Y3 ) @ R )
       => ( ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ Y3 @ Z ) @ R )
         => ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ X5 @ Z ) @ R ) ) ) ) ).

% transE
thf(fact_569_transE,axiom,
    ! [R: set_Pr6132369733443093455rm_c_d,X5: term_a_b > term_c_d,Y3: term_a_b > term_c_d,Z: term_a_b > term_c_d] :
      ( ( trans_7810550044775523879rm_c_d @ top_to44634570330034989rm_c_d @ R )
     => ( ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ X5 @ Y3 ) @ R )
       => ( ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ Y3 @ Z ) @ R )
         => ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ X5 @ Z ) @ R ) ) ) ) ).

% transE
thf(fact_570_transE,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X5: nat,Y3: nat,Z: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Z ) @ R ) ) ) ) ).

% transE
thf(fact_571_transD,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d,X5: term_c_d,Y3: term_c_d,Z: term_c_d] :
      ( ( trans_on_term_c_d @ top_top_set_term_c_d @ R )
     => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Y3 ) @ R )
       => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ Y3 @ Z ) @ R )
         => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Z ) @ R ) ) ) ) ).

% transD
thf(fact_572_transD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X5: term_a_b,Y3: term_a_b,Z: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y3 ) @ R )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Z ) @ R ) ) ) ) ).

% transD
thf(fact_573_transD,axiom,
    ! [R: set_Pr8693737435421807431at_nat,X5: product_prod_nat_nat,Y3: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) @ R )
       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z ) @ R )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Z ) @ R ) ) ) ) ).

% transD
thf(fact_574_transD,axiom,
    ! [R: set_Pr1076074158304066111at_nat,X5: sum_sum_nat_nat,Y3: sum_sum_nat_nat,Z: sum_sum_nat_nat] :
      ( ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R )
     => ( ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ X5 @ Y3 ) @ R )
       => ( ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ Y3 @ Z ) @ R )
         => ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ X5 @ Z ) @ R ) ) ) ) ).

% transD
thf(fact_575_transD,axiom,
    ! [R: set_Pr5488025237498180813et_nat,X5: set_nat,Y3: set_nat,Z: set_nat] :
      ( ( trans_on_set_nat @ top_top_set_set_nat @ R )
     => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y3 ) @ R )
       => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y3 @ Z ) @ R )
         => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Z ) @ R ) ) ) ) ).

% transD
thf(fact_576_transD,axiom,
    ! [R: set_Pr8788715879341982671rm_a_b,X5: term_c_d > term_a_b,Y3: term_c_d > term_a_b,Z: term_c_d > term_a_b] :
      ( ( trans_5020182533697283623rm_a_b @ top_to220653634128270125rm_a_b @ R )
     => ( ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ X5 @ Y3 ) @ R )
       => ( ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ Y3 @ Z ) @ R )
         => ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ X5 @ Z ) @ R ) ) ) ) ).

% transD
thf(fact_577_transD,axiom,
    ! [R: set_Pr6132369733443093455rm_c_d,X5: term_a_b > term_c_d,Y3: term_a_b > term_c_d,Z: term_a_b > term_c_d] :
      ( ( trans_7810550044775523879rm_c_d @ top_to44634570330034989rm_c_d @ R )
     => ( ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ X5 @ Y3 ) @ R )
       => ( ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ Y3 @ Z ) @ R )
         => ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ X5 @ Z ) @ R ) ) ) ) ).

% transD
thf(fact_578_transD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X5: nat,Y3: nat,Z: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Z ) @ R ) ) ) ) ).

% transD
thf(fact_579_imageE,axiom,
    ! [B2: $o,F: $o > $o,A3: set_o] :
      ( ( member_o @ B2 @ ( image_o_o @ F @ A3 ) )
     => ~ ! [X3: $o] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_o @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_580_imageE,axiom,
    ! [B2: $o,F: nat > $o,A3: set_nat] :
      ( ( member_o @ B2 @ ( image_nat_o @ F @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_581_imageE,axiom,
    ! [B2: nat,F: $o > nat,A3: set_o] :
      ( ( member_nat @ B2 @ ( image_o_nat @ F @ A3 ) )
     => ~ ! [X3: $o] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_o @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_582_imageE,axiom,
    ! [B2: nat,F: nat > nat,A3: set_nat] :
      ( ( member_nat @ B2 @ ( image_nat_nat2 @ F @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_583_imageE,axiom,
    ! [B2: $o,F: set_nat > $o,A3: set_set_nat] :
      ( ( member_o @ B2 @ ( image_set_nat_o @ F @ A3 ) )
     => ~ ! [X3: set_nat] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_set_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_584_imageE,axiom,
    ! [B2: set_nat,F: $o > set_nat,A3: set_o] :
      ( ( member_set_nat @ B2 @ ( image_o_set_nat @ F @ A3 ) )
     => ~ ! [X3: $o] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_o @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_585_imageE,axiom,
    ! [B2: set_nat,F: nat > set_nat,A3: set_nat] :
      ( ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_586_imageE,axiom,
    ! [B2: nat,F: set_nat > nat,A3: set_set_nat] :
      ( ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A3 ) )
     => ~ ! [X3: set_nat] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_set_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_587_imageE,axiom,
    ! [B2: product_prod_nat_nat,F: $o > product_prod_nat_nat,A3: set_o] :
      ( ( member8440522571783428010at_nat @ B2 @ ( image_3855930084881510382at_nat @ F @ A3 ) )
     => ~ ! [X3: $o] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_o @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_588_imageE,axiom,
    ! [B2: product_prod_nat_nat,F: nat > product_prod_nat_nat,A3: set_nat] :
      ( ( member8440522571783428010at_nat @ B2 @ ( image_5846123807819985514at_nat @ F @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_589_image__image,axiom,
    ! [F: set_nat > set_nat,G: ( nat > $o ) > set_nat,A3: set_nat_o] :
      ( ( image_7916887816326733075et_nat @ F @ ( image_nat_o_set_nat @ G @ A3 ) )
      = ( image_nat_o_set_nat
        @ ^ [X: nat > $o] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_590_image__image,axiom,
    ! [F: ( nat > $o ) > set_nat,G: ( nat > $o ) > nat > $o,A3: set_nat_o] :
      ( ( image_nat_o_set_nat @ F @ ( image_nat_o_nat_o @ G @ A3 ) )
      = ( image_nat_o_set_nat
        @ ^ [X: nat > $o] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_591_image__image,axiom,
    ! [F: set_Pr4821568463224695273rm_a_b > set_Pr4821568463224695273rm_a_b,G: ( produc5661168290637732233rm_a_b > $o ) > set_Pr4821568463224695273rm_a_b,A3: set_Pr6472531492093999850_a_b_o] :
      ( ( image_5477805164141817157rm_a_b @ F @ ( image_540309731347893486rm_a_b @ G @ A3 ) )
      = ( image_540309731347893486rm_a_b
        @ ^ [X: produc5661168290637732233rm_a_b > $o] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_592_image__image,axiom,
    ! [F: set_Pr4645549399426460137rm_c_d > set_Pr4645549399426460137rm_c_d,G: ( produc8451535801715972489rm_c_d > $o ) > set_Pr4645549399426460137rm_c_d,A3: set_Pr2504828417160952554_c_d_o] :
      ( ( image_3137723552505605445rm_c_d @ F @ ( image_2252888096169626862rm_c_d @ G @ A3 ) )
      = ( image_2252888096169626862rm_c_d
        @ ^ [X: produc8451535801715972489rm_c_d > $o] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_593_image__image,axiom,
    ! [F: ( produc5661168290637732233rm_a_b > $o ) > set_Pr4821568463224695273rm_a_b,G: ( produc5661168290637732233rm_a_b > $o ) > produc5661168290637732233rm_a_b > $o,A3: set_Pr6472531492093999850_a_b_o] :
      ( ( image_540309731347893486rm_a_b @ F @ ( image_655970553839572847_a_b_o @ G @ A3 ) )
      = ( image_540309731347893486rm_a_b
        @ ^ [X: produc5661168290637732233rm_a_b > $o] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_594_image__image,axiom,
    ! [F: ( produc5661168290637732233rm_a_b > $o ) > set_Pr4821568463224695273rm_a_b,G: ( term_c_d > term_a_b > $o ) > produc5661168290637732233rm_a_b > $o,A3: set_te7068155017529424062_a_b_o] :
      ( ( image_540309731347893486rm_a_b @ F @ ( image_7540181528456194509_a_b_o @ G @ A3 ) )
      = ( image_5041726775426359568rm_a_b
        @ ^ [X: term_c_d > term_a_b > $o] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_595_image__image,axiom,
    ! [F: ( produc8451535801715972489rm_c_d > $o ) > set_Pr4645549399426460137rm_c_d,G: ( produc8451535801715972489rm_c_d > $o ) > produc8451535801715972489rm_c_d > $o,A3: set_Pr2504828417160952554_c_d_o] :
      ( ( image_2252888096169626862rm_c_d @ F @ ( image_4299811549171767151_c_d_o @ G @ A3 ) )
      = ( image_2252888096169626862rm_c_d
        @ ^ [X: produc8451535801715972489rm_c_d > $o] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_596_image__image,axiom,
    ! [F: ( produc8451535801715972489rm_c_d > $o ) > set_Pr4645549399426460137rm_c_d,G: ( term_a_b > term_c_d > $o ) > produc8451535801715972489rm_c_d > $o,A3: set_te8875761578035853502_c_d_o] :
      ( ( image_2252888096169626862rm_c_d @ F @ ( image_4899642137928444365_c_d_o @ G @ A3 ) )
      = ( image_5453966010614964496rm_c_d
        @ ^ [X: term_a_b > term_c_d > $o] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_597_image__image,axiom,
    ! [F: ( produc5661168290637732233rm_a_b > $o ) > produc5661168290637732233rm_a_b > $o,G: ( term_c_d > term_a_b > $o ) > produc5661168290637732233rm_a_b > $o,A3: set_te7068155017529424062_a_b_o] :
      ( ( image_655970553839572847_a_b_o @ F @ ( image_7540181528456194509_a_b_o @ G @ A3 ) )
      = ( image_7540181528456194509_a_b_o
        @ ^ [X: term_c_d > term_a_b > $o] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_598_image__image,axiom,
    ! [F: ( produc8451535801715972489rm_c_d > $o ) > produc8451535801715972489rm_c_d > $o,G: ( term_a_b > term_c_d > $o ) > produc8451535801715972489rm_c_d > $o,A3: set_te8875761578035853502_c_d_o] :
      ( ( image_4299811549171767151_c_d_o @ F @ ( image_4899642137928444365_c_d_o @ G @ A3 ) )
      = ( image_4899642137928444365_c_d_o
        @ ^ [X: term_a_b > term_c_d > $o] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_599_Compr__image__eq,axiom,
    ! [F: nat > nat,A3: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_nat_nat2 @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_nat_nat2 @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_600_Compr__image__eq,axiom,
    ! [F: $o > $o,A3: set_o,P: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( image_o_o @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_o_o @ F
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_601_Compr__image__eq,axiom,
    ! [F: nat > $o,A3: set_nat,P: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( image_nat_o @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_nat_o @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_602_Compr__image__eq,axiom,
    ! [F: $o > nat,A3: set_o,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_o_nat @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_o_nat @ F
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_603_Compr__image__eq,axiom,
    ! [F: set_nat > $o,A3: set_set_nat,P: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( image_set_nat_o @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_set_nat_o @ F
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( member_set_nat @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_604_Compr__image__eq,axiom,
    ! [F: set_nat > nat,A3: set_set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_set_nat_nat @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_set_nat_nat @ F
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( member_set_nat @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_605_Compr__image__eq,axiom,
    ! [F: $o > set_nat,A3: set_o,P: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( member_set_nat @ X @ ( image_o_set_nat @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_o_set_nat @ F
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_606_Compr__image__eq,axiom,
    ! [F: nat > set_nat,A3: set_nat,P: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( member_set_nat @ X @ ( image_nat_set_nat @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_nat_set_nat @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_607_Compr__image__eq,axiom,
    ! [F: product_prod_nat_nat > $o,A3: set_Pr1261947904930325089at_nat,P: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( image_3693632289388996572_nat_o @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_3693632289388996572_nat_o @ F
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_608_Compr__image__eq,axiom,
    ! [F: term_c_d > $o,A3: set_term_c_d,P: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( image_term_c_d_o @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_term_c_d_o @ F
        @ ( collect_term_c_d
          @ ^ [X: term_c_d] :
              ( ( member_term_c_d @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_609_conversep_Osimps,axiom,
    ( conver8884510217699480863rm_c_d
    = ( ^ [R2: term_a_b > term_c_d > $o,A12: term_c_d,A23: term_a_b] :
        ? [A: term_a_b,B: term_c_d] :
          ( ( A12 = B )
          & ( A23 = A )
          & ( R2 @ A @ B ) ) ) ) ).

% conversep.simps
thf(fact_610_conversep_Osimps,axiom,
    ( conver1316451903988077855rm_a_b
    = ( ^ [R2: term_c_d > term_a_b > $o,A12: term_a_b,A23: term_c_d] :
        ? [A: term_c_d,B: term_a_b] :
          ( ( A12 = B )
          & ( A23 = A )
          & ( R2 @ A @ B ) ) ) ) ).

% conversep.simps
thf(fact_611_imageI,axiom,
    ! [X5: $o,A3: set_o,F: $o > $o] :
      ( ( member_o @ X5 @ A3 )
     => ( member_o @ ( F @ X5 ) @ ( image_o_o @ F @ A3 ) ) ) ).

% imageI
thf(fact_612_imageI,axiom,
    ! [X5: $o,A3: set_o,F: $o > nat] :
      ( ( member_o @ X5 @ A3 )
     => ( member_nat @ ( F @ X5 ) @ ( image_o_nat @ F @ A3 ) ) ) ).

% imageI
thf(fact_613_imageI,axiom,
    ! [X5: nat,A3: set_nat,F: nat > $o] :
      ( ( member_nat @ X5 @ A3 )
     => ( member_o @ ( F @ X5 ) @ ( image_nat_o @ F @ A3 ) ) ) ).

% imageI
thf(fact_614_imageI,axiom,
    ! [X5: nat,A3: set_nat,F: nat > nat] :
      ( ( member_nat @ X5 @ A3 )
     => ( member_nat @ ( F @ X5 ) @ ( image_nat_nat2 @ F @ A3 ) ) ) ).

% imageI
thf(fact_615_imageI,axiom,
    ! [X5: $o,A3: set_o,F: $o > set_nat] :
      ( ( member_o @ X5 @ A3 )
     => ( member_set_nat @ ( F @ X5 ) @ ( image_o_set_nat @ F @ A3 ) ) ) ).

% imageI
thf(fact_616_imageI,axiom,
    ! [X5: set_nat,A3: set_set_nat,F: set_nat > $o] :
      ( ( member_set_nat @ X5 @ A3 )
     => ( member_o @ ( F @ X5 ) @ ( image_set_nat_o @ F @ A3 ) ) ) ).

% imageI
thf(fact_617_imageI,axiom,
    ! [X5: set_nat,A3: set_set_nat,F: set_nat > nat] :
      ( ( member_set_nat @ X5 @ A3 )
     => ( member_nat @ ( F @ X5 ) @ ( image_set_nat_nat @ F @ A3 ) ) ) ).

% imageI
thf(fact_618_imageI,axiom,
    ! [X5: nat,A3: set_nat,F: nat > set_nat] :
      ( ( member_nat @ X5 @ A3 )
     => ( member_set_nat @ ( F @ X5 ) @ ( image_nat_set_nat @ F @ A3 ) ) ) ).

% imageI
thf(fact_619_imageI,axiom,
    ! [X5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X5 @ A3 )
     => ( member_o @ ( F @ X5 ) @ ( image_3693632289388996572_nat_o @ F @ A3 ) ) ) ).

% imageI
thf(fact_620_imageI,axiom,
    ! [X5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( member8440522571783428010at_nat @ X5 @ A3 )
     => ( member_nat @ ( F @ X5 ) @ ( image_2486076414777270412at_nat @ F @ A3 ) ) ) ).

% imageI
thf(fact_621_rangeE,axiom,
    ! [B2: $o,F: nat > $o] :
      ( ( member_o @ B2 @ ( image_nat_o @ F @ top_top_set_nat ) )
     => ~ ! [X3: nat] :
            ( B2
            = ( ~ ( F @ X3 ) ) ) ) ).

% rangeE
thf(fact_622_rangeE,axiom,
    ! [B2: nat,F: nat > nat] :
      ( ( member_nat @ B2 @ ( image_nat_nat2 @ F @ top_top_set_nat ) )
     => ~ ! [X3: nat] :
            ( B2
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_623_rangeE,axiom,
    ! [B2: set_nat,F: nat > set_nat] :
      ( ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ top_top_set_nat ) )
     => ~ ! [X3: nat] :
            ( B2
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_624_rangeE,axiom,
    ! [B2: $o,F: set_nat > $o] :
      ( ( member_o @ B2 @ ( image_set_nat_o @ F @ top_top_set_set_nat ) )
     => ~ ! [X3: set_nat] :
            ( B2
            = ( ~ ( F @ X3 ) ) ) ) ).

% rangeE
thf(fact_625_rangeE,axiom,
    ! [B2: nat,F: set_nat > nat] :
      ( ( member_nat @ B2 @ ( image_set_nat_nat @ F @ top_top_set_set_nat ) )
     => ~ ! [X3: set_nat] :
            ( B2
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_626_rangeE,axiom,
    ! [B2: product_prod_nat_nat,F: nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B2 @ ( image_5846123807819985514at_nat @ F @ top_top_set_nat ) )
     => ~ ! [X3: nat] :
            ( B2
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_627_rangeE,axiom,
    ! [B2: $o,F: sum_sum_nat_nat > $o] :
      ( ( member_o @ B2 @ ( image_6056915518860092946_nat_o @ F @ top_to6661820994512907621at_nat ) )
     => ~ ! [X3: sum_sum_nat_nat] :
            ( B2
            = ( ~ ( F @ X3 ) ) ) ) ).

% rangeE
thf(fact_628_rangeE,axiom,
    ! [B2: nat,F: sum_sum_nat_nat > nat] :
      ( ( member_nat @ B2 @ ( image_1320371278474632150at_nat @ F @ top_to6661820994512907621at_nat ) )
     => ~ ! [X3: sum_sum_nat_nat] :
            ( B2
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_629_rangeE,axiom,
    ! [B2: set_nat,F: set_nat > set_nat] :
      ( ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat ) )
     => ~ ! [X3: set_nat] :
            ( B2
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_630_rangeE,axiom,
    ! [B2: set_nat,F: ( nat > $o ) > set_nat] :
      ( ( member_set_nat @ B2 @ ( image_nat_o_set_nat @ F @ top_top_set_nat_o2 ) )
     => ~ ! [X3: nat > $o] :
            ( B2
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_631_rangeI,axiom,
    ! [F: nat > $o,X5: nat] : ( member_o @ ( F @ X5 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).

% rangeI
thf(fact_632_rangeI,axiom,
    ! [F: nat > nat,X5: nat] : ( member_nat @ ( F @ X5 ) @ ( image_nat_nat2 @ F @ top_top_set_nat ) ) ).

% rangeI
thf(fact_633_rangeI,axiom,
    ! [F: nat > set_nat,X5: nat] : ( member_set_nat @ ( F @ X5 ) @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ).

% rangeI
thf(fact_634_rangeI,axiom,
    ! [F: set_nat > $o,X5: set_nat] : ( member_o @ ( F @ X5 ) @ ( image_set_nat_o @ F @ top_top_set_set_nat ) ) ).

% rangeI
thf(fact_635_rangeI,axiom,
    ! [F: set_nat > nat,X5: set_nat] : ( member_nat @ ( F @ X5 ) @ ( image_set_nat_nat @ F @ top_top_set_set_nat ) ) ).

% rangeI
thf(fact_636_rangeI,axiom,
    ! [F: nat > product_prod_nat_nat,X5: nat] : ( member8440522571783428010at_nat @ ( F @ X5 ) @ ( image_5846123807819985514at_nat @ F @ top_top_set_nat ) ) ).

% rangeI
thf(fact_637_rangeI,axiom,
    ! [F: sum_sum_nat_nat > $o,X5: sum_sum_nat_nat] : ( member_o @ ( F @ X5 ) @ ( image_6056915518860092946_nat_o @ F @ top_to6661820994512907621at_nat ) ) ).

% rangeI
thf(fact_638_rangeI,axiom,
    ! [F: sum_sum_nat_nat > nat,X5: sum_sum_nat_nat] : ( member_nat @ ( F @ X5 ) @ ( image_1320371278474632150at_nat @ F @ top_to6661820994512907621at_nat ) ) ).

% rangeI
thf(fact_639_rangeI,axiom,
    ! [F: set_nat > set_nat,X5: set_nat] : ( member_set_nat @ ( F @ X5 ) @ ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat ) ) ).

% rangeI
thf(fact_640_rangeI,axiom,
    ! [F: ( nat > $o ) > set_nat,X5: nat > $o] : ( member_set_nat @ ( F @ X5 ) @ ( image_nat_o_set_nat @ F @ top_top_set_nat_o2 ) ) ).

% rangeI
thf(fact_641_UNIV__eq__I,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ A3 )
     => ( top_to1314267278846557113rm_a_b = A3 ) ) ).

% UNIV_eq_I
thf(fact_642_UNIV__eq__I,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A3 )
     => ( top_to4669805908274784177at_nat = A3 ) ) ).

% UNIV_eq_I
thf(fact_643_UNIV__eq__I,axiom,
    ! [A3: set_o] :
      ( ! [X3: $o] : ( member_o @ X3 @ A3 )
     => ( top_top_set_o = A3 ) ) ).

% UNIV_eq_I
thf(fact_644_UNIV__eq__I,axiom,
    ! [A3: set_Sum_sum_nat_nat] :
      ( ! [X3: sum_sum_nat_nat] : ( member8583185029347631382at_nat @ X3 @ A3 )
     => ( top_to6661820994512907621at_nat = A3 ) ) ).

% UNIV_eq_I
thf(fact_645_UNIV__eq__I,axiom,
    ! [A3: set_set_nat] :
      ( ! [X3: set_nat] : ( member_set_nat @ X3 @ A3 )
     => ( top_top_set_set_nat = A3 ) ) ).

% UNIV_eq_I
thf(fact_646_UNIV__eq__I,axiom,
    ! [A3: set_te1997870499059199101rm_a_b] :
      ( ! [X3: term_c_d > term_a_b] : ( member3140172230840508126rm_a_b @ X3 @ A3 )
     => ( top_to220653634128270125rm_a_b = A3 ) ) ).

% UNIV_eq_I
thf(fact_647_UNIV__eq__I,axiom,
    ! [A3: set_te1821851435260963965rm_c_d] :
      ( ! [X3: term_a_b > term_c_d] : ( member5930539741918748382rm_c_d @ X3 @ A3 )
     => ( top_to44634570330034989rm_c_d = A3 ) ) ).

% UNIV_eq_I
thf(fact_648_UNIV__eq__I,axiom,
    ! [A3: set_nat] :
      ( ! [X3: nat] : ( member_nat @ X3 @ A3 )
     => ( top_top_set_nat = A3 ) ) ).

% UNIV_eq_I
thf(fact_649_image__iff,axiom,
    ! [Z: set_Pr4821568463224695273rm_a_b,F: ( produc5661168290637732233rm_a_b > $o ) > set_Pr4821568463224695273rm_a_b,A3: set_Pr6472531492093999850_a_b_o] :
      ( ( member35261728838302610rm_a_b @ Z @ ( image_540309731347893486rm_a_b @ F @ A3 ) )
      = ( ? [X: produc5661168290637732233rm_a_b > $o] :
            ( ( member656444953465433547_a_b_o @ X @ A3 )
            & ( Z
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_650_image__iff,axiom,
    ! [Z: set_Pr4645549399426460137rm_c_d,F: ( produc8451535801715972489rm_c_d > $o ) > set_Pr4645549399426460137rm_c_d,A3: set_Pr2504828417160952554_c_d_o] :
      ( ( member9082614701894843282rm_c_d @ Z @ ( image_2252888096169626862rm_c_d @ F @ A3 ) )
      = ( ? [X: produc8451535801715972489rm_c_d > $o] :
            ( ( member7278092155855023563_c_d_o @ X @ A3 )
            & ( Z
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_651_image__iff,axiom,
    ! [Z: produc5661168290637732233rm_a_b > $o,F: ( term_c_d > term_a_b > $o ) > produc5661168290637732233rm_a_b > $o,A3: set_te7068155017529424062_a_b_o] :
      ( ( member656444953465433547_a_b_o @ Z @ ( image_7540181528456194509_a_b_o @ F @ A3 ) )
      = ( ? [X: term_c_d > term_a_b > $o] :
            ( ( member7495156513829265543_a_b_o @ X @ A3 )
            & ( Z
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_652_image__iff,axiom,
    ! [Z: produc8451535801715972489rm_c_d > $o,F: ( term_a_b > term_c_d > $o ) > produc8451535801715972489rm_c_d > $o,A3: set_te8875761578035853502_c_d_o] :
      ( ( member7278092155855023563_c_d_o @ Z @ ( image_4899642137928444365_c_d_o @ F @ A3 ) )
      = ( ? [X: term_a_b > term_c_d > $o] :
            ( ( member8627074640717963399_c_d_o @ X @ A3 )
            & ( Z
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_653_image__iff,axiom,
    ! [Z: set_nat,F: ( nat > $o ) > set_nat,A3: set_nat_o] :
      ( ( member_set_nat @ Z @ ( image_nat_o_set_nat @ F @ A3 ) )
      = ( ? [X: nat > $o] :
            ( ( member_nat_o @ X @ A3 )
            & ( Z
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_654_range__eqI,axiom,
    ! [B2: $o,F: nat > $o,X5: nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( member_o @ B2 @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_655_range__eqI,axiom,
    ! [B2: nat,F: nat > nat,X5: nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( member_nat @ B2 @ ( image_nat_nat2 @ F @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_656_range__eqI,axiom,
    ! [B2: set_nat,F: nat > set_nat,X5: nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_657_range__eqI,axiom,
    ! [B2: $o,F: set_nat > $o,X5: set_nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( member_o @ B2 @ ( image_set_nat_o @ F @ top_top_set_set_nat ) ) ) ).

% range_eqI
thf(fact_658_range__eqI,axiom,
    ! [B2: nat,F: set_nat > nat,X5: set_nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( member_nat @ B2 @ ( image_set_nat_nat @ F @ top_top_set_set_nat ) ) ) ).

% range_eqI
thf(fact_659_range__eqI,axiom,
    ! [B2: product_prod_nat_nat,F: nat > product_prod_nat_nat,X5: nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( member8440522571783428010at_nat @ B2 @ ( image_5846123807819985514at_nat @ F @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_660_range__eqI,axiom,
    ! [B2: $o,F: sum_sum_nat_nat > $o,X5: sum_sum_nat_nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( member_o @ B2 @ ( image_6056915518860092946_nat_o @ F @ top_to6661820994512907621at_nat ) ) ) ).

% range_eqI
thf(fact_661_range__eqI,axiom,
    ! [B2: nat,F: sum_sum_nat_nat > nat,X5: sum_sum_nat_nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( member_nat @ B2 @ ( image_1320371278474632150at_nat @ F @ top_to6661820994512907621at_nat ) ) ) ).

% range_eqI
thf(fact_662_range__eqI,axiom,
    ! [B2: set_nat,F: set_nat > set_nat,X5: set_nat] :
      ( ( B2
        = ( F @ X5 ) )
     => ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat ) ) ) ).

% range_eqI
thf(fact_663_range__eqI,axiom,
    ! [B2: set_nat,F: ( nat > $o ) > set_nat,X5: nat > $o] :
      ( ( B2
        = ( F @ X5 ) )
     => ( member_set_nat @ B2 @ ( image_nat_o_set_nat @ F @ top_top_set_nat_o2 ) ) ) ).

% range_eqI
thf(fact_664_bex__imageD,axiom,
    ! [F: ( produc5661168290637732233rm_a_b > $o ) > set_Pr4821568463224695273rm_a_b,A3: set_Pr6472531492093999850_a_b_o,P: set_Pr4821568463224695273rm_a_b > $o] :
      ( ? [X4: set_Pr4821568463224695273rm_a_b] :
          ( ( member35261728838302610rm_a_b @ X4 @ ( image_540309731347893486rm_a_b @ F @ A3 ) )
          & ( P @ X4 ) )
     => ? [X3: produc5661168290637732233rm_a_b > $o] :
          ( ( member656444953465433547_a_b_o @ X3 @ A3 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_665_bex__imageD,axiom,
    ! [F: ( produc8451535801715972489rm_c_d > $o ) > set_Pr4645549399426460137rm_c_d,A3: set_Pr2504828417160952554_c_d_o,P: set_Pr4645549399426460137rm_c_d > $o] :
      ( ? [X4: set_Pr4645549399426460137rm_c_d] :
          ( ( member9082614701894843282rm_c_d @ X4 @ ( image_2252888096169626862rm_c_d @ F @ A3 ) )
          & ( P @ X4 ) )
     => ? [X3: produc8451535801715972489rm_c_d > $o] :
          ( ( member7278092155855023563_c_d_o @ X3 @ A3 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_666_bex__imageD,axiom,
    ! [F: ( term_c_d > term_a_b > $o ) > produc5661168290637732233rm_a_b > $o,A3: set_te7068155017529424062_a_b_o,P: ( produc5661168290637732233rm_a_b > $o ) > $o] :
      ( ? [X4: produc5661168290637732233rm_a_b > $o] :
          ( ( member656444953465433547_a_b_o @ X4 @ ( image_7540181528456194509_a_b_o @ F @ A3 ) )
          & ( P @ X4 ) )
     => ? [X3: term_c_d > term_a_b > $o] :
          ( ( member7495156513829265543_a_b_o @ X3 @ A3 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_667_bex__imageD,axiom,
    ! [F: ( term_a_b > term_c_d > $o ) > produc8451535801715972489rm_c_d > $o,A3: set_te8875761578035853502_c_d_o,P: ( produc8451535801715972489rm_c_d > $o ) > $o] :
      ( ? [X4: produc8451535801715972489rm_c_d > $o] :
          ( ( member7278092155855023563_c_d_o @ X4 @ ( image_4899642137928444365_c_d_o @ F @ A3 ) )
          & ( P @ X4 ) )
     => ? [X3: term_a_b > term_c_d > $o] :
          ( ( member8627074640717963399_c_d_o @ X3 @ A3 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_668_bex__imageD,axiom,
    ! [F: ( nat > $o ) > set_nat,A3: set_nat_o,P: set_nat > $o] :
      ( ? [X4: set_nat] :
          ( ( member_set_nat @ X4 @ ( image_nat_o_set_nat @ F @ A3 ) )
          & ( P @ X4 ) )
     => ? [X3: nat > $o] :
          ( ( member_nat_o @ X3 @ A3 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_669_image__cong,axiom,
    ! [M: set_Pr6472531492093999850_a_b_o,N: set_Pr6472531492093999850_a_b_o,F: ( produc5661168290637732233rm_a_b > $o ) > set_Pr4821568463224695273rm_a_b,G: ( produc5661168290637732233rm_a_b > $o ) > set_Pr4821568463224695273rm_a_b] :
      ( ( M = N )
     => ( ! [X3: produc5661168290637732233rm_a_b > $o] :
            ( ( member656444953465433547_a_b_o @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_540309731347893486rm_a_b @ F @ M )
          = ( image_540309731347893486rm_a_b @ G @ N ) ) ) ) ).

% image_cong
thf(fact_670_image__cong,axiom,
    ! [M: set_Pr2504828417160952554_c_d_o,N: set_Pr2504828417160952554_c_d_o,F: ( produc8451535801715972489rm_c_d > $o ) > set_Pr4645549399426460137rm_c_d,G: ( produc8451535801715972489rm_c_d > $o ) > set_Pr4645549399426460137rm_c_d] :
      ( ( M = N )
     => ( ! [X3: produc8451535801715972489rm_c_d > $o] :
            ( ( member7278092155855023563_c_d_o @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_2252888096169626862rm_c_d @ F @ M )
          = ( image_2252888096169626862rm_c_d @ G @ N ) ) ) ) ).

% image_cong
thf(fact_671_image__cong,axiom,
    ! [M: set_te7068155017529424062_a_b_o,N: set_te7068155017529424062_a_b_o,F: ( term_c_d > term_a_b > $o ) > produc5661168290637732233rm_a_b > $o,G: ( term_c_d > term_a_b > $o ) > produc5661168290637732233rm_a_b > $o] :
      ( ( M = N )
     => ( ! [X3: term_c_d > term_a_b > $o] :
            ( ( member7495156513829265543_a_b_o @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_7540181528456194509_a_b_o @ F @ M )
          = ( image_7540181528456194509_a_b_o @ G @ N ) ) ) ) ).

% image_cong
thf(fact_672_image__cong,axiom,
    ! [M: set_te8875761578035853502_c_d_o,N: set_te8875761578035853502_c_d_o,F: ( term_a_b > term_c_d > $o ) > produc8451535801715972489rm_c_d > $o,G: ( term_a_b > term_c_d > $o ) > produc8451535801715972489rm_c_d > $o] :
      ( ( M = N )
     => ( ! [X3: term_a_b > term_c_d > $o] :
            ( ( member8627074640717963399_c_d_o @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_4899642137928444365_c_d_o @ F @ M )
          = ( image_4899642137928444365_c_d_o @ G @ N ) ) ) ) ).

% image_cong
thf(fact_673_image__cong,axiom,
    ! [M: set_nat_o,N: set_nat_o,F: ( nat > $o ) > set_nat,G: ( nat > $o ) > set_nat] :
      ( ( M = N )
     => ( ! [X3: nat > $o] :
            ( ( member_nat_o @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_nat_o_set_nat @ F @ M )
          = ( image_nat_o_set_nat @ G @ N ) ) ) ) ).

% image_cong
thf(fact_674_ball__imageD,axiom,
    ! [F: ( produc5661168290637732233rm_a_b > $o ) > set_Pr4821568463224695273rm_a_b,A3: set_Pr6472531492093999850_a_b_o,P: set_Pr4821568463224695273rm_a_b > $o] :
      ( ! [X3: set_Pr4821568463224695273rm_a_b] :
          ( ( member35261728838302610rm_a_b @ X3 @ ( image_540309731347893486rm_a_b @ F @ A3 ) )
         => ( P @ X3 ) )
     => ! [X4: produc5661168290637732233rm_a_b > $o] :
          ( ( member656444953465433547_a_b_o @ X4 @ A3 )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_675_ball__imageD,axiom,
    ! [F: ( produc8451535801715972489rm_c_d > $o ) > set_Pr4645549399426460137rm_c_d,A3: set_Pr2504828417160952554_c_d_o,P: set_Pr4645549399426460137rm_c_d > $o] :
      ( ! [X3: set_Pr4645549399426460137rm_c_d] :
          ( ( member9082614701894843282rm_c_d @ X3 @ ( image_2252888096169626862rm_c_d @ F @ A3 ) )
         => ( P @ X3 ) )
     => ! [X4: produc8451535801715972489rm_c_d > $o] :
          ( ( member7278092155855023563_c_d_o @ X4 @ A3 )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_676_ball__imageD,axiom,
    ! [F: ( term_c_d > term_a_b > $o ) > produc5661168290637732233rm_a_b > $o,A3: set_te7068155017529424062_a_b_o,P: ( produc5661168290637732233rm_a_b > $o ) > $o] :
      ( ! [X3: produc5661168290637732233rm_a_b > $o] :
          ( ( member656444953465433547_a_b_o @ X3 @ ( image_7540181528456194509_a_b_o @ F @ A3 ) )
         => ( P @ X3 ) )
     => ! [X4: term_c_d > term_a_b > $o] :
          ( ( member7495156513829265543_a_b_o @ X4 @ A3 )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_677_ball__imageD,axiom,
    ! [F: ( term_a_b > term_c_d > $o ) > produc8451535801715972489rm_c_d > $o,A3: set_te8875761578035853502_c_d_o,P: ( produc8451535801715972489rm_c_d > $o ) > $o] :
      ( ! [X3: produc8451535801715972489rm_c_d > $o] :
          ( ( member7278092155855023563_c_d_o @ X3 @ ( image_4899642137928444365_c_d_o @ F @ A3 ) )
         => ( P @ X3 ) )
     => ! [X4: term_a_b > term_c_d > $o] :
          ( ( member8627074640717963399_c_d_o @ X4 @ A3 )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_678_ball__imageD,axiom,
    ! [F: ( nat > $o ) > set_nat,A3: set_nat_o,P: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ ( image_nat_o_set_nat @ F @ A3 ) )
         => ( P @ X3 ) )
     => ! [X4: nat > $o] :
          ( ( member_nat_o @ X4 @ A3 )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_679_UNIV__witness,axiom,
    ? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ top_to1314267278846557113rm_a_b ) ).

% UNIV_witness
thf(fact_680_UNIV__witness,axiom,
    ? [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ top_to4669805908274784177at_nat ) ).

% UNIV_witness
thf(fact_681_UNIV__witness,axiom,
    ? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).

% UNIV_witness
thf(fact_682_UNIV__witness,axiom,
    ? [X3: sum_sum_nat_nat] : ( member8583185029347631382at_nat @ X3 @ top_to6661820994512907621at_nat ) ).

% UNIV_witness
thf(fact_683_UNIV__witness,axiom,
    ? [X3: set_nat] : ( member_set_nat @ X3 @ top_top_set_set_nat ) ).

% UNIV_witness
thf(fact_684_UNIV__witness,axiom,
    ? [X3: term_c_d > term_a_b] : ( member3140172230840508126rm_a_b @ X3 @ top_to220653634128270125rm_a_b ) ).

% UNIV_witness
thf(fact_685_UNIV__witness,axiom,
    ? [X3: term_a_b > term_c_d] : ( member5930539741918748382rm_c_d @ X3 @ top_to44634570330034989rm_c_d ) ).

% UNIV_witness
thf(fact_686_UNIV__witness,axiom,
    ? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_687_rev__image__eqI,axiom,
    ! [X5: $o,A3: set_o,B2: $o,F: $o > $o] :
      ( ( member_o @ X5 @ A3 )
     => ( ( B2
          = ( F @ X5 ) )
       => ( member_o @ B2 @ ( image_o_o @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_688_rev__image__eqI,axiom,
    ! [X5: $o,A3: set_o,B2: nat,F: $o > nat] :
      ( ( member_o @ X5 @ A3 )
     => ( ( B2
          = ( F @ X5 ) )
       => ( member_nat @ B2 @ ( image_o_nat @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_689_rev__image__eqI,axiom,
    ! [X5: nat,A3: set_nat,B2: $o,F: nat > $o] :
      ( ( member_nat @ X5 @ A3 )
     => ( ( B2
          = ( F @ X5 ) )
       => ( member_o @ B2 @ ( image_nat_o @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_690_rev__image__eqI,axiom,
    ! [X5: nat,A3: set_nat,B2: nat,F: nat > nat] :
      ( ( member_nat @ X5 @ A3 )
     => ( ( B2
          = ( F @ X5 ) )
       => ( member_nat @ B2 @ ( image_nat_nat2 @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_691_rev__image__eqI,axiom,
    ! [X5: $o,A3: set_o,B2: set_nat,F: $o > set_nat] :
      ( ( member_o @ X5 @ A3 )
     => ( ( B2
          = ( F @ X5 ) )
       => ( member_set_nat @ B2 @ ( image_o_set_nat @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_692_rev__image__eqI,axiom,
    ! [X5: set_nat,A3: set_set_nat,B2: $o,F: set_nat > $o] :
      ( ( member_set_nat @ X5 @ A3 )
     => ( ( B2
          = ( F @ X5 ) )
       => ( member_o @ B2 @ ( image_set_nat_o @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_693_rev__image__eqI,axiom,
    ! [X5: set_nat,A3: set_set_nat,B2: nat,F: set_nat > nat] :
      ( ( member_set_nat @ X5 @ A3 )
     => ( ( B2
          = ( F @ X5 ) )
       => ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_694_rev__image__eqI,axiom,
    ! [X5: nat,A3: set_nat,B2: set_nat,F: nat > set_nat] :
      ( ( member_nat @ X5 @ A3 )
     => ( ( B2
          = ( F @ X5 ) )
       => ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_695_rev__image__eqI,axiom,
    ! [X5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: $o,F: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X5 @ A3 )
     => ( ( B2
          = ( F @ X5 ) )
       => ( member_o @ B2 @ ( image_3693632289388996572_nat_o @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_696_rev__image__eqI,axiom,
    ! [X5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: nat,F: product_prod_nat_nat > nat] :
      ( ( member8440522571783428010at_nat @ X5 @ A3 )
     => ( ( B2
          = ( F @ X5 ) )
       => ( member_nat @ B2 @ ( image_2486076414777270412at_nat @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_697_conversepD,axiom,
    ! [R: term_a_b > term_c_d > $o,B2: term_c_d,A4: term_a_b] :
      ( ( conver8884510217699480863rm_c_d @ R @ B2 @ A4 )
     => ( R @ A4 @ B2 ) ) ).

% conversepD
thf(fact_698_conversepD,axiom,
    ! [R: term_c_d > term_a_b > $o,B2: term_a_b,A4: term_c_d] :
      ( ( conver1316451903988077855rm_a_b @ R @ B2 @ A4 )
     => ( R @ A4 @ B2 ) ) ).

% conversepD
thf(fact_699_conversepE,axiom,
    ! [R: term_a_b > term_c_d > $o,A1: term_c_d,A22: term_a_b] :
      ( ( conver8884510217699480863rm_c_d @ R @ A1 @ A22 )
     => ( R @ A22 @ A1 ) ) ).

% conversepE
thf(fact_700_conversepE,axiom,
    ! [R: term_c_d > term_a_b > $o,A1: term_a_b,A22: term_c_d] :
      ( ( conver1316451903988077855rm_a_b @ R @ A1 @ A22 )
     => ( R @ A22 @ A1 ) ) ).

% conversepE
thf(fact_701_conversepI,axiom,
    ! [R: term_a_b > term_c_d > $o,A4: term_a_b,B2: term_c_d] :
      ( ( R @ A4 @ B2 )
     => ( conver8884510217699480863rm_c_d @ R @ B2 @ A4 ) ) ).

% conversepI
thf(fact_702_conversepI,axiom,
    ! [R: term_c_d > term_a_b > $o,A4: term_c_d,B2: term_a_b] :
      ( ( R @ A4 @ B2 )
     => ( conver1316451903988077855rm_a_b @ R @ B2 @ A4 ) ) ).

% conversepI
thf(fact_703_range__composition,axiom,
    ! [F: ( nat > $o ) > set_nat,G: nat > nat > $o] :
      ( ( image_nat_set_nat
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ top_top_set_nat )
      = ( image_nat_o_set_nat @ F @ ( image_nat_nat_o @ G @ top_top_set_nat ) ) ) ).

% range_composition
thf(fact_704_range__composition,axiom,
    ! [F: set_nat > set_nat,G: ( nat > $o ) > set_nat] :
      ( ( image_nat_o_set_nat
        @ ^ [X: nat > $o] : ( F @ ( G @ X ) )
        @ top_top_set_nat_o2 )
      = ( image_7916887816326733075et_nat @ F @ ( image_nat_o_set_nat @ G @ top_top_set_nat_o2 ) ) ) ).

% range_composition
thf(fact_705_range__composition,axiom,
    ! [F: ( nat > $o ) > set_nat,G: set_nat > nat > $o] :
      ( ( image_7916887816326733075et_nat
        @ ^ [X: set_nat] : ( F @ ( G @ X ) )
        @ top_top_set_set_nat )
      = ( image_nat_o_set_nat @ F @ ( image_set_nat_nat_o @ G @ top_top_set_set_nat ) ) ) ).

% range_composition
thf(fact_706_range__composition,axiom,
    ! [F: ( nat > $o ) > set_nat,G: ( nat > $o ) > nat > $o] :
      ( ( image_nat_o_set_nat
        @ ^ [X: nat > $o] : ( F @ ( G @ X ) )
        @ top_top_set_nat_o2 )
      = ( image_nat_o_set_nat @ F @ ( image_nat_o_nat_o @ G @ top_top_set_nat_o2 ) ) ) ).

% range_composition
thf(fact_707_range__composition,axiom,
    ! [F: ( nat > $o ) > set_nat,G: sum_sum_nat_nat > nat > $o] :
      ( ( image_5891067694031531916et_nat
        @ ^ [X: sum_sum_nat_nat] : ( F @ ( G @ X ) )
        @ top_to6661820994512907621at_nat )
      = ( image_nat_o_set_nat @ F @ ( image_4407986529083857913_nat_o @ G @ top_to6661820994512907621at_nat ) ) ) ).

% range_composition
thf(fact_708_range__composition,axiom,
    ! [F: ( nat > $o ) > set_nat,G: ( term_c_d > term_a_b ) > nat > $o] :
      ( ( image_1393055134168367700et_nat
        @ ^ [X: term_c_d > term_a_b] : ( F @ ( G @ X ) )
        @ top_to220653634128270125rm_a_b )
      = ( image_nat_o_set_nat @ F @ ( image_2035932386948199217_nat_o @ G @ top_to220653634128270125rm_a_b ) ) ) ).

% range_composition
thf(fact_709_range__composition,axiom,
    ! [F: ( nat > $o ) > set_nat,G: ( term_a_b > term_c_d ) > nat > $o] :
      ( ( image_5265032364538446420et_nat
        @ ^ [X: term_a_b > term_c_d] : ( F @ ( G @ X ) )
        @ top_to44634570330034989rm_c_d )
      = ( image_nat_o_set_nat @ F @ ( image_7954562766338437937_nat_o @ G @ top_to44634570330034989rm_c_d ) ) ) ).

% range_composition
thf(fact_710_range__composition,axiom,
    ! [F: ( produc5661168290637732233rm_a_b > $o ) > set_Pr4821568463224695273rm_a_b,G: nat > produc5661168290637732233rm_a_b > $o] :
      ( ( image_4120006726334158418rm_a_b
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ top_top_set_nat )
      = ( image_540309731347893486rm_a_b @ F @ ( image_7219171546257965835_a_b_o @ G @ top_top_set_nat ) ) ) ).

% range_composition
thf(fact_711_range__composition,axiom,
    ! [F: ( produc8451535801715972489rm_c_d > $o ) > set_Pr4645549399426460137rm_c_d,G: nat > produc8451535801715972489rm_c_d > $o] :
      ( ( image_3943987662535923282rm_c_d
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ top_top_set_nat )
      = ( image_2252888096169626862rm_c_d @ F @ ( image_4617446711792780043_c_d_o @ G @ top_top_set_nat ) ) ) ).

% range_composition
thf(fact_712_range__composition,axiom,
    ! [F: ( term_c_d > term_a_b > $o ) > produc5661168290637732233rm_a_b > $o,G: nat > term_c_d > term_a_b > $o] :
      ( ( image_7219171546257965835_a_b_o
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ top_top_set_nat )
      = ( image_7540181528456194509_a_b_o @ F @ ( image_4201813665361132359_a_b_o @ G @ top_top_set_nat ) ) ) ).

% range_composition
thf(fact_713_irrefl__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R_A )
     => ( ( irrefl_on_nat @ top_top_set_nat @ R_B )
       => ( irrefl7408258699751485431at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_lex_prod
thf(fact_714_irrefl__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr5488025237498180813et_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R_A )
     => ( ( irrefl_on_set_nat @ top_top_set_set_nat @ R_B )
       => ( irrefl3552268651162415277et_nat @ top_to173688896754200935et_nat @ ( lex_prod_nat_set_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_lex_prod
thf(fact_715_irrefl__lex__prod,axiom,
    ! [R_A: set_Pr5488025237498180813et_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_set_nat @ top_top_set_set_nat @ R_A )
     => ( ( irrefl_on_nat @ top_top_set_nat @ R_B )
       => ( irrefl8643532438522300333at_nat @ top_to3374598108630517863at_nat @ ( lex_prod_set_nat_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_lex_prod
thf(fact_716_irrefl__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( irrefl7408258699751485431at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( irrefl_on_nat @ top_top_set_nat @ R_B )
       => ( irrefl1416403456917577354at_nat @ top_to2567003362141017830at_nat @ ( lex_pr4669217892513370978at_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_lex_prod
thf(fact_717_irrefl__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R_A )
     => ( ( irrefl7408258699751485431at_nat @ top_to4669805908274784177at_nat @ R_B )
       => ( irrefl290916472950276272at_nat @ top_to7745748144976627212at_nat @ ( lex_pr8029265285556086080at_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_lex_prod
thf(fact_718_irrefl__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1076074158304066111at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R_A )
     => ( ( irrefl688463893611329673at_nat @ top_to6661820994512907621at_nat @ R_B )
       => ( irrefl3943603201710458278at_nat @ top_to678699698364323680at_nat @ ( lex_pr6930960718592407552at_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_lex_prod
thf(fact_719_irrefl__lex__prod,axiom,
    ! [R_A: set_Pr1076074158304066111at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( irrefl688463893611329673at_nat @ top_to6661820994512907621at_nat @ R_A )
     => ( ( irrefl_on_nat @ top_top_set_nat @ R_B )
       => ( irrefl1193366675185269926at_nat @ top_to8411399450232720992at_nat @ ( lex_pr7572635211855035776at_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_lex_prod
thf(fact_720_irrefl__lex__prod,axiom,
    ! [R_A: set_Pr5488025237498180813et_nat,R_B: set_Pr5488025237498180813et_nat] :
      ( ( irrefl_on_set_nat @ top_top_set_set_nat @ R_A )
     => ( ( irrefl_on_set_nat @ top_top_set_set_nat @ R_B )
       => ( irrefl1622284741316079715et_nat @ top_to3040159065575275037et_nat @ ( lex_pr6462853727283027389et_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_lex_prod
thf(fact_721_irrefl__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr5488025237498180813et_nat] :
      ( ( irrefl7408258699751485431at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( irrefl_on_set_nat @ top_top_set_set_nat @ R_B )
       => ( irrefl5411402643388076864et_nat @ top_to4194386802282351772et_nat @ ( lex_pr6326417796491097880et_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_lex_prod
thf(fact_722_irrefl__lex__prod,axiom,
    ! [R_A: set_Pr1076074158304066111at_nat,R_B: set_Pr5488025237498180813et_nat] :
      ( ( irrefl688463893611329673at_nat @ top_to6661820994512907621at_nat @ R_A )
     => ( ( irrefl_on_set_nat @ top_top_set_set_nat @ R_B )
       => ( irrefl3112842807125672284et_nat @ top_to6784681116597541910et_nat @ ( lex_pr277683831282854198et_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_lex_prod
thf(fact_723_UNIV__def,axiom,
    ( top_to1573239103265677113rm_c_d
    = ( collec8193319511663418484rm_c_d
      @ ^ [X: produc8451535801715972489rm_c_d] : $true ) ) ).

% UNIV_def
thf(fact_724_UNIV__def,axiom,
    ( top_to4669805908274784177at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X: product_prod_nat_nat] : $true ) ) ).

% UNIV_def
thf(fact_725_UNIV__def,axiom,
    ( top_top_set_term_c_d
    = ( collect_term_c_d
      @ ^ [X: term_c_d] : $true ) ) ).

% UNIV_def
thf(fact_726_UNIV__def,axiom,
    ( top_top_set_term_a_b
    = ( collect_term_a_b
      @ ^ [X: term_a_b] : $true ) ) ).

% UNIV_def
thf(fact_727_UNIV__def,axiom,
    ( top_to6661820994512907621at_nat
    = ( collec1753597841410690900at_nat
      @ ^ [X: sum_sum_nat_nat] : $true ) ) ).

% UNIV_def
thf(fact_728_UNIV__def,axiom,
    ( top_top_set_set_nat
    = ( collect_set_nat
      @ ^ [X: set_nat] : $true ) ) ).

% UNIV_def
thf(fact_729_UNIV__def,axiom,
    ( top_to220653634128270125rm_a_b
    = ( collec7633585144569403164rm_a_b
      @ ^ [X: term_c_d > term_a_b] : $true ) ) ).

% UNIV_def
thf(fact_730_UNIV__def,axiom,
    ( top_to44634570330034989rm_c_d
    = ( collec1200580618792867612rm_c_d
      @ ^ [X: term_a_b > term_c_d] : $true ) ) ).

% UNIV_def
thf(fact_731_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $true ) ) ).

% UNIV_def
thf(fact_732_trans__inv__image,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( trans_on_nat @ top_top_set_nat @ ( inv_image_nat_nat @ R @ F ) ) ) ).

% trans_inv_image
thf(fact_733_trans__inv__image,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F: set_nat > nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( trans_on_set_nat @ top_top_set_set_nat @ ( inv_im5077416110856988202et_nat @ R @ F ) ) ) ).

% trans_inv_image
thf(fact_734_trans__inv__image,axiom,
    ! [R: set_Pr5488025237498180813et_nat,F: nat > set_nat] :
      ( ( trans_on_set_nat @ top_top_set_set_nat @ R )
     => ( trans_on_nat @ top_top_set_nat @ ( inv_im1511781206201618986at_nat @ R @ F ) ) ) ).

% trans_inv_image
thf(fact_735_trans__inv__image,axiom,
    ! [R: set_Pr8693737435421807431at_nat,F: nat > product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R )
     => ( trans_on_nat @ top_top_set_nat @ ( inv_im8721129524585065343at_nat @ R @ F ) ) ) ).

% trans_inv_image
thf(fact_736_trans__inv__image,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ ( inv_im2857804880773004637at_nat @ R @ F ) ) ) ).

% trans_inv_image
thf(fact_737_trans__inv__image,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F: sum_sum_nat_nat > nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ ( inv_im9000632410610895779at_nat @ R @ F ) ) ) ).

% trans_inv_image
thf(fact_738_trans__inv__image,axiom,
    ! [R: set_Pr1076074158304066111at_nat,F: nat > sum_sum_nat_nat] :
      ( ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R )
     => ( trans_on_nat @ top_top_set_nat @ ( inv_im418934867018748195at_nat @ R @ F ) ) ) ).

% trans_inv_image
thf(fact_739_trans__inv__image,axiom,
    ! [R: set_Pr5488025237498180813et_nat,F: set_nat > set_nat] :
      ( ( trans_on_set_nat @ top_top_set_set_nat @ R )
     => ( trans_on_set_nat @ top_top_set_set_nat @ ( inv_im3688519544425059808et_nat @ R @ F ) ) ) ).

% trans_inv_image
thf(fact_740_trans__inv__image,axiom,
    ! [R: set_Pr8693737435421807431at_nat,F: set_nat > product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R )
     => ( trans_on_set_nat @ top_top_set_set_nat @ ( inv_im8811740315869305653et_nat @ R @ F ) ) ) ).

% trans_inv_image
thf(fact_741_trans__inv__image,axiom,
    ! [R: set_Pr1076074158304066111at_nat,F: set_nat > sum_sum_nat_nat] :
      ( ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R )
     => ( trans_on_set_nat @ top_top_set_set_nat @ ( inv_im7339408600134414553et_nat @ R @ F ) ) ) ).

% trans_inv_image
thf(fact_742_trans__onD,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X5: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
      ( ( trans_5404704112166290345rm_a_b @ A3 @ R )
     => ( ( member5869715511025134514rm_a_b @ X5 @ A3 )
       => ( ( member5869715511025134514rm_a_b @ Y3 @ A3 )
         => ( ( member5869715511025134514rm_a_b @ Z @ A3 )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X5 @ Y3 ) @ R )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ R )
               => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X5 @ Z ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_743_trans__onD,axiom,
    ! [A3: set_o,R: set_Product_prod_o_o,X5: $o,Y3: $o,Z: $o] :
      ( ( trans_on_o @ A3 @ R )
     => ( ( member_o @ X5 @ A3 )
       => ( ( member_o @ Y3 @ A3 )
         => ( ( member_o @ Z @ A3 )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y3 ) @ R )
             => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ Z ) @ R )
               => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Z ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_744_trans__onD,axiom,
    ! [A3: set_set_nat,R: set_Pr5488025237498180813et_nat,X5: set_nat,Y3: set_nat,Z: set_nat] :
      ( ( trans_on_set_nat @ A3 @ R )
     => ( ( member_set_nat @ X5 @ A3 )
       => ( ( member_set_nat @ Y3 @ A3 )
         => ( ( member_set_nat @ Z @ A3 )
           => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y3 ) @ R )
             => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y3 @ Z ) @ R )
               => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Z ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_745_trans__onD,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d,X5: term_c_d,Y3: term_c_d,Z: term_c_d] :
      ( ( trans_on_term_c_d @ A3 @ R )
     => ( ( member_term_c_d @ X5 @ A3 )
       => ( ( member_term_c_d @ Y3 @ A3 )
         => ( ( member_term_c_d @ Z @ A3 )
           => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Y3 ) @ R )
             => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ Y3 @ Z ) @ R )
               => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ Z ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_746_trans__onD,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X5: term_a_b,Y3: term_a_b,Z: term_a_b] :
      ( ( trans_on_term_a_b @ A3 @ R )
     => ( ( member_term_a_b @ X5 @ A3 )
       => ( ( member_term_a_b @ Y3 @ A3 )
         => ( ( member_term_a_b @ Z @ A3 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y3 ) @ R )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Z ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_747_trans__onD,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat,X5: nat,Y3: nat,Z: nat] :
      ( ( trans_on_nat @ A3 @ R )
     => ( ( member_nat @ X5 @ A3 )
       => ( ( member_nat @ Y3 @ A3 )
         => ( ( member_nat @ Z @ A3 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Z ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_748_trans__onD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X5: product_prod_nat_nat,Y3: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ A3 @ R )
     => ( ( member8440522571783428010at_nat @ X5 @ A3 )
       => ( ( member8440522571783428010at_nat @ Y3 @ A3 )
         => ( ( member8440522571783428010at_nat @ Z @ A3 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) @ R )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z ) @ R )
               => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Z ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_749_trans__onI,axiom,
    ! [A3: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ A3 )
         => ( ( member5869715511025134514rm_a_b @ Y4 @ A3 )
           => ( ( member5869715511025134514rm_a_b @ Z3 @ A3 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y4 ) @ R )
               => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ Z3 ) @ R )
                 => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_5404704112166290345rm_a_b @ A3 @ R ) ) ).

% trans_onI
thf(fact_750_trans__onI,axiom,
    ! [A3: set_o,R: set_Product_prod_o_o] :
      ( ! [X3: $o,Y4: $o,Z3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ( member_o @ Y4 @ A3 )
           => ( ( member_o @ Z3 @ A3 )
             => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y4 ) @ R )
               => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ Z3 ) @ R )
                 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_on_o @ A3 @ R ) ) ).

% trans_onI
thf(fact_751_trans__onI,axiom,
    ! [A3: set_set_nat,R: set_Pr5488025237498180813et_nat] :
      ( ! [X3: set_nat,Y4: set_nat,Z3: set_nat] :
          ( ( member_set_nat @ X3 @ A3 )
         => ( ( member_set_nat @ Y4 @ A3 )
           => ( ( member_set_nat @ Z3 @ A3 )
             => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ Y4 ) @ R )
               => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y4 @ Z3 ) @ R )
                 => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_on_set_nat @ A3 @ R ) ) ).

% trans_onI
thf(fact_752_trans__onI,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ! [X3: term_c_d,Y4: term_c_d,Z3: term_c_d] :
          ( ( member_term_c_d @ X3 @ A3 )
         => ( ( member_term_c_d @ Y4 @ A3 )
           => ( ( member_term_c_d @ Z3 @ A3 )
             => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X3 @ Y4 ) @ R )
               => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ Y4 @ Z3 ) @ R )
                 => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_on_term_c_d @ A3 @ R ) ) ).

% trans_onI
thf(fact_753_trans__onI,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y4: term_a_b,Z3: term_a_b] :
          ( ( member_term_a_b @ X3 @ A3 )
         => ( ( member_term_a_b @ Y4 @ A3 )
           => ( ( member_term_a_b @ Z3 @ A3 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z3 ) @ R )
                 => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_on_term_a_b @ A3 @ R ) ) ).

% trans_onI
thf(fact_754_trans__onI,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y4: nat,Z3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( member_nat @ Y4 @ A3 )
           => ( ( member_nat @ Z3 @ A3 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z3 ) @ R )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_on_nat @ A3 @ R ) ) ).

% trans_onI
thf(fact_755_trans__onI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y4: product_prod_nat_nat,Z3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A3 )
         => ( ( member8440522571783428010at_nat @ Y4 @ A3 )
           => ( ( member8440522571783428010at_nat @ Z3 @ A3 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R )
               => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ Z3 ) @ R )
                 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_3241703717017304737at_nat @ A3 @ R ) ) ).

% trans_onI
thf(fact_756_trans__on__def,axiom,
    ( trans_on_term_c_d
    = ( ^ [A2: set_term_c_d,R2: set_Pr5080540287643815273rm_c_d] :
        ! [X: term_c_d] :
          ( ( member_term_c_d @ X @ A2 )
         => ! [Y: term_c_d] :
              ( ( member_term_c_d @ Y @ A2 )
             => ! [Z2: term_c_d] :
                  ( ( member_term_c_d @ Z2 @ A2 )
                 => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X @ Y ) @ R2 )
                   => ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ Y @ Z2 ) @ R2 )
                     => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_757_trans__on__def,axiom,
    ( trans_on_term_a_b
    = ( ^ [A2: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
        ! [X: term_a_b] :
          ( ( member_term_a_b @ X @ A2 )
         => ! [Y: term_a_b] :
              ( ( member_term_a_b @ Y @ A2 )
             => ! [Z2: term_a_b] :
                  ( ( member_term_a_b @ Z2 @ A2 )
                 => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
                   => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R2 )
                     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_758_trans__on__def,axiom,
    ( trans_on_nat
    = ( ^ [A2: set_nat,R2: set_Pr1261947904930325089at_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A2 )
         => ! [Y: nat] :
              ( ( member_nat @ Y @ A2 )
             => ! [Z2: nat] :
                  ( ( member_nat @ Z2 @ A2 )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
                   => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ R2 )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_759_trans__on__def,axiom,
    ( trans_3241703717017304737at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
        ! [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ A2 )
         => ! [Y: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y @ A2 )
             => ! [Z2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Z2 @ A2 )
                 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
                   => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z2 ) @ R2 )
                     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_760_irreflpD,axiom,
    ! [R3: sum_sum_nat_nat > sum_sum_nat_nat > $o,X5: sum_sum_nat_nat] :
      ( ( irrefl3240489695051909285at_nat @ top_to6661820994512907621at_nat @ R3 )
     => ~ ( R3 @ X5 @ X5 ) ) ).

% irreflpD
thf(fact_761_irreflpD,axiom,
    ! [R3: set_nat > set_nat > $o,X5: set_nat] :
      ( ( irreflp_on_set_nat @ top_top_set_set_nat @ R3 )
     => ~ ( R3 @ X5 @ X5 ) ) ).

% irreflpD
thf(fact_762_irreflpD,axiom,
    ! [R3: ( term_c_d > term_a_b ) > ( term_c_d > term_a_b ) > $o,X5: term_c_d > term_a_b] :
      ( ( irrefl4183777908429839213rm_a_b @ top_to220653634128270125rm_a_b @ R3 )
     => ~ ( R3 @ X5 @ X5 ) ) ).

% irreflpD
thf(fact_763_irreflpD,axiom,
    ! [R3: ( term_a_b > term_c_d ) > ( term_a_b > term_c_d ) > $o,X5: term_a_b > term_c_d] :
      ( ( irrefl6974145419508079469rm_c_d @ top_to44634570330034989rm_c_d @ R3 )
     => ~ ( R3 @ X5 @ X5 ) ) ).

% irreflpD
thf(fact_764_irreflpD,axiom,
    ! [R3: nat > nat > $o,X5: nat] :
      ( ( irreflp_on_nat @ top_top_set_nat @ R3 )
     => ~ ( R3 @ X5 @ X5 ) ) ).

% irreflpD
thf(fact_765_irreflpI,axiom,
    ! [R3: sum_sum_nat_nat > sum_sum_nat_nat > $o] :
      ( ! [A6: sum_sum_nat_nat] :
          ~ ( R3 @ A6 @ A6 )
     => ( irrefl3240489695051909285at_nat @ top_to6661820994512907621at_nat @ R3 ) ) ).

% irreflpI
thf(fact_766_irreflpI,axiom,
    ! [R3: set_nat > set_nat > $o] :
      ( ! [A6: set_nat] :
          ~ ( R3 @ A6 @ A6 )
     => ( irreflp_on_set_nat @ top_top_set_set_nat @ R3 ) ) ).

% irreflpI
thf(fact_767_irreflpI,axiom,
    ! [R3: ( term_c_d > term_a_b ) > ( term_c_d > term_a_b ) > $o] :
      ( ! [A6: term_c_d > term_a_b] :
          ~ ( R3 @ A6 @ A6 )
     => ( irrefl4183777908429839213rm_a_b @ top_to220653634128270125rm_a_b @ R3 ) ) ).

% irreflpI
thf(fact_768_irreflpI,axiom,
    ! [R3: ( term_a_b > term_c_d ) > ( term_a_b > term_c_d ) > $o] :
      ( ! [A6: term_a_b > term_c_d] :
          ~ ( R3 @ A6 @ A6 )
     => ( irrefl6974145419508079469rm_c_d @ top_to44634570330034989rm_c_d @ R3 ) ) ).

% irreflpI
thf(fact_769_irreflpI,axiom,
    ! [R3: nat > nat > $o] :
      ( ! [A6: nat] :
          ~ ( R3 @ A6 @ A6 )
     => ( irreflp_on_nat @ top_top_set_nat @ R3 ) ) ).

% irreflpI
thf(fact_770_trans__join,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d] :
      ( ( trans_on_term_c_d @ top_top_set_term_c_d @ R )
      = ( ! [X: produc4531938369520450825rm_c_d] :
            ( ( member820888157712331442rm_c_d @ X @ R )
           => ( produc6447316180697048258_c_d_o
              @ ^ [Y: term_c_d,Y12: term_c_d] :
                ! [Z2: produc4531938369520450825rm_c_d] :
                  ( ( member820888157712331442rm_c_d @ Z2 @ R )
                 => ( produc6447316180697048258_c_d_o
                    @ ^ [Y22: term_c_d,Aa: term_c_d] :
                        ( ( Y12 = Y22 )
                       => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ Y @ Aa ) @ R ) )
                    @ Z2 ) )
              @ X ) ) ) ) ).

% trans_join
thf(fact_771_trans__join,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R )
      = ( ! [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ R )
           => ( produc6590410687421337004_nat_o
              @ ^ [Y: product_prod_nat_nat,Y12: product_prod_nat_nat] :
                ! [Z2: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ Z2 @ R )
                 => ( produc6590410687421337004_nat_o
                    @ ^ [Y22: product_prod_nat_nat,Aa: product_prod_nat_nat] :
                        ( ( Y12 = Y22 )
                       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Aa ) @ R ) )
                    @ Z2 ) )
              @ X ) ) ) ) ).

% trans_join
thf(fact_772_trans__join,axiom,
    ! [R: set_Pr1076074158304066111at_nat] :
      ( ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R )
      = ( ! [X: produc3819362883047884511at_nat] :
            ( ( member4902808451920011912at_nat @ X @ R )
           => ( produc6670131743076204012_nat_o
              @ ^ [Y: sum_sum_nat_nat,Y12: sum_sum_nat_nat] :
                ! [Z2: produc3819362883047884511at_nat] :
                  ( ( member4902808451920011912at_nat @ Z2 @ R )
                 => ( produc6670131743076204012_nat_o
                    @ ^ [Y22: sum_sum_nat_nat,Aa: sum_sum_nat_nat] :
                        ( ( Y12 = Y22 )
                       => ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ Y @ Aa ) @ R ) )
                    @ Z2 ) )
              @ X ) ) ) ) ).

% trans_join
thf(fact_773_trans__join,axiom,
    ! [R: set_Pr5488025237498180813et_nat] :
      ( ( trans_on_set_nat @ top_top_set_set_nat @ R )
      = ( ! [X: produc7819656566062154093et_nat] :
            ( ( member8277197624267554838et_nat @ X @ R )
           => ( produc6247414631856714078_nat_o
              @ ^ [Y: set_nat,Y12: set_nat] :
                ! [Z2: produc7819656566062154093et_nat] :
                  ( ( member8277197624267554838et_nat @ Z2 @ R )
                 => ( produc6247414631856714078_nat_o
                    @ ^ [Y22: set_nat,Aa: set_nat] :
                        ( ( Y12 = Y22 )
                       => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y @ Aa ) @ R ) )
                    @ Z2 ) )
              @ X ) ) ) ) ).

% trans_join
thf(fact_774_trans__join,axiom,
    ! [R: set_Pr8788715879341982671rm_a_b] :
      ( ( trans_5020182533697283623rm_a_b @ top_to220653634128270125rm_a_b @ R )
      = ( ! [X: produc1668039282281753199rm_a_b] :
            ( ( member4454048570373794840rm_a_b @ X @ R )
           => ( produc3909228560075629660_a_b_o
              @ ^ [Y: term_c_d > term_a_b,Y12: term_c_d > term_a_b] :
                ! [Z2: produc1668039282281753199rm_a_b] :
                  ( ( member4454048570373794840rm_a_b @ Z2 @ R )
                 => ( produc3909228560075629660_a_b_o
                    @ ^ [Y22: term_c_d > term_a_b,Aa: term_c_d > term_a_b] :
                        ( ( Y12 = Y22 )
                       => ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ Y @ Aa ) @ R ) )
                    @ Z2 ) )
              @ X ) ) ) ) ).

% trans_join
thf(fact_775_trans__join,axiom,
    ! [R: set_Pr6132369733443093455rm_c_d] :
      ( ( trans_7810550044775523879rm_c_d @ top_to44634570330034989rm_c_d @ R )
      = ( ! [X: produc4272247648830171759rm_c_d] :
            ( ( member7058256936922213400rm_c_d @ X @ R )
           => ( produc8827469034848429148_c_d_o
              @ ^ [Y: term_a_b > term_c_d,Y12: term_a_b > term_c_d] :
                ! [Z2: produc4272247648830171759rm_c_d] :
                  ( ( member7058256936922213400rm_c_d @ Z2 @ R )
                 => ( produc8827469034848429148_c_d_o
                    @ ^ [Y22: term_a_b > term_c_d,Aa: term_a_b > term_c_d] :
                        ( ( Y12 = Y22 )
                       => ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ Y @ Aa ) @ R ) )
                    @ Z2 ) )
              @ X ) ) ) ) ).

% trans_join
thf(fact_776_trans__join,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R )
      = ( ! [X: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X @ R )
           => ( produc6251846659952913346_a_b_o
              @ ^ [Y: term_a_b,Y12: term_a_b] :
                ! [Z2: produc357393685978478089rm_a_b] :
                  ( ( member5869715511025134514rm_a_b @ Z2 @ R )
                 => ( produc6251846659952913346_a_b_o
                    @ ^ [Y22: term_a_b,Aa: term_a_b] :
                        ( ( Y12 = Y22 )
                       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Aa ) @ R ) )
                    @ Z2 ) )
              @ X ) ) ) ) ).

% trans_join
thf(fact_777_trans__join,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
      = ( ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ R )
           => ( produc6081775807080527818_nat_o
              @ ^ [Y: nat,Y12: nat] :
                ! [Z2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Z2 @ R )
                 => ( produc6081775807080527818_nat_o
                    @ ^ [Y22: nat,Aa: nat] :
                        ( ( Y12 = Y22 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Aa ) @ R ) )
                    @ Z2 ) )
              @ X ) ) ) ) ).

% trans_join
thf(fact_778_conversep__converse__eq,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d] :
      ( ( conver4964912785503959199rm_c_d
        @ ^ [X: term_c_d,Y: term_c_d] : ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X @ Y ) @ R ) )
      = ( ^ [X: term_c_d,Y: term_c_d] : ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X @ Y ) @ ( conver7337423426258204773rm_c_d @ R ) ) ) ) ).

% conversep_converse_eq
thf(fact_779_conversep__converse__eq,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( conver5236049336183599519rm_a_b
        @ ^ [X: term_a_b,Y: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R ) )
      = ( ^ [X: term_a_b,Y: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).

% conversep_converse_eq
thf(fact_780_conversep__converse__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( conversep_nat_nat
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) )
      = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( converse_nat_nat @ R ) ) ) ) ).

% conversep_converse_eq
thf(fact_781_conversep__converse__eq,axiom,
    ! [R: set_Pr4645549399426460137rm_c_d] :
      ( ( conver8884510217699480863rm_c_d
        @ ^ [X: term_a_b,Y: term_c_d] : ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ X @ Y ) @ R ) )
      = ( ^ [X: term_c_d,Y: term_a_b] : ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ X @ Y ) @ ( conver2033648821598950629rm_c_d @ R ) ) ) ) ).

% conversep_converse_eq
thf(fact_782_conversep__converse__eq,axiom,
    ! [R: set_Pr4821568463224695273rm_a_b] :
      ( ( conver1316451903988077855rm_a_b
        @ ^ [X: term_c_d,Y: term_a_b] : ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ X @ Y ) @ R ) )
      = ( ^ [X: term_a_b,Y: term_c_d] : ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ X @ Y ) @ ( conver3688962544742323429rm_a_b @ R ) ) ) ) ).

% conversep_converse_eq
thf(fact_783_reflD,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d,A4: term_c_d] :
      ( ( refl_on_term_c_d @ top_top_set_term_c_d @ R )
     => ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ A4 ) @ R ) ) ).

% reflD
thf(fact_784_reflD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A4: term_a_b] :
      ( ( refl_on_term_a_b @ top_top_set_term_a_b @ R )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R ) ) ).

% reflD
thf(fact_785_reflD,axiom,
    ! [R: set_Pr1076074158304066111at_nat,A4: sum_sum_nat_nat] :
      ( ( refl_o9084340785372409280at_nat @ top_to6661820994512907621at_nat @ R )
     => ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ A4 @ A4 ) @ R ) ) ).

% reflD
thf(fact_786_reflD,axiom,
    ! [R: set_Pr5488025237498180813et_nat,A4: set_nat] :
      ( ( refl_on_set_nat @ top_top_set_set_nat @ R )
     => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A4 @ A4 ) @ R ) ) ).

% reflD
thf(fact_787_reflD,axiom,
    ! [R: set_Pr8788715879341982671rm_a_b,A4: term_c_d > term_a_b] :
      ( ( refl_o3744124953998825864rm_a_b @ top_to220653634128270125rm_a_b @ R )
     => ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ A4 @ A4 ) @ R ) ) ).

% reflD
thf(fact_788_reflD,axiom,
    ! [R: set_Pr6132369733443093455rm_c_d,A4: term_a_b > term_c_d] :
      ( ( refl_o6534492465077066120rm_c_d @ top_to44634570330034989rm_c_d @ R )
     => ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ A4 @ A4 ) @ R ) ) ).

% reflD
thf(fact_789_reflD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A4: nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R ) ) ).

% reflD
thf(fact_790_reflI,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d] :
      ( ! [X3: term_c_d] : ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X3 @ X3 ) @ R )
     => ( refl_on_term_c_d @ top_top_set_term_c_d @ R ) ) ).

% reflI
thf(fact_791_reflI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R )
     => ( refl_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).

% reflI
thf(fact_792_reflI,axiom,
    ! [R: set_Pr1076074158304066111at_nat] :
      ( ! [X3: sum_sum_nat_nat] : ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ X3 @ X3 ) @ R )
     => ( refl_o9084340785372409280at_nat @ top_to6661820994512907621at_nat @ R ) ) ).

% reflI
thf(fact_793_reflI,axiom,
    ! [R: set_Pr5488025237498180813et_nat] :
      ( ! [X3: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ X3 ) @ R )
     => ( refl_on_set_nat @ top_top_set_set_nat @ R ) ) ).

% reflI
thf(fact_794_reflI,axiom,
    ! [R: set_Pr8788715879341982671rm_a_b] :
      ( ! [X3: term_c_d > term_a_b] : ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ X3 @ X3 ) @ R )
     => ( refl_o3744124953998825864rm_a_b @ top_to220653634128270125rm_a_b @ R ) ) ).

% reflI
thf(fact_795_reflI,axiom,
    ! [R: set_Pr6132369733443093455rm_c_d] :
      ( ! [X3: term_a_b > term_c_d] : ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ X3 @ X3 ) @ R )
     => ( refl_o6534492465077066120rm_c_d @ top_to44634570330034989rm_c_d @ R ) ) ).

% reflI
thf(fact_796_reflI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
     => ( refl_on_nat @ top_top_set_nat @ R ) ) ).

% reflI
thf(fact_797_irreflI,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d] :
      ( ! [A6: term_c_d] :
          ~ ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A6 @ A6 ) @ R )
     => ( irrefl_on_term_c_d @ top_top_set_term_c_d @ R ) ) ).

% irreflI
thf(fact_798_irreflI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [A6: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ A6 ) @ R )
     => ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).

% irreflI
thf(fact_799_irreflI,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ! [A6: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A6 @ A6 ) @ R )
     => ( irrefl7408258699751485431at_nat @ top_to4669805908274784177at_nat @ R ) ) ).

% irreflI
thf(fact_800_irreflI,axiom,
    ! [R: set_Pr1076074158304066111at_nat] :
      ( ! [A6: sum_sum_nat_nat] :
          ~ ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ A6 @ A6 ) @ R )
     => ( irrefl688463893611329673at_nat @ top_to6661820994512907621at_nat @ R ) ) ).

% irreflI
thf(fact_801_irreflI,axiom,
    ! [R: set_Pr5488025237498180813et_nat] :
      ( ! [A6: set_nat] :
          ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A6 @ A6 ) @ R )
     => ( irrefl_on_set_nat @ top_top_set_set_nat @ R ) ) ).

% irreflI
thf(fact_802_irreflI,axiom,
    ! [R: set_Pr8788715879341982671rm_a_b] :
      ( ! [A6: term_c_d > term_a_b] :
          ~ ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ A6 @ A6 ) @ R )
     => ( irrefl7486889657381885777rm_a_b @ top_to220653634128270125rm_a_b @ R ) ) ).

% irreflI
thf(fact_803_irreflI,axiom,
    ! [R: set_Pr6132369733443093455rm_c_d] :
      ( ! [A6: term_a_b > term_c_d] :
          ~ ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ A6 @ A6 ) @ R )
     => ( irrefl1053885131605350225rm_c_d @ top_to44634570330034989rm_c_d @ R ) ) ).

% irreflI
thf(fact_804_irreflI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [A6: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ A6 ) @ R )
     => ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).

% irreflI
thf(fact_805_irreflD,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d,X5: term_c_d] :
      ( ( irrefl_on_term_c_d @ top_top_set_term_c_d @ R )
     => ~ ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X5 @ X5 ) @ R ) ) ).

% irreflD
thf(fact_806_irreflD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X5: term_a_b] :
      ( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ X5 ) @ R ) ) ).

% irreflD
thf(fact_807_irreflD,axiom,
    ! [R: set_Pr8693737435421807431at_nat,X5: product_prod_nat_nat] :
      ( ( irrefl7408258699751485431at_nat @ top_to4669805908274784177at_nat @ R )
     => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ X5 ) @ R ) ) ).

% irreflD
thf(fact_808_irreflD,axiom,
    ! [R: set_Pr1076074158304066111at_nat,X5: sum_sum_nat_nat] :
      ( ( irrefl688463893611329673at_nat @ top_to6661820994512907621at_nat @ R )
     => ~ ( member4902808451920011912at_nat @ ( produc3532782441974027415at_nat @ X5 @ X5 ) @ R ) ) ).

% irreflD
thf(fact_809_irreflD,axiom,
    ! [R: set_Pr5488025237498180813et_nat,X5: set_nat] :
      ( ( irrefl_on_set_nat @ top_top_set_set_nat @ R )
     => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ X5 ) @ R ) ) ).

% irreflD
thf(fact_810_irreflD,axiom,
    ! [R: set_Pr8788715879341982671rm_a_b,X5: term_c_d > term_a_b] :
      ( ( irrefl7486889657381885777rm_a_b @ top_to220653634128270125rm_a_b @ R )
     => ~ ( member4454048570373794840rm_a_b @ ( produc6696390399768932903rm_a_b @ X5 @ X5 ) @ R ) ) ).

% irreflD
thf(fact_811_irreflD,axiom,
    ! [R: set_Pr6132369733443093455rm_c_d,X5: term_a_b > term_c_d] :
      ( ( irrefl1053885131605350225rm_c_d @ top_to44634570330034989rm_c_d @ R )
     => ~ ( member7058256936922213400rm_c_d @ ( produc3043613304339051047rm_c_d @ X5 @ X5 ) @ R ) ) ).

% irreflD
thf(fact_812_irreflD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X5: nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R )
     => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ R ) ) ).

% irreflD
thf(fact_813_refl__inv__image,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R3 )
     => ( refl_on_nat @ top_top_set_nat @ ( inv_image_nat_nat @ R3 @ F ) ) ) ).

% refl_inv_image
thf(fact_814_refl__inv__image,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F: set_nat > nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R3 )
     => ( refl_on_set_nat @ top_top_set_set_nat @ ( inv_im5077416110856988202et_nat @ R3 @ F ) ) ) ).

% refl_inv_image
thf(fact_815_refl__inv__image,axiom,
    ! [R3: set_Pr5488025237498180813et_nat,F: nat > set_nat] :
      ( ( refl_on_set_nat @ top_top_set_set_nat @ R3 )
     => ( refl_on_nat @ top_top_set_nat @ ( inv_im1511781206201618986at_nat @ R3 @ F ) ) ) ).

% refl_inv_image
thf(fact_816_refl__inv__image,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F: sum_sum_nat_nat > nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R3 )
     => ( refl_o9084340785372409280at_nat @ top_to6661820994512907621at_nat @ ( inv_im9000632410610895779at_nat @ R3 @ F ) ) ) ).

% refl_inv_image
thf(fact_817_refl__inv__image,axiom,
    ! [R3: set_Pr1076074158304066111at_nat,F: nat > sum_sum_nat_nat] :
      ( ( refl_o9084340785372409280at_nat @ top_to6661820994512907621at_nat @ R3 )
     => ( refl_on_nat @ top_top_set_nat @ ( inv_im418934867018748195at_nat @ R3 @ F ) ) ) ).

% refl_inv_image
thf(fact_818_refl__inv__image,axiom,
    ! [R3: set_Pr5488025237498180813et_nat,F: set_nat > set_nat] :
      ( ( refl_on_set_nat @ top_top_set_set_nat @ R3 )
     => ( refl_on_set_nat @ top_top_set_set_nat @ ( inv_im3688519544425059808et_nat @ R3 @ F ) ) ) ).

% refl_inv_image
thf(fact_819_refl__inv__image,axiom,
    ! [R3: set_Pr1076074158304066111at_nat,F: set_nat > sum_sum_nat_nat] :
      ( ( refl_o9084340785372409280at_nat @ top_to6661820994512907621at_nat @ R3 )
     => ( refl_on_set_nat @ top_top_set_set_nat @ ( inv_im7339408600134414553et_nat @ R3 @ F ) ) ) ).

% refl_inv_image
thf(fact_820_refl__inv__image,axiom,
    ! [R3: set_Pr5488025237498180813et_nat,F: sum_sum_nat_nat > set_nat] :
      ( ( refl_on_set_nat @ top_top_set_set_nat @ R3 )
     => ( refl_o9084340785372409280at_nat @ top_to6661820994512907621at_nat @ ( inv_im1879972631959990873at_nat @ R3 @ F ) ) ) ).

% refl_inv_image
thf(fact_821_refl__inv__image,axiom,
    ! [R3: set_Pr1076074158304066111at_nat,F: sum_sum_nat_nat > sum_sum_nat_nat] :
      ( ( refl_o9084340785372409280at_nat @ top_to6661820994512907621at_nat @ R3 )
     => ( refl_o9084340785372409280at_nat @ top_to6661820994512907621at_nat @ ( inv_im5732205804868199378at_nat @ R3 @ F ) ) ) ).

% refl_inv_image
thf(fact_822_refl__inv__image,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F: ( term_c_d > term_a_b ) > nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R3 )
     => ( refl_o3744124953998825864rm_a_b @ top_to220653634128270125rm_a_b @ ( inv_im908594706308016235rm_a_b @ R3 @ F ) ) ) ).

% refl_inv_image
thf(fact_823_the__sym__eq__trivial,axiom,
    ! [X5: produc5661168290637732233rm_a_b] :
      ( ( the_Pr9157697819281395472rm_a_b
        @ ( ^ [Y7: produc5661168290637732233rm_a_b,Z4: produc5661168290637732233rm_a_b] : ( Y7 = Z4 )
          @ X5 ) )
      = X5 ) ).

% the_sym_eq_trivial
thf(fact_824_the__sym__eq__trivial,axiom,
    ! [X5: produc8451535801715972489rm_c_d] :
      ( ( the_Pr2724693293504859920rm_c_d
        @ ( ^ [Y7: produc8451535801715972489rm_c_d,Z4: produc8451535801715972489rm_c_d] : ( Y7 = Z4 )
          @ X5 ) )
      = X5 ) ).

% the_sym_eq_trivial
thf(fact_825_the__eq__trivial,axiom,
    ! [A4: produc5661168290637732233rm_a_b] :
      ( ( the_Pr9157697819281395472rm_a_b
        @ ^ [X: produc5661168290637732233rm_a_b] : ( X = A4 ) )
      = A4 ) ).

% the_eq_trivial
thf(fact_826_the__eq__trivial,axiom,
    ! [A4: produc8451535801715972489rm_c_d] :
      ( ( the_Pr2724693293504859920rm_c_d
        @ ^ [X: produc8451535801715972489rm_c_d] : ( X = A4 ) )
      = A4 ) ).

% the_eq_trivial
thf(fact_827_the__equality,axiom,
    ! [P: produc5661168290637732233rm_a_b > $o,A4: produc5661168290637732233rm_a_b] :
      ( ( P @ A4 )
     => ( ! [X3: produc5661168290637732233rm_a_b] :
            ( ( P @ X3 )
           => ( X3 = A4 ) )
       => ( ( the_Pr9157697819281395472rm_a_b @ P )
          = A4 ) ) ) ).

% the_equality
thf(fact_828_the__equality,axiom,
    ! [P: produc8451535801715972489rm_c_d > $o,A4: produc8451535801715972489rm_c_d] :
      ( ( P @ A4 )
     => ( ! [X3: produc8451535801715972489rm_c_d] :
            ( ( P @ X3 )
           => ( X3 = A4 ) )
       => ( ( the_Pr2724693293504859920rm_c_d @ P )
          = A4 ) ) ) ).

% the_equality
thf(fact_829_iso__tuple__UNIV__I,axiom,
    ! [X5: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X5 @ top_to1314267278846557113rm_a_b ) ).

% iso_tuple_UNIV_I
thf(fact_830_iso__tuple__UNIV__I,axiom,
    ! [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ top_to4669805908274784177at_nat ) ).

% iso_tuple_UNIV_I
thf(fact_831_iso__tuple__UNIV__I,axiom,
    ! [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ).

% iso_tuple_UNIV_I
thf(fact_832_iso__tuple__UNIV__I,axiom,
    ! [X5: sum_sum_nat_nat] : ( member8583185029347631382at_nat @ X5 @ top_to6661820994512907621at_nat ) ).

% iso_tuple_UNIV_I
thf(fact_833_iso__tuple__UNIV__I,axiom,
    ! [X5: set_nat] : ( member_set_nat @ X5 @ top_top_set_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_834_iso__tuple__UNIV__I,axiom,
    ! [X5: term_c_d > term_a_b] : ( member3140172230840508126rm_a_b @ X5 @ top_to220653634128270125rm_a_b ) ).

% iso_tuple_UNIV_I
thf(fact_835_iso__tuple__UNIV__I,axiom,
    ! [X5: term_a_b > term_c_d] : ( member5930539741918748382rm_c_d @ X5 @ top_to44634570330034989rm_c_d ) ).

% iso_tuple_UNIV_I
thf(fact_836_iso__tuple__UNIV__I,axiom,
    ! [X5: nat] : ( member_nat @ X5 @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_837_top__apply,axiom,
    ( top_top_nat_o
    = ( ^ [X: nat] : top_top_o ) ) ).

% top_apply
thf(fact_838_surj__def,axiom,
    ! [F: nat > nat] :
      ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
        = top_top_set_nat )
      = ( ! [Y: nat] :
          ? [X: nat] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_839_surj__def,axiom,
    ! [F: nat > set_nat] :
      ( ( ( image_nat_set_nat @ F @ top_top_set_nat )
        = top_top_set_set_nat )
      = ( ! [Y: set_nat] :
          ? [X: nat] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_840_surj__def,axiom,
    ! [F: set_nat > nat] :
      ( ( ( image_set_nat_nat @ F @ top_top_set_set_nat )
        = top_top_set_nat )
      = ( ! [Y: nat] :
          ? [X: set_nat] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_841_surj__def,axiom,
    ! [F: nat > sum_sum_nat_nat] :
      ( ( ( image_678696785212003926at_nat @ F @ top_top_set_nat )
        = top_to6661820994512907621at_nat )
      = ( ! [Y: sum_sum_nat_nat] :
          ? [X: nat] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_842_surj__def,axiom,
    ! [F: sum_sum_nat_nat > nat] :
      ( ( ( image_1320371278474632150at_nat @ F @ top_to6661820994512907621at_nat )
        = top_top_set_nat )
      = ( ! [Y: nat] :
          ? [X: sum_sum_nat_nat] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_843_surj__def,axiom,
    ! [F: set_nat > set_nat] :
      ( ( ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat )
        = top_top_set_set_nat )
      = ( ! [Y: set_nat] :
          ? [X: set_nat] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_844_surj__def,axiom,
    ! [F: ( nat > $o ) > set_nat] :
      ( ( ( image_nat_o_set_nat @ F @ top_top_set_nat_o2 )
        = top_top_set_set_nat )
      = ( ! [Y: set_nat] :
          ? [X: nat > $o] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_845_surj__def,axiom,
    ! [F: sum_sum_nat_nat > set_nat] :
      ( ( ( image_5891067694031531916et_nat @ F @ top_to6661820994512907621at_nat )
        = top_top_set_set_nat )
      = ( ! [Y: set_nat] :
          ? [X: sum_sum_nat_nat] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_846_surj__def,axiom,
    ! [F: set_nat > sum_sum_nat_nat] :
      ( ( ( image_431631725857108236at_nat @ F @ top_top_set_set_nat )
        = top_to6661820994512907621at_nat )
      = ( ! [Y: sum_sum_nat_nat] :
          ? [X: set_nat] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_847_surj__def,axiom,
    ! [F: sum_sum_nat_nat > sum_sum_nat_nat] :
      ( ( ( image_7938014925024745989at_nat @ F @ top_to6661820994512907621at_nat )
        = top_to6661820994512907621at_nat )
      = ( ! [Y: sum_sum_nat_nat] :
          ? [X: sum_sum_nat_nat] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_848_surjI,axiom,
    ! [G: nat > nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_nat_nat2 @ G @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_849_surjI,axiom,
    ! [G: nat > set_nat,F: set_nat > nat] :
      ( ! [X3: set_nat] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_nat_set_nat @ G @ top_top_set_nat )
        = top_top_set_set_nat ) ) ).

% surjI
thf(fact_850_surjI,axiom,
    ! [G: set_nat > nat,F: nat > set_nat] :
      ( ! [X3: nat] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_set_nat_nat @ G @ top_top_set_set_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_851_surjI,axiom,
    ! [G: nat > sum_sum_nat_nat,F: sum_sum_nat_nat > nat] :
      ( ! [X3: sum_sum_nat_nat] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_678696785212003926at_nat @ G @ top_top_set_nat )
        = top_to6661820994512907621at_nat ) ) ).

% surjI
thf(fact_852_surjI,axiom,
    ! [G: sum_sum_nat_nat > nat,F: nat > sum_sum_nat_nat] :
      ( ! [X3: nat] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_1320371278474632150at_nat @ G @ top_to6661820994512907621at_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_853_surjI,axiom,
    ! [G: set_nat > set_nat,F: set_nat > set_nat] :
      ( ! [X3: set_nat] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_7916887816326733075et_nat @ G @ top_top_set_set_nat )
        = top_top_set_set_nat ) ) ).

% surjI
thf(fact_854_surjI,axiom,
    ! [G: ( nat > $o ) > set_nat,F: set_nat > nat > $o] :
      ( ! [X3: set_nat] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_nat_o_set_nat @ G @ top_top_set_nat_o2 )
        = top_top_set_set_nat ) ) ).

% surjI
thf(fact_855_surjI,axiom,
    ! [G: sum_sum_nat_nat > set_nat,F: set_nat > sum_sum_nat_nat] :
      ( ! [X3: set_nat] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_5891067694031531916et_nat @ G @ top_to6661820994512907621at_nat )
        = top_top_set_set_nat ) ) ).

% surjI
thf(fact_856_surjI,axiom,
    ! [G: set_nat > sum_sum_nat_nat,F: sum_sum_nat_nat > set_nat] :
      ( ! [X3: sum_sum_nat_nat] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_431631725857108236at_nat @ G @ top_top_set_set_nat )
        = top_to6661820994512907621at_nat ) ) ).

% surjI
thf(fact_857_surjI,axiom,
    ! [G: sum_sum_nat_nat > sum_sum_nat_nat,F: sum_sum_nat_nat > sum_sum_nat_nat] :
      ( ! [X3: sum_sum_nat_nat] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_7938014925024745989at_nat @ G @ top_to6661820994512907621at_nat )
        = top_to6661820994512907621at_nat ) ) ).

% surjI
thf(fact_858_top__empty__eq,axiom,
    ( top_to1262317115727556708_a_b_o
    = ( ^ [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ top_to1314267278846557113rm_a_b ) ) ) ).

% top_empty_eq
thf(fact_859_top__empty__eq,axiom,
    ( top_to3137496036531963500_nat_o
    = ( ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq
thf(fact_860_top__empty__eq,axiom,
    ( top_top_o_o
    = ( ^ [X: $o] : ( member_o @ X @ top_top_set_o ) ) ) ).

% top_empty_eq
thf(fact_861_top__empty__eq,axiom,
    ( top_to8066396485052406624_nat_o
    = ( ^ [X: sum_sum_nat_nat] : ( member8583185029347631382at_nat @ X @ top_to6661820994512907621at_nat ) ) ) ).

% top_empty_eq
thf(fact_862_top__empty__eq,axiom,
    ( top_top_set_nat_o
    = ( ^ [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ) ) ).

% top_empty_eq
thf(fact_863_top__empty__eq,axiom,
    ( top_to2333469598226860952_a_b_o
    = ( ^ [X: term_c_d > term_a_b] : ( member3140172230840508126rm_a_b @ X @ top_to220653634128270125rm_a_b ) ) ) ).

% top_empty_eq
thf(fact_864_top__empty__eq,axiom,
    ( top_to8955116800616450968_c_d_o
    = ( ^ [X: term_a_b > term_c_d] : ( member5930539741918748382rm_c_d @ X @ top_to44634570330034989rm_c_d ) ) ) ).

% top_empty_eq
thf(fact_865_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_866_top__set__def,axiom,
    ( top_to1573239103265677113rm_c_d
    = ( collec8193319511663418484rm_c_d @ top_to1960084308039068900_c_d_o ) ) ).

% top_set_def
thf(fact_867_top__set__def,axiom,
    ( top_to4669805908274784177at_nat
    = ( collec3392354462482085612at_nat @ top_to3137496036531963500_nat_o ) ) ).

% top_set_def
thf(fact_868_top__set__def,axiom,
    ( top_top_set_term_c_d
    = ( collect_term_c_d @ top_top_term_c_d_o ) ) ).

% top_set_def
thf(fact_869_top__set__def,axiom,
    ( top_top_set_term_a_b
    = ( collect_term_a_b @ top_top_term_a_b_o ) ) ).

% top_set_def
thf(fact_870_top__set__def,axiom,
    ( top_to6661820994512907621at_nat
    = ( collec1753597841410690900at_nat @ top_to8066396485052406624_nat_o ) ) ).

% top_set_def
thf(fact_871_top__set__def,axiom,
    ( top_top_set_set_nat
    = ( collect_set_nat @ top_top_set_nat_o ) ) ).

% top_set_def
thf(fact_872_top__set__def,axiom,
    ( top_to220653634128270125rm_a_b
    = ( collec7633585144569403164rm_a_b @ top_to2333469598226860952_a_b_o ) ) ).

% top_set_def
thf(fact_873_top__set__def,axiom,
    ( top_to44634570330034989rm_c_d
    = ( collec1200580618792867612rm_c_d @ top_to8955116800616450968_c_d_o ) ) ).

% top_set_def
thf(fact_874_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_875_top__empty__eq2,axiom,
    ( top_to7769959678420217390_c_d_o
    = ( ^ [X: term_a_b,Y: term_c_d] : ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ X @ Y ) @ top_to1573239103265677113rm_c_d ) ) ) ).

% top_empty_eq2
thf(fact_876_top__empty__eq2,axiom,
    ( top_to6638041551531519534_a_b_o
    = ( ^ [X: term_c_d,Y: term_a_b] : ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ X @ Y ) @ top_to1749258167063912249rm_a_b ) ) ) ).

% top_empty_eq2
thf(fact_877_top__empty__eq2,axiom,
    ( top_to7335808743843031726_c_d_o
    = ( ^ [X: term_c_d,Y: term_c_d] : ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ X @ Y ) @ top_to2008229991483032249rm_c_d ) ) ) ).

% top_empty_eq2
thf(fact_878_top__empty__eq2,axiom,
    ( top_to7072192486108705198_a_b_o
    = ( ^ [X: term_a_b,Y: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ top_to1314267278846557113rm_a_b ) ) ) ).

% top_empty_eq2
thf(fact_879_top__empty__eq2,axiom,
    ( top_top_nat_nat_o
    = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq2
thf(fact_880_theI,axiom,
    ! [P: produc5661168290637732233rm_a_b > $o,A4: produc5661168290637732233rm_a_b] :
      ( ( P @ A4 )
     => ( ! [X3: produc5661168290637732233rm_a_b] :
            ( ( P @ X3 )
           => ( X3 = A4 ) )
       => ( P @ ( the_Pr9157697819281395472rm_a_b @ P ) ) ) ) ).

% theI
thf(fact_881_theI,axiom,
    ! [P: produc8451535801715972489rm_c_d > $o,A4: produc8451535801715972489rm_c_d] :
      ( ( P @ A4 )
     => ( ! [X3: produc8451535801715972489rm_c_d] :
            ( ( P @ X3 )
           => ( X3 = A4 ) )
       => ( P @ ( the_Pr2724693293504859920rm_c_d @ P ) ) ) ) ).

% theI
thf(fact_882_theI_H,axiom,
    ! [P: produc5661168290637732233rm_a_b > $o] :
      ( ? [X4: produc5661168290637732233rm_a_b] :
          ( ( P @ X4 )
          & ! [Y4: produc5661168290637732233rm_a_b] :
              ( ( P @ Y4 )
             => ( Y4 = X4 ) ) )
     => ( P @ ( the_Pr9157697819281395472rm_a_b @ P ) ) ) ).

% theI'
thf(fact_883_theI_H,axiom,
    ! [P: produc8451535801715972489rm_c_d > $o] :
      ( ? [X4: produc8451535801715972489rm_c_d] :
          ( ( P @ X4 )
          & ! [Y4: produc8451535801715972489rm_c_d] :
              ( ( P @ Y4 )
             => ( Y4 = X4 ) ) )
     => ( P @ ( the_Pr2724693293504859920rm_c_d @ P ) ) ) ).

% theI'
thf(fact_884_theI2,axiom,
    ! [P: produc5661168290637732233rm_a_b > $o,A4: produc5661168290637732233rm_a_b,Q: produc5661168290637732233rm_a_b > $o] :
      ( ( P @ A4 )
     => ( ! [X3: produc5661168290637732233rm_a_b] :
            ( ( P @ X3 )
           => ( X3 = A4 ) )
       => ( ! [X3: produc5661168290637732233rm_a_b] :
              ( ( P @ X3 )
             => ( Q @ X3 ) )
         => ( Q @ ( the_Pr9157697819281395472rm_a_b @ P ) ) ) ) ) ).

% theI2
thf(fact_885_theI2,axiom,
    ! [P: produc8451535801715972489rm_c_d > $o,A4: produc8451535801715972489rm_c_d,Q: produc8451535801715972489rm_c_d > $o] :
      ( ( P @ A4 )
     => ( ! [X3: produc8451535801715972489rm_c_d] :
            ( ( P @ X3 )
           => ( X3 = A4 ) )
       => ( ! [X3: produc8451535801715972489rm_c_d] :
              ( ( P @ X3 )
             => ( Q @ X3 ) )
         => ( Q @ ( the_Pr2724693293504859920rm_c_d @ P ) ) ) ) ) ).

% theI2
thf(fact_886_If__def,axiom,
    ( if_Pro6628055518645656911rm_a_b
    = ( ^ [P2: $o,X: produc5661168290637732233rm_a_b,Y: produc5661168290637732233rm_a_b] :
          ( the_Pr9157697819281395472rm_a_b
          @ ^ [Z2: produc5661168290637732233rm_a_b] :
              ( ( P2
               => ( Z2 = X ) )
              & ( ~ P2
               => ( Z2 = Y ) ) ) ) ) ) ).

% If_def
thf(fact_887_If__def,axiom,
    ( if_Pro195050992869121359rm_c_d
    = ( ^ [P2: $o,X: produc8451535801715972489rm_c_d,Y: produc8451535801715972489rm_c_d] :
          ( the_Pr2724693293504859920rm_c_d
          @ ^ [Z2: produc8451535801715972489rm_c_d] :
              ( ( P2
               => ( Z2 = X ) )
              & ( ~ P2
               => ( Z2 = Y ) ) ) ) ) ) ).

% If_def
thf(fact_888_the1I2,axiom,
    ! [P: produc5661168290637732233rm_a_b > $o,Q: produc5661168290637732233rm_a_b > $o] :
      ( ? [X4: produc5661168290637732233rm_a_b] :
          ( ( P @ X4 )
          & ! [Y4: produc5661168290637732233rm_a_b] :
              ( ( P @ Y4 )
             => ( Y4 = X4 ) ) )
     => ( ! [X3: produc5661168290637732233rm_a_b] :
            ( ( P @ X3 )
           => ( Q @ X3 ) )
       => ( Q @ ( the_Pr9157697819281395472rm_a_b @ P ) ) ) ) ).

% the1I2
thf(fact_889_the1I2,axiom,
    ! [P: produc8451535801715972489rm_c_d > $o,Q: produc8451535801715972489rm_c_d > $o] :
      ( ? [X4: produc8451535801715972489rm_c_d] :
          ( ( P @ X4 )
          & ! [Y4: produc8451535801715972489rm_c_d] :
              ( ( P @ Y4 )
             => ( Y4 = X4 ) ) )
     => ( ! [X3: produc8451535801715972489rm_c_d] :
            ( ( P @ X3 )
           => ( Q @ X3 ) )
       => ( Q @ ( the_Pr2724693293504859920rm_c_d @ P ) ) ) ) ).

% the1I2
thf(fact_890_the1__equality,axiom,
    ! [P: produc5661168290637732233rm_a_b > $o,A4: produc5661168290637732233rm_a_b] :
      ( ? [X4: produc5661168290637732233rm_a_b] :
          ( ( P @ X4 )
          & ! [Y4: produc5661168290637732233rm_a_b] :
              ( ( P @ Y4 )
             => ( Y4 = X4 ) ) )
     => ( ( P @ A4 )
       => ( ( the_Pr9157697819281395472rm_a_b @ P )
          = A4 ) ) ) ).

% the1_equality
thf(fact_891_the1__equality,axiom,
    ! [P: produc8451535801715972489rm_c_d > $o,A4: produc8451535801715972489rm_c_d] :
      ( ? [X4: produc8451535801715972489rm_c_d] :
          ( ( P @ X4 )
          & ! [Y4: produc8451535801715972489rm_c_d] :
              ( ( P @ Y4 )
             => ( Y4 = X4 ) ) )
     => ( ( P @ A4 )
       => ( ( the_Pr2724693293504859920rm_c_d @ P )
          = A4 ) ) ) ).

% the1_equality
thf(fact_892_surjD,axiom,
    ! [F: nat > nat,Y3: nat] :
      ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ? [X3: nat] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_893_surjD,axiom,
    ! [F: nat > set_nat,Y3: set_nat] :
      ( ( ( image_nat_set_nat @ F @ top_top_set_nat )
        = top_top_set_set_nat )
     => ? [X3: nat] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_894_surjD,axiom,
    ! [F: set_nat > nat,Y3: nat] :
      ( ( ( image_set_nat_nat @ F @ top_top_set_set_nat )
        = top_top_set_nat )
     => ? [X3: set_nat] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_895_surjD,axiom,
    ! [F: nat > sum_sum_nat_nat,Y3: sum_sum_nat_nat] :
      ( ( ( image_678696785212003926at_nat @ F @ top_top_set_nat )
        = top_to6661820994512907621at_nat )
     => ? [X3: nat] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_896_surjD,axiom,
    ! [F: sum_sum_nat_nat > nat,Y3: nat] :
      ( ( ( image_1320371278474632150at_nat @ F @ top_to6661820994512907621at_nat )
        = top_top_set_nat )
     => ? [X3: sum_sum_nat_nat] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_897_surjD,axiom,
    ! [F: set_nat > set_nat,Y3: set_nat] :
      ( ( ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat )
        = top_top_set_set_nat )
     => ? [X3: set_nat] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_898_surjD,axiom,
    ! [F: ( nat > $o ) > set_nat,Y3: set_nat] :
      ( ( ( image_nat_o_set_nat @ F @ top_top_set_nat_o2 )
        = top_top_set_set_nat )
     => ? [X3: nat > $o] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_899_surjD,axiom,
    ! [F: sum_sum_nat_nat > set_nat,Y3: set_nat] :
      ( ( ( image_5891067694031531916et_nat @ F @ top_to6661820994512907621at_nat )
        = top_top_set_set_nat )
     => ? [X3: sum_sum_nat_nat] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_900_surjD,axiom,
    ! [F: set_nat > sum_sum_nat_nat,Y3: sum_sum_nat_nat] :
      ( ( ( image_431631725857108236at_nat @ F @ top_top_set_set_nat )
        = top_to6661820994512907621at_nat )
     => ? [X3: set_nat] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_901_surjD,axiom,
    ! [F: sum_sum_nat_nat > sum_sum_nat_nat,Y3: sum_sum_nat_nat] :
      ( ( ( image_7938014925024745989at_nat @ F @ top_to6661820994512907621at_nat )
        = top_to6661820994512907621at_nat )
     => ? [X3: sum_sum_nat_nat] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_902_surjE,axiom,
    ! [F: nat > nat,Y3: nat] :
      ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ~ ! [X3: nat] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_903_surjE,axiom,
    ! [F: nat > set_nat,Y3: set_nat] :
      ( ( ( image_nat_set_nat @ F @ top_top_set_nat )
        = top_top_set_set_nat )
     => ~ ! [X3: nat] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_904_surjE,axiom,
    ! [F: set_nat > nat,Y3: nat] :
      ( ( ( image_set_nat_nat @ F @ top_top_set_set_nat )
        = top_top_set_nat )
     => ~ ! [X3: set_nat] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_905_surjE,axiom,
    ! [F: nat > sum_sum_nat_nat,Y3: sum_sum_nat_nat] :
      ( ( ( image_678696785212003926at_nat @ F @ top_top_set_nat )
        = top_to6661820994512907621at_nat )
     => ~ ! [X3: nat] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_906_surjE,axiom,
    ! [F: sum_sum_nat_nat > nat,Y3: nat] :
      ( ( ( image_1320371278474632150at_nat @ F @ top_to6661820994512907621at_nat )
        = top_top_set_nat )
     => ~ ! [X3: sum_sum_nat_nat] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_907_surjE,axiom,
    ! [F: set_nat > set_nat,Y3: set_nat] :
      ( ( ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat )
        = top_top_set_set_nat )
     => ~ ! [X3: set_nat] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_908_surjE,axiom,
    ! [F: ( nat > $o ) > set_nat,Y3: set_nat] :
      ( ( ( image_nat_o_set_nat @ F @ top_top_set_nat_o2 )
        = top_top_set_set_nat )
     => ~ ! [X3: nat > $o] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_909_surjE,axiom,
    ! [F: sum_sum_nat_nat > set_nat,Y3: set_nat] :
      ( ( ( image_5891067694031531916et_nat @ F @ top_to6661820994512907621at_nat )
        = top_top_set_set_nat )
     => ~ ! [X3: sum_sum_nat_nat] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_910_surjE,axiom,
    ! [F: set_nat > sum_sum_nat_nat,Y3: sum_sum_nat_nat] :
      ( ( ( image_431631725857108236at_nat @ F @ top_top_set_set_nat )
        = top_to6661820994512907621at_nat )
     => ~ ! [X3: set_nat] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_911_surjE,axiom,
    ! [F: sum_sum_nat_nat > sum_sum_nat_nat,Y3: sum_sum_nat_nat] :
      ( ( ( image_7938014925024745989at_nat @ F @ top_to6661820994512907621at_nat )
        = top_to6661820994512907621at_nat )
     => ~ ! [X3: sum_sum_nat_nat] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_912_preorder__on__def,axiom,
    ( order_9011181989990678936at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
          ( ( refl_o7122697186039111552at_nat @ A2 @ R2 )
          & ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R2 ) ) ) ) ).

% preorder_on_def
thf(fact_913_preorder__on__def,axiom,
    ( order_4276180158213025192at_nat
    = ( ^ [A2: set_Sum_sum_nat_nat,R2: set_Pr1076074158304066111at_nat] :
          ( ( refl_o9084340785372409280at_nat @ A2 @ R2 )
          & ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R2 ) ) ) ) ).

% preorder_on_def
thf(fact_914_preorder__on__def,axiom,
    ( order_4261761335103714223et_nat
    = ( ^ [A2: set_set_nat,R2: set_Pr5488025237498180813et_nat] :
          ( ( refl_on_set_nat @ A2 @ R2 )
          & ( trans_on_set_nat @ top_top_set_set_nat @ R2 ) ) ) ) ).

% preorder_on_def
thf(fact_915_preorder__on__def,axiom,
    ( order_7309662683583697264rm_a_b
    = ( ^ [A2: set_te1997870499059199101rm_a_b,R2: set_Pr8788715879341982671rm_a_b] :
          ( ( refl_o3744124953998825864rm_a_b @ A2 @ R2 )
          & ( trans_5020182533697283623rm_a_b @ top_to220653634128270125rm_a_b @ R2 ) ) ) ) ).

% preorder_on_def
thf(fact_916_preorder__on__def,axiom,
    ( order_876658157807161712rm_c_d
    = ( ^ [A2: set_te1821851435260963965rm_c_d,R2: set_Pr6132369733443093455rm_c_d] :
          ( ( refl_o6534492465077066120rm_c_d @ A2 @ R2 )
          & ( trans_7810550044775523879rm_c_d @ top_to44634570330034989rm_c_d @ R2 ) ) ) ) ).

% preorder_on_def
thf(fact_917_preorder__on__def,axiom,
    ( order_4861654808422542329on_nat
    = ( ^ [A2: set_nat,R2: set_Pr1261947904930325089at_nat] :
          ( ( refl_on_nat @ A2 @ R2 )
          & ( trans_on_nat @ top_top_set_nat @ R2 ) ) ) ) ).

% preorder_on_def
thf(fact_918_equivE,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R )
     => ~ ( ( refl_o7122697186039111552at_nat @ A3 @ R )
         => ( ( sym_on6289306843426450082at_nat @ top_to4669805908274784177at_nat @ R )
           => ~ ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R ) ) ) ) ).

% equivE
thf(fact_919_equivE,axiom,
    ! [A3: set_Sum_sum_nat_nat,R: set_Pr1076074158304066111at_nat] :
      ( ( equiv_7517305544406289097at_nat @ A3 @ R )
     => ~ ( ( refl_o9084340785372409280at_nat @ A3 @ R )
         => ( ( sym_on5063332426876622878at_nat @ top_to6661820994512907621at_nat @ R )
           => ~ ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R ) ) ) ) ).

% equivE
thf(fact_920_equivE,axiom,
    ! [A3: set_set_nat,R: set_Pr5488025237498180813et_nat] :
      ( ( equiv_equiv_set_nat @ A3 @ R )
     => ~ ( ( refl_on_set_nat @ A3 @ R )
         => ( ( sym_on_set_nat @ top_top_set_set_nat @ R )
           => ~ ( trans_on_set_nat @ top_top_set_set_nat @ R ) ) ) ) ).

% equivE
thf(fact_921_equivE,axiom,
    ! [A3: set_te1997870499059199101rm_a_b,R: set_Pr8788715879341982671rm_a_b] :
      ( ( equiv_214858368621111697rm_a_b @ A3 @ R )
     => ~ ( ( refl_o3744124953998825864rm_a_b @ A3 @ R )
         => ( ( sym_on724379605478305254rm_a_b @ top_to220653634128270125rm_a_b @ R )
           => ~ ( trans_5020182533697283623rm_a_b @ top_to220653634128270125rm_a_b @ R ) ) ) ) ).

% equivE
thf(fact_922_equivE,axiom,
    ! [A3: set_te1821851435260963965rm_c_d,R: set_Pr6132369733443093455rm_c_d] :
      ( ( equiv_3005225879699351953rm_c_d @ A3 @ R )
     => ~ ( ( refl_o6534492465077066120rm_c_d @ A3 @ R )
         => ( ( sym_on3514747116556545510rm_c_d @ top_to44634570330034989rm_c_d @ R )
           => ~ ( trans_7810550044775523879rm_c_d @ top_to44634570330034989rm_c_d @ R ) ) ) ) ).

% equivE
thf(fact_923_equivE,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( equiv_equiv_nat @ A3 @ R )
     => ~ ( ( refl_on_nat @ A3 @ R )
         => ( ( sym_on_nat @ top_top_set_nat @ R )
           => ~ ( trans_on_nat @ top_top_set_nat @ R ) ) ) ) ).

% equivE
thf(fact_924_equivI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R )
     => ( ( sym_on6289306843426450082at_nat @ top_to4669805908274784177at_nat @ R )
       => ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R )
         => ( equiv_5407799809235660215at_nat @ A3 @ R ) ) ) ) ).

% equivI
thf(fact_925_equivI,axiom,
    ! [A3: set_Sum_sum_nat_nat,R: set_Pr1076074158304066111at_nat] :
      ( ( refl_o9084340785372409280at_nat @ A3 @ R )
     => ( ( sym_on5063332426876622878at_nat @ top_to6661820994512907621at_nat @ R )
       => ( ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R )
         => ( equiv_7517305544406289097at_nat @ A3 @ R ) ) ) ) ).

% equivI
thf(fact_926_equivI,axiom,
    ! [A3: set_set_nat,R: set_Pr5488025237498180813et_nat] :
      ( ( refl_on_set_nat @ A3 @ R )
     => ( ( sym_on_set_nat @ top_top_set_set_nat @ R )
       => ( ( trans_on_set_nat @ top_top_set_set_nat @ R )
         => ( equiv_equiv_set_nat @ A3 @ R ) ) ) ) ).

% equivI
thf(fact_927_equivI,axiom,
    ! [A3: set_te1997870499059199101rm_a_b,R: set_Pr8788715879341982671rm_a_b] :
      ( ( refl_o3744124953998825864rm_a_b @ A3 @ R )
     => ( ( sym_on724379605478305254rm_a_b @ top_to220653634128270125rm_a_b @ R )
       => ( ( trans_5020182533697283623rm_a_b @ top_to220653634128270125rm_a_b @ R )
         => ( equiv_214858368621111697rm_a_b @ A3 @ R ) ) ) ) ).

% equivI
thf(fact_928_equivI,axiom,
    ! [A3: set_te1821851435260963965rm_c_d,R: set_Pr6132369733443093455rm_c_d] :
      ( ( refl_o6534492465077066120rm_c_d @ A3 @ R )
     => ( ( sym_on3514747116556545510rm_c_d @ top_to44634570330034989rm_c_d @ R )
       => ( ( trans_7810550044775523879rm_c_d @ top_to44634570330034989rm_c_d @ R )
         => ( equiv_3005225879699351953rm_c_d @ A3 @ R ) ) ) ) ).

% equivI
thf(fact_929_equivI,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( refl_on_nat @ A3 @ R )
     => ( ( sym_on_nat @ top_top_set_nat @ R )
       => ( ( trans_on_nat @ top_top_set_nat @ R )
         => ( equiv_equiv_nat @ A3 @ R ) ) ) ) ).

% equivI
thf(fact_930_equiv__def,axiom,
    ( equiv_5407799809235660215at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
          ( ( refl_o7122697186039111552at_nat @ A2 @ R2 )
          & ( sym_on6289306843426450082at_nat @ top_to4669805908274784177at_nat @ R2 )
          & ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R2 ) ) ) ) ).

% equiv_def
thf(fact_931_equiv__def,axiom,
    ( equiv_7517305544406289097at_nat
    = ( ^ [A2: set_Sum_sum_nat_nat,R2: set_Pr1076074158304066111at_nat] :
          ( ( refl_o9084340785372409280at_nat @ A2 @ R2 )
          & ( sym_on5063332426876622878at_nat @ top_to6661820994512907621at_nat @ R2 )
          & ( trans_2659081265866093919at_nat @ top_to6661820994512907621at_nat @ R2 ) ) ) ) ).

% equiv_def
thf(fact_932_equiv__def,axiom,
    ( equiv_equiv_set_nat
    = ( ^ [A2: set_set_nat,R2: set_Pr5488025237498180813et_nat] :
          ( ( refl_on_set_nat @ A2 @ R2 )
          & ( sym_on_set_nat @ top_top_set_set_nat @ R2 )
          & ( trans_on_set_nat @ top_top_set_set_nat @ R2 ) ) ) ) ).

% equiv_def
thf(fact_933_equiv__def,axiom,
    ( equiv_214858368621111697rm_a_b
    = ( ^ [A2: set_te1997870499059199101rm_a_b,R2: set_Pr8788715879341982671rm_a_b] :
          ( ( refl_o3744124953998825864rm_a_b @ A2 @ R2 )
          & ( sym_on724379605478305254rm_a_b @ top_to220653634128270125rm_a_b @ R2 )
          & ( trans_5020182533697283623rm_a_b @ top_to220653634128270125rm_a_b @ R2 ) ) ) ) ).

% equiv_def
thf(fact_934_equiv__def,axiom,
    ( equiv_3005225879699351953rm_c_d
    = ( ^ [A2: set_te1821851435260963965rm_c_d,R2: set_Pr6132369733443093455rm_c_d] :
          ( ( refl_o6534492465077066120rm_c_d @ A2 @ R2 )
          & ( sym_on3514747116556545510rm_c_d @ top_to44634570330034989rm_c_d @ R2 )
          & ( trans_7810550044775523879rm_c_d @ top_to44634570330034989rm_c_d @ R2 ) ) ) ) ).

% equiv_def
thf(fact_935_equiv__def,axiom,
    ( equiv_equiv_nat
    = ( ^ [A2: set_nat,R2: set_Pr1261947904930325089at_nat] :
          ( ( refl_on_nat @ A2 @ R2 )
          & ( sym_on_nat @ top_top_set_nat @ R2 )
          & ( trans_on_nat @ top_top_set_nat @ R2 ) ) ) ) ).

% equiv_def
thf(fact_936_aboveS__def,axiom,
    ( order_8635148640901146612rm_c_d
    = ( ^ [R2: set_Pr8929823917597237063rm_c_d,A: produc8451535801715972489rm_c_d] :
          ( collec8193319511663418484rm_c_d
          @ ^ [B: produc8451535801715972489rm_c_d] :
              ( ( B != A )
              & ( member3585784829053206928rm_c_d @ ( produc4766828187215050455rm_c_d @ A @ B ) @ R2 ) ) ) ) ) ).

% aboveS_def
thf(fact_937_aboveS__def,axiom,
    ( order_4940263126897443436at_nat
    = ( ^ [R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B: product_prod_nat_nat] :
              ( ( B != A )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% aboveS_def
thf(fact_938_aboveS__def,axiom,
    ( order_aboveS_set_nat
    = ( ^ [R2: set_Pr5488025237498180813et_nat,A: set_nat] :
          ( collect_set_nat
          @ ^ [B: set_nat] :
              ( ( B != A )
              & ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% aboveS_def
thf(fact_939_aboveS__def,axiom,
    ( order_2507179780302717801rm_c_d
    = ( ^ [R2: set_Pr5080540287643815273rm_c_d,A: term_c_d] :
          ( collect_term_c_d
          @ ^ [B: term_c_d] :
              ( ( B != A )
              & ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A @ B ) @ R2 ) ) ) ) ) ).

% aboveS_def
thf(fact_940_aboveS__def,axiom,
    ( order_8082090935641612265rm_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
          ( collect_term_a_b
          @ ^ [B: term_a_b] :
              ( ( B != A )
              & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 ) ) ) ) ) ).

% aboveS_def
thf(fact_941_aboveS__def,axiom,
    ( order_aboveS_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat,A: nat] :
          ( collect_nat
          @ ^ [B: nat] :
              ( ( B != A )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% aboveS_def
thf(fact_942_irrefl__on__lex__prod,axiom,
    ! [A3: set_term_c_d,R_A: set_Pr5080540287643815273rm_c_d,B6: set_term_c_d,R_B: set_Pr5080540287643815273rm_c_d] :
      ( ( irrefl_on_term_c_d @ A3 @ R_A )
     => ( ( irrefl_on_term_c_d @ B6 @ R_B )
       => ( irrefl1571329246850119167rm_c_d
          @ ( produc1682081246768872562rm_c_d @ A3
            @ ^ [Uu: term_c_d] : B6 )
          @ ( lex_pr5892420715117795161rm_c_d @ R_A @ R_B ) ) ) ) ).

% irrefl_on_lex_prod
thf(fact_943_irrefl__on__lex__prod,axiom,
    ! [A3: set_term_a_b,R_A: set_Pr4386577575007340137rm_a_b,B6: set_term_a_b,R_B: set_Pr4386577575007340137rm_a_b] :
      ( ( irrefl_on_term_a_b @ A3 @ R_A )
     => ( ( irrefl_on_term_a_b @ B6 @ R_B )
       => ( irrefl6620156600162922239rm_a_b
          @ ( produc1953217797448512882rm_a_b @ A3
            @ ^ [Uu: term_a_b] : B6 )
          @ ( lex_pr6163557265797435481rm_a_b @ R_A @ R_B ) ) ) ) ).

% irrefl_on_lex_prod
thf(fact_944_irrefl__on__lex__prod,axiom,
    ! [A3: set_nat,R_A: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( irrefl_on_nat @ A3 @ R_A )
     => ( ( irrefl7408258699751485431at_nat @ B6 @ R_B )
       => ( irrefl290916472950276272at_nat
          @ ( produc1809337555817847783at_nat @ A3
            @ ^ [Uu: nat] : B6 )
          @ ( lex_pr8029265285556086080at_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_on_lex_prod
thf(fact_945_irrefl__on__lex__prod,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R_A: set_Pr8693737435421807431at_nat,B6: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( irrefl7408258699751485431at_nat @ A3 @ R_A )
     => ( ( irrefl_on_nat @ B6 @ R_B )
       => ( irrefl1416403456917577354at_nat
          @ ( produc7672662199629908489at_nat @ A3
            @ ^ [Uu: product_prod_nat_nat] : B6 )
          @ ( lex_pr4669217892513370978at_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_on_lex_prod
thf(fact_946_irrefl__on__lex__prod,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R_A: set_Pr8693737435421807431at_nat,B6: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( irrefl7408258699751485431at_nat @ A3 @ R_A )
     => ( ( irrefl7408258699751485431at_nat @ B6 @ R_B )
       => ( irrefl4089853883888266845at_nat
          @ ( produc2761391749766926216at_nat @ A3
            @ ^ [Uu: product_prod_nat_nat] : B6 )
          @ ( lex_pr8801849515957261039at_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_on_lex_prod
thf(fact_947_irrefl__on__lex__prod,axiom,
    ! [A3: set_term_c_d,R_A: set_Pr5080540287643815273rm_c_d,B6: set_term_a_b,R_B: set_Pr4386577575007340137rm_a_b] :
      ( ( irrefl_on_term_c_d @ A3 @ R_A )
     => ( ( irrefl_on_term_a_b @ B6 @ R_B )
       => ( irrefl2700559167967400575rm_a_b
          @ ( produc7256992402107767026rm_a_b @ A3
            @ ^ [Uu: term_c_d] : B6 )
          @ ( lex_pr2243959833601913817rm_a_b @ R_A @ R_B ) ) ) ) ).

% irrefl_on_lex_prod
thf(fact_948_irrefl__on__lex__prod,axiom,
    ! [A3: set_term_a_b,R_A: set_Pr4386577575007340137rm_a_b,B6: set_term_c_d,R_B: set_Pr5080540287643815273rm_c_d] :
      ( ( irrefl_on_term_a_b @ A3 @ R_A )
     => ( ( irrefl_on_term_c_d @ B6 @ R_B )
       => ( irrefl5490926679045640831rm_c_d
          @ ( produc5601678678964394226rm_c_d @ A3
            @ ^ [Uu: term_a_b] : B6 )
          @ ( lex_pr588646110458541017rm_c_d @ R_A @ R_B ) ) ) ) ).

% irrefl_on_lex_prod
thf(fact_949_irrefl__on__lex__prod,axiom,
    ! [A3: set_nat,R_A: set_Pr1261947904930325089at_nat,B6: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ A3 @ R_A )
     => ( ( irrefl_on_nat @ B6 @ R_B )
       => ( irrefl7408258699751485431at_nat
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : B6 )
          @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_on_lex_prod
thf(fact_950_top1I,axiom,
    ! [X5: nat] : ( top_top_nat_o @ X5 ) ).

% top1I
thf(fact_951_SigmaI,axiom,
    ! [A4: $o,A3: set_o,B2: $o,B6: $o > set_o] :
      ( ( member_o @ A4 @ A3 )
     => ( ( member_o @ B2 @ ( B6 @ A4 ) )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A4 @ B2 ) @ ( product_Sigma_o_o @ A3 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_952_SigmaI,axiom,
    ! [A4: $o,A3: set_o,B2: nat,B6: $o > set_nat] :
      ( ( member_o @ A4 @ A3 )
     => ( ( member_nat @ B2 @ ( B6 @ A4 ) )
       => ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A4 @ B2 ) @ ( product_Sigma_o_nat @ A3 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_953_SigmaI,axiom,
    ! [A4: nat,A3: set_nat,B2: $o,B6: nat > set_o] :
      ( ( member_nat @ A4 @ A3 )
     => ( ( member_o @ B2 @ ( B6 @ A4 ) )
       => ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A4 @ B2 ) @ ( product_Sigma_nat_o @ A3 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_954_SigmaI,axiom,
    ! [A4: nat,A3: set_nat,B2: nat,B6: nat > set_nat] :
      ( ( member_nat @ A4 @ A3 )
     => ( ( member_nat @ B2 @ ( B6 @ A4 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( produc457027306803732586at_nat @ A3 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_955_SigmaI,axiom,
    ! [A4: $o,A3: set_o,B2: set_nat,B6: $o > set_set_nat] :
      ( ( member_o @ A4 @ A3 )
     => ( ( member_set_nat @ B2 @ ( B6 @ A4 ) )
       => ( member4905127681128465508et_nat @ ( produc818605102413008839et_nat @ A4 @ B2 ) @ ( produc7056857006710681430et_nat @ A3 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_956_SigmaI,axiom,
    ! [A4: set_nat,A3: set_set_nat,B2: $o,B6: set_nat > set_o] :
      ( ( member_set_nat @ A4 @ A3 )
     => ( ( member_o @ B2 @ ( B6 @ A4 ) )
       => ( member1847065620493055006_nat_o @ ( produc7144525260524587961_nat_o @ A4 @ B2 ) @ ( produc4159405127967484744_nat_o @ A3 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_957_SigmaI,axiom,
    ! [A4: set_nat,A3: set_set_nat,B2: nat,B6: set_nat > set_nat] :
      ( ( member_set_nat @ A4 @ A3 )
     => ( ( member_nat @ B2 @ ( B6 @ A4 ) )
       => ( member8873588116083876704at_nat @ ( produc641871753055645167at_nat @ A4 @ B2 ) @ ( produc5318310618559172640at_nat @ A3 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_958_SigmaI,axiom,
    ! [A4: nat,A3: set_nat,B2: set_nat,B6: nat > set_set_nat] :
      ( ( member_nat @ A4 @ A3 )
     => ( ( member_set_nat @ B2 @ ( B6 @ A4 ) )
       => ( member3782324328723991648et_nat @ ( produc4207506657711014383et_nat @ A4 @ B2 ) @ ( produc8883945523214541856et_nat @ A3 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_959_SigmaI,axiom,
    ! [A4: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: $o,B6: product_prod_nat_nat > set_o] :
      ( ( member8440522571783428010at_nat @ A4 @ A3 )
     => ( ( member_o @ B2 @ ( B6 @ A4 ) )
       => ( member2200362271989506215_nat_o @ ( produc855456346497908590_nat_o @ A4 @ B2 ) @ ( produc203973861852599583_nat_o @ A3 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_960_SigmaI,axiom,
    ! [A4: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: nat,B6: product_prod_nat_nat > set_nat] :
      ( ( member8440522571783428010at_nat @ A4 @ A3 )
     => ( ( member_nat @ B2 @ ( B6 @ A4 ) )
       => ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A4 @ B2 ) @ ( produc7672662199629908489at_nat @ A3 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_961_mem__Sigma__iff,axiom,
    ! [A4: $o,B2: $o,A3: set_o,B6: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A4 @ B2 ) @ ( product_Sigma_o_o @ A3 @ B6 ) )
      = ( ( member_o @ A4 @ A3 )
        & ( member_o @ B2 @ ( B6 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_962_mem__Sigma__iff,axiom,
    ! [A4: $o,B2: nat,A3: set_o,B6: $o > set_nat] :
      ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A4 @ B2 ) @ ( product_Sigma_o_nat @ A3 @ B6 ) )
      = ( ( member_o @ A4 @ A3 )
        & ( member_nat @ B2 @ ( B6 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_963_mem__Sigma__iff,axiom,
    ! [A4: nat,B2: $o,A3: set_nat,B6: nat > set_o] :
      ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A4 @ B2 ) @ ( product_Sigma_nat_o @ A3 @ B6 ) )
      = ( ( member_nat @ A4 @ A3 )
        & ( member_o @ B2 @ ( B6 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_964_mem__Sigma__iff,axiom,
    ! [A4: nat,B2: nat,A3: set_nat,B6: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( produc457027306803732586at_nat @ A3 @ B6 ) )
      = ( ( member_nat @ A4 @ A3 )
        & ( member_nat @ B2 @ ( B6 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_965_mem__Sigma__iff,axiom,
    ! [A4: $o,B2: set_nat,A3: set_o,B6: $o > set_set_nat] :
      ( ( member4905127681128465508et_nat @ ( produc818605102413008839et_nat @ A4 @ B2 ) @ ( produc7056857006710681430et_nat @ A3 @ B6 ) )
      = ( ( member_o @ A4 @ A3 )
        & ( member_set_nat @ B2 @ ( B6 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_966_mem__Sigma__iff,axiom,
    ! [A4: set_nat,B2: $o,A3: set_set_nat,B6: set_nat > set_o] :
      ( ( member1847065620493055006_nat_o @ ( produc7144525260524587961_nat_o @ A4 @ B2 ) @ ( produc4159405127967484744_nat_o @ A3 @ B6 ) )
      = ( ( member_set_nat @ A4 @ A3 )
        & ( member_o @ B2 @ ( B6 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_967_mem__Sigma__iff,axiom,
    ! [A4: set_nat,B2: nat,A3: set_set_nat,B6: set_nat > set_nat] :
      ( ( member8873588116083876704at_nat @ ( produc641871753055645167at_nat @ A4 @ B2 ) @ ( produc5318310618559172640at_nat @ A3 @ B6 ) )
      = ( ( member_set_nat @ A4 @ A3 )
        & ( member_nat @ B2 @ ( B6 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_968_mem__Sigma__iff,axiom,
    ! [A4: nat,B2: set_nat,A3: set_nat,B6: nat > set_set_nat] :
      ( ( member3782324328723991648et_nat @ ( produc4207506657711014383et_nat @ A4 @ B2 ) @ ( produc8883945523214541856et_nat @ A3 @ B6 ) )
      = ( ( member_nat @ A4 @ A3 )
        & ( member_set_nat @ B2 @ ( B6 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_969_mem__Sigma__iff,axiom,
    ! [A4: product_prod_nat_nat,B2: $o,A3: set_Pr1261947904930325089at_nat,B6: product_prod_nat_nat > set_o] :
      ( ( member2200362271989506215_nat_o @ ( produc855456346497908590_nat_o @ A4 @ B2 ) @ ( produc203973861852599583_nat_o @ A3 @ B6 ) )
      = ( ( member8440522571783428010at_nat @ A4 @ A3 )
        & ( member_o @ B2 @ ( B6 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_970_mem__Sigma__iff,axiom,
    ! [A4: product_prod_nat_nat,B2: nat,A3: set_Pr1261947904930325089at_nat,B6: product_prod_nat_nat > set_nat] :
      ( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A4 @ B2 ) @ ( produc7672662199629908489at_nat @ A3 @ B6 ) )
      = ( ( member8440522571783428010at_nat @ A4 @ A3 )
        & ( member_nat @ B2 @ ( B6 @ A4 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_971_sym__on__converse,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( sym_on6289306843426450082at_nat @ A3 @ ( conver5251360641855610363at_nat @ R ) )
      = ( sym_on6289306843426450082at_nat @ A3 @ R ) ) ).

% sym_on_converse
thf(fact_972_sym__on__converse,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ( sym_on_term_c_d @ A3 @ ( conver7337423426258204773rm_c_d @ R ) )
      = ( sym_on_term_c_d @ A3 @ R ) ) ).

% sym_on_converse
thf(fact_973_sym__on__converse,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( sym_on_term_a_b @ A3 @ ( conver7608559976937845093rm_a_b @ R ) )
      = ( sym_on_term_a_b @ A3 @ R ) ) ).

% sym_on_converse
thf(fact_974_sym__on__converse,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( sym_on_nat @ A3 @ ( converse_nat_nat @ R ) )
      = ( sym_on_nat @ A3 @ R ) ) ).

% sym_on_converse
thf(fact_975_preorder__on__converse,axiom,
    ! [A3: set_term_c_d,R: set_Pr5080540287643815273rm_c_d] :
      ( ( order_1478562143648067901rm_c_d @ A3 @ ( conver7337423426258204773rm_c_d @ R ) )
      = ( order_1478562143648067901rm_c_d @ A3 @ R ) ) ).

% preorder_on_converse
thf(fact_976_preorder__on__converse,axiom,
    ! [A3: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( order_7053473298986962365rm_a_b @ A3 @ ( conver7608559976937845093rm_a_b @ R ) )
      = ( order_7053473298986962365rm_a_b @ A3 @ R ) ) ).

% preorder_on_converse
thf(fact_977_preorder__on__converse,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( order_4861654808422542329on_nat @ A3 @ ( converse_nat_nat @ R ) )
      = ( order_4861654808422542329on_nat @ A3 @ R ) ) ).

% preorder_on_converse
thf(fact_978_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [A: nat,B: nat] :
              ( ( P @ A )
              & ( Q @ B ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_979_Collect__case__prod,axiom,
    ! [P: term_a_b > $o,Q: term_c_d > $o] :
      ( ( collec8193319511663418484rm_c_d
        @ ( produc3147824153920350274_c_d_o
          @ ^ [A: term_a_b,B: term_c_d] :
              ( ( P @ A )
              & ( Q @ B ) ) ) )
      = ( produc5601678678964394226rm_c_d @ ( collect_term_a_b @ P )
        @ ^ [Uu: term_a_b] : ( collect_term_c_d @ Q ) ) ) ).

% Collect_case_prod
thf(fact_980_Collect__case__prod,axiom,
    ! [P: term_c_d > $o,Q: term_a_b > $o] :
      ( ( collec5402952000585178228rm_a_b
        @ ( produc327966649874835522_a_b_o
          @ ^ [A: term_c_d,B: term_a_b] :
              ( ( P @ A )
              & ( Q @ B ) ) ) )
      = ( produc7256992402107767026rm_a_b @ ( collect_term_c_d @ P )
        @ ^ [Uu: term_c_d] : ( collect_term_a_b @ Q ) ) ) ).

% Collect_case_prod
thf(fact_981_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: set_nat > $o] :
      ( ( collec6652498225584317346et_nat
        @ ( produc6720110361610378900_nat_o
          @ ^ [A: nat,B: set_nat] :
              ( ( P @ A )
              & ( Q @ B ) ) ) )
      = ( produc8883945523214541856et_nat @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collect_set_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_982_Collect__case__prod,axiom,
    ! [P: set_nat > $o,Q: nat > $o] :
      ( ( collec2520389976089426594at_nat
        @ ( produc8429754692982665876_nat_o
          @ ^ [A: set_nat,B: nat] :
              ( ( P @ A )
              & ( Q @ B ) ) ) )
      = ( produc5318310618559172640at_nat @ ( collect_set_nat @ P )
        @ ^ [Uu: set_nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_983_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec5903703980526211963at_nat
        @ ( produc5864757623865647827_nat_o
          @ ^ [A: nat,B: product_prod_nat_nat] :
              ( ( P @ A )
              & ( Q @ B ) ) ) )
      = ( produc1809337555817847783at_nat @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_984_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: term_c_d > $o] :
      ( ( collec6661519183098615344rm_c_d
        @ ( produc4132180821779813254_c_d_o
          @ ^ [A: nat,B: term_c_d] :
              ( ( P @ A )
              & ( Q @ B ) ) ) )
      = ( produc2110156549022185774rm_c_d @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collect_term_c_d @ Q ) ) ) ).

% Collect_case_prod
thf(fact_985_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: term_a_b > $o] :
      ( ( collec7790749104215896752rm_a_b
        @ ( produc7236203327812376326_a_b_o
          @ ^ [A: nat,B: term_a_b] :
              ( ( P @ A )
              & ( Q @ B ) ) ) )
      = ( produc7685067704361080238rm_a_b @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collect_term_a_b @ Q ) ) ) ).

% Collect_case_prod
thf(fact_986_Collect__case__prod,axiom,
    ! [P: product_prod_nat_nat > $o,Q: nat > $o] :
      ( ( collec7029190964493513045at_nat
        @ ( produc8758269395934548017_nat_o
          @ ^ [A: product_prod_nat_nat,B: nat] :
              ( ( P @ A )
              & ( Q @ B ) ) ) )
      = ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P )
        @ ^ [Uu: product_prod_nat_nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_987_Collect__case__prod,axiom,
    ! [P: term_c_d > $o,Q: nat > $o] :
      ( ( collec4710900250797789872_d_nat
        @ ( produc8272582700005955334_nat_o
          @ ^ [A: term_c_d,B: nat] :
              ( ( P @ A )
              & ( Q @ B ) ) ) )
      = ( produc4081145061811738030_d_nat @ ( collect_term_c_d @ P )
        @ ^ [Uu: term_c_d] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_988_UNIV__Times__UNIV,axiom,
    ( ( produc457027306803732586at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% UNIV_Times_UNIV
thf(fact_989_UNIV__Times__UNIV,axiom,
    ( ( produc8883945523214541856et_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_set_nat )
    = top_to173688896754200935et_nat ) ).

% UNIV_Times_UNIV
thf(fact_990_UNIV__Times__UNIV,axiom,
    ( ( produc5318310618559172640at_nat @ top_top_set_set_nat
      @ ^ [Uu: set_nat] : top_top_set_nat )
    = top_to3374598108630517863at_nat ) ).

% UNIV_Times_UNIV
thf(fact_991_UNIV__Times__UNIV,axiom,
    ( ( produc1920824200941101209at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_to6661820994512907621at_nat )
    = top_to678699698364323680at_nat ) ).

% UNIV_Times_UNIV
thf(fact_992_UNIV__Times__UNIV,axiom,
    ( ( produc2562498694203729433at_nat @ top_to6661820994512907621at_nat
      @ ^ [Uu: sum_sum_nat_nat] : top_top_set_nat )
    = top_to8411399450232720992at_nat ) ).

% UNIV_Times_UNIV
thf(fact_993_UNIV__Times__UNIV,axiom,
    ( ( produc105401717395143638et_nat @ top_top_set_set_nat
      @ ^ [Uu: set_nat] : top_top_set_set_nat )
    = top_to3040159065575275037et_nat ) ).

% UNIV_Times_UNIV
thf(fact_994_UNIV__Times__UNIV,axiom,
    ( ( produc6761753240170360271et_nat @ top_to6661820994512907621at_nat
      @ ^ [Uu: sum_sum_nat_nat] : top_top_set_set_nat )
    = top_to6784681116597541910et_nat ) ).

% UNIV_Times_UNIV
thf(fact_995_UNIV__Times__UNIV,axiom,
    ( ( produc1302317271995936591at_nat @ top_top_set_set_nat
      @ ^ [Uu: set_nat] : top_to6661820994512907621at_nat )
    = top_to1463716400325515030at_nat ) ).

% UNIV_Times_UNIV
thf(fact_996_UNIV__Times__UNIV,axiom,
    ( ( produc7256992402107767026rm_a_b @ top_top_set_term_c_d
      @ ^ [Uu: term_c_d] : top_top_set_term_a_b )
    = top_to1749258167063912249rm_a_b ) ).

% UNIV_Times_UNIV
thf(fact_997_UNIV__Times__UNIV,axiom,
    ( ( produc5601678678964394226rm_c_d @ top_top_set_term_a_b
      @ ^ [Uu: term_a_b] : top_top_set_term_c_d )
    = top_to1573239103265677113rm_c_d ) ).

% UNIV_Times_UNIV
thf(fact_998_top__conj_I2_J,axiom,
    ! [P: $o,X5: nat] :
      ( ( P
        & ( top_top_nat_o @ X5 ) )
      = P ) ).

% top_conj(2)
thf(fact_999_top__conj_I1_J,axiom,
    ! [X5: nat,P: $o] :
      ( ( ( top_top_nat_o @ X5 )
        & P )
      = P ) ).

% top_conj(1)
thf(fact_1000_sym__on__lex__prod,axiom,
    ! [A3: set_nat,R_A: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( sym_on_nat @ A3 @ R_A )
     => ( ( sym_on6289306843426450082at_nat @ B6 @ R_B )
       => ( sym_on4734970415044109381at_nat
          @ ( produc1809337555817847783at_nat @ A3
            @ ^ [Uu: nat] : B6 )
          @ ( lex_pr8029265285556086080at_nat @ R_A @ R_B ) ) ) ) ).

% sym_on_lex_prod
thf(fact_1001_sym__on__lex__prod,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R_A: set_Pr8693737435421807431at_nat,B6: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( sym_on6289306843426450082at_nat @ A3 @ R_A )
     => ( ( sym_on_nat @ B6 @ R_B )
       => ( sym_on5860457399011410463at_nat
          @ ( produc7672662199629908489at_nat @ A3
            @ ^ [Uu: product_prod_nat_nat] : B6 )
          @ ( lex_pr4669217892513370978at_nat @ R_A @ R_B ) ) ) ) ).

% sym_on_lex_prod
thf(fact_1002_sym__on__lex__prod,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R_A: set_Pr8693737435421807431at_nat,B6: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( sym_on6289306843426450082at_nat @ A3 @ R_A )
     => ( ( sym_on6289306843426450082at_nat @ B6 @ R_B )
       => ( sym_on6757258482634480776at_nat
          @ ( produc2761391749766926216at_nat @ A3
            @ ^ [Uu: product_prod_nat_nat] : B6 )
          @ ( lex_pr8801849515957261039at_nat @ R_A @ R_B ) ) ) ) ).

% sym_on_lex_prod
thf(fact_1003_sym__on__lex__prod,axiom,
    ! [A3: set_term_c_d,R_A: set_Pr5080540287643815273rm_c_d,B6: set_term_c_d,R_B: set_Pr5080540287643815273rm_c_d] :
      ( ( sym_on_term_c_d @ A3 @ R_A )
     => ( ( sym_on_term_c_d @ B6 @ R_B )
       => ( sym_on440321080665279914rm_c_d
          @ ( produc1682081246768872562rm_c_d @ A3
            @ ^ [Uu: term_c_d] : B6 )
          @ ( lex_pr5892420715117795161rm_c_d @ R_A @ R_B ) ) ) ) ).

% sym_on_lex_prod
thf(fact_1004_sym__on__lex__prod,axiom,
    ! [A3: set_term_a_b,R_A: set_Pr4386577575007340137rm_a_b,B6: set_term_a_b,R_B: set_Pr4386577575007340137rm_a_b] :
      ( ( sym_on_term_a_b @ A3 @ R_A )
     => ( ( sym_on_term_a_b @ B6 @ R_B )
       => ( sym_on5489148433978082986rm_a_b
          @ ( produc1953217797448512882rm_a_b @ A3
            @ ^ [Uu: term_a_b] : B6 )
          @ ( lex_pr6163557265797435481rm_a_b @ R_A @ R_B ) ) ) ) ).

% sym_on_lex_prod
thf(fact_1005_sym__on__lex__prod,axiom,
    ! [A3: set_term_c_d,R_A: set_Pr5080540287643815273rm_c_d,B6: set_term_a_b,R_B: set_Pr4386577575007340137rm_a_b] :
      ( ( sym_on_term_c_d @ A3 @ R_A )
     => ( ( sym_on_term_a_b @ B6 @ R_B )
       => ( sym_on1569551001782561322rm_a_b
          @ ( produc7256992402107767026rm_a_b @ A3
            @ ^ [Uu: term_c_d] : B6 )
          @ ( lex_pr2243959833601913817rm_a_b @ R_A @ R_B ) ) ) ) ).

% sym_on_lex_prod
thf(fact_1006_sym__on__lex__prod,axiom,
    ! [A3: set_term_a_b,R_A: set_Pr4386577575007340137rm_a_b,B6: set_term_c_d,R_B: set_Pr5080540287643815273rm_c_d] :
      ( ( sym_on_term_a_b @ A3 @ R_A )
     => ( ( sym_on_term_c_d @ B6 @ R_B )
       => ( sym_on4359918512860801578rm_c_d
          @ ( produc5601678678964394226rm_c_d @ A3
            @ ^ [Uu: term_a_b] : B6 )
          @ ( lex_pr588646110458541017rm_c_d @ R_A @ R_B ) ) ) ) ).

% sym_on_lex_prod
thf(fact_1007_sym__on__lex__prod,axiom,
    ! [A3: set_nat,R_A: set_Pr1261947904930325089at_nat,B6: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( sym_on_nat @ A3 @ R_A )
     => ( ( sym_on_nat @ B6 @ R_B )
       => ( sym_on6289306843426450082at_nat
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : B6 )
          @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% sym_on_lex_prod
thf(fact_1008_Times__eq__cancel2,axiom,
    ! [X5: term_a_b,C3: set_term_a_b,A3: set_term_c_d,B6: set_term_c_d] :
      ( ( member_term_a_b @ X5 @ C3 )
     => ( ( ( produc7256992402107767026rm_a_b @ A3
            @ ^ [Uu: term_c_d] : C3 )
          = ( produc7256992402107767026rm_a_b @ B6
            @ ^ [Uu: term_c_d] : C3 ) )
        = ( A3 = B6 ) ) ) ).

% Times_eq_cancel2
thf(fact_1009_Times__eq__cancel2,axiom,
    ! [X5: term_c_d,C3: set_term_c_d,A3: set_term_a_b,B6: set_term_a_b] :
      ( ( member_term_c_d @ X5 @ C3 )
     => ( ( ( produc5601678678964394226rm_c_d @ A3
            @ ^ [Uu: term_a_b] : C3 )
          = ( produc5601678678964394226rm_c_d @ B6
            @ ^ [Uu: term_a_b] : C3 ) )
        = ( A3 = B6 ) ) ) ).

% Times_eq_cancel2
thf(fact_1010_Times__eq__cancel2,axiom,
    ! [X5: nat,C3: set_nat,A3: set_nat,B6: set_nat] :
      ( ( member_nat @ X5 @ C3 )
     => ( ( ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : C3 )
          = ( produc457027306803732586at_nat @ B6
            @ ^ [Uu: nat] : C3 ) )
        = ( A3 = B6 ) ) ) ).

% Times_eq_cancel2
thf(fact_1011_Times__eq__cancel2,axiom,
    ! [X5: term_c_d,C3: set_term_c_d,A3: set_term_c_d,B6: set_term_c_d] :
      ( ( member_term_c_d @ X5 @ C3 )
     => ( ( ( produc1682081246768872562rm_c_d @ A3
            @ ^ [Uu: term_c_d] : C3 )
          = ( produc1682081246768872562rm_c_d @ B6
            @ ^ [Uu: term_c_d] : C3 ) )
        = ( A3 = B6 ) ) ) ).

% Times_eq_cancel2
thf(fact_1012_Times__eq__cancel2,axiom,
    ! [X5: term_a_b,C3: set_term_a_b,A3: set_term_a_b,B6: set_term_a_b] :
      ( ( member_term_a_b @ X5 @ C3 )
     => ( ( ( produc1953217797448512882rm_a_b @ A3
            @ ^ [Uu: term_a_b] : C3 )
          = ( produc1953217797448512882rm_a_b @ B6
            @ ^ [Uu: term_a_b] : C3 ) )
        = ( A3 = B6 ) ) ) ).

% Times_eq_cancel2
thf(fact_1013_Sigma__cong,axiom,
    ! [A3: set_term_c_d,B6: set_term_c_d,C3: term_c_d > set_term_a_b,D: term_c_d > set_term_a_b] :
      ( ( A3 = B6 )
     => ( ! [X3: term_c_d] :
            ( ( member_term_c_d @ X3 @ B6 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( produc7256992402107767026rm_a_b @ A3 @ C3 )
          = ( produc7256992402107767026rm_a_b @ B6 @ D ) ) ) ) ).

% Sigma_cong
thf(fact_1014_Sigma__cong,axiom,
    ! [A3: set_term_a_b,B6: set_term_a_b,C3: term_a_b > set_term_c_d,D: term_a_b > set_term_c_d] :
      ( ( A3 = B6 )
     => ( ! [X3: term_a_b] :
            ( ( member_term_a_b @ X3 @ B6 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( produc5601678678964394226rm_c_d @ A3 @ C3 )
          = ( produc5601678678964394226rm_c_d @ B6 @ D ) ) ) ) ).

% Sigma_cong
thf(fact_1015_Sigma__cong,axiom,
    ! [A3: set_nat,B6: set_nat,C3: nat > set_nat,D: nat > set_nat] :
      ( ( A3 = B6 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B6 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( produc457027306803732586at_nat @ A3 @ C3 )
          = ( produc457027306803732586at_nat @ B6 @ D ) ) ) ) ).

% Sigma_cong
thf(fact_1016_Sigma__cong,axiom,
    ! [A3: set_term_c_d,B6: set_term_c_d,C3: term_c_d > set_term_c_d,D: term_c_d > set_term_c_d] :
      ( ( A3 = B6 )
     => ( ! [X3: term_c_d] :
            ( ( member_term_c_d @ X3 @ B6 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( produc1682081246768872562rm_c_d @ A3 @ C3 )
          = ( produc1682081246768872562rm_c_d @ B6 @ D ) ) ) ) ).

% Sigma_cong
thf(fact_1017_Sigma__cong,axiom,
    ! [A3: set_term_a_b,B6: set_term_a_b,C3: term_a_b > set_term_a_b,D: term_a_b > set_term_a_b] :
      ( ( A3 = B6 )
     => ( ! [X3: term_a_b] :
            ( ( member_term_a_b @ X3 @ B6 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( produc1953217797448512882rm_a_b @ A3 @ C3 )
          = ( produc1953217797448512882rm_a_b @ B6 @ D ) ) ) ) ).

% Sigma_cong
thf(fact_1018_SigmaE,axiom,
    ! [C: product_prod_o_o,A3: set_o,B6: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ C @ ( product_Sigma_o_o @ A3 @ B6 ) )
     => ~ ! [X3: $o] :
            ( ( member_o @ X3 @ A3 )
           => ! [Y4: $o] :
                ( ( member_o @ Y4 @ ( B6 @ X3 ) )
               => ( C
                 != ( product_Pair_o_o @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1019_SigmaE,axiom,
    ! [C: product_prod_o_nat,A3: set_o,B6: $o > set_nat] :
      ( ( member2802428098988154798_o_nat @ C @ ( product_Sigma_o_nat @ A3 @ B6 ) )
     => ~ ! [X3: $o] :
            ( ( member_o @ X3 @ A3 )
           => ! [Y4: nat] :
                ( ( member_nat @ Y4 @ ( B6 @ X3 ) )
               => ( C
                 != ( product_Pair_o_nat @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1020_SigmaE,axiom,
    ! [C: product_prod_nat_o,A3: set_nat,B6: nat > set_o] :
      ( ( member6310962623043647828_nat_o @ C @ ( product_Sigma_nat_o @ A3 @ B6 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ! [Y4: $o] :
                ( ( member_o @ Y4 @ ( B6 @ X3 ) )
               => ( C
                 != ( product_Pair_nat_o @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1021_SigmaE,axiom,
    ! [C: product_prod_nat_nat,A3: set_nat,B6: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( produc457027306803732586at_nat @ A3 @ B6 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ! [Y4: nat] :
                ( ( member_nat @ Y4 @ ( B6 @ X3 ) )
               => ( C
                 != ( product_Pair_nat_nat @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1022_SigmaE,axiom,
    ! [C: produc573827679308723021et_nat,A3: set_o,B6: $o > set_set_nat] :
      ( ( member4905127681128465508et_nat @ C @ ( produc7056857006710681430et_nat @ A3 @ B6 ) )
     => ~ ! [X3: $o] :
            ( ( member_o @ X3 @ A3 )
           => ! [Y4: set_nat] :
                ( ( member_set_nat @ Y4 @ ( B6 @ X3 ) )
               => ( C
                 != ( produc818605102413008839et_nat @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1023_SigmaE,axiom,
    ! [C: produc6739137655528088327_nat_o,A3: set_set_nat,B6: set_nat > set_o] :
      ( ( member1847065620493055006_nat_o @ C @ ( produc4159405127967484744_nat_o @ A3 @ B6 ) )
     => ~ ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ! [Y4: $o] :
                ( ( member_o @ Y4 @ ( B6 @ X3 ) )
               => ( C
                 != ( produc7144525260524587961_nat_o @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1024_SigmaE,axiom,
    ! [C: produc7491599851749785783at_nat,A3: set_set_nat,B6: set_nat > set_nat] :
      ( ( member8873588116083876704at_nat @ C @ ( produc5318310618559172640at_nat @ A3 @ B6 ) )
     => ~ ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ! [Y4: nat] :
                ( ( member_nat @ Y4 @ ( B6 @ X3 ) )
               => ( C
                 != ( produc641871753055645167at_nat @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1025_SigmaE,axiom,
    ! [C: produc2400336064389900727et_nat,A3: set_nat,B6: nat > set_set_nat] :
      ( ( member3782324328723991648et_nat @ C @ ( produc8883945523214541856et_nat @ A3 @ B6 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ! [Y4: set_nat] :
                ( ( member_set_nat @ Y4 @ ( B6 @ X3 ) )
               => ( C
                 != ( produc4207506657711014383et_nat @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1026_SigmaE,axiom,
    ! [C: produc652263959640188542_nat_o,A3: set_Pr1261947904930325089at_nat,B6: product_prod_nat_nat > set_o] :
      ( ( member2200362271989506215_nat_o @ C @ ( produc203973861852599583_nat_o @ A3 @ B6 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A3 )
           => ! [Y4: $o] :
                ( ( member_o @ Y4 @ ( B6 @ X3 ) )
               => ( C
                 != ( produc855456346497908590_nat_o @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1027_SigmaE,axiom,
    ! [C: produc8373899037510109440at_nat,A3: set_Pr1261947904930325089at_nat,B6: product_prod_nat_nat > set_nat] :
      ( ( member3348759134392003351at_nat @ C @ ( produc7672662199629908489at_nat @ A3 @ B6 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A3 )
           => ! [Y4: nat] :
                ( ( member_nat @ Y4 @ ( B6 @ X3 ) )
               => ( C
                 != ( produc6350711070570205562at_nat @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1028_SigmaD1,axiom,
    ! [A4: term_c_d,B2: term_a_b,A3: set_term_c_d,B6: term_c_d > set_term_a_b] :
      ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) @ ( produc7256992402107767026rm_a_b @ A3 @ B6 ) )
     => ( member_term_c_d @ A4 @ A3 ) ) ).

% SigmaD1
thf(fact_1029_SigmaD1,axiom,
    ! [A4: term_a_b,B2: term_c_d,A3: set_term_a_b,B6: term_a_b > set_term_c_d] :
      ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) @ ( produc5601678678964394226rm_c_d @ A3 @ B6 ) )
     => ( member_term_a_b @ A4 @ A3 ) ) ).

% SigmaD1
thf(fact_1030_SigmaD1,axiom,
    ! [A4: nat,B2: nat,A3: set_nat,B6: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( produc457027306803732586at_nat @ A3 @ B6 ) )
     => ( member_nat @ A4 @ A3 ) ) ).

% SigmaD1
thf(fact_1031_SigmaD1,axiom,
    ! [A4: term_c_d,B2: term_c_d,A3: set_term_c_d,B6: term_c_d > set_term_c_d] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) @ ( produc1682081246768872562rm_c_d @ A3 @ B6 ) )
     => ( member_term_c_d @ A4 @ A3 ) ) ).

% SigmaD1
thf(fact_1032_SigmaD1,axiom,
    ! [A4: term_a_b,B2: term_a_b,A3: set_term_a_b,B6: term_a_b > set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) @ ( produc1953217797448512882rm_a_b @ A3 @ B6 ) )
     => ( member_term_a_b @ A4 @ A3 ) ) ).

% SigmaD1
thf(fact_1033_SigmaD2,axiom,
    ! [A4: term_c_d,B2: term_a_b,A3: set_term_c_d,B6: term_c_d > set_term_a_b] :
      ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) @ ( produc7256992402107767026rm_a_b @ A3 @ B6 ) )
     => ( member_term_a_b @ B2 @ ( B6 @ A4 ) ) ) ).

% SigmaD2
thf(fact_1034_SigmaD2,axiom,
    ! [A4: term_a_b,B2: term_c_d,A3: set_term_a_b,B6: term_a_b > set_term_c_d] :
      ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) @ ( produc5601678678964394226rm_c_d @ A3 @ B6 ) )
     => ( member_term_c_d @ B2 @ ( B6 @ A4 ) ) ) ).

% SigmaD2
thf(fact_1035_SigmaD2,axiom,
    ! [A4: nat,B2: nat,A3: set_nat,B6: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( produc457027306803732586at_nat @ A3 @ B6 ) )
     => ( member_nat @ B2 @ ( B6 @ A4 ) ) ) ).

% SigmaD2
thf(fact_1036_SigmaD2,axiom,
    ! [A4: term_c_d,B2: term_c_d,A3: set_term_c_d,B6: term_c_d > set_term_c_d] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) @ ( produc1682081246768872562rm_c_d @ A3 @ B6 ) )
     => ( member_term_c_d @ B2 @ ( B6 @ A4 ) ) ) ).

% SigmaD2
thf(fact_1037_SigmaD2,axiom,
    ! [A4: term_a_b,B2: term_a_b,A3: set_term_a_b,B6: term_a_b > set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) @ ( produc1953217797448512882rm_a_b @ A3 @ B6 ) )
     => ( member_term_a_b @ B2 @ ( B6 @ A4 ) ) ) ).

% SigmaD2
thf(fact_1038_SigmaE2,axiom,
    ! [A4: nat,B2: $o,A3: set_nat,B6: nat > set_o] :
      ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A4 @ B2 ) @ ( product_Sigma_nat_o @ A3 @ B6 ) )
     => ~ ( ( member_nat @ A4 @ A3 )
         => ~ ( member_o @ B2 @ ( B6 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_1039_SigmaE2,axiom,
    ! [A4: nat,B2: set_nat,A3: set_nat,B6: nat > set_set_nat] :
      ( ( member3782324328723991648et_nat @ ( produc4207506657711014383et_nat @ A4 @ B2 ) @ ( produc8883945523214541856et_nat @ A3 @ B6 ) )
     => ~ ( ( member_nat @ A4 @ A3 )
         => ~ ( member_set_nat @ B2 @ ( B6 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_1040_SigmaE2,axiom,
    ! [A4: term_c_d,B2: term_a_b,A3: set_term_c_d,B6: term_c_d > set_term_a_b] :
      ( ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ A4 @ B2 ) @ ( produc7256992402107767026rm_a_b @ A3 @ B6 ) )
     => ~ ( ( member_term_c_d @ A4 @ A3 )
         => ~ ( member_term_a_b @ B2 @ ( B6 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_1041_SigmaE2,axiom,
    ! [A4: term_a_b,B2: term_c_d,A3: set_term_a_b,B6: term_a_b > set_term_c_d] :
      ( ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ A4 @ B2 ) @ ( produc5601678678964394226rm_c_d @ A3 @ B6 ) )
     => ~ ( ( member_term_a_b @ A4 @ A3 )
         => ~ ( member_term_c_d @ B2 @ ( B6 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_1042_SigmaE2,axiom,
    ! [A4: nat,B2: nat,A3: set_nat,B6: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( produc457027306803732586at_nat @ A3 @ B6 ) )
     => ~ ( ( member_nat @ A4 @ A3 )
         => ~ ( member_nat @ B2 @ ( B6 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_1043_SigmaE2,axiom,
    ! [A4: term_c_d,B2: term_c_d,A3: set_term_c_d,B6: term_c_d > set_term_c_d] :
      ( ( member820888157712331442rm_c_d @ ( produc6749061249757032257rm_c_d @ A4 @ B2 ) @ ( produc1682081246768872562rm_c_d @ A3 @ B6 ) )
     => ~ ( ( member_term_c_d @ A4 @ A3 )
         => ~ ( member_term_c_d @ B2 @ ( B6 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_1044_SigmaE2,axiom,
    ! [A4: term_a_b,B2: term_a_b,A3: set_term_a_b,B6: term_a_b > set_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) @ ( produc1953217797448512882rm_a_b @ A3 @ B6 ) )
     => ~ ( ( member_term_a_b @ A4 @ A3 )
         => ~ ( member_term_a_b @ B2 @ ( B6 @ A4 ) ) ) ) ).

% SigmaE2
thf(fact_1045_Collect__case__prod__Sigma,axiom,
    ! [P: nat > $o,Q: nat > nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y: nat] :
              ( ( P @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P )
        @ ^ [X: nat] : ( collect_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1046_Collect__case__prod__Sigma,axiom,
    ! [P: term_a_b > $o,Q: term_a_b > term_c_d > $o] :
      ( ( collec8193319511663418484rm_c_d
        @ ( produc3147824153920350274_c_d_o
          @ ^ [X: term_a_b,Y: term_c_d] :
              ( ( P @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc5601678678964394226rm_c_d @ ( collect_term_a_b @ P )
        @ ^ [X: term_a_b] : ( collect_term_c_d @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1047_Collect__case__prod__Sigma,axiom,
    ! [P: term_c_d > $o,Q: term_c_d > term_a_b > $o] :
      ( ( collec5402952000585178228rm_a_b
        @ ( produc327966649874835522_a_b_o
          @ ^ [X: term_c_d,Y: term_a_b] :
              ( ( P @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc7256992402107767026rm_a_b @ ( collect_term_c_d @ P )
        @ ^ [X: term_c_d] : ( collect_term_a_b @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1048_sym__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( sym_on_nat @ top_top_set_nat @ R_A )
     => ( ( sym_on_nat @ top_top_set_nat @ R_B )
       => ( sym_on6289306843426450082at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% sym_lex_prod
thf(fact_1049_symI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R ) )
     => ( sym_on_nat @ top_top_set_nat @ R ) ) ).

% symI
thf(fact_1050_symE,axiom,
    ! [R: set_Pr1261947904930325089at_nat,B2: nat,A4: nat] :
      ( ( sym_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R ) ) ) ).

% symE
thf(fact_1051_symD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X5: nat,Y3: nat] :
      ( ( sym_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X5 ) @ R ) ) ) ).

% symD
thf(fact_1052_sym__conv__converse__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( sym_on_nat @ top_top_set_nat @ R )
      = ( ( converse_nat_nat @ R )
        = R ) ) ).

% sym_conv_converse_eq
thf(fact_1053_sym__inv__image,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( sym_on_nat @ top_top_set_nat @ R )
     => ( sym_on_nat @ top_top_set_nat @ ( inv_image_nat_nat @ R @ F ) ) ) ).

% sym_inv_image
thf(fact_1054_converse__Times,axiom,
    ! [A3: set_term_a_b,B6: set_term_c_d] :
      ( ( conver2033648821598950629rm_c_d
        @ ( produc5601678678964394226rm_c_d @ A3
          @ ^ [Uu: term_a_b] : B6 ) )
      = ( produc7256992402107767026rm_a_b @ B6
        @ ^ [Uu: term_c_d] : A3 ) ) ).

% converse_Times
thf(fact_1055_converse__Times,axiom,
    ! [A3: set_term_c_d,B6: set_term_a_b] :
      ( ( conver3688962544742323429rm_a_b
        @ ( produc7256992402107767026rm_a_b @ A3
          @ ^ [Uu: term_c_d] : B6 ) )
      = ( produc5601678678964394226rm_c_d @ B6
        @ ^ [Uu: term_a_b] : A3 ) ) ).

% converse_Times
thf(fact_1056_flip__pred,axiom,
    ! [A3: set_Pr4645549399426460137rm_c_d,R3: term_c_d > term_a_b > $o] :
      ( ( ord_le377442527001235849rm_c_d @ A3 @ ( collec8193319511663418484rm_c_d @ ( produc3147824153920350274_c_d_o @ ( conver1316451903988077855rm_a_b @ R3 ) ) ) )
     => ( ord_le553461590799470985rm_a_b
        @ ( image_3339315644087553349rm_a_b
          @ ( produc2919470507846847027rm_a_b
            @ ^ [X: term_a_b,Y: term_c_d] : ( produc3100600368241150913rm_a_b @ Y @ X ) )
          @ A3 )
        @ ( collec5402952000585178228rm_a_b @ ( produc327966649874835522_a_b_o @ R3 ) ) ) ) ).

% flip_pred
thf(fact_1057_flip__pred,axiom,
    ! [A3: set_Pr4821568463224695273rm_a_b,R3: term_a_b > term_c_d > $o] :
      ( ( ord_le553461590799470985rm_a_b @ A3 @ ( collec5402952000585178228rm_a_b @ ( produc327966649874835522_a_b_o @ ( conver8884510217699480863rm_c_d @ R3 ) ) ) )
     => ( ord_le377442527001235849rm_c_d
        @ ( image_6407262405037280581rm_c_d
          @ ( produc5370438780080785971rm_c_d
            @ ^ [X: term_c_d,Y: term_a_b] : ( produc1445286645097778113rm_c_d @ Y @ X ) )
          @ A3 )
        @ ( collec8193319511663418484rm_c_d @ ( produc3147824153920350274_c_d_o @ R3 ) ) ) ) ).

% flip_pred
thf(fact_1058_map__prod__surj,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_nat2 @ G @ top_top_set_nat )
          = top_top_set_nat )
       => ( ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ top_to4669805908274784177at_nat )
          = top_to4669805908274784177at_nat ) ) ) ).

% map_prod_surj
thf(fact_1059_equivp__equiv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( equiv_equiv_nat @ top_top_set_nat @ A3 )
      = ( equiv_equivp_nat
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A3 ) ) ) ).

% equivp_equiv
thf(fact_1060_dual__order_Orefl,axiom,
    ! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).

% dual_order.refl
thf(fact_1061_order__refl,axiom,
    ! [X5: nat] : ( ord_less_eq_nat @ X5 @ X5 ) ).

% order_refl
thf(fact_1062_converse__mono,axiom,
    ! [R: set_Pr4645549399426460137rm_c_d,S: set_Pr4645549399426460137rm_c_d] :
      ( ( ord_le553461590799470985rm_a_b @ ( conver2033648821598950629rm_c_d @ R ) @ ( conver2033648821598950629rm_c_d @ S ) )
      = ( ord_le377442527001235849rm_c_d @ R @ S ) ) ).

% converse_mono
thf(fact_1063_converse__mono,axiom,
    ! [R: set_Pr4821568463224695273rm_a_b,S: set_Pr4821568463224695273rm_a_b] :
      ( ( ord_le377442527001235849rm_c_d @ ( conver3688962544742323429rm_a_b @ R ) @ ( conver3688962544742323429rm_a_b @ S ) )
      = ( ord_le553461590799470985rm_a_b @ R @ S ) ) ).

% converse_mono
thf(fact_1064_subset__Collect__iff,axiom,
    ! [B6: set_nat,A3: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B6 @ A3 )
     => ( ( ord_less_eq_set_nat @ B6
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ B6 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1065_subset__CollectI,axiom,
    ! [B6: set_nat,A3: set_nat,Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B6 @ A3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B6 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ B6 )
                & ( Q @ X ) ) )
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1066_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_1067_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1068_Ball__Collect,axiom,
    ( ball_nat
    = ( ^ [A2: set_nat,P2: nat > $o] : ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ P2 ) ) ) ) ).

% Ball_Collect
thf(fact_1069_Collect__restrict,axiom,
    ! [X7: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_1070_prop__restrict,axiom,
    ! [X5: nat,Z5: set_nat,X7: set_nat,P: nat > $o] :
      ( ( member_nat @ X5 @ Z5 )
     => ( ( ord_less_eq_set_nat @ Z5
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_1071_Collect__subset,axiom,
    ! [A3: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_1072_order__antisym__conv,axiom,
    ! [Y3: nat,X5: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X5 )
     => ( ( ord_less_eq_nat @ X5 @ Y3 )
        = ( X5 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_1073_linorder__le__cases,axiom,
    ! [X5: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X5 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X5 ) ) ).

% linorder_le_cases
thf(fact_1074_ord__le__eq__subst,axiom,
    ! [A4: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A4 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1075_ord__eq__le__subst,axiom,
    ! [A4: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( A4
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1076_linorder__linear,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X5 ) ) ).

% linorder_linear
thf(fact_1077_order__eq__refl,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( X5 = Y3 )
     => ( ord_less_eq_nat @ X5 @ Y3 ) ) ).

% order_eq_refl
thf(fact_1078_order__subst2,axiom,
    ! [A4: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A4 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1079_order__subst1,axiom,
    ! [A4: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1080_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y7: nat,Z4: nat] : ( Y7 = Z4 ) )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ A @ B )
          & ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1081_antisym,axiom,
    ! [A4: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A4 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A4 )
       => ( A4 = B2 ) ) ) ).

% antisym
thf(fact_1082_dual__order_Otrans,axiom,
    ! [B2: nat,A4: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A4 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A4 ) ) ) ).

% dual_order.trans
thf(fact_1083_dual__order_Oantisym,axiom,
    ! [B2: nat,A4: nat] :
      ( ( ord_less_eq_nat @ B2 @ A4 )
     => ( ( ord_less_eq_nat @ A4 @ B2 )
       => ( A4 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_1084_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y7: nat,Z4: nat] : ( Y7 = Z4 ) )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ B @ A )
          & ( ord_less_eq_nat @ A @ B ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1085_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A4: nat,B2: nat] :
      ( ! [A6: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A6 @ B4 )
         => ( P @ A6 @ B4 ) )
     => ( ! [A6: nat,B4: nat] :
            ( ( P @ B4 @ A6 )
           => ( P @ A6 @ B4 ) )
       => ( P @ A4 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_1086_order__trans,axiom,
    ! [X5: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z )
       => ( ord_less_eq_nat @ X5 @ Z ) ) ) ).

% order_trans
thf(fact_1087_order_Otrans,axiom,
    ! [A4: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A4 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A4 @ C ) ) ) ).

% order.trans
thf(fact_1088_order__antisym,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X5 )
       => ( X5 = Y3 ) ) ) ).

% order_antisym
thf(fact_1089_ord__le__eq__trans,axiom,
    ! [A4: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A4 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A4 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1090_ord__eq__le__trans,axiom,
    ! [A4: nat,B2: nat,C: nat] :
      ( ( A4 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A4 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1091_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y7: nat,Z4: nat] : ( Y7 = Z4 ) )
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1092_le__cases3,axiom,
    ! [X5: nat,Y3: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X5 @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X5 )
         => ~ ( ord_less_eq_nat @ X5 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X5 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X5 ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X5 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X5 )
                 => ~ ( ord_less_eq_nat @ X5 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1093_nle__le,axiom,
    ! [A4: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A4 @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A4 )
        & ( B2 != A4 ) ) ) ).

% nle_le
thf(fact_1094_top__greatest,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).

% top_greatest
thf(fact_1095_top_Oextremum__unique,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
      = ( A4 = top_top_set_nat ) ) ).

% top.extremum_unique
thf(fact_1096_top_Oextremum__uniqueI,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
     => ( A4 = top_top_set_nat ) ) ).

% top.extremum_uniqueI
thf(fact_1097_subset__UNIV,axiom,
    ! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_1098_converse__subset__swap,axiom,
    ! [R: set_Pr4821568463224695273rm_a_b,S: set_Pr4645549399426460137rm_c_d] :
      ( ( ord_le553461590799470985rm_a_b @ R @ ( conver2033648821598950629rm_c_d @ S ) )
      = ( ord_le377442527001235849rm_c_d @ ( conver3688962544742323429rm_a_b @ R ) @ S ) ) ).

% converse_subset_swap
thf(fact_1099_converse__subset__swap,axiom,
    ! [R: set_Pr4645549399426460137rm_c_d,S: set_Pr4821568463224695273rm_a_b] :
      ( ( ord_le377442527001235849rm_c_d @ R @ ( conver3688962544742323429rm_a_b @ S ) )
      = ( ord_le553461590799470985rm_a_b @ ( conver2033648821598950629rm_c_d @ R ) @ S ) ) ).

% converse_subset_swap
thf(fact_1100_case__prod__map__prod,axiom,
    ! [H: term_a_b > term_c_d > $o,F: term_a_b > term_a_b,G: term_c_d > term_c_d,X5: produc8451535801715972489rm_c_d] :
      ( ( produc3147824153920350274_c_d_o @ H @ ( produc4701833573052254282rm_c_d @ F @ G @ X5 ) )
      = ( produc3147824153920350274_c_d_o
        @ ^ [L: term_a_b,R2: term_c_d] : ( H @ ( F @ L ) @ ( G @ R2 ) )
        @ X5 ) ) ).

% case_prod_map_prod
thf(fact_1101_case__prod__map__prod,axiom,
    ! [H: term_a_b > term_c_d > $o,F: term_c_d > term_a_b,G: term_a_b > term_c_d,X5: produc5661168290637732233rm_a_b] :
      ( ( produc3147824153920350274_c_d_o @ H @ ( produc7098524691748194378rm_c_d @ F @ G @ X5 ) )
      = ( produc327966649874835522_a_b_o
        @ ^ [L: term_c_d,R2: term_a_b] : ( H @ ( F @ L ) @ ( G @ R2 ) )
        @ X5 ) ) ).

% case_prod_map_prod
thf(fact_1102_case__prod__map__prod,axiom,
    ! [H: term_c_d > term_a_b > $o,F: term_a_b > term_c_d,G: term_c_d > term_a_b,X5: produc8451535801715972489rm_c_d] :
      ( ( produc327966649874835522_a_b_o @ H @ ( produc7211074937178392650rm_a_b @ F @ G @ X5 ) )
      = ( produc3147824153920350274_c_d_o
        @ ^ [L: term_a_b,R2: term_c_d] : ( H @ ( F @ L ) @ ( G @ R2 ) )
        @ X5 ) ) ).

% case_prod_map_prod
thf(fact_1103_case__prod__map__prod,axiom,
    ! [H: term_c_d > term_a_b > $o,F: term_c_d > term_c_d,G: term_a_b > term_a_b,X5: produc5661168290637732233rm_a_b] :
      ( ( produc327966649874835522_a_b_o @ H @ ( produc384394019019556938rm_a_b @ F @ G @ X5 ) )
      = ( produc327966649874835522_a_b_o
        @ ^ [L: term_c_d,R2: term_a_b] : ( H @ ( F @ L ) @ ( G @ R2 ) )
        @ X5 ) ) ).

% case_prod_map_prod
thf(fact_1104_sym__trans__comp__subset,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( sym_on_nat @ top_top_set_nat @ R )
     => ( ( trans_on_nat @ top_top_set_nat @ R )
       => ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ ( converse_nat_nat @ R ) @ R ) @ R ) ) ) ).

% sym_trans_comp_subset
thf(fact_1105_under__incr,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A4: nat,B2: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R )
       => ( ord_less_eq_set_nat @ ( order_under_nat @ R @ A4 ) @ ( order_under_nat @ R @ B2 ) ) ) ) ).

% under_incr
thf(fact_1106_conj__subset__def,axiom,
    ! [A3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A3
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ P ) )
        & ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_1107_converse__relcomp,axiom,
    ! [R: set_Pr4821568463224695273rm_a_b,S: set_Pr4645549399426460137rm_c_d] :
      ( ( conver7337423426258204773rm_c_d @ ( relcom952951045985826111rm_c_d @ R @ S ) )
      = ( relcom952951045985826111rm_c_d @ ( conver2033648821598950629rm_c_d @ S ) @ ( conver3688962544742323429rm_a_b @ R ) ) ) ).

% converse_relcomp
thf(fact_1108_converse__relcomp,axiom,
    ! [R: set_Pr4645549399426460137rm_c_d,S: set_Pr4821568463224695273rm_a_b] :
      ( ( conver7608559976937845093rm_a_b @ ( relcom5673934560341955007rm_a_b @ R @ S ) )
      = ( relcom5673934560341955007rm_a_b @ ( conver3688962544742323429rm_a_b @ S ) @ ( conver2033648821598950629rm_c_d @ R ) ) ) ).

% converse_relcomp
thf(fact_1109_converse__relcomp,axiom,
    ! [R: set_Pr4645549399426460137rm_c_d,S: set_Pr5080540287643815273rm_c_d] :
      ( ( conver2033648821598950629rm_c_d @ ( relcom99023405003060543rm_c_d @ R @ S ) )
      = ( relcom2608264769129198911rm_a_b @ ( conver7337423426258204773rm_c_d @ S ) @ ( conver2033648821598950629rm_c_d @ R ) ) ) ).

% converse_relcomp
thf(fact_1110_converse__relcomp,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4645549399426460137rm_c_d] :
      ( ( conver2033648821598950629rm_c_d @ ( relcom4018620837198582207rm_c_d @ R @ S ) )
      = ( relcom6527862201324720575rm_a_b @ ( conver2033648821598950629rm_c_d @ S ) @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).

% converse_relcomp
thf(fact_1111_converse__relcomp,axiom,
    ! [R: set_Pr4821568463224695273rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( conver3688962544742323429rm_a_b @ ( relcom6527862201324720575rm_a_b @ R @ S ) )
      = ( relcom4018620837198582207rm_c_d @ ( conver7608559976937845093rm_a_b @ S ) @ ( conver3688962544742323429rm_a_b @ R ) ) ) ).

% converse_relcomp
thf(fact_1112_converse__relcomp,axiom,
    ! [R: set_Pr5080540287643815273rm_c_d,S: set_Pr4821568463224695273rm_a_b] :
      ( ( conver3688962544742323429rm_a_b @ ( relcom2608264769129198911rm_a_b @ R @ S ) )
      = ( relcom99023405003060543rm_c_d @ ( conver3688962544742323429rm_a_b @ S ) @ ( conver7337423426258204773rm_c_d @ R ) ) ) ).

% converse_relcomp
thf(fact_1113_under__def,axiom,
    ( order_under_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat,A: nat] :
          ( collect_nat
          @ ^ [B: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R2 ) ) ) ) ).

% under_def
thf(fact_1114_Collect__case__prod__mono,axiom,
    ! [A3: term_a_b > term_c_d > $o,B6: term_a_b > term_c_d > $o] :
      ( ( ord_le101682103841947774_c_d_o @ A3 @ B6 )
     => ( ord_le377442527001235849rm_c_d @ ( collec8193319511663418484rm_c_d @ ( produc3147824153920350274_c_d_o @ A3 ) ) @ ( collec8193319511663418484rm_c_d @ ( produc3147824153920350274_c_d_o @ B6 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_1115_Collect__case__prod__mono,axiom,
    ! [A3: term_c_d > term_a_b > $o,B6: term_c_d > term_a_b > $o] :
      ( ( ord_le8193136013808025726_a_b_o @ A3 @ B6 )
     => ( ord_le553461590799470985rm_a_b @ ( collec5402952000585178228rm_a_b @ ( produc327966649874835522_a_b_o @ A3 ) ) @ ( collec5402952000585178228rm_a_b @ ( produc327966649874835522_a_b_o @ B6 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_1116_relcomp3__transI,axiom,
    ! [B6: set_Pr1261947904930325089at_nat,T: nat,U: nat,A3: set_Pr1261947904930325089at_nat,S: nat,V: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ B6 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ ( relcomp_nat_nat_nat @ B6 @ ( relcomp_nat_nat_nat @ A3 @ B6 ) ) )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ B6 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ B6 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ V ) @ ( relcomp_nat_nat_nat @ B6 @ ( relcomp_nat_nat_nat @ A3 @ B6 ) ) ) ) ) ) ) ).

% relcomp3_transI
thf(fact_1117_trans__O__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ A3 )
      = ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ A3 @ A3 ) @ A3 ) ) ).

% trans_O_iff
thf(fact_1118_trans__O__subset,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R @ R ) @ R ) ) ).

% trans_O_subset
thf(fact_1119_trans__refl__imp__O__id,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( refl_on_nat @ top_top_set_nat @ R )
       => ( ( relcomp_nat_nat_nat @ R @ R )
          = R ) ) ) ).

% trans_refl_imp_O_id
thf(fact_1120_relImage__Gr,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A3: set_term_a_b,F: term_a_b > term_c_d] :
      ( ( ord_le118470702582115849rm_a_b @ R3
        @ ( produc1953217797448512882rm_a_b @ A3
          @ ^ [Uu: term_a_b] : A3 ) )
     => ( ( bNF_Gr6034662968427871246rm_c_d @ R3 @ F )
        = ( relcom952951045985826111rm_c_d @ ( conver2033648821598950629rm_c_d @ ( bNF_Gr3285426624312675404rm_c_d @ A3 @ F ) ) @ ( relcom4018620837198582207rm_c_d @ R3 @ ( bNF_Gr3285426624312675404rm_c_d @ A3 @ F ) ) ) ) ) ).

% relImage_Gr
thf(fact_1121_relImage__Gr,axiom,
    ! [R3: set_Pr5080540287643815273rm_c_d,A3: set_term_c_d,F: term_c_d > term_a_b] :
      ( ( ord_le812433415218590985rm_c_d @ R3
        @ ( produc1682081246768872562rm_c_d @ A3
          @ ^ [Uu: term_c_d] : A3 ) )
     => ( ( bNF_Gr7689976691571244046rm_a_b @ R3 @ F )
        = ( relcom5673934560341955007rm_a_b @ ( conver3688962544742323429rm_a_b @ ( bNF_Gr4940740347456048204rm_a_b @ A3 @ F ) ) @ ( relcom2608264769129198911rm_a_b @ R3 @ ( bNF_Gr4940740347456048204rm_a_b @ A3 @ F ) ) ) ) ) ).

% relImage_Gr
thf(fact_1122_relInvImage__Gr,axiom,
    ! [R3: set_Pr5080540287643815273rm_c_d,B6: set_term_c_d,A3: set_term_a_b,F: term_a_b > term_c_d] :
      ( ( ord_le812433415218590985rm_c_d @ R3
        @ ( produc1682081246768872562rm_c_d @ B6
          @ ^ [Uu: term_c_d] : B6 ) )
     => ( ( bNF_Gr7274730409055677163rm_c_d @ A3 @ R3 @ F )
        = ( relcom5673934560341955007rm_a_b @ ( bNF_Gr3285426624312675404rm_c_d @ A3 @ F ) @ ( relcom2608264769129198911rm_a_b @ R3 @ ( conver2033648821598950629rm_c_d @ ( bNF_Gr3285426624312675404rm_c_d @ A3 @ F ) ) ) ) ) ) ).

% relInvImage_Gr
thf(fact_1123_relInvImage__Gr,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,B6: set_term_a_b,A3: set_term_c_d,F: term_c_d > term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R3
        @ ( produc1953217797448512882rm_a_b @ B6
          @ ^ [Uu: term_a_b] : B6 ) )
     => ( ( bNF_Gr8930044132199049963rm_a_b @ A3 @ R3 @ F )
        = ( relcom952951045985826111rm_c_d @ ( bNF_Gr4940740347456048204rm_a_b @ A3 @ F ) @ ( relcom4018620837198582207rm_c_d @ R3 @ ( conver3688962544742323429rm_a_b @ ( bNF_Gr4940740347456048204rm_a_b @ A3 @ F ) ) ) ) ) ) ).

% relInvImage_Gr
thf(fact_1124_above__def,axiom,
    ( order_above_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat,A: nat] :
          ( collect_nat
          @ ^ [B: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) ) ) ) ).

% above_def
thf(fact_1125_conversep__in__rel,axiom,
    ! [R3: set_Pr4645549399426460137rm_c_d] :
      ( ( conver8884510217699480863rm_c_d @ ( fun_in8391593592550394988rm_c_d @ R3 ) )
      = ( fun_in823535278838991980rm_a_b @ ( conver2033648821598950629rm_c_d @ R3 ) ) ) ).

% conversep_in_rel
thf(fact_1126_conversep__in__rel,axiom,
    ! [R3: set_Pr4821568463224695273rm_a_b] :
      ( ( conver1316451903988077855rm_a_b @ ( fun_in823535278838991980rm_a_b @ R3 ) )
      = ( fun_in8391593592550394988rm_c_d @ ( conver3688962544742323429rm_a_b @ R3 ) ) ) ).

% conversep_in_rel
thf(fact_1127_wf__imp__irrefl,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R )
     => ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).

% wf_imp_irrefl
thf(fact_1128_in__rel__Collect__case__prod__eq,axiom,
    ! [X7: term_a_b > term_c_d > $o] :
      ( ( fun_in8391593592550394988rm_c_d @ ( collec8193319511663418484rm_c_d @ ( produc3147824153920350274_c_d_o @ X7 ) ) )
      = X7 ) ).

% in_rel_Collect_case_prod_eq
thf(fact_1129_in__rel__Collect__case__prod__eq,axiom,
    ! [X7: term_c_d > term_a_b > $o] :
      ( ( fun_in823535278838991980rm_a_b @ ( collec5402952000585178228rm_a_b @ ( produc327966649874835522_a_b_o @ X7 ) ) )
      = X7 ) ).

% in_rel_Collect_case_prod_eq
thf(fact_1130_Collect__case__prod__in__rel__leE,axiom,
    ! [X7: set_Pr4645549399426460137rm_c_d,Y8: set_Pr4645549399426460137rm_c_d] :
      ( ( ord_le377442527001235849rm_c_d @ X7 @ ( collec8193319511663418484rm_c_d @ ( produc3147824153920350274_c_d_o @ ( fun_in8391593592550394988rm_c_d @ Y8 ) ) ) )
     => ( ord_le377442527001235849rm_c_d @ X7 @ Y8 ) ) ).

% Collect_case_prod_in_rel_leE
thf(fact_1131_Collect__case__prod__in__rel__leE,axiom,
    ! [X7: set_Pr4821568463224695273rm_a_b,Y8: set_Pr4821568463224695273rm_a_b] :
      ( ( ord_le553461590799470985rm_a_b @ X7 @ ( collec5402952000585178228rm_a_b @ ( produc327966649874835522_a_b_o @ ( fun_in823535278838991980rm_a_b @ Y8 ) ) ) )
     => ( ord_le553461590799470985rm_a_b @ X7 @ Y8 ) ) ).

% Collect_case_prod_in_rel_leE
thf(fact_1132_Collect__case__prod__in__rel__leI,axiom,
    ! [X7: set_Pr4645549399426460137rm_c_d,Y8: set_Pr4645549399426460137rm_c_d] :
      ( ( ord_le377442527001235849rm_c_d @ X7 @ Y8 )
     => ( ord_le377442527001235849rm_c_d @ X7 @ ( collec8193319511663418484rm_c_d @ ( produc3147824153920350274_c_d_o @ ( fun_in8391593592550394988rm_c_d @ Y8 ) ) ) ) ) ).

% Collect_case_prod_in_rel_leI
thf(fact_1133_Collect__case__prod__in__rel__leI,axiom,
    ! [X7: set_Pr4821568463224695273rm_a_b,Y8: set_Pr4821568463224695273rm_a_b] :
      ( ( ord_le553461590799470985rm_a_b @ X7 @ Y8 )
     => ( ord_le553461590799470985rm_a_b @ X7 @ ( collec5402952000585178228rm_a_b @ ( produc327966649874835522_a_b_o @ ( fun_in823535278838991980rm_a_b @ Y8 ) ) ) ) ) ).

% Collect_case_prod_in_rel_leI
thf(fact_1134_wf__finite__segments,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R )
     => ( ( trans_on_nat @ top_top_set_nat @ R )
       => ( ! [X3: nat] :
              ( finite_finite_nat
              @ ( collect_nat
                @ ^ [Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X3 ) @ R ) ) )
         => ( wf_nat @ R ) ) ) ) ).

% wf_finite_segments
thf(fact_1135_wf__converse,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R )
     => ( ( trans_on_nat @ top_top_set_nat @ R )
       => ( ( finite6177210948735845034at_nat @ R )
         => ( wf_nat @ ( converse_nat_nat @ R ) ) ) ) ) ).

% wf_converse
thf(fact_1136_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_1137_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_1138_finite__imageI,axiom,
    ! [F4: set_nat,H: nat > nat] :
      ( ( finite_finite_nat @ F4 )
     => ( finite_finite_nat @ ( image_nat_nat2 @ H @ F4 ) ) ) ).

% finite_imageI
thf(fact_1139_finite__Plus__UNIV__iff,axiom,
    ( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_1140_finite__converse,axiom,
    ! [R: set_Pr4645549399426460137rm_c_d] :
      ( ( finite3743762059973507634rm_a_b @ ( conver2033648821598950629rm_c_d @ R ) )
      = ( finite6534129571051747890rm_c_d @ R ) ) ).

% finite_converse
thf(fact_1141_finite__converse,axiom,
    ! [R: set_Pr4821568463224695273rm_a_b] :
      ( ( finite6534129571051747890rm_c_d @ ( conver3688962544742323429rm_a_b @ R ) )
      = ( finite3743762059973507634rm_a_b @ R ) ) ).

% finite_converse
thf(fact_1142_finite__Collect__not,axiom,
    ! [P: nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ~ ( P @ X ) ) )
        = ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Collect_not
thf(fact_1143_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B7: set_nat] : ( ord_less_eq_set_nat @ B7 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_1144_finite__SigmaI,axiom,
    ! [A3: set_nat,B6: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A6: nat] :
            ( ( member_nat @ A6 @ A3 )
           => ( finite_finite_nat @ ( B6 @ A6 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A3 @ B6 ) ) ) ) ).

% finite_SigmaI
thf(fact_1145_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1146_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B6: set_nat,R3: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B6 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B6 )
                  & ( R3 @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B6 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A: nat] :
                        ( ( member_nat @ A @ A3 )
                        & ( R3 @ A @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1147_finite__inverse__image__gen,axiom,
    ! [A3: set_nat,F: nat > nat,D: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on_nat_nat @ F @ D )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D )
                & ( member_nat @ ( F @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_1148_finite__cartesian__product,axiom,
    ! [A3: set_nat,B6: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B6 )
       => ( finite6177210948735845034at_nat
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : B6 ) ) ) ) ).

% finite_cartesian_product
thf(fact_1149_finite__UNIV__surj__inj,axiom,
    ! [F: nat > nat] :
      ( ( finite_finite_nat @ top_top_set_nat )
     => ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
          = top_top_set_nat )
       => ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ) ).

% finite_UNIV_surj_inj
thf(fact_1150_finite__UNIV__inj__surj,axiom,
    ! [F: nat > nat] :
      ( ( finite_finite_nat @ top_top_set_nat )
     => ( ( inj_on_nat_nat @ F @ top_top_set_nat )
       => ( ( image_nat_nat2 @ F @ top_top_set_nat )
          = top_top_set_nat ) ) ) ).

% finite_UNIV_inj_surj
thf(fact_1151_Finite__Set_Ofinite__set,axiom,
    ( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% Finite_Set.finite_set
thf(fact_1152_finite__imageD,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A3 ) )
     => ( ( inj_on_nat_nat @ F @ A3 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% finite_imageD
thf(fact_1153_finite__image__iff,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( inj_on_nat_nat @ F @ A3 )
     => ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A3 ) )
        = ( finite_finite_nat @ A3 ) ) ) ).

% finite_image_iff
thf(fact_1154_finite__inverse__image,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on_nat_nat @ F @ top_top_set_nat )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] : ( member_nat @ ( F @ J ) @ A3 ) ) ) ) ) ).

% finite_inverse_image
thf(fact_1155_infinite__imp__elem,axiom,
    ! [A3: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ? [X3: nat] : ( member_nat @ X3 @ A3 ) ) ).

% infinite_imp_elem
thf(fact_1156_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_1157_finite__prod,axiom,
    ( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_1158_endo__inj__surj,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A3 ) @ A3 )
       => ( ( inj_on_nat_nat @ F @ A3 )
         => ( ( image_nat_nat2 @ F @ A3 )
            = A3 ) ) ) ) ).

% endo_inj_surj
thf(fact_1159_inj__on__finite,axiom,
    ! [F: nat > nat,A3: set_nat,B6: set_nat] :
      ( ( inj_on_nat_nat @ F @ A3 )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A3 ) @ B6 )
       => ( ( finite_finite_nat @ B6 )
         => ( finite_finite_nat @ A3 ) ) ) ) ).

% inj_on_finite
thf(fact_1160_finite__surj__inj,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ A3 @ ( image_nat_nat2 @ F @ A3 ) )
       => ( inj_on_nat_nat @ F @ A3 ) ) ) ).

% finite_surj_inj
thf(fact_1161_infinite__UNIV__char__0,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_char_0
thf(fact_1162_ex__new__if__finite,axiom,
    ! [A3: set_nat] :
      ( ~ ( finite_finite_nat @ top_top_set_nat )
     => ( ( finite_finite_nat @ A3 )
       => ? [A6: nat] :
            ~ ( member_nat @ A6 @ A3 ) ) ) ).

% ex_new_if_finite
thf(fact_1163_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A3 ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A: nat] :
                      ( ( member_nat @ A @ A3 )
                      & ( ( F @ A )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1164_inj__graph,axiom,
    ( inj_on8823995452884580655rm_c_d
    @ ^ [F2: term_a_b > term_c_d] :
        ( collec8193319511663418484rm_c_d
        @ ( produc3147824153920350274_c_d_o
          @ ^ [X: term_a_b,Y: term_c_d] :
              ( Y
              = ( F2 @ X ) ) ) )
    @ top_to44634570330034989rm_c_d ) ).

% inj_graph
thf(fact_1165_inj__graph,axiom,
    ( inj_on2179106752557545775rm_a_b
    @ ^ [F2: term_c_d > term_a_b] :
        ( collec5402952000585178228rm_a_b
        @ ( produc327966649874835522_a_b_o
          @ ^ [X: term_c_d,Y: term_a_b] :
              ( Y
              = ( F2 @ X ) ) ) )
    @ top_to220653634128270125rm_a_b ) ).

% inj_graph
thf(fact_1166_all__finite__subset__image,axiom,
    ! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
      ( ( ! [B7: set_nat] :
            ( ( ( finite_finite_nat @ B7 )
              & ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat2 @ F @ A3 ) ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set_nat] :
            ( ( ( finite_finite_nat @ B7 )
              & ( ord_less_eq_set_nat @ B7 @ A3 ) )
           => ( P @ ( image_nat_nat2 @ F @ B7 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1167_ex__finite__subset__image,axiom,
    ! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
      ( ( ? [B7: set_nat] :
            ( ( finite_finite_nat @ B7 )
            & ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat2 @ F @ A3 ) )
            & ( P @ B7 ) ) )
      = ( ? [B7: set_nat] :
            ( ( finite_finite_nat @ B7 )
            & ( ord_less_eq_set_nat @ B7 @ A3 )
            & ( P @ ( image_nat_nat2 @ F @ B7 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1168_finite__subset__image,axiom,
    ! [B6: set_nat,F: nat > nat,A3: set_nat] :
      ( ( finite_finite_nat @ B6 )
     => ( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat2 @ F @ A3 ) )
       => ? [C4: set_nat] :
            ( ( ord_less_eq_set_nat @ C4 @ A3 )
            & ( finite_finite_nat @ C4 )
            & ( B6
              = ( image_nat_nat2 @ F @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1169_finite__surj,axiom,
    ! [A3: set_nat,B6: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat2 @ F @ A3 ) )
       => ( finite_finite_nat @ B6 ) ) ) ).

% finite_surj
thf(fact_1170_finite__range__imageI,axiom,
    ! [G: nat > nat,F: nat > nat] :
      ( ( finite_finite_nat @ ( image_nat_nat2 @ G @ top_top_set_nat ) )
     => ( finite_finite_nat
        @ ( image_nat_nat2
          @ ^ [X: nat] : ( F @ ( G @ X ) )
          @ top_top_set_nat ) ) ) ).

% finite_range_imageI
thf(fact_1171_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_1172_infinite__cartesian__product,axiom,
    ! [A3: set_nat,B6: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ~ ( finite_finite_nat @ B6 )
       => ~ ( finite6177210948735845034at_nat
            @ ( produc457027306803732586at_nat @ A3
              @ ^ [Uu: nat] : B6 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_1173_sorted__list__of__set_Oinj__on,axiom,
    ( inj_on_nat_nat
    @ ^ [X: nat] : X
    @ top_top_set_nat ) ).

% sorted_list_of_set.inj_on
thf(fact_1174_infinite__countable__subset,axiom,
    ! [S2: set_nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ? [F5: nat > nat] :
          ( ( inj_on_nat_nat @ F5 @ top_top_set_nat )
          & ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F5 @ top_top_set_nat ) @ S2 ) ) ) ).

% infinite_countable_subset
thf(fact_1175_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_1176_infinite__iff__countable__subset,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ? [F2: nat > nat] :
            ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
            & ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F2 @ top_top_set_nat ) @ S2 ) ) ) ) ).

% infinite_iff_countable_subset
thf(fact_1177_range__inj__infinite,axiom,
    ! [F: nat > nat] :
      ( ( inj_on_nat_nat @ F @ top_top_set_nat )
     => ~ ( finite_finite_nat @ ( image_nat_nat2 @ F @ top_top_set_nat ) ) ) ).

% range_inj_infinite
thf(fact_1178_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_1179_nat__not__finite,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% nat_not_finite
thf(fact_1180_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_1181_finite__quotient,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ R
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : A3 ) )
       => ( finite1152437895449049373et_nat @ ( equiv_quotient_nat @ A3 @ R ) ) ) ) ).

% finite_quotient
thf(fact_1182_Sup__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
    = top_top_set_nat ) ).

% Sup_UNIV
thf(fact_1183_finite__UN,axiom,
    ! [A3: set_nat,B6: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B6 @ A3 ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A3 )
             => ( finite_finite_nat @ ( B6 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1184_finite__UN__I,axiom,
    ! [A3: set_nat,B6: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A6: nat] :
            ( ( member_nat @ A6 @ A3 )
           => ( finite_finite_nat @ ( B6 @ A6 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B6 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1185_Union__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
    = top_top_set_nat ) ).

% Union_UNIV
thf(fact_1186_finite__equiv__class,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat,X7: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ R
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : A3 ) )
       => ( ( member_set_nat @ X7 @ ( equiv_quotient_nat @ A3 @ R ) )
         => ( finite_finite_nat @ X7 ) ) ) ) ).

% finite_equiv_class
thf(fact_1187_Chains__subset_H,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R )
     => ( ord_le6893508408891458716et_nat
        @ ( collect_set_nat
          @ ( pred_chain_nat @ top_top_set_nat
            @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) )
        @ ( chains_nat @ R ) ) ) ).

% Chains_subset'
thf(fact_1188_Chains__alt__def,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R )
     => ( ( chains_nat @ R )
        = ( collect_set_nat
          @ ( pred_chain_nat @ top_top_set_nat
            @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ) ).

% Chains_alt_def
thf(fact_1189_Sup__SUP__eq,axiom,
    ( comple8317665133742190828_nat_o
    = ( ^ [S4: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S4 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_1190_Sup__set__def,axiom,
    ( comple7399068483239264473et_nat
    = ( ^ [A2: set_set_nat] :
          ( collect_nat
          @ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A2 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1191_Sup__SUP__eq2,axiom,
    ( comple4776141392677706955_c_d_o
    = ( ^ [S4: set_te8875761578035853502_c_d_o,X: term_a_b,Y: term_c_d] : ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ X @ Y ) @ ( comple5229724214311509974rm_c_d @ ( image_2252888096169626862rm_c_d @ collec8193319511663418484rm_c_d @ ( image_4899642137928444365_c_d_o @ produc3147824153920350274_c_d_o @ S4 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_1192_Sup__SUP__eq2,axiom,
    ( comple3644223265789009099_a_b_o
    = ( ^ [S4: set_te7068155017529424062_a_b_o,X: term_c_d,Y: term_a_b] : ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ X @ Y ) @ ( comple5405743278109745110rm_a_b @ ( image_540309731347893486rm_a_b @ collec5402952000585178228rm_a_b @ ( image_7540181528456194509_a_b_o @ produc327966649874835522_a_b_o @ S4 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_1193_Chains__subset,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ord_le6893508408891458716et_nat @ ( chains_nat @ R )
      @ ( collect_set_nat
        @ ( pred_chain_nat @ top_top_set_nat
          @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ).

% Chains_subset
thf(fact_1194_chain__subset__trans__Union,axiom,
    ! [R3: set_se7855581050983116737at_nat] :
      ( ( chain_3023788373544792282at_nat @ R3 )
     => ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ R3 )
           => ( trans_on_nat @ top_top_set_nat @ X3 ) )
       => ( trans_on_nat @ top_top_set_nat @ ( comple5685304695842803022at_nat @ R3 ) ) ) ) ).

% chain_subset_trans_Union
thf(fact_1195_Fpow__def,axiom,
    ( finite_Fpow_nat
    = ( ^ [A2: set_nat] :
          ( collect_set_nat
          @ ^ [X8: set_nat] :
              ( ( ord_less_eq_set_nat @ X8 @ A2 )
              & ( finite_finite_nat @ X8 ) ) ) ) ) ).

% Fpow_def
thf(fact_1196_Inter__UNIV__conv_I2_J,axiom,
    ! [A3: set_set_nat] :
      ( ( top_top_set_nat
        = ( comple7806235888213564991et_nat @ A3 ) )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A3 )
           => ( X = top_top_set_nat ) ) ) ) ).

% Inter_UNIV_conv(2)
thf(fact_1197_Inter__UNIV__conv_I1_J,axiom,
    ! [A3: set_set_nat] :
      ( ( ( comple7806235888213564991et_nat @ A3 )
        = top_top_set_nat )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A3 )
           => ( X = top_top_set_nat ) ) ) ) ).

% Inter_UNIV_conv(1)
thf(fact_1198_Inf__top__conv_I2_J,axiom,
    ! [A3: set_set_nat] :
      ( ( top_top_set_nat
        = ( comple7806235888213564991et_nat @ A3 ) )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A3 )
           => ( X = top_top_set_nat ) ) ) ) ).

% Inf_top_conv(2)
thf(fact_1199_Inf__top__conv_I1_J,axiom,
    ! [A3: set_set_nat] :
      ( ( ( comple7806235888213564991et_nat @ A3 )
        = top_top_set_nat )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A3 )
           => ( X = top_top_set_nat ) ) ) ) ).

% Inf_top_conv(1)
thf(fact_1200_Inf__INT__eq,axiom,
    ( comple6214475593288795910_nat_o
    = ( ^ [S4: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7806235888213564991et_nat @ ( image_nat_o_set_nat @ collect_nat @ S4 ) ) ) ) ) ).

% Inf_INT_eq
thf(fact_1201_Inf__set__def,axiom,
    ( comple7806235888213564991et_nat
    = ( ^ [A2: set_set_nat] :
          ( collect_nat
          @ ^ [X: nat] : ( complete_Inf_Inf_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A2 ) ) ) ) ) ).

% Inf_set_def
thf(fact_1202_Inf__INT__eq2,axiom,
    ( comple7831901308807748069_c_d_o
    = ( ^ [S4: set_te8875761578035853502_c_d_o,X: term_a_b,Y: term_c_d] : ( member4740485589907853106rm_c_d @ ( produc1445286645097778113rm_c_d @ X @ Y ) @ ( comple146610104219517168rm_c_d @ ( image_2252888096169626862rm_c_d @ collec8193319511663418484rm_c_d @ ( image_4899642137928444365_c_d_o @ produc3147824153920350274_c_d_o @ S4 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_1203_Inf__INT__eq2,axiom,
    ( comple6699983181919050213_a_b_o
    = ( ^ [S4: set_te7068155017529424062_a_b_o,X: term_c_d,Y: term_a_b] : ( member1950118078829612850rm_a_b @ ( produc3100600368241150913rm_a_b @ X @ Y ) @ ( comple322629168017752304rm_a_b @ ( image_540309731347893486rm_a_b @ collec5402952000585178228rm_a_b @ ( image_7540181528456194509_a_b_o @ produc327966649874835522_a_b_o @ S4 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_1204_chain__subset__antisym__Union,axiom,
    ! [R3: set_se7855581050983116737at_nat] :
      ( ( chain_3023788373544792282at_nat @ R3 )
     => ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ R3 )
           => ( antisym_on_nat @ top_top_set_nat @ X3 ) )
       => ( antisym_on_nat @ top_top_set_nat @ ( comple5685304695842803022at_nat @ R3 ) ) ) ) ).

% chain_subset_antisym_Union
thf(fact_1205_card__le__inj,axiom,
    ! [A3: set_nat,B6: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B6 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B6 ) )
         => ? [F5: nat > nat] :
              ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F5 @ A3 ) @ B6 )
              & ( inj_on_nat_nat @ F5 @ A3 ) ) ) ) ) ).

% card_le_inj
thf(fact_1206_card__inverse,axiom,
    ! [R3: set_Pr4645549399426460137rm_c_d] :
      ( ( finite2153137035963156913rm_a_b @ ( conver2033648821598950629rm_c_d @ R3 ) )
      = ( finite4943504547041397169rm_c_d @ R3 ) ) ).

% card_inverse
thf(fact_1207_card__inverse,axiom,
    ! [R3: set_Pr4821568463224695273rm_a_b] :
      ( ( finite4943504547041397169rm_c_d @ ( conver3688962544742323429rm_a_b @ R3 ) )
      = ( finite2153137035963156913rm_a_b @ R3 ) ) ).

% card_inverse
thf(fact_1208_Inter__eq,axiom,
    ( comple7806235888213564991et_nat
    = ( ^ [A2: set_set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
            ! [Y: set_nat] :
              ( ( member_set_nat @ Y @ A2 )
             => ( member_nat @ X @ Y ) ) ) ) ) ).

% Inter_eq
thf(fact_1209_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ top_top_set_nat )
     => ( ( ( finite_card_nat @ A3 )
          = ( finite_card_nat @ top_top_set_nat ) )
       => ( A3 = top_top_set_nat ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
thf(fact_1210_antisymD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X5: nat,Y3: nat] :
      ( ( antisym_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X5 ) @ R )
         => ( X5 = Y3 ) ) ) ) ).

% antisymD
thf(fact_1211_antisymI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R )
           => ( X3 = Y4 ) ) )
     => ( antisym_on_nat @ top_top_set_nat @ R ) ) ).

% antisymI
thf(fact_1212_antisym__subset,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ R @ S )
     => ( ( antisym_on_nat @ top_top_set_nat @ S )
       => ( antisym_on_nat @ top_top_set_nat @ R ) ) ) ).

% antisym_subset
thf(fact_1213_surjective__iff__injective__gen,axiom,
    ! [S2: set_nat,T3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_nat @ T3 )
       => ( ( ( finite_card_nat @ S2 )
            = ( finite_card_nat @ T3 ) )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ S2 ) @ T3 )
           => ( ( ! [X: nat] :
                    ( ( member_nat @ X @ T3 )
                   => ? [Y: nat] :
                        ( ( member_nat @ Y @ S2 )
                        & ( ( F @ Y )
                          = X ) ) ) )
              = ( inj_on_nat_nat @ F @ S2 ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_1214_card__bij__eq,axiom,
    ! [F: nat > nat,A3: set_nat,B6: set_nat,G: nat > nat] :
      ( ( inj_on_nat_nat @ F @ A3 )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A3 ) @ B6 )
       => ( ( inj_on_nat_nat @ G @ B6 )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ G @ B6 ) @ A3 )
           => ( ( finite_finite_nat @ A3 )
             => ( ( finite_finite_nat @ B6 )
               => ( ( finite_card_nat @ A3 )
                  = ( finite_card_nat @ B6 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_1215_inj__on__iff__card__le,axiom,
    ! [A3: set_nat,B6: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B6 )
       => ( ( ? [F2: nat > nat] :
                ( ( inj_on_nat_nat @ F2 @ A3 )
                & ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F2 @ A3 ) @ B6 ) ) )
          = ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B6 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_1216_underS__incr,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A4: nat,B2: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( antisym_on_nat @ top_top_set_nat @ R )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R )
         => ( ord_less_eq_set_nat @ ( order_underS_nat @ R @ A4 ) @ ( order_underS_nat @ R @ B2 ) ) ) ) ) ).

% underS_incr
thf(fact_1217_partial__order__on__def,axiom,
    ( order_5251275573222108571on_nat
    = ( ^ [A2: set_nat,R2: set_Pr1261947904930325089at_nat] :
          ( ( order_4861654808422542329on_nat @ A2 @ R2 )
          & ( antisym_on_nat @ top_top_set_nat @ R2 ) ) ) ) ).

% partial_order_on_def
thf(fact_1218_underS__def,axiom,
    ( order_underS_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat,A: nat] :
          ( collect_nat
          @ ^ [B: nat] :
              ( ( B != A )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% underS_def
thf(fact_1219_Id__on__def_H,axiom,
    ! [A3: nat > $o] :
      ( ( id_on_nat @ ( collect_nat @ A3 ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y: nat] :
              ( ( X = Y )
              & ( A3 @ X ) ) ) ) ) ).

% Id_on_def'
thf(fact_1220_trans__Id__on,axiom,
    ! [A3: set_nat] : ( trans_on_nat @ top_top_set_nat @ ( id_on_nat @ A3 ) ) ).

% trans_Id_on
thf(fact_1221_sym__Id__on,axiom,
    ! [A3: set_nat] : ( sym_on_nat @ top_top_set_nat @ ( id_on_nat @ A3 ) ) ).

% sym_Id_on
thf(fact_1222_antisym__Id__on,axiom,
    ! [A3: set_nat] : ( antisym_on_nat @ top_top_set_nat @ ( id_on_nat @ A3 ) ) ).

% antisym_Id_on
thf(fact_1223_partial__order__onD_I2_J,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( order_5251275573222108571on_nat @ A3 @ R )
     => ( trans_on_nat @ top_top_set_nat @ R ) ) ).

% partial_order_onD(2)
thf(fact_1224_partial__order__onD_I3_J,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( order_5251275573222108571on_nat @ A3 @ R )
     => ( antisym_on_nat @ top_top_set_nat @ R ) ) ).

% partial_order_onD(3)
thf(fact_1225_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_1226_finite__image__set2,axiom,
    ! [P: nat > $o,Q: nat > $o,F: nat > nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: nat,Y: nat] :
                ( ( Uu
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1227_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X: nat] :
              ( ( Uu
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_1228_setcompr__eq__image,axiom,
    ! [F: nat > nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: nat] :
            ( ( Uu
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_nat_nat2 @ F @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1229_full__SetCompr__eq,axiom,
    ! [F: nat > nat] :
      ( ( collect_nat
        @ ^ [U2: nat] :
          ? [X: nat] :
            ( U2
            = ( F @ X ) ) )
      = ( image_nat_nat2 @ F @ top_top_set_nat ) ) ).

% full_SetCompr_eq
thf(fact_1230_Collect__ex__eq,axiom,
    ! [P: nat > nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
          ? [X8: nat] : ( P @ X @ X8 ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [Y: nat] :
              ( collect_nat
              @ ^ [X: nat] : ( P @ X @ Y ) )
          @ top_top_set_nat ) ) ) ).

% Collect_ex_eq
thf(fact_1231_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_1232_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_1233_converse__empty,axiom,
    ( ( conver2033648821598950629rm_c_d @ bot_bo456493045772458581rm_c_d )
    = bot_bo632512109570693717rm_a_b ) ).

% converse_empty
thf(fact_1234_converse__empty,axiom,
    ( ( conver3688962544742323429rm_a_b @ bot_bo632512109570693717rm_a_b )
    = bot_bo456493045772458581rm_c_d ) ).

% converse_empty
thf(fact_1235_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_nat
            @ ^ [S3: nat] : P )
          = top_top_set_nat ) )
      & ( ~ P
       => ( ( collect_nat
            @ ^ [S3: nat] : P )
          = bot_bot_set_nat ) ) ) ).

% Collect_const
thf(fact_1236_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec8193319511663418484rm_c_d
            @ ( produc3147824153920350274_c_d_o
              @ ^ [A: term_a_b,B: term_c_d] : P ) )
          = top_to1573239103265677113rm_c_d ) )
      & ( ~ P
       => ( ( collec8193319511663418484rm_c_d
            @ ( produc3147824153920350274_c_d_o
              @ ^ [A: term_a_b,B: term_c_d] : P ) )
          = bot_bo456493045772458581rm_c_d ) ) ) ).

% Collect_const_case_prod
thf(fact_1237_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec5402952000585178228rm_a_b
            @ ( produc327966649874835522_a_b_o
              @ ^ [A: term_c_d,B: term_a_b] : P ) )
          = top_to1749258167063912249rm_a_b ) )
      & ( ~ P
       => ( ( collec5402952000585178228rm_a_b
            @ ( produc327966649874835522_a_b_o
              @ ^ [A: term_c_d,B: term_a_b] : P ) )
          = bot_bo632512109570693717rm_a_b ) ) ) ).

% Collect_const_case_prod
thf(fact_1238_Inf__empty,axiom,
    ( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
    = top_top_set_nat ) ).

% Inf_empty
thf(fact_1239_Pow__UNIV,axiom,
    ( ( pow_nat @ top_top_set_nat )
    = top_top_set_set_nat ) ).

% Pow_UNIV
thf(fact_1240_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
thf(fact_1241_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% empty_def
thf(fact_1242_bot_Oextremum,axiom,
    ! [A4: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A4 ) ).

% bot.extremum
thf(fact_1243_bot_Oextremum__unique,axiom,
    ! [A4: nat] :
      ( ( ord_less_eq_nat @ A4 @ bot_bot_nat )
      = ( A4 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1244_bot_Oextremum__uniqueI,axiom,
    ! [A4: nat] :
      ( ( ord_less_eq_nat @ A4 @ bot_bot_nat )
     => ( A4 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1245_Inter__empty,axiom,
    ( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
    = top_top_set_nat ) ).

% Inter_empty
thf(fact_1246_trans__empty,axiom,
    trans_on_nat @ top_top_set_nat @ bot_bo2099793752762293965at_nat ).

% trans_empty
thf(fact_1247_antisym__empty,axiom,
    antisym_on_nat @ top_top_set_nat @ bot_bo2099793752762293965at_nat ).

% antisym_empty
thf(fact_1248_finite__cartesian__product__iff,axiom,
    ! [A3: set_nat,B6: set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B6 ) )
      = ( ( A3 = bot_bot_set_nat )
        | ( B6 = bot_bot_set_nat )
        | ( ( finite_finite_nat @ A3 )
          & ( finite_finite_nat @ B6 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_1249_finite__SigmaI2,axiom,
    ! [A3: set_nat,B6: nat > set_nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( ( B6 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A6: nat] :
            ( ( member_nat @ A6 @ A3 )
           => ( finite_finite_nat @ ( B6 @ A6 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A3 @ B6 ) ) ) ) ).

% finite_SigmaI2
thf(fact_1250_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_1251_singleton__conv,axiom,
    ! [A4: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( X = A4 ) )
      = ( insert_nat @ A4 @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_1252_singleton__conv2,axiom,
    ! [A4: nat] :
      ( ( collect_nat
        @ ( ^ [Y7: nat,Z4: nat] : ( Y7 = Z4 )
          @ A4 ) )
      = ( insert_nat @ A4 @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_1253_Collect__conv__if,axiom,
    ! [P: nat > $o,A4: nat] :
      ( ( ( P @ A4 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A4 )
                & ( P @ X ) ) )
          = ( insert_nat @ A4 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A4 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A4 )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_1254_Collect__conv__if2,axiom,
    ! [P: nat > $o,A4: nat] :
      ( ( ( P @ A4 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A4 = X )
                & ( P @ X ) ) )
          = ( insert_nat @ A4 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A4 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A4 = X )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_1255_insert__UNIV,axiom,
    ! [X5: nat] :
      ( ( insert_nat @ X5 @ top_top_set_nat )
      = top_top_set_nat ) ).

% insert_UNIV
thf(fact_1256_insert__Collect,axiom,
    ! [A4: nat,P: nat > $o] :
      ( ( insert_nat @ A4 @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A4 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_1257_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A: nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A )
              | ( member_nat @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_1258_antisym__singleton,axiom,
    ! [X5: product_prod_nat_nat] : ( antisym_on_nat @ top_top_set_nat @ ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) ).

% antisym_singleton
thf(fact_1259_trans__singleton,axiom,
    ! [A4: nat] : ( trans_on_nat @ top_top_set_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ bot_bo2099793752762293965at_nat ) ) ).

% trans_singleton
thf(fact_1260_card__quotient__disjoint,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on8105003582846801791et_nat
          @ ^ [X: nat] : ( equiv_quotient_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ R )
          @ A3 )
       => ( ( finite_card_set_nat @ ( equiv_quotient_nat @ A3 @ R ) )
          = ( finite_card_nat @ A3 ) ) ) ) ).

% card_quotient_disjoint
thf(fact_1261_trans__wf__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( wf_nat @ R )
        = ( ! [A: nat] :
              ( wf_nat
              @ ( inf_in2572325071724192079at_nat @ R
                @ ( produc457027306803732586at_nat @ ( image_nat_nat @ ( converse_nat_nat @ R ) @ ( insert_nat @ A @ bot_bot_set_nat ) )
                  @ ^ [Uu: nat] : ( image_nat_nat @ ( converse_nat_nat @ R ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% trans_wf_iff
thf(fact_1262_Int__UNIV,axiom,
    ! [A3: set_nat,B6: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B6 )
        = top_top_set_nat )
      = ( ( A3 = top_top_set_nat )
        & ( B6 = top_top_set_nat ) ) ) ).

% Int_UNIV
thf(fact_1263_trans__Int,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( trans_on_nat @ top_top_set_nat @ S )
       => ( trans_on_nat @ top_top_set_nat @ ( inf_in2572325071724192079at_nat @ R @ S ) ) ) ) ).

% trans_Int
thf(fact_1264_sym__Int,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sym_on_nat @ top_top_set_nat @ R )
     => ( ( sym_on_nat @ top_top_set_nat @ S )
       => ( sym_on_nat @ top_top_set_nat @ ( inf_in2572325071724192079at_nat @ R @ S ) ) ) ) ).

% sym_Int
thf(fact_1265_Int__UNIV__right,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ top_top_set_nat )
      = A3 ) ).

% Int_UNIV_right
thf(fact_1266_Int__UNIV__left,axiom,
    ! [B6: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ B6 )
      = B6 ) ).

% Int_UNIV_left
thf(fact_1267_Collect__conj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1268_Int__Collect,axiom,
    ! [X5: nat,A3: set_nat,P: nat > $o] :
      ( ( member_nat @ X5 @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) )
      = ( ( member_nat @ X5 @ A3 )
        & ( P @ X5 ) ) ) ).

% Int_Collect
thf(fact_1269_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A2: set_nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( member_nat @ X @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_1270_converse__Int,axiom,
    ! [R: set_Pr4645549399426460137rm_c_d,S: set_Pr4645549399426460137rm_c_d] :
      ( ( conver2033648821598950629rm_c_d @ ( inf_in1952793154879802071rm_c_d @ R @ S ) )
      = ( inf_in2128812218678037207rm_a_b @ ( conver2033648821598950629rm_c_d @ R ) @ ( conver2033648821598950629rm_c_d @ S ) ) ) ).

% converse_Int
thf(fact_1271_converse__Int,axiom,
    ! [R: set_Pr4821568463224695273rm_a_b,S: set_Pr4821568463224695273rm_a_b] :
      ( ( conver3688962544742323429rm_a_b @ ( inf_in2128812218678037207rm_a_b @ R @ S ) )
      = ( inf_in1952793154879802071rm_c_d @ ( conver3688962544742323429rm_a_b @ R ) @ ( conver3688962544742323429rm_a_b @ S ) ) ) ).

% converse_Int
thf(fact_1272_Int__Collect__mono,axiom,
    ! [A3: set_nat,B6: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A3 @ B6 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B6 @ ( collect_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1273_sym__Int__converse,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : ( sym_on_nat @ top_top_set_nat @ ( inf_in2572325071724192079at_nat @ R @ ( converse_nat_nat @ R ) ) ) ).

% sym_Int_converse
thf(fact_1274_Image__Int__eq,axiom,
    ! [R3: set_Pr4645549399426460137rm_c_d,A3: set_term_a_b,B6: set_term_a_b] :
      ( ( single6888608300608217517rm_a_b @ ( conver2033648821598950629rm_c_d @ R3 ) )
     => ( ( image_1348876462811891197rm_c_d @ R3 @ ( inf_inf_set_term_a_b @ A3 @ B6 ) )
        = ( inf_inf_set_term_c_d @ ( image_1348876462811891197rm_c_d @ R3 @ A3 ) @ ( image_1348876462811891197rm_c_d @ R3 @ B6 ) ) ) ) ).

% Image_Int_eq
thf(fact_1275_Image__Int__eq,axiom,
    ! [R3: set_Pr4821568463224695273rm_a_b,A3: set_term_c_d,B6: set_term_c_d] :
      ( ( single5233294577464844717rm_c_d @ ( conver3688962544742323429rm_a_b @ R3 ) )
     => ( ( image_3004190185955263997rm_a_b @ R3 @ ( inf_inf_set_term_c_d @ A3 @ B6 ) )
        = ( inf_inf_set_term_a_b @ ( image_3004190185955263997rm_a_b @ R3 @ A3 ) @ ( image_3004190185955263997rm_a_b @ R3 @ B6 ) ) ) ) ).

% Image_Int_eq

% Helper facts (5)
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_T,axiom,
    ! [X5: produc8451535801715972489rm_c_d,Y3: produc8451535801715972489rm_c_d] :
      ( ( if_Pro195050992869121359rm_c_d @ $false @ X5 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__c_Mtf__d_J_J_T,axiom,
    ! [X5: produc8451535801715972489rm_c_d,Y3: produc8451535801715972489rm_c_d] :
      ( ( if_Pro195050992869121359rm_c_d @ $true @ X5 @ Y3 )
      = X5 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_T,axiom,
    ! [X5: produc5661168290637732233rm_a_b,Y3: produc5661168290637732233rm_a_b] :
      ( ( if_Pro6628055518645656911rm_a_b @ $false @ X5 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Term__Oterm_Itf__c_Mtf__d_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_T,axiom,
    ! [X5: produc5661168290637732233rm_a_b,Y3: produc5661168290637732233rm_a_b] :
      ( ( if_Pro6628055518645656911rm_a_b @ $true @ X5 @ Y3 )
      = X5 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [X4: produc8451535801715972489rm_c_d] :
      ( ( member4740485589907853106rm_c_d @ X4 @ r )
     => ( produc3147824153920350274_c_d_o
        @ ^ [S3: term_a_b,T2: term_c_d] :
            ( ( term_ground_a_b @ S3 )
            & ( term_ground_c_d @ T2 ) )
        @ X4 ) ) ).

thf(conj_1,conjecture,
    ! [X3: produc5661168290637732233rm_a_b] :
      ( ( member1950118078829612850rm_a_b @ X3 @ ( conver2033648821598950629rm_c_d @ r ) )
     => ( produc327966649874835522_a_b_o
        @ ^ [S3: term_c_d,T2: term_a_b] :
            ( ( term_ground_c_d @ S3 )
            & ( term_ground_a_b @ T2 ) )
        @ X3 ) ) ).

%------------------------------------------------------------------------------