%------------------------------------------------------------------------------
% 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    : Fishers_Inequality/0033_Matrix_Vector_Extras/prob_00411_016355__27946424_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1871 ( 622 unt; 584 typ;   0 def)
%            Number of atoms       : 2996 (1769 equ;   0 cnn)
%            Maximal formula atoms :    9 (   2 avg)
%            Number of connectives : 10892 (  68   ~;   1   |; 146   &;9372   @)
%                                         (   0 <=>;1305  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   6 avg)
%            Number of types       :   38 (  37 usr)
%            Number of type conns  : 5682 (5682   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  550 ( 547 usr;  35 con; 0-6 aty)
%            Number of variables   : 4145 ( 252   ^;3797   !;  96   ?;4145   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 15:48:06.774
%------------------------------------------------------------------------------
% Could-be-implicit typings (37)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
    set_Pr8928021450653196913at_rat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Rat__Orat_Mt__Nat__Onat_J_J,type,
    set_Pr6084635751276098665at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    set_Pr4105333604307423337at_rat: $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__Set__Oset_It__Sum____Type__Osum_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
    set_Sum_sum_rat_rat: $tType ).

thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Rat__Orat_Mt__Nat__Onat_J_J,type,
    set_Sum_sum_rat_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    set_Sum_sum_nat_rat: $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__Option__Ooption_It__Rat__Orat_J_J,type,
    set_option_rat: $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__Set__Oset_It__Rat__Orat_J_J,type,
    set_set_rat: $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__Rat__Orat_M_Eo_J_J,type,
    set_rat_o: $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__Set__Oset_It__Set__Oset_Itf__c_J_J,type,
    set_set_c: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    set_set_b: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    set_set_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__c_J_J,type,
    set_c_c: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__b_J_J,type,
    set_c_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
    set_c_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
    set_b_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
    set_b_a: $tType ).

thf(ty_n_t__Matrix__Ovec_It__Rat__Orat_J,type,
    vec_rat: $tType ).

thf(ty_n_t__Matrix__Ovec_It__Nat__Onat_J,type,
    vec_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
    set_rat: $tType ).

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

thf(ty_n_t__Matrix__Ovec_Itf__c_J,type,
    vec_c: $tType ).

thf(ty_n_t__Matrix__Ovec_Itf__b_J,type,
    vec_b: $tType ).

thf(ty_n_t__Matrix__Ovec_Itf__a_J,type,
    vec_a: $tType ).

thf(ty_n_t__Set__Oset_Itf__c_J,type,
    set_c: $tType ).

thf(ty_n_t__Set__Oset_Itf__b_J,type,
    set_b: $tType ).

thf(ty_n_t__Set__Oset_Itf__a_J,type,
    set_a: $tType ).

thf(ty_n_t__Rat__Orat,type,
    rat: $tType ).

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

thf(ty_n_tf__c,type,
    c: $tType ).

thf(ty_n_tf__b,type,
    b: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (547)
thf(sy_c_BNF__Composition_ODEADID_Opred__DEADID_001_062_Itf__b_Mtf__a_J,type,
    bNF_pred_DEADID_b_a: ( b > a ) > $o ).

thf(sy_c_BNF__Composition_ODEADID_Opred__DEADID_001_062_Itf__c_Mtf__a_J,type,
    bNF_pred_DEADID_c_a: ( c > a ) > $o ).

thf(sy_c_BNF__Composition_ODEADID_Opred__DEADID_001tf__a,type,
    bNF_pred_DEADID_a: a > $o ).

thf(sy_c_BNF__Composition_ODEADID_Opred__DEADID_001tf__b,type,
    bNF_pred_DEADID_b: b > $o ).

thf(sy_c_BNF__Composition_ODEADID_Opred__DEADID_001tf__c,type,
    bNF_pred_DEADID_c: c > $o ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_It__Nat__Onat_J,type,
    boolea778851993438741648et_nat: ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat ) > set_nat > set_nat > $o ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_It__Rat__Orat_J,type,
    boolea6493995471900120728et_rat: ( set_rat > set_rat > set_rat ) > ( set_rat > set_rat > set_rat ) > ( set_rat > set_rat ) > set_rat > set_rat > $o ).

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__Rat__Orat_M_Eo_J,type,
    comple2477142665972227838_rat_o: set_rat_o > rat > $o ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
    complete_Inf_Inf_nat: set_nat > nat ).

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__Rat__Orat_J,type,
    comple4298007329820168263et_rat: set_set_rat > set_rat ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__a_J,type,
    comple6135023378680113637_set_a: set_set_a > set_a ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__b_J,type,
    comple6135023382983342438_set_b: set_set_b > set_b ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__c_J,type,
    comple6135023387286571239_set_c: set_set_c > set_c ).

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__Rat__Orat_M_Eo_J,type,
    comple4580332206425622756_rat_o: set_rat_o > rat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: set_nat > nat ).

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__Rat__Orat_J,type,
    comple3890839924845867745et_rat: set_set_rat > set_rat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
    comple2307003609928055243_set_a: set_set_a > set_a ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__b_J,type,
    comple2307003614231284044_set_b: set_set_b > set_b ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__c_J,type,
    comple2307003618534512845_set_c: set_set_c > set_c ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Rat__Orat,type,
    condit1579696412822616692ve_rat: set_rat > $o ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Rat__Orat,type,
    condit1103211067700513672ow_rat: set_rat > $o ).

thf(sy_c_Countable_Ofrom__nat_001t__Nat__Onat,type,
    from_nat_nat: nat > nat ).

thf(sy_c_Countable_Ofrom__nat_001t__Rat__Orat,type,
    from_nat_rat: nat > rat ).

thf(sy_c_Countable_Onat__to__rat__surj,type,
    nat_to_rat_surj: nat > rat ).

thf(sy_c_Countable_Oto__nat_001t__Nat__Onat,type,
    to_nat_nat: nat > nat ).

thf(sy_c_Countable_Oto__nat_001t__Rat__Orat,type,
    to_nat_rat: rat > nat ).

thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001t__Nat__Onat_001t__Nat__Onat,type,
    disjoi6798895846410478970at_nat: ( nat > set_nat ) > set_nat > $o ).

thf(sy_c_Disjoint__Sets_Opartition__on_001t__Nat__Onat,type,
    disjoi4774308525696689793on_nat: set_nat > set_set_nat > $o ).

thf(sy_c_Disjoint__Sets_Opartition__on_001t__Rat__Orat,type,
    disjoi4139178465610194057on_rat: set_rat > set_set_rat > $o ).

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

thf(sy_c_Finite__Set_OFpow_001t__Rat__Orat,type,
    finite_Fpow_rat: set_rat > set_set_rat ).

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

thf(sy_c_Finite__Set_Ocard_001t__Rat__Orat,type,
    finite_card_rat: set_rat > 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__Option__Ooption_It__Rat__Orat_J,type,
    finite2014924581280026175on_rat: set_option_rat > $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__Nat__Onat_Mt__Rat__Orat_J,type,
    finite2668982390342448306at_rat: set_Pr4105333604307423337at_rat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Rat__Orat_Mt__Nat__Onat_J,type,
    finite5860656304800979122at_nat: set_Pr6084635751276098665at_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J,type,
    finite2352427746407582394at_rat: set_Pr8928021450653196913at_rat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
    finite_finite_rat: set_rat > $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__Set__Oset_It__Rat__Orat_J,type,
    finite6867581373910428453et_rat: set_set_rat > $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_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Rat__Orat_J,type,
    finite2679478125380364318at_rat: set_Sum_sum_nat_rat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Rat__Orat_Mt__Nat__Onat_J,type,
    finite5871152039838895134at_nat: set_Sum_sum_rat_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Rat__Orat_Mt__Rat__Orat_J,type,
    finite2362923481445498406at_rat: set_Sum_sum_rat_rat > $o ).

thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    finite5529483035118572448et_nat: ( nat > set_nat > set_nat ) > set_nat > set_nat > set_nat ).

thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Rat__Orat_J,type,
    finite2021254476725175720et_rat: ( nat > set_rat > set_rat ) > set_rat > set_nat > set_rat ).

thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    finite677925301803182934et_nat: ( set_nat > set_nat > set_nat ) > set_nat > set_set_nat > set_nat ).

thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
    finite5923809206502920038et_rat: ( set_rat > set_rat > set_rat ) > set_rat > set_set_rat > set_rat ).

thf(sy_c_Fun_Ocomp_001_062_Itf__b_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__a_Mtf__b_J,type,
    comp_b_a_c_a_a_b: ( ( b > a ) > c > a ) > ( ( a > b ) > b > a ) > ( a > b ) > c > a ).

thf(sy_c_Fun_Ocomp_001_062_Itf__b_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__a_Mtf__c_J,type,
    comp_b_a_c_a_a_c: ( ( b > a ) > c > a ) > ( ( a > c ) > b > a ) > ( a > c ) > c > a ).

thf(sy_c_Fun_Ocomp_001_062_Itf__b_Mtf__b_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__a_Mtf__b_J,type,
    comp_b_b_c_a_a_b: ( ( b > b ) > c > a ) > ( ( a > b ) > b > b ) > ( a > b ) > c > a ).

thf(sy_c_Fun_Ocomp_001_062_Itf__b_Mtf__b_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__a_Mtf__c_J,type,
    comp_b_b_c_a_a_c: ( ( b > b ) > c > a ) > ( ( a > c ) > b > b ) > ( a > c ) > c > a ).

thf(sy_c_Fun_Ocomp_001_062_Itf__b_Mtf__b_J_001_062_Itf__c_Mtf__b_J_001_062_Itf__a_Mtf__c_J,type,
    comp_b_b_c_b_a_c: ( ( b > b ) > c > b ) > ( ( a > c ) > b > b ) > ( a > c ) > c > b ).

thf(sy_c_Fun_Ocomp_001_062_Itf__b_Mtf__c_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__a_Mtf__c_J,type,
    comp_b_c_c_a_a_c: ( ( b > c ) > c > a ) > ( ( a > c ) > b > c ) > ( a > c ) > c > a ).

thf(sy_c_Fun_Ocomp_001_062_Itf__c_Mtf__a_J_001tf__c_001tf__c,type,
    comp_c_a_c_c: ( ( c > a ) > c ) > ( c > c > a ) > c > c ).

thf(sy_c_Fun_Ocomp_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__a_Mtf__b_J,type,
    comp_c_b_c_a_a_b: ( ( c > b ) > c > a ) > ( ( a > b ) > c > b ) > ( a > b ) > c > a ).

thf(sy_c_Fun_Ocomp_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__a_Mtf__c_J,type,
    comp_c_b_c_a_a_c: ( ( c > b ) > c > a ) > ( ( a > c ) > c > b ) > ( a > c ) > c > a ).

thf(sy_c_Fun_Ocomp_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__b_J_001_062_Itf__a_Mtf__c_J,type,
    comp_c_b_c_b_a_c: ( ( c > b ) > c > b ) > ( ( a > c ) > c > b ) > ( a > c ) > c > b ).

thf(sy_c_Fun_Ocomp_001_062_Itf__c_Mtf__c_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__a_Mtf__c_J,type,
    comp_c_c_c_a_a_c: ( ( c > c ) > c > a ) > ( ( a > c ) > c > c ) > ( a > c ) > c > a ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
    comp_nat_nat_rat: ( nat > nat ) > ( rat > nat ) > rat > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Rat__Orat_001t__Nat__Onat,type,
    comp_nat_rat_nat: ( nat > rat ) > ( nat > nat ) > nat > rat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Rat__Orat_001t__Rat__Orat,type,
    comp_nat_rat_rat: ( nat > rat ) > ( rat > nat ) > rat > rat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    comp_nat_set_nat_nat: ( nat > set_nat ) > ( nat > nat ) > nat > set_nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Set__Oset_It__Rat__Orat_J_001tf__a,type,
    comp_nat_set_rat_a: ( nat > set_rat ) > ( a > nat ) > a > set_rat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Set__Oset_It__Rat__Orat_J_001tf__b,type,
    comp_nat_set_rat_b: ( nat > set_rat ) > ( b > nat ) > b > set_rat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Set__Oset_It__Rat__Orat_J_001tf__c,type,
    comp_nat_set_rat_c: ( nat > set_rat ) > ( c > nat ) > c > set_rat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__a_001tf__b,type,
    comp_nat_a_b: ( nat > a ) > ( b > nat ) > b > a ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__a_001tf__c,type,
    comp_nat_a_c: ( nat > a ) > ( c > nat ) > c > a ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__b_001tf__b,type,
    comp_nat_b_b: ( nat > b ) > ( b > nat ) > b > b ).

thf(sy_c_Fun_Ocomp_001t__Rat__Orat_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_rat_nat_nat: ( rat > nat ) > ( nat > rat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Rat__Orat_001t__Rat__Orat_001t__Nat__Onat,type,
    comp_rat_rat_nat: ( rat > rat ) > ( nat > rat ) > nat > rat ).

thf(sy_c_Fun_Ocomp_001t__Rat__Orat_001t__Rat__Orat_001t__Rat__Orat,type,
    comp_rat_rat_rat: ( rat > rat ) > ( rat > rat ) > rat > rat ).

thf(sy_c_Fun_Ocomp_001t__Rat__Orat_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    comp_rat_set_nat_nat: ( rat > set_nat ) > ( nat > rat ) > nat > set_nat ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
    comp_s2351873599094224870at_nat: ( set_nat > set_nat > set_nat ) > ( nat > set_nat ) > nat > set_nat > set_nat ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    comp_s3433241188411525313at_nat: ( set_nat > set_nat ) > ( nat > set_nat ) > nat > set_nat ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    comp_s8964582002068861047et_nat: ( set_nat > set_nat ) > ( set_nat > set_nat ) > set_nat > set_nat ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Rat__Orat_J_001t__Nat__Onat,type,
    comp_s8255929034757298889at_nat: ( set_nat > set_rat ) > ( nat > set_nat ) > nat > set_rat ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    comp_s8495322428307219071et_nat: ( set_nat > set_rat ) > ( set_nat > set_nat ) > set_nat > set_rat ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Rat__Orat_J_001tf__b,type,
    comp_s1772850473432672326_rat_b: ( set_nat > set_rat ) > ( b > set_nat ) > b > set_rat ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Rat__Orat_J_001tf__c,type,
    comp_s1772850473432672327_rat_c: ( set_nat > set_rat ) > ( c > set_nat ) > c > set_rat ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Rat__Orat_J_001_062_It__Set__Oset_It__Rat__Orat_J_Mt__Set__Oset_It__Rat__Orat_J_J_001t__Nat__Onat,type,
    comp_s6252137336010071550at_nat: ( set_rat > set_rat > set_rat ) > ( nat > set_rat ) > nat > set_rat > set_rat ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    comp_s1665167385571447943et_nat: ( set_rat > set_rat ) > ( set_nat > set_rat ) > set_nat > set_rat ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
    comp_s7380310864032827023et_rat: ( set_rat > set_rat ) > ( set_rat > set_rat ) > set_rat > set_rat ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__b_J,type,
    comp_s9136116061129617852_set_b: ( set_a > set_a ) > ( set_b > set_a ) > set_b > set_a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__c_J,type,
    comp_s9136116065432846653_set_c: ( set_a > set_a ) > ( set_c > set_a ) > set_c > set_a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__a_J,type,
    comp_s1400706057266411770_set_a: ( set_a > set_b ) > ( set_a > set_a ) > set_a > set_b ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__a_J,type,
    comp_s2888668094561210297_set_a: ( set_a > set_c ) > ( set_a > set_a ) > set_a > set_c ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__b_J,type,
    comp_s7019688003139142845_set_b: ( set_b > set_a ) > ( set_b > set_b ) > set_b > set_a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__c_J,type,
    comp_s7019688007442371646_set_c: ( set_b > set_a ) > ( set_c > set_b ) > set_c > set_a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__a_J,type,
    comp_s8507650036130712571_set_a: ( set_b > set_b ) > ( set_a > set_b ) > set_a > set_b ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__c_J,type,
    comp_s8507650044737170173_set_c: ( set_b > set_b ) > ( set_c > set_b ) > set_c > set_b ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__a_J,type,
    comp_s772240036570735290_set_a: ( set_b > set_c ) > ( set_a > set_b ) > set_a > set_c ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__b_J,type,
    comp_s772240040873964091_set_b: ( set_b > set_c ) > ( set_b > set_b ) > set_b > set_c ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
    comp_s4903259940845439037_set_a: ( set_c > set_a ) > ( set_a > set_c ) > set_a > set_a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__c_J,type,
    comp_s4903259949451896639_set_c: ( set_c > set_a ) > ( set_c > set_c ) > set_c > set_a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__a_J,type,
    comp_s6391221978140237564_set_a: ( set_c > set_b ) > ( set_a > set_c ) > set_a > set_b ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
    comp_s6391221982443466365_set_b: ( set_c > set_b ) > ( set_b > set_c ) > set_b > set_b ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__a_J,type,
    comp_s7879184015435036091_set_a: ( set_c > set_c ) > ( set_a > set_c ) > set_a > set_c ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__b_J,type,
    comp_s7879184019738264892_set_b: ( set_c > set_c ) > ( set_b > set_c ) > set_b > set_c ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Set__Oset_It__Rat__Orat_J_001tf__a,type,
    comp_a_set_rat_a: ( a > set_rat ) > ( a > a ) > a > set_rat ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Set__Oset_It__Rat__Orat_J_001tf__b,type,
    comp_a_set_rat_b: ( a > set_rat ) > ( b > a ) > b > set_rat ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Set__Oset_It__Rat__Orat_J_001tf__c,type,
    comp_a_set_rat_c: ( a > set_rat ) > ( c > a ) > c > set_rat ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Nat__Onat,type,
    comp_a_a_nat: ( a > a ) > ( nat > a ) > nat > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Rat__Orat,type,
    comp_a_a_rat: ( a > a ) > ( rat > a ) > rat > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
    comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__b,type,
    comp_a_a_b: ( a > a ) > ( b > a ) > b > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__c,type,
    comp_a_a_c: ( a > a ) > ( c > a ) > c > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__a,type,
    comp_a_b_a: ( a > b ) > ( a > a ) > a > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__b,type,
    comp_a_b_b: ( a > b ) > ( b > a ) > b > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__c,type,
    comp_a_b_c: ( a > b ) > ( c > a ) > c > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__a,type,
    comp_a_c_a: ( a > c ) > ( a > a ) > a > c ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__b,type,
    comp_a_c_b: ( a > c ) > ( b > a ) > b > c ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__c,type,
    comp_a_c_c: ( a > c ) > ( c > a ) > c > c ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Set__Oset_It__Nat__Onat_J_001tf__b,type,
    comp_b_set_nat_b: ( b > set_nat ) > ( b > b ) > b > set_nat ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Set__Oset_It__Nat__Onat_J_001tf__c,type,
    comp_b_set_nat_c: ( b > set_nat ) > ( c > b ) > c > set_nat ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Set__Oset_It__Rat__Orat_J_001tf__a,type,
    comp_b_set_rat_a: ( b > set_rat ) > ( a > b ) > a > set_rat ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Set__Oset_It__Rat__Orat_J_001tf__b,type,
    comp_b_set_rat_b: ( b > set_rat ) > ( b > b ) > b > set_rat ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Set__Oset_It__Rat__Orat_J_001tf__c,type,
    comp_b_set_rat_c: ( b > set_rat ) > ( c > b ) > c > set_rat ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__a,type,
    comp_b_a_a: ( b > a ) > ( a > b ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__b,type,
    comp_b_a_b: ( b > a ) > ( b > b ) > b > a ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__c,type,
    comp_b_a_c: ( b > a ) > ( c > b ) > c > a ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001t__Nat__Onat,type,
    comp_b_b_nat: ( b > b ) > ( nat > b ) > nat > b ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__a,type,
    comp_b_b_a: ( b > b ) > ( a > b ) > a > b ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__b,type,
    comp_b_b_b: ( b > b ) > ( b > b ) > b > b ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__c,type,
    comp_b_b_c: ( b > b ) > ( c > b ) > c > b ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__c_001tf__a,type,
    comp_b_c_a: ( b > c ) > ( a > b ) > a > c ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__c_001tf__b,type,
    comp_b_c_b: ( b > c ) > ( b > b ) > b > c ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__c_001tf__c,type,
    comp_b_c_c: ( b > c ) > ( c > b ) > c > c ).

thf(sy_c_Fun_Ocomp_001tf__c_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
    comp_c_c_a_c_a: ( c > c > a ) > ( ( c > a ) > c ) > ( c > a ) > c > a ).

thf(sy_c_Fun_Ocomp_001tf__c_001t__Set__Oset_It__Nat__Onat_J_001tf__c,type,
    comp_c_set_nat_c: ( c > set_nat ) > ( c > c ) > c > set_nat ).

thf(sy_c_Fun_Ocomp_001tf__c_001t__Set__Oset_It__Rat__Orat_J_001tf__a,type,
    comp_c_set_rat_a: ( c > set_rat ) > ( a > c ) > a > set_rat ).

thf(sy_c_Fun_Ocomp_001tf__c_001t__Set__Oset_It__Rat__Orat_J_001tf__c,type,
    comp_c_set_rat_c: ( c > set_rat ) > ( c > c ) > c > set_rat ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__a,type,
    comp_c_a_a: ( c > a ) > ( a > c ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__b,type,
    comp_c_a_b: ( c > a ) > ( b > c ) > b > a ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__c,type,
    comp_c_a_c: ( c > a ) > ( c > c ) > c > a ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__a,type,
    comp_c_b_a: ( c > b ) > ( a > c ) > a > b ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__b,type,
    comp_c_b_b: ( c > b ) > ( b > c ) > b > b ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__c,type,
    comp_c_b_c: ( c > b ) > ( c > c ) > c > b ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001t__Nat__Onat,type,
    comp_c_c_nat: ( c > c ) > ( nat > c ) > nat > c ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001tf__a,type,
    comp_c_c_a: ( c > c ) > ( a > c ) > a > c ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001tf__b,type,
    comp_c_c_b: ( c > c ) > ( b > c ) > b > c ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001tf__c,type,
    comp_c_c_c: ( c > c ) > ( c > c ) > c > c ).

thf(sy_c_Fun_Ofcomp_001t__Nat__Onat_001tf__a_001tf__a,type,
    fcomp_nat_a_a: ( nat > a ) > ( a > a ) > nat > a ).

thf(sy_c_Fun_Ofcomp_001t__Rat__Orat_001tf__a_001tf__a,type,
    fcomp_rat_a_a: ( rat > a ) > ( a > a ) > rat > a ).

thf(sy_c_Fun_Ofcomp_001tf__a_001tf__a_001tf__a,type,
    fcomp_a_a_a: ( a > a ) > ( a > a ) > a > a ).

thf(sy_c_Fun_Ofcomp_001tf__a_001tf__a_001tf__b,type,
    fcomp_a_a_b: ( a > a ) > ( a > b ) > a > b ).

thf(sy_c_Fun_Ofcomp_001tf__a_001tf__b_001tf__a,type,
    fcomp_a_b_a: ( a > b ) > ( b > a ) > a > a ).

thf(sy_c_Fun_Ofcomp_001tf__b_001tf__a_001tf__a,type,
    fcomp_b_a_a: ( b > a ) > ( a > a ) > b > a ).

thf(sy_c_Fun_Ofcomp_001tf__b_001tf__a_001tf__b,type,
    fcomp_b_a_b: ( b > a ) > ( a > b ) > b > b ).

thf(sy_c_Fun_Ofcomp_001tf__b_001tf__b_001tf__a,type,
    fcomp_b_b_a: ( b > b ) > ( b > a ) > b > a ).

thf(sy_c_Fun_Ofcomp_001tf__b_001tf__b_001tf__b,type,
    fcomp_b_b_b: ( b > b ) > ( b > b ) > b > b ).

thf(sy_c_Fun_Ofcomp_001tf__b_001tf__c_001tf__a,type,
    fcomp_b_c_a: ( b > c ) > ( c > a ) > b > a ).

thf(sy_c_Fun_Ofcomp_001tf__b_001tf__c_001tf__b,type,
    fcomp_b_c_b: ( b > c ) > ( c > b ) > b > b ).

thf(sy_c_Fun_Ofcomp_001tf__c_001tf__a_001tf__a,type,
    fcomp_c_a_a: ( c > a ) > ( a > a ) > c > a ).

thf(sy_c_Fun_Ofcomp_001tf__c_001tf__a_001tf__b,type,
    fcomp_c_a_b: ( c > a ) > ( a > b ) > c > b ).

thf(sy_c_Fun_Ofcomp_001tf__c_001tf__b_001tf__a,type,
    fcomp_c_b_a: ( c > b ) > ( b > a ) > c > a ).

thf(sy_c_Fun_Ofcomp_001tf__c_001tf__b_001tf__b,type,
    fcomp_c_b_b: ( c > b ) > ( b > b ) > c > b ).

thf(sy_c_Fun_Ofcomp_001tf__c_001tf__c_001tf__a,type,
    fcomp_c_c_a: ( c > c ) > ( c > a ) > c > a ).

thf(sy_c_Fun_Ofcomp_001tf__c_001tf__c_001tf__b,type,
    fcomp_c_c_b: ( c > c ) > ( c > b ) > c > b ).

thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Rat__Orat,type,
    fun_upd_nat_rat: ( nat > rat ) > nat > rat > nat > rat ).

thf(sy_c_Fun_Ofun__upd_001tf__b_001tf__a,type,
    fun_upd_b_a: ( b > a ) > b > a > b > a ).

thf(sy_c_Fun_Ofun__upd_001tf__b_001tf__b,type,
    fun_upd_b_b: ( b > b ) > b > b > b > b ).

thf(sy_c_Fun_Ofun__upd_001tf__c_001tf__a,type,
    fun_upd_c_a: ( c > a ) > c > a > c > a ).

thf(sy_c_Fun_Ofun__upd_001tf__c_001tf__b,type,
    fun_upd_c_b: ( c > b ) > c > b > c > b ).

thf(sy_c_Fun_Ofun__upd_001tf__c_001tf__c,type,
    fun_upd_c_c: ( c > c ) > c > c > c > c ).

thf(sy_c_Fun_Oid_001_062_I_062_Itf__c_Mtf__a_J_Mtf__c_J,type,
    id_c_a_c: ( ( c > a ) > c ) > ( c > a ) > c ).

thf(sy_c_Fun_Oid_001_062_It__Nat__Onat_Mtf__a_J,type,
    id_nat_a: ( nat > a ) > nat > a ).

thf(sy_c_Fun_Oid_001_062_It__Rat__Orat_Mtf__a_J,type,
    id_rat_a: ( rat > a ) > rat > a ).

thf(sy_c_Fun_Oid_001_062_Itf__a_Mtf__a_J,type,
    id_a_a: ( a > a ) > a > a ).

thf(sy_c_Fun_Oid_001_062_Itf__a_Mtf__b_J,type,
    id_a_b: ( a > b ) > a > b ).

thf(sy_c_Fun_Oid_001_062_Itf__a_Mtf__c_J,type,
    id_a_c: ( a > c ) > a > c ).

thf(sy_c_Fun_Oid_001_062_Itf__b_Mtf__a_J,type,
    id_b_a: ( b > a ) > b > a ).

thf(sy_c_Fun_Oid_001_062_Itf__b_Mtf__b_J,type,
    id_b_b: ( b > b ) > b > b ).

thf(sy_c_Fun_Oid_001_062_Itf__b_Mtf__c_J,type,
    id_b_c: ( b > c ) > b > c ).

thf(sy_c_Fun_Oid_001_062_Itf__c_Mtf__a_J,type,
    id_c_a: ( c > a ) > c > a ).

thf(sy_c_Fun_Oid_001_062_Itf__c_Mtf__b_J,type,
    id_c_b: ( c > b ) > c > b ).

thf(sy_c_Fun_Oid_001_062_Itf__c_Mtf__c_J,type,
    id_c_c: ( c > c ) > c > c ).

thf(sy_c_Fun_Oid_001_Eo,type,
    id_o: $o > $o ).

thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
    id_nat: nat > nat ).

thf(sy_c_Fun_Oid_001t__Rat__Orat,type,
    id_rat: rat > rat ).

thf(sy_c_Fun_Oid_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
    id_set_b_a: set_b_a > set_b_a ).

thf(sy_c_Fun_Oid_001t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
    id_set_c_a: set_c_a > set_c_a ).

thf(sy_c_Fun_Oid_001t__Set__Oset_It__Nat__Onat_J,type,
    id_set_nat: set_nat > set_nat ).

thf(sy_c_Fun_Oid_001t__Set__Oset_It__Rat__Orat_J,type,
    id_set_rat: set_rat > set_rat ).

thf(sy_c_Fun_Oid_001t__Set__Oset_Itf__a_J,type,
    id_set_a: set_a > set_a ).

thf(sy_c_Fun_Oid_001t__Set__Oset_Itf__b_J,type,
    id_set_b: set_b > set_b ).

thf(sy_c_Fun_Oid_001t__Set__Oset_Itf__c_J,type,
    id_set_c: set_c > set_c ).

thf(sy_c_Fun_Oid_001tf__a,type,
    id_a: a > a ).

thf(sy_c_Fun_Oid_001tf__b,type,
    id_b: b > b ).

thf(sy_c_Fun_Oid_001tf__c,type,
    id_c: c > c ).

thf(sy_c_Fun_Oinj__on_001_062_Itf__b_Mtf__a_J_001_062_Itf__b_Mtf__a_J,type,
    inj_on_b_a_b_a: ( ( b > a ) > b > a ) > set_b_a > $o ).

thf(sy_c_Fun_Oinj__on_001_062_Itf__b_Mtf__b_J_001_062_Itf__b_Mtf__a_J,type,
    inj_on_b_b_b_a: ( ( b > b ) > b > a ) > set_b_b > $o ).

thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
    inj_on_c_a_c_a: ( ( c > a ) > c > a ) > set_c_a > $o ).

thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__a_J,type,
    inj_on_c_b_c_a: ( ( c > b ) > c > a ) > set_c_b > $o ).

thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__b_J,type,
    inj_on_c_b_c_b: ( ( c > b ) > c > b ) > set_c_b > $o ).

thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__c_J_001_062_Itf__c_Mtf__a_J,type,
    inj_on_c_c_c_a: ( ( c > c ) > c > a ) > set_c_c > $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__Rat__Orat,type,
    inj_on_nat_rat: ( nat > rat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Rat__Orat_001t__Nat__Onat,type,
    inj_on_rat_nat: ( rat > nat ) > set_rat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Rat__Orat_001t__Rat__Orat,type,
    inj_on_rat_rat: ( rat > rat ) > set_rat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Rat__Orat_J,type,
    inj_on1096178645466186887et_rat: ( set_nat > set_rat ) > set_set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
    inj_on_a_a: ( a > a ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001tf__b,type,
    inj_on_a_b: ( a > b ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001tf__c,type,
    inj_on_a_c: ( a > c ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__b_001tf__a,type,
    inj_on_b_a: ( b > a ) > set_b > $o ).

thf(sy_c_Fun_Oinj__on_001tf__b_001tf__b,type,
    inj_on_b_b: ( b > b ) > set_b > $o ).

thf(sy_c_Fun_Oinj__on_001tf__b_001tf__c,type,
    inj_on_b_c: ( b > c ) > set_b > $o ).

thf(sy_c_Fun_Oinj__on_001tf__c_001tf__a,type,
    inj_on_c_a: ( c > a ) > set_c > $o ).

thf(sy_c_Fun_Oinj__on_001tf__c_001tf__b,type,
    inj_on_c_b: ( c > b ) > set_c > $o ).

thf(sy_c_Fun_Oinj__on_001tf__c_001tf__c,type,
    inj_on_c_c: ( c > c ) > set_c > $o ).

thf(sy_c_Fun_Omap__fun_001_062_It__Nat__Onat_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
    map_fu8093432620143282062_nat_a: ( ( nat > a ) > c > a ) > ( ( c > a ) > nat > a ) > ( ( c > a ) > c > a ) > ( nat > a ) > nat > a ).

thf(sy_c_Fun_Omap__fun_001_062_It__Rat__Orat_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_It__Rat__Orat_Mtf__a_J,type,
    map_fu5905973340530010510_rat_a: ( ( rat > a ) > c > a ) > ( ( c > a ) > rat > a ) > ( ( c > a ) > c > a ) > ( rat > a ) > rat > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__a_J_001_062_I_062_Itf__c_Mtf__a_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_062_It__Nat__Onat_Mtf__a_J_J,type,
    map_fu6764128878074885870_nat_a: ( ( a > a ) > a > a ) > ( ( ( c > a ) > c > a ) > ( nat > a ) > nat > a ) > ( ( a > a ) > ( c > a ) > c > a ) > ( a > a ) > ( nat > a ) > nat > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__a_J_001_062_I_062_Itf__c_Mtf__a_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_It__Rat__Orat_Mtf__a_J_M_062_It__Rat__Orat_Mtf__a_J_J,type,
    map_fu1803736159731858670_rat_a: ( ( a > a ) > a > a ) > ( ( ( c > a ) > c > a ) > ( rat > a ) > rat > a ) > ( ( a > a ) > ( c > a ) > c > a ) > ( a > a ) > ( rat > a ) > rat > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__a_J_001_062_I_062_Itf__c_Mtf__a_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__a_Mtf__a_J_M_062_Itf__a_Mtf__a_J_J,type,
    map_fu1239118397357385138_a_a_a: ( ( a > a ) > a > a ) > ( ( ( c > a ) > c > a ) > ( a > a ) > a > a ) > ( ( a > a ) > ( c > a ) > c > a ) > ( a > a ) > ( a > a ) > a > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__a_J_001_062_I_062_Itf__c_Mtf__a_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__b_Mtf__a_J_M_062_Itf__b_Mtf__a_J_J,type,
    map_fu4804686371920740528_a_b_a: ( ( a > a ) > a > a ) > ( ( ( c > a ) > c > a ) > ( b > a ) > b > a ) > ( ( a > a ) > ( c > a ) > c > a ) > ( a > a ) > ( b > a ) > b > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__a_J_001_062_I_062_Itf__c_Mtf__a_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__c_Mtf__a_J_M_062_Itf__c_Mtf__a_J_J,type,
    map_fu8370254346484095918_a_c_a: ( ( a > a ) > a > a ) > ( ( ( c > a ) > c > a ) > ( c > a ) > c > a ) > ( ( a > a ) > ( c > a ) > c > a ) > ( a > a ) > ( c > a ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__a_Mtf__a_J_001_062_Itf__b_Mtf__a_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__b_Mtf__a_J_M_062_Itf__b_Mtf__a_J_J,type,
    map_fu3792799338330679792_a_b_a: ( ( a > a ) > b > a ) > ( ( ( c > b ) > c > a ) > ( b > a ) > b > a ) > ( ( b > a ) > ( c > b ) > c > a ) > ( a > a ) > ( b > a ) > b > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__a_Mtf__a_J_001_062_Itf__b_Mtf__a_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__c_Mtf__a_J_M_062_Itf__c_Mtf__a_J_J,type,
    map_fu7358367312894035182_a_c_a: ( ( a > a ) > b > a ) > ( ( ( c > b ) > c > a ) > ( c > a ) > c > a ) > ( ( b > a ) > ( c > b ) > c > a ) > ( a > a ) > ( c > a ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__a_Mtf__a_J_001_062_Itf__b_Mtf__b_J_001tf__b_001tf__a,type,
    map_fun_a_a_b_b_b_a: ( ( a > a ) > b > b ) > ( b > a ) > ( ( b > b ) > b ) > ( a > a ) > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__a_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__a_Mtf__a_J,type,
    map_fu4997489139926905770_a_a_a: ( ( a > a ) > c > a ) > ( ( c > a ) > a > a ) > ( ( c > a ) > c > a ) > ( a > a ) > a > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__a_Mtf__b_J_001_062_Itf__a_Mtf__b_J_001tf__b_001tf__b,type,
    map_fun_a_b_a_b_b_b: ( ( a > b ) > a > b ) > ( b > b ) > ( ( a > b ) > b ) > ( a > b ) > b ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__a_Mtf__b_J_001_062_Itf__b_Mtf__b_J_001tf__b_001tf__b,type,
    map_fun_a_b_b_b_b_b: ( ( a > b ) > b > b ) > ( b > b ) > ( ( b > b ) > b ) > ( a > b ) > b ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__a_Mtf__c_J_001_062_Itf__a_Mtf__b_J_001tf__b_001tf__c,type,
    map_fun_a_c_a_b_b_c: ( ( a > c ) > a > b ) > ( b > c ) > ( ( a > b ) > b ) > ( a > c ) > c ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__a_J_001_062_Itf__a_Mtf__b_J_001tf__b_001tf__a,type,
    map_fun_b_a_a_b_b_a: ( ( b > a ) > a > b ) > ( b > a ) > ( ( a > b ) > b ) > ( b > a ) > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__a_J_001_062_Itf__b_Mtf__a_J_001_062_I_062_Itf__b_Mtf__b_J_M_062_Itf__b_Mtf__a_J_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__a_J_J,type,
    map_fu593469440952318896_b_c_a: ( ( b > a ) > b > a ) > ( ( ( b > b ) > b > a ) > ( c > b ) > c > a ) > ( ( b > a ) > ( b > b ) > b > a ) > ( b > a ) > ( c > b ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__a_J_001_062_Itf__b_Mtf__a_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__b_Mtf__b_J_M_062_Itf__b_Mtf__a_J_J,type,
    map_fu2001909935855630000_b_b_a: ( ( b > a ) > b > a ) > ( ( ( c > b ) > c > a ) > ( b > b ) > b > a ) > ( ( b > a ) > ( c > b ) > c > a ) > ( b > a ) > ( b > b ) > b > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__a_J_001_062_Itf__b_Mtf__a_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__a_J_J,type,
    map_fu5567477910418985390_b_c_a: ( ( b > a ) > b > a ) > ( ( ( c > b ) > c > a ) > ( c > b ) > c > a ) > ( ( b > a ) > ( c > b ) > c > a ) > ( b > a ) > ( c > b ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__a_J_001_062_Itf__b_Mtf__b_J_001tf__b_001tf__a,type,
    map_fun_b_a_b_b_b_a: ( ( b > a ) > b > b ) > ( b > a ) > ( ( b > b ) > b ) > ( b > a ) > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_I_062_Itf__c_Mtf__c_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__b_Mtf__b_J_M_062_Itf__b_Mtf__a_J_J,type,
    map_fu990022902265569264_b_b_a: ( ( b > a ) > c > a ) > ( ( ( c > c ) > c > a ) > ( b > b ) > b > a ) > ( ( c > a ) > ( c > c ) > c > a ) > ( b > a ) > ( b > b ) > b > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_I_062_Itf__c_Mtf__c_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__a_J_J,type,
    map_fu4555590876828924654_b_c_a: ( ( b > a ) > c > a ) > ( ( ( c > c ) > c > a ) > ( c > b ) > c > a ) > ( ( c > a ) > ( c > c ) > c > a ) > ( b > a ) > ( c > b ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__b_Mtf__a_J,type,
    map_fu7382088187427195432_a_b_a: ( ( b > a ) > c > a ) > ( ( c > a ) > b > a ) > ( ( c > a ) > c > a ) > ( b > a ) > b > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__a_J_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__b_Mtf__a_J,type,
    map_fu7046846374025825705_a_b_a: ( ( b > a ) > c > b ) > ( ( c > a ) > b > a ) > ( ( c > b ) > c > a ) > ( b > a ) > b > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__b_J_001_062_Itf__a_Mtf__b_J_001tf__b_001tf__b,type,
    map_fun_b_b_a_b_b_b: ( ( b > b ) > a > b ) > ( b > b ) > ( ( a > b ) > b ) > ( b > b ) > b ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__b_J_001_062_Itf__b_Mtf__a_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__b_J_J,type,
    map_fu5478692276376878896_b_c_b: ( ( b > b ) > b > a ) > ( ( ( c > b ) > c > a ) > ( c > b ) > c > b ) > ( ( b > a ) > ( c > b ) > c > a ) > ( b > b ) > ( c > b ) > c > b ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__b_J_001_062_Itf__b_Mtf__b_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__b_J_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__b_J_J,type,
    map_fu6441608359614649650_b_c_b: ( ( b > b ) > b > b ) > ( ( ( c > b ) > c > b ) > ( c > b ) > c > b ) > ( ( b > b ) > ( c > b ) > c > b ) > ( b > b ) > ( c > b ) > c > b ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__b_J_001_062_Itf__b_Mtf__b_J_001tf__b_001tf__b,type,
    map_fun_b_b_b_b_b_b: ( ( b > b ) > b > b ) > ( b > b ) > ( ( b > b ) > b ) > ( b > b ) > b ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__b_J_001_062_Itf__c_Mtf__a_J_001_062_I_062_Itf__c_Mtf__c_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__b_J_J,type,
    map_fu4466805242786818160_b_c_b: ( ( b > b ) > c > a ) > ( ( ( c > c ) > c > a ) > ( c > b ) > c > b ) > ( ( c > a ) > ( c > c ) > c > a ) > ( b > b ) > ( c > b ) > c > b ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__b_J_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__b_Mtf__a_J,type,
    map_fu1196190707875695402_a_b_a: ( ( b > b ) > c > b ) > ( ( c > a ) > b > a ) > ( ( c > b ) > c > a ) > ( b > b ) > b > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__b_Mtf__b_J_001_062_Itf__c_Mtf__c_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__b_Mtf__a_J,type,
    map_fu860948894474325675_a_b_a: ( ( b > b ) > c > c ) > ( ( c > a ) > b > a ) > ( ( c > c ) > c > a ) > ( b > b ) > b > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__a_J_001_062_Itf__a_Mtf__b_J_001tf__b_001tf__a,type,
    map_fun_c_a_a_b_b_a: ( ( c > a ) > a > b ) > ( b > a ) > ( ( a > b ) > b ) > ( c > a ) > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__a_J_001_062_Itf__b_Mtf__a_J_001_062_I_062_Itf__c_Mtf__b_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__c_Mtf__c_J_M_062_Itf__c_Mtf__a_J_J,type,
    map_fu3776588507943935598_c_c_a: ( ( c > a ) > b > a ) > ( ( ( c > b ) > c > a ) > ( c > c ) > c > a ) > ( ( b > a ) > ( c > b ) > c > a ) > ( c > a ) > ( c > c ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_I_062_Itf__c_Mtf__c_J_M_062_Itf__c_Mtf__a_J_J_001_062_I_062_Itf__c_Mtf__c_J_M_062_Itf__c_Mtf__a_J_J,type,
    map_fu2764701474353874862_c_c_a: ( ( c > a ) > c > a ) > ( ( ( c > c ) > c > a ) > ( c > c ) > c > a ) > ( ( c > a ) > ( c > c ) > c > a ) > ( c > a ) > ( c > c ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
    map_fu543315198072709286_a_c_a: ( ( c > a ) > c > a ) > ( ( c > a ) > c > a ) > ( ( c > a ) > c > a ) > ( c > a ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001tf__c_001tf__c,type,
    map_fun_c_a_c_a_c_c: ( ( c > a ) > c > a ) > ( c > c ) > ( ( c > a ) > c ) > ( c > a ) > c ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
    map_fu208073384671339559_a_c_a: ( ( c > a ) > c > b ) > ( ( c > a ) > c > a ) > ( ( c > b ) > c > a ) > ( c > a ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__b_J_001_062_Itf__a_Mtf__b_J_001tf__b_001tf__b,type,
    map_fun_c_b_a_b_b_b: ( ( c > b ) > a > b ) > ( b > b ) > ( ( a > b ) > b ) > ( c > b ) > b ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__b_J_001_062_Itf__b_Mtf__b_J_001_062_Itf__b_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
    map_fu2703170745860999978_a_c_a: ( ( c > b ) > b > b ) > ( ( b > a ) > c > a ) > ( ( b > b ) > b > a ) > ( c > b ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
    map_fu3580789755375985064_a_c_a: ( ( c > b ) > c > b ) > ( ( c > a ) > c > a ) > ( ( c > b ) > c > a ) > ( c > b ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__b_J,type,
    map_fu3580789759679213865_a_c_b: ( ( c > b ) > c > b ) > ( ( c > a ) > c > b ) > ( ( c > b ) > c > a ) > ( c > b ) > c > b ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__b_J,type,
    map_fu5259323019911985834_b_c_b: ( ( c > b ) > c > b ) > ( ( c > b ) > c > b ) > ( ( c > b ) > c > b ) > ( c > b ) > c > b ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__c_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
    map_fu3245547941974615337_a_c_a: ( ( c > b ) > c > c ) > ( ( c > a ) > c > a ) > ( ( c > c ) > c > a ) > ( c > b ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__c_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__b_J,type,
    map_fu3245547946277844138_a_c_b: ( ( c > b ) > c > c ) > ( ( c > a ) > c > b ) > ( ( c > c ) > c > a ) > ( c > b ) > c > b ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__c_J_001_062_Itf__c_Mtf__b_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
    map_fu6953506126080630569_a_c_a: ( ( c > c ) > c > b ) > ( ( c > a ) > c > a ) > ( ( c > b ) > c > a ) > ( c > c ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_062_Itf__c_Mtf__c_J_001_062_Itf__c_Mtf__c_J_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
    map_fu6618264312679260842_a_c_a: ( ( c > c ) > c > c ) > ( ( c > a ) > c > a ) > ( ( c > c ) > c > a ) > ( c > c ) > c > a ).

thf(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
    map_fu1481860808675471894_a_a_a: ( $o > $o ) > ( ( a > a > a ) > a > a > a ) > ( $o > a > a > a ) > $o > a > a > a ).

thf(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_001_062_Itf__b_M_062_Itf__b_Mtf__b_J_J,type,
    map_fu6771103618013453783_b_b_b: ( $o > $o ) > ( ( a > a > a ) > b > b > b ) > ( $o > a > a > a ) > $o > b > b > b ).

thf(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_001_062_Itf__c_M_062_Itf__c_Mtf__c_J_J,type,
    map_fu2836974390496659864_c_c_c: ( $o > $o ) > ( ( a > a > a ) > c > c > c ) > ( $o > a > a > a ) > $o > c > c > c ).

thf(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001_062_Itf__b_M_062_Itf__b_Mtf__b_J_J_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
    map_fu2226394946436255765_a_a_a: ( $o > $o ) > ( ( b > b > b ) > a > a > a ) > ( $o > b > b > b ) > $o > a > a > a ).

thf(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001_062_Itf__b_M_062_Itf__b_Mtf__b_J_J_001_062_Itf__b_M_062_Itf__b_Mtf__b_J_J,type,
    map_fu7515637755774237654_b_b_b: ( $o > $o ) > ( ( b > b > b ) > b > b > b ) > ( $o > b > b > b ) > $o > b > b > b ).

thf(sy_c_Fun_Omap__fun_001t__Nat__Onat_001tf__c_001tf__a_001tf__a,type,
    map_fun_nat_c_a_a: ( nat > c ) > ( a > a ) > ( c > a ) > nat > a ).

thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001tf__c_001tf__a_001tf__a,type,
    map_fun_rat_c_a_a: ( rat > c ) > ( a > a ) > ( c > a ) > rat > a ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__a_001_062_I_062_Itf__a_Mtf__b_J_Mtf__b_J_001_062_I_062_Itf__a_Mtf__b_J_Mtf__b_J,type,
    map_fu8102468183056746006_a_b_b: ( a > a ) > ( ( ( a > b ) > b ) > ( a > b ) > b ) > ( a > ( a > b ) > b ) > a > ( a > b ) > b ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__a_001_062_I_062_Itf__a_Mtf__b_J_Mtf__b_J_001_062_I_062_Itf__a_Mtf__c_J_Mtf__c_J,type,
    map_fu7466819186754988118_a_c_c: ( a > a ) > ( ( ( a > b ) > b ) > ( a > c ) > c ) > ( a > ( a > b ) > b ) > a > ( a > c ) > c ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__a_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__a_J,type,
    map_fun_a_a_a_a_a_a: ( a > a ) > ( ( a > a ) > a > a ) > ( a > a > a ) > a > a > a ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__a_001tf__a_001tf__a,type,
    map_fun_a_a_a_a: ( a > a ) > ( a > a ) > ( a > a ) > a > a ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__a_001tf__b_001tf__a,type,
    map_fun_a_a_b_a: ( a > a ) > ( b > a ) > ( a > b ) > a > a ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__a_001tf__b_001tf__b,type,
    map_fun_a_a_b_b: ( a > a ) > ( b > b ) > ( a > b ) > a > b ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__a_001tf__c_001tf__a,type,
    map_fun_a_a_c_a: ( a > a ) > ( c > a ) > ( a > c ) > a > a ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__a_001tf__c_001tf__b,type,
    map_fun_a_a_c_b: ( a > a ) > ( c > b ) > ( a > c ) > a > b ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__a_001tf__c_001tf__c,type,
    map_fun_a_a_c_c: ( a > a ) > ( c > c ) > ( a > c ) > a > c ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__b_001_062_I_062_Itf__b_Mtf__b_J_Mtf__b_J_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
    map_fu4840009800104922774_a_a_a: ( a > b ) > ( ( ( b > b ) > b ) > ( a > a ) > a ) > ( b > ( b > b ) > b ) > a > ( a > a ) > a ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__b_001_062_I_062_Itf__b_Mtf__b_J_Mtf__b_J_001_062_I_062_Itf__a_Mtf__b_J_Mtf__b_J,type,
    map_fu4204360803803164886_a_b_b: ( a > b ) > ( ( ( b > b ) > b ) > ( a > b ) > b ) > ( b > ( b > b ) > b ) > a > ( a > b ) > b ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__b_001_062_Itf__b_Mtf__b_J_001_062_Itf__a_Mtf__a_J,type,
    map_fun_a_b_b_b_a_a: ( a > b ) > ( ( b > b ) > a > a ) > ( b > b > b ) > a > a > a ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__b_001tf__a_001tf__b,type,
    map_fun_a_b_a_b: ( a > b ) > ( a > b ) > ( b > a ) > a > b ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__b_001tf__b_001tf__a,type,
    map_fun_a_b_b_a: ( a > b ) > ( b > a ) > ( b > b ) > a > a ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__b_001tf__b_001tf__b,type,
    map_fun_a_b_b_b: ( a > b ) > ( b > b ) > ( b > b ) > a > b ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__c_001tf__a_001tf__a,type,
    map_fun_a_c_a_a: ( a > c ) > ( a > a ) > ( c > a ) > a > a ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__c_001tf__a_001tf__b,type,
    map_fun_a_c_a_b: ( a > c ) > ( a > b ) > ( c > a ) > a > b ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__c_001tf__b_001tf__b,type,
    map_fun_a_c_b_b: ( a > c ) > ( b > b ) > ( c > b ) > a > b ).

thf(sy_c_Fun_Omap__fun_001tf__a_001tf__c_001tf__c_001tf__a,type,
    map_fun_a_c_c_a: ( a > c ) > ( c > a ) > ( c > c ) > a > a ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__a_001_062_I_062_Itf__a_Mtf__b_J_Mtf__b_J_001_062_I_062_Itf__b_Mtf__a_J_Mtf__a_J,type,
    map_fu9089132640597423382_b_a_a: ( b > a ) > ( ( ( a > b ) > b ) > ( b > a ) > a ) > ( a > ( a > b ) > b ) > b > ( b > a ) > a ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__a_001_062_I_062_Itf__a_Mtf__b_J_Mtf__b_J_001_062_I_062_Itf__b_Mtf__b_J_Mtf__b_J,type,
    map_fu8453483644295665494_b_b_b: ( b > a ) > ( ( ( a > b ) > b ) > ( b > b ) > b ) > ( a > ( a > b ) > b ) > b > ( b > b ) > b ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__a_001_062_Itf__a_Mtf__a_J_001_062_Itf__b_Mtf__b_J,type,
    map_fun_b_a_a_a_b_b: ( b > a ) > ( ( a > a ) > b > b ) > ( a > a > a ) > b > b > b ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__a_001tf__a_001tf__a,type,
    map_fun_b_a_a_a: ( b > a ) > ( a > a ) > ( a > a ) > b > a ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__a_001tf__a_001tf__b,type,
    map_fun_b_a_a_b: ( b > a ) > ( a > b ) > ( a > a ) > b > b ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__a_001tf__b_001tf__a,type,
    map_fun_b_a_b_a: ( b > a ) > ( b > a ) > ( a > b ) > b > a ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__a_001tf__b_001tf__b,type,
    map_fun_b_a_b_b: ( b > a ) > ( b > b ) > ( a > b ) > b > b ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__a_001tf__c_001tf__a,type,
    map_fun_b_a_c_a: ( b > a ) > ( c > a ) > ( a > c ) > b > a ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__a_001tf__c_001tf__b,type,
    map_fun_b_a_c_b: ( b > a ) > ( c > b ) > ( a > c ) > b > b ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__a_001tf__c_001tf__c,type,
    map_fun_b_a_c_c: ( b > a ) > ( c > c ) > ( a > c ) > b > c ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__b_001_062_I_062_Itf__b_Mtf__b_J_Mtf__b_J_001_062_I_062_Itf__b_Mtf__a_J_Mtf__a_J,type,
    map_fu5191025261343842262_b_a_a: ( b > b ) > ( ( ( b > b ) > b ) > ( b > a ) > a ) > ( b > ( b > b ) > b ) > b > ( b > a ) > a ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__b_001_062_I_062_Itf__b_Mtf__b_J_Mtf__b_J_001_062_I_062_Itf__b_Mtf__b_J_Mtf__b_J,type,
    map_fu4555376265042084374_b_b_b: ( b > b ) > ( ( ( b > b ) > b ) > ( b > b ) > b ) > ( b > ( b > b ) > b ) > b > ( b > b ) > b ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__b_001_062_Itf__b_Mtf__b_J_001_062_Itf__b_Mtf__b_J,type,
    map_fun_b_b_b_b_b_b2: ( b > b ) > ( ( b > b ) > b > b ) > ( b > b > b ) > b > b > b ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__b_001tf__a_001tf__a,type,
    map_fun_b_b_a_a: ( b > b ) > ( a > a ) > ( b > a ) > b > a ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__b_001tf__a_001tf__b,type,
    map_fun_b_b_a_b: ( b > b ) > ( a > b ) > ( b > a ) > b > b ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__b_001tf__b_001tf__a,type,
    map_fun_b_b_b_a: ( b > b ) > ( b > a ) > ( b > b ) > b > a ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__b_001tf__b_001tf__b,type,
    map_fun_b_b_b_b: ( b > b ) > ( b > b ) > ( b > b ) > b > b ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__b_001tf__c_001tf__c,type,
    map_fun_b_b_c_c: ( b > b ) > ( c > c ) > ( b > c ) > b > c ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__c_001tf__a_001tf__a,type,
    map_fun_b_c_a_a: ( b > c ) > ( a > a ) > ( c > a ) > b > a ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__c_001tf__b_001tf__b,type,
    map_fun_b_c_b_b: ( b > c ) > ( b > b ) > ( c > b ) > b > b ).

thf(sy_c_Fun_Omap__fun_001tf__b_001tf__c_001tf__c_001tf__b,type,
    map_fun_b_c_c_b: ( b > c ) > ( c > b ) > ( c > c ) > b > b ).

thf(sy_c_Fun_Omap__fun_001tf__c_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J_001tf__c,type,
    map_fun_c_c_a_c_a_c: ( c > c > a ) > ( ( c > a ) > c ) > ( ( c > a ) > c > a ) > c > c ).

thf(sy_c_Fun_Omap__fun_001tf__c_001t__Nat__Onat_001tf__a_001tf__a,type,
    map_fun_c_nat_a_a: ( c > nat ) > ( a > a ) > ( nat > a ) > c > a ).

thf(sy_c_Fun_Omap__fun_001tf__c_001t__Rat__Orat_001tf__a_001tf__a,type,
    map_fun_c_rat_a_a: ( c > rat ) > ( a > a ) > ( rat > a ) > c > a ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__a_001_062_I_062_Itf__a_Mtf__b_J_Mtf__b_J_001_062_I_062_Itf__c_Mtf__a_J_Mtf__a_J,type,
    map_fu216776064981567062_c_a_a: ( c > a ) > ( ( ( a > b ) > b ) > ( c > a ) > a ) > ( a > ( a > b ) > b ) > c > ( c > a ) > a ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__a_001_062_I_062_Itf__a_Mtf__b_J_Mtf__b_J_001_062_I_062_Itf__c_Mtf__b_J_Mtf__b_J,type,
    map_fu8804499105534584982_c_b_b: ( c > a ) > ( ( ( a > b ) > b ) > ( c > b ) > b ) > ( a > ( a > b ) > b ) > c > ( c > b ) > b ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__a_001_062_Itf__a_Mtf__a_J_001_062_Itf__c_Mtf__c_J,type,
    map_fun_c_a_a_a_c_c: ( c > a ) > ( ( a > a ) > c > c ) > ( a > a > a ) > c > c > c ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__a_001tf__a_001tf__a,type,
    map_fun_c_a_a_a: ( c > a ) > ( a > a ) > ( a > a ) > c > a ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__a_001tf__a_001tf__c,type,
    map_fun_c_a_a_c: ( c > a ) > ( a > c ) > ( a > a ) > c > c ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__a_001tf__b_001tf__a,type,
    map_fun_c_a_b_a: ( c > a ) > ( b > a ) > ( a > b ) > c > a ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__a_001tf__b_001tf__b,type,
    map_fun_c_a_b_b: ( c > a ) > ( b > b ) > ( a > b ) > c > b ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__a_001tf__c_001tf__a,type,
    map_fun_c_a_c_a: ( c > a ) > ( c > a ) > ( a > c ) > c > a ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__a_001tf__c_001tf__b,type,
    map_fun_c_a_c_b: ( c > a ) > ( c > b ) > ( a > c ) > c > b ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__a_001tf__c_001tf__c,type,
    map_fun_c_a_c_c: ( c > a ) > ( c > c ) > ( a > c ) > c > c ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__b_001tf__a_001tf__a,type,
    map_fun_c_b_a_a: ( c > b ) > ( a > a ) > ( b > a ) > c > a ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__b_001tf__a_001tf__b,type,
    map_fun_c_b_a_b: ( c > b ) > ( a > b ) > ( b > a ) > c > b ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__b_001tf__b_001tf__a,type,
    map_fun_c_b_b_a: ( c > b ) > ( b > a ) > ( b > b ) > c > a ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__b_001tf__b_001tf__b,type,
    map_fun_c_b_b_b: ( c > b ) > ( b > b ) > ( b > b ) > c > b ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__b_001tf__b_001tf__c,type,
    map_fun_c_b_b_c: ( c > b ) > ( b > c ) > ( b > b ) > c > c ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__b_001tf__c_001tf__a,type,
    map_fun_c_b_c_a: ( c > b ) > ( c > a ) > ( b > c ) > c > a ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__c_001tf__a_001tf__a,type,
    map_fun_c_c_a_a: ( c > c ) > ( a > a ) > ( c > a ) > c > a ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__c_001tf__a_001tf__b,type,
    map_fun_c_c_a_b: ( c > c ) > ( a > b ) > ( c > a ) > c > b ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__c_001tf__b_001tf__a,type,
    map_fun_c_c_b_a: ( c > c ) > ( b > a ) > ( c > b ) > c > a ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__c_001tf__b_001tf__b,type,
    map_fun_c_c_b_b: ( c > c ) > ( b > b ) > ( c > b ) > c > b ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__c_001tf__b_001tf__c,type,
    map_fun_c_c_b_c: ( c > c ) > ( b > c ) > ( c > b ) > c > c ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__c_001tf__c_001tf__a,type,
    map_fun_c_c_c_a: ( c > c ) > ( c > a ) > ( c > c ) > c > a ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__c_001tf__c_001tf__b,type,
    map_fun_c_c_c_b: ( c > c ) > ( c > b ) > ( c > c ) > c > b ).

thf(sy_c_Fun_Omap__fun_001tf__c_001tf__c_001tf__c_001tf__c,type,
    map_fun_c_c_c_c: ( c > c ) > ( c > c ) > ( c > c ) > c > c ).

thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Rat__Orat,type,
    the_inv_into_nat_rat: set_nat > ( nat > rat ) > rat > nat ).

thf(sy_c_Fun_Othe__inv__into_001t__Rat__Orat_001t__Nat__Onat,type,
    the_inv_into_rat_nat: set_rat > ( rat > nat ) > nat > rat ).

thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__a,type,
    the_inv_into_a_a: set_a > ( a > a ) > a > a ).

thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__b,type,
    the_inv_into_a_b: set_a > ( a > b ) > b > a ).

thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__c,type,
    the_inv_into_a_c: set_a > ( a > c ) > c > a ).

thf(sy_c_Fun_Othe__inv__into_001tf__b_001tf__a,type,
    the_inv_into_b_a: set_b > ( b > a ) > a > b ).

thf(sy_c_Fun_Othe__inv__into_001tf__b_001tf__b,type,
    the_inv_into_b_b: set_b > ( b > b ) > b > b ).

thf(sy_c_Fun_Othe__inv__into_001tf__b_001tf__c,type,
    the_inv_into_b_c: set_b > ( b > c ) > c > b ).

thf(sy_c_Fun_Othe__inv__into_001tf__c_001tf__a,type,
    the_inv_into_c_a: set_c > ( c > a ) > a > c ).

thf(sy_c_Fun_Othe__inv__into_001tf__c_001tf__b,type,
    the_inv_into_c_b: set_c > ( c > b ) > b > c ).

thf(sy_c_Fun_Othe__inv__into_001tf__c_001tf__c,type,
    the_inv_into_c_c: set_c > ( c > c ) > c > c ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
    minus_minus_rat: rat > rat > rat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Rat__Orat_J,type,
    minus_minus_set_rat: set_rat > set_rat > set_rat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
    minus_5007325069933123869et_rat: set_set_rat > set_set_rat > set_set_rat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: nat > nat > nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
    uminus_uminus_rat: rat > rat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Rat__Orat_J,type,
    uminus2201863774496077783et_rat: set_rat > set_rat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
    uminus_uminus_set_a: set_a > set_a ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__b_J,type,
    uminus_uminus_set_b: set_b > set_b ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__c_J,type,
    uminus_uminus_set_c: set_c > set_c ).

thf(sy_c_If_001tf__a,type,
    if_a: $o > a > a > a ).

thf(sy_c_If_001tf__b,type,
    if_b: $o > b > b > b ).

thf(sy_c_If_001tf__c,type,
    if_c: $o > c > c > c ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
    inf_inf_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Rat__Orat_J,type,
    inf_inf_set_rat: set_rat > set_rat > set_rat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Rat__Orat_J,type,
    sup_sup_set_rat: set_rat > set_rat > set_rat ).

thf(sy_c_Matrix_Omap__vec_001t__Nat__Onat_001t__Rat__Orat,type,
    map_vec_nat_rat: ( nat > rat ) > vec_nat > vec_rat ).

thf(sy_c_Matrix_Omap__vec_001t__Nat__Onat_001tf__c,type,
    map_vec_nat_c: ( nat > c ) > vec_nat > vec_c ).

thf(sy_c_Matrix_Omap__vec_001t__Rat__Orat_001tf__c,type,
    map_vec_rat_c: ( rat > c ) > vec_rat > vec_c ).

thf(sy_c_Matrix_Omap__vec_001tf__a_001t__Rat__Orat,type,
    map_vec_a_rat: ( a > rat ) > vec_a > vec_rat ).

thf(sy_c_Matrix_Omap__vec_001tf__a_001tf__a,type,
    map_vec_a_a: ( a > a ) > vec_a > vec_a ).

thf(sy_c_Matrix_Omap__vec_001tf__a_001tf__b,type,
    map_vec_a_b: ( a > b ) > vec_a > vec_b ).

thf(sy_c_Matrix_Omap__vec_001tf__a_001tf__c,type,
    map_vec_a_c: ( a > c ) > vec_a > vec_c ).

thf(sy_c_Matrix_Omap__vec_001tf__b_001tf__a,type,
    map_vec_b_a: ( b > a ) > vec_b > vec_a ).

thf(sy_c_Matrix_Omap__vec_001tf__b_001tf__c,type,
    map_vec_b_c: ( b > c ) > vec_b > vec_c ).

thf(sy_c_Matrix_Omap__vec_001tf__c_001t__Nat__Onat,type,
    map_vec_c_nat: ( c > nat ) > vec_c > vec_nat ).

thf(sy_c_Matrix_Omap__vec_001tf__c_001t__Rat__Orat,type,
    map_vec_c_rat: ( c > rat ) > vec_c > vec_rat ).

thf(sy_c_Matrix_Omap__vec_001tf__c_001tf__a,type,
    map_vec_c_a: ( c > a ) > vec_c > vec_a ).

thf(sy_c_Matrix_Omap__vec_001tf__c_001tf__b,type,
    map_vec_c_b: ( c > b ) > vec_c > vec_b ).

thf(sy_c_Matrix_Omap__vec_001tf__c_001tf__c,type,
    map_vec_c_c: ( c > c ) > vec_c > vec_c ).

thf(sy_c_Matrix_Ovec__set_001t__Nat__Onat,type,
    vec_set_nat: vec_nat > set_nat ).

thf(sy_c_Matrix_Ovec__set_001t__Rat__Orat,type,
    vec_set_rat: vec_rat > set_rat ).

thf(sy_c_Matrix_Ovec__set_001tf__a,type,
    vec_set_a: vec_a > set_a ).

thf(sy_c_Matrix_Ovec__set_001tf__b,type,
    vec_set_b: vec_b > set_b ).

thf(sy_c_Matrix_Ovec__set_001tf__c,type,
    vec_set_c: vec_c > set_c ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
    bot_bot_set_set_rat: set_set_rat ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: rat > rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_eq_set_rat: set_rat > set_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
    ord_le513522071413781156et_rat: set_set_rat > set_set_rat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
    top_top_nat_o: nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Rat__Orat_M_Eo_J,type,
    top_top_rat_o: rat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
    top_top_set_nat_o: set_nat_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Rat__Orat_M_Eo_J_J,type,
    top_top_set_rat_o: set_rat_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
    top_top_set_b_a: set_b_a ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
    top_top_set_b_b: set_b_b ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
    top_top_set_c_a: set_c_a ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__c_Mtf__b_J_J,type,
    top_top_set_c_b: set_c_b ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__c_Mtf__c_J_J,type,
    top_top_set_c_c: set_c_c ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
    top_to8920198386146353926on_nat: set_option_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Rat__Orat_J_J,type,
    top_to2540212048668676366on_rat: set_option_rat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    top_to7513191607651882425at_rat: set_Pr4105333604307423337at_rat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Rat__Orat_Mt__Nat__Onat_J_J,type,
    top_to269121717765781945at_nat: set_Pr6084635751276098665at_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
    top_to3112507417142880193at_rat: set_Pr8928021450653196913at_rat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Rat__Orat_J,type,
    top_top_set_rat: set_rat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    top_top_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
    top_top_set_set_rat: set_set_rat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    top_top_set_set_a: set_set_a ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    top_top_set_set_b: set_set_b ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__c_J_J,type,
    top_top_set_set_c: set_set_c ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    top_to281834657035230061at_rat: set_Sum_sum_nat_rat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Rat__Orat_Mt__Nat__Onat_J_J,type,
    top_to2261136804003905389at_nat: set_Sum_sum_rat_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
    top_to5104522503381003637at_rat: set_Sum_sum_rat_rat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
    top_top_set_a: set_a ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__b_J,type,
    top_top_set_b: set_b ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__c_J,type,
    top_top_set_c: set_c ).

thf(sy_c_Quotient_OQuotient3_001_062_Itf__b_Mtf__a_J_001_062_Itf__b_Mtf__a_J,type,
    quotient3_b_a_b_a: ( ( b > a ) > ( b > a ) > $o ) > ( ( b > a ) > b > a ) > ( ( b > a ) > b > a ) > $o ).

thf(sy_c_Quotient_OQuotient3_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
    quotient3_c_a_c_a: ( ( c > a ) > ( c > a ) > $o ) > ( ( c > a ) > c > a ) > ( ( c > a ) > c > a ) > $o ).

thf(sy_c_Quotient_OQuotient3_001_062_Itf__c_Mtf__a_J_001tf__c,type,
    quotient3_c_a_c: ( ( c > a ) > ( c > a ) > $o ) > ( ( c > a ) > c ) > ( c > c > a ) > $o ).

thf(sy_c_Quotient_OQuotient3_001tf__a_001tf__a,type,
    quotient3_a_a: ( a > a > $o ) > ( a > a ) > ( a > a ) > $o ).

thf(sy_c_Quotient_OQuotient3_001tf__a_001tf__b,type,
    quotient3_a_b: ( a > a > $o ) > ( a > b ) > ( b > a ) > $o ).

thf(sy_c_Quotient_OQuotient3_001tf__a_001tf__c,type,
    quotient3_a_c: ( a > a > $o ) > ( a > c ) > ( c > a ) > $o ).

thf(sy_c_Quotient_OQuotient3_001tf__b_001tf__a,type,
    quotient3_b_a: ( b > b > $o ) > ( b > a ) > ( a > b ) > $o ).

thf(sy_c_Quotient_OQuotient3_001tf__b_001tf__b,type,
    quotient3_b_b: ( b > b > $o ) > ( b > b ) > ( b > b ) > $o ).

thf(sy_c_Quotient_OQuotient3_001tf__b_001tf__c,type,
    quotient3_b_c: ( b > b > $o ) > ( b > c ) > ( c > b ) > $o ).

thf(sy_c_Quotient_OQuotient3_001tf__c_001t__Nat__Onat,type,
    quotient3_c_nat: ( c > c > $o ) > ( c > nat ) > ( nat > c ) > $o ).

thf(sy_c_Quotient_OQuotient3_001tf__c_001t__Rat__Orat,type,
    quotient3_c_rat: ( c > c > $o ) > ( c > rat ) > ( rat > c ) > $o ).

thf(sy_c_Quotient_OQuotient3_001tf__c_001tf__a,type,
    quotient3_c_a: ( c > c > $o ) > ( c > a ) > ( a > c ) > $o ).

thf(sy_c_Quotient_OQuotient3_001tf__c_001tf__b,type,
    quotient3_c_b: ( c > c > $o ) > ( c > b ) > ( b > c ) > $o ).

thf(sy_c_Quotient_OQuotient3_001tf__c_001tf__c,type,
    quotient3_c_c: ( c > c > $o ) > ( c > c ) > ( c > c ) > $o ).

thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Rat__Orat,type,
    field_6020823756834552118ts_rat: set_rat ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: nat > nat > $o ).

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
    collect_rat: ( rat > $o ) > set_rat ).

thf(sy_c_Set_OCollect_001tf__a,type,
    collect_a: ( a > $o ) > set_a ).

thf(sy_c_Set_OCollect_001tf__b,type,
    collect_b: ( b > $o ) > set_b ).

thf(sy_c_Set_OCollect_001tf__c,type,
    collect_c: ( c > $o ) > set_c ).

thf(sy_c_Set_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

thf(sy_c_Set_OPow_001t__Rat__Orat,type,
    pow_rat: set_rat > set_set_rat ).

thf(sy_c_Set_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
    bind_nat_nat: set_nat > ( nat > set_nat ) > set_nat ).

thf(sy_c_Set_Obind_001t__Nat__Onat_001t__Rat__Orat,type,
    bind_nat_rat: set_nat > ( nat > set_rat ) > set_rat ).

thf(sy_c_Set_Obind_001t__Rat__Orat_001t__Nat__Onat,type,
    bind_rat_nat: set_rat > ( rat > set_nat ) > set_nat ).

thf(sy_c_Set_Odisjnt_001t__Nat__Onat,type,
    disjnt_nat: set_nat > set_nat > $o ).

thf(sy_c_Set_Odisjnt_001t__Rat__Orat,type,
    disjnt_rat: set_rat > set_rat > $o ).

thf(sy_c_Set_Oimage_001_062_Itf__b_Mtf__a_J_001_062_Itf__b_Mtf__a_J,type,
    image_b_a_b_a: ( ( b > a ) > b > a ) > set_b_a > set_b_a ).

thf(sy_c_Set_Oimage_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
    image_c_a_c_a: ( ( c > a ) > c > a ) > set_c_a > set_c_a ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Rat__Orat,type,
    image_nat_rat: ( nat > rat ) > set_nat > set_rat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Rat__Orat_J,type,
    image_nat_set_rat: ( nat > set_rat ) > set_nat > set_set_rat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
    image_nat_a: ( nat > a ) > set_nat > set_a ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__b,type,
    image_nat_b: ( nat > b ) > set_nat > set_b ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__c,type,
    image_nat_c: ( nat > c ) > set_nat > set_c ).

thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Nat__Onat,type,
    image_rat_nat: ( rat > nat ) > set_rat > set_nat ).

thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Rat__Orat,type,
    image_rat_rat: ( rat > rat ) > set_rat > set_rat ).

thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Set__Oset_It__Rat__Orat_J,type,
    image_rat_set_rat: ( rat > set_rat ) > set_rat > set_set_rat ).

thf(sy_c_Set_Oimage_001t__Rat__Orat_001tf__a,type,
    image_rat_a: ( rat > a ) > set_rat > set_a ).

thf(sy_c_Set_Oimage_001t__Rat__Orat_001tf__b,type,
    image_rat_b: ( rat > b ) > set_rat > set_b ).

thf(sy_c_Set_Oimage_001t__Rat__Orat_001tf__c,type,
    image_rat_c: ( rat > c ) > set_rat > set_c ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Rat__Orat_J,type,
    image_4408659257933336347et_rat: ( set_nat > set_rat ) > set_set_nat > set_set_rat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
    image_3939399684171694371et_rat: ( set_rat > set_rat ) > set_set_rat > set_set_rat ).

thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
    image_a_nat: ( a > nat ) > set_a > set_nat ).

thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
    image_a_set_nat: ( a > set_nat ) > set_a > set_set_nat ).

thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Rat__Orat_J,type,
    image_a_set_rat: ( a > set_rat ) > set_a > set_set_rat ).

thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
    image_a_a: ( a > a ) > set_a > set_a ).

thf(sy_c_Set_Oimage_001tf__a_001tf__b,type,
    image_a_b: ( a > b ) > set_a > set_b ).

thf(sy_c_Set_Oimage_001tf__a_001tf__c,type,
    image_a_c: ( a > c ) > set_a > set_c ).

thf(sy_c_Set_Oimage_001tf__b_001t__Nat__Onat,type,
    image_b_nat: ( b > nat ) > set_b > set_nat ).

thf(sy_c_Set_Oimage_001tf__b_001t__Set__Oset_It__Rat__Orat_J,type,
    image_b_set_rat: ( b > set_rat ) > set_b > set_set_rat ).

thf(sy_c_Set_Oimage_001tf__b_001tf__a,type,
    image_b_a: ( b > a ) > set_b > set_a ).

thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
    image_b_b: ( b > b ) > set_b > set_b ).

thf(sy_c_Set_Oimage_001tf__b_001tf__c,type,
    image_b_c: ( b > c ) > set_b > set_c ).

thf(sy_c_Set_Oimage_001tf__c_001t__Nat__Onat,type,
    image_c_nat: ( c > nat ) > set_c > set_nat ).

thf(sy_c_Set_Oimage_001tf__c_001t__Set__Oset_It__Rat__Orat_J,type,
    image_c_set_rat: ( c > set_rat ) > set_c > set_set_rat ).

thf(sy_c_Set_Oimage_001tf__c_001tf__a,type,
    image_c_a: ( c > a ) > set_c > set_a ).

thf(sy_c_Set_Oimage_001tf__c_001tf__b,type,
    image_c_b: ( c > b ) > set_c > set_b ).

thf(sy_c_Set_Oimage_001tf__c_001tf__c,type,
    image_c_c: ( c > c ) > set_c > set_c ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Rat__Orat,type,
    insert_rat: rat > set_rat > set_rat ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Rat__Orat_J,type,
    insert_set_rat: set_rat > set_set_rat > set_set_rat ).

thf(sy_c_Set_Opairwise_001t__Rat__Orat,type,
    pairwise_rat: ( rat > rat > $o ) > set_rat > $o ).

thf(sy_c_Set_Opairwise_001t__Set__Oset_It__Nat__Onat_J,type,
    pairwise_set_nat: ( set_nat > set_nat > $o ) > set_set_nat > $o ).

thf(sy_c_Set_Opairwise_001t__Set__Oset_It__Rat__Orat_J,type,
    pairwise_set_rat: ( set_rat > set_rat > $o ) > set_set_rat > $o ).

thf(sy_c_Set_Othe__elem_001t__Rat__Orat,type,
    the_elem_rat: set_rat > rat ).

thf(sy_c_Set_Ovimage_001_062_Itf__b_Mtf__a_J_001_062_Itf__b_Mtf__a_J,type,
    vimage_b_a_b_a: ( ( b > a ) > b > a ) > set_b_a > set_b_a ).

thf(sy_c_Set_Ovimage_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
    vimage_c_a_c_a: ( ( c > a ) > c > a ) > set_c_a > set_c_a ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
    vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Rat__Orat,type,
    vimage_nat_rat: ( nat > rat ) > set_rat > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__a,type,
    vimage_nat_a: ( nat > a ) > set_a > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__b,type,
    vimage_nat_b: ( nat > b ) > set_b > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__c,type,
    vimage_nat_c: ( nat > c ) > set_c > set_nat ).

thf(sy_c_Set_Ovimage_001t__Rat__Orat_001t__Nat__Onat,type,
    vimage_rat_nat: ( rat > nat ) > set_nat > set_rat ).

thf(sy_c_Set_Ovimage_001t__Rat__Orat_001t__Rat__Orat,type,
    vimage_rat_rat: ( rat > rat ) > set_rat > set_rat ).

thf(sy_c_Set_Ovimage_001t__Rat__Orat_001tf__a,type,
    vimage_rat_a: ( rat > a ) > set_a > set_rat ).

thf(sy_c_Set_Ovimage_001t__Rat__Orat_001tf__b,type,
    vimage_rat_b: ( rat > b ) > set_b > set_rat ).

thf(sy_c_Set_Ovimage_001t__Rat__Orat_001tf__c,type,
    vimage_rat_c: ( rat > c ) > set_c > set_rat ).

thf(sy_c_Set_Ovimage_001tf__a_001t__Nat__Onat,type,
    vimage_a_nat: ( a > nat ) > set_nat > set_a ).

thf(sy_c_Set_Ovimage_001tf__a_001t__Rat__Orat,type,
    vimage_a_rat: ( a > rat ) > set_rat > set_a ).

thf(sy_c_Set_Ovimage_001tf__a_001tf__a,type,
    vimage_a_a: ( a > a ) > set_a > set_a ).

thf(sy_c_Set_Ovimage_001tf__a_001tf__c,type,
    vimage_a_c: ( a > c ) > set_c > set_a ).

thf(sy_c_Set_Ovimage_001tf__b_001t__Nat__Onat,type,
    vimage_b_nat: ( b > nat ) > set_nat > set_b ).

thf(sy_c_Set_Ovimage_001tf__b_001t__Rat__Orat,type,
    vimage_b_rat: ( b > rat ) > set_rat > set_b ).

thf(sy_c_Set_Ovimage_001tf__b_001tf__a,type,
    vimage_b_a: ( b > a ) > set_a > set_b ).

thf(sy_c_Set_Ovimage_001tf__b_001tf__b,type,
    vimage_b_b: ( b > b ) > set_b > set_b ).

thf(sy_c_Set_Ovimage_001tf__b_001tf__c,type,
    vimage_b_c: ( b > c ) > set_c > set_b ).

thf(sy_c_Set_Ovimage_001tf__c_001t__Nat__Onat,type,
    vimage_c_nat: ( c > nat ) > set_nat > set_c ).

thf(sy_c_Set_Ovimage_001tf__c_001t__Rat__Orat,type,
    vimage_c_rat: ( c > rat ) > set_rat > set_c ).

thf(sy_c_Set_Ovimage_001tf__c_001tf__a,type,
    vimage_c_a: ( c > a ) > set_a > set_c ).

thf(sy_c_Set_Ovimage_001tf__c_001tf__b,type,
    vimage_c_b: ( c > b ) > set_b > set_c ).

thf(sy_c_Set_Ovimage_001tf__c_001tf__c,type,
    vimage_c_c: ( c > c ) > set_c > set_c ).

thf(sy_c_Typedef_Otype__definition_001t__Nat__Onat_001t__Rat__Orat,type,
    type_d5615363888691252950at_rat: ( nat > rat ) > ( rat > nat ) > set_rat > $o ).

thf(sy_c_Typedef_Otype__definition_001t__Rat__Orat_001t__Nat__Onat,type,
    type_d5933939304842882774at_nat: ( rat > nat ) > ( nat > rat ) > set_nat > $o ).

thf(sy_c_Typedef_Otype__definition_001tf__a_001t__Nat__Onat,type,
    type_d7819358849495216162_a_nat: ( a > nat ) > ( nat > a ) > set_nat > $o ).

thf(sy_c_Typedef_Otype__definition_001tf__a_001tf__a,type,
    type_definition_a_a: ( a > a ) > ( a > a ) > set_a > $o ).

thf(sy_c_Typedef_Otype__definition_001tf__a_001tf__b,type,
    type_definition_a_b: ( a > b ) > ( b > a ) > set_b > $o ).

thf(sy_c_Typedef_Otype__definition_001tf__a_001tf__c,type,
    type_definition_a_c: ( a > c ) > ( c > a ) > set_c > $o ).

thf(sy_c_Typedef_Otype__definition_001tf__b_001t__Nat__Onat,type,
    type_d9054803178451610659_b_nat: ( b > nat ) > ( nat > b ) > set_nat > $o ).

thf(sy_c_Typedef_Otype__definition_001tf__b_001tf__a,type,
    type_definition_b_a: ( b > a ) > ( a > b ) > set_a > $o ).

thf(sy_c_Typedef_Otype__definition_001tf__b_001tf__b,type,
    type_definition_b_b: ( b > b ) > ( b > b ) > set_b > $o ).

thf(sy_c_Typedef_Otype__definition_001tf__b_001tf__c,type,
    type_definition_b_c: ( b > c ) > ( c > b ) > set_c > $o ).

thf(sy_c_Typedef_Otype__definition_001tf__c_001tf__a,type,
    type_definition_c_a: ( c > a ) > ( a > c ) > set_a > $o ).

thf(sy_c_Typedef_Otype__definition_001tf__c_001tf__c,type,
    type_definition_c_c: ( c > c ) > ( c > c ) > set_c > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
    member_nat_o: ( nat > $o ) > set_nat_o > $o ).

thf(sy_c_member_001_062_It__Rat__Orat_M_Eo_J,type,
    member_rat_o: ( rat > $o ) > set_rat_o > $o ).

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

thf(sy_c_member_001t__Rat__Orat,type,
    member_rat: rat > set_rat > $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__Rat__Orat_J,type,
    member_set_rat: set_rat > set_set_rat > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a: set_a > set_set_a > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
    member_set_b: set_b > set_set_b > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__c_J,type,
    member_set_c: set_c > set_set_c > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_c_member_001tf__b,type,
    member_b: b > set_b > $o ).

thf(sy_c_member_001tf__c,type,
    member_c: c > set_c > $o ).

thf(sy_v_f,type,
    f: b > a ).

thf(sy_v_g,type,
    g: c > b ).

thf(sy_v_v,type,
    v: vec_c ).

% Relevant facts (1279)
thf(fact_0_comp__apply,axiom,
    ( comp_b_a_c
    = ( ^ [F: b > a,G: c > b,X: c] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_1_comp__apply,axiom,
    ( comp_c_a_c
    = ( ^ [F: c > a,G: c > c,X: c] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_2_comp__apply,axiom,
    ( comp_b_b_c
    = ( ^ [F: b > b,G: c > b,X: c] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_3_comp__apply,axiom,
    ( comp_b_a_b
    = ( ^ [F: b > a,G: b > b,X: b] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_4_comp__apply,axiom,
    ( comp_a_a_c
    = ( ^ [F: a > a,G: c > a,X: c] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_5_comp__apply,axiom,
    ( comp_a_a_b
    = ( ^ [F: a > a,G: b > a,X: b] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_6_comp__apply,axiom,
    ( comp_a_b_b
    = ( ^ [F: a > b,G: b > a,X: b] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_7_comp__apply,axiom,
    ( comp_a_b_a
    = ( ^ [F: a > b,G: a > a,X: a] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_8_comp__apply,axiom,
    ( comp_a_a_rat
    = ( ^ [F: a > a,G: rat > a,X: rat] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_9_comp__apply,axiom,
    ( comp_a_a_nat
    = ( ^ [F: a > a,G: nat > a,X: nat] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_10_fun_Omap__comp,axiom,
    ! [G2: a > a,F2: b > a,V: c > b] :
      ( ( comp_a_a_c @ G2 @ ( comp_b_a_c @ F2 @ V ) )
      = ( comp_b_a_c @ ( comp_a_a_b @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_11_fun_Omap__comp,axiom,
    ! [G2: b > a,F2: c > b,V: c > c] :
      ( ( comp_b_a_c @ G2 @ ( comp_c_b_c @ F2 @ V ) )
      = ( comp_c_a_c @ ( comp_b_a_c @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_12_fun_Omap__comp,axiom,
    ! [G2: b > a,F2: b > b,V: c > b] :
      ( ( comp_b_a_c @ G2 @ ( comp_b_b_c @ F2 @ V ) )
      = ( comp_b_a_c @ ( comp_b_a_b @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_13_fun_Omap__comp,axiom,
    ! [G2: a > b,F2: b > a,V: c > b] :
      ( ( comp_a_b_c @ G2 @ ( comp_b_a_c @ F2 @ V ) )
      = ( comp_b_b_c @ ( comp_a_b_b @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_14_fun_Omap__comp,axiom,
    ! [G2: b > a,F2: a > b,V: c > a] :
      ( ( comp_b_a_c @ G2 @ ( comp_a_b_c @ F2 @ V ) )
      = ( comp_a_a_c @ ( comp_b_a_a @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_15_fun_Omap__comp,axiom,
    ! [G2: c > a,F2: b > c,V: c > b] :
      ( ( comp_c_a_c @ G2 @ ( comp_b_c_c @ F2 @ V ) )
      = ( comp_b_a_c @ ( comp_c_a_b @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_16_fun_Omap__comp,axiom,
    ! [G2: c > a,F2: c > c,V: c > c] :
      ( ( comp_c_a_c @ G2 @ ( comp_c_c_c @ F2 @ V ) )
      = ( comp_c_a_c @ ( comp_c_a_c @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_17_fun_Omap__comp,axiom,
    ! [G2: c > a,F2: a > c,V: c > a] :
      ( ( comp_c_a_c @ G2 @ ( comp_a_c_c @ F2 @ V ) )
      = ( comp_a_a_c @ ( comp_c_a_a @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_18_fun_Omap__comp,axiom,
    ! [G2: b > b,F2: c > b,V: c > c] :
      ( ( comp_b_b_c @ G2 @ ( comp_c_b_c @ F2 @ V ) )
      = ( comp_c_b_c @ ( comp_b_b_c @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_19_fun_Omap__comp,axiom,
    ! [G2: b > b,F2: b > b,V: c > b] :
      ( ( comp_b_b_c @ G2 @ ( comp_b_b_c @ F2 @ V ) )
      = ( comp_b_b_c @ ( comp_b_b_b @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_20_comp__def,axiom,
    ( comp_b_a_c
    = ( ^ [F: b > a,G: c > b,X: c] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_21_comp__def,axiom,
    ( comp_c_a_c
    = ( ^ [F: c > a,G: c > c,X: c] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_22_comp__def,axiom,
    ( comp_b_b_c
    = ( ^ [F: b > b,G: c > b,X: c] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_23_comp__def,axiom,
    ( comp_b_a_b
    = ( ^ [F: b > a,G: b > b,X: b] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_24_comp__def,axiom,
    ( comp_a_a_c
    = ( ^ [F: a > a,G: c > a,X: c] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_25_comp__def,axiom,
    ( comp_a_a_b
    = ( ^ [F: a > a,G: b > a,X: b] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_26_comp__def,axiom,
    ( comp_a_b_b
    = ( ^ [F: a > b,G: b > a,X: b] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_27_comp__def,axiom,
    ( comp_a_b_a
    = ( ^ [F: a > b,G: a > a,X: a] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_28_comp__def,axiom,
    ( comp_a_a_rat
    = ( ^ [F: a > a,G: rat > a,X: rat] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_29_comp__def,axiom,
    ( comp_a_a_nat
    = ( ^ [F: a > a,G: nat > a,X: nat] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_30_comp__assoc,axiom,
    ! [F2: b > a,G2: c > b,H: c > c] :
      ( ( comp_c_a_c @ ( comp_b_a_c @ F2 @ G2 ) @ H )
      = ( comp_b_a_c @ F2 @ ( comp_c_b_c @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_31_comp__assoc,axiom,
    ! [F2: a > a,G2: b > a,H: c > b] :
      ( ( comp_b_a_c @ ( comp_a_a_b @ F2 @ G2 ) @ H )
      = ( comp_a_a_c @ F2 @ ( comp_b_a_c @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_32_comp__assoc,axiom,
    ! [F2: b > a,G2: b > b,H: c > b] :
      ( ( comp_b_a_c @ ( comp_b_a_b @ F2 @ G2 ) @ H )
      = ( comp_b_a_c @ F2 @ ( comp_b_b_c @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_33_comp__assoc,axiom,
    ! [F2: b > a,G2: c > b,H: b > c] :
      ( ( comp_c_a_b @ ( comp_b_a_c @ F2 @ G2 ) @ H )
      = ( comp_b_a_b @ F2 @ ( comp_c_b_b @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_34_comp__assoc,axiom,
    ! [F2: b > b,G2: c > b,H: c > c] :
      ( ( comp_c_b_c @ ( comp_b_b_c @ F2 @ G2 ) @ H )
      = ( comp_b_b_c @ F2 @ ( comp_c_b_c @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_35_comp__assoc,axiom,
    ! [F2: a > a,G2: c > a,H: b > c] :
      ( ( comp_c_a_b @ ( comp_a_a_c @ F2 @ G2 ) @ H )
      = ( comp_a_a_b @ F2 @ ( comp_c_a_b @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_36_comp__assoc,axiom,
    ! [F2: c > a,G2: b > c,H: c > b] :
      ( ( comp_b_a_c @ ( comp_c_a_b @ F2 @ G2 ) @ H )
      = ( comp_c_a_c @ F2 @ ( comp_b_c_c @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_37_comp__assoc,axiom,
    ! [F2: c > a,G2: c > c,H: c > c] :
      ( ( comp_c_a_c @ ( comp_c_a_c @ F2 @ G2 ) @ H )
      = ( comp_c_a_c @ F2 @ ( comp_c_c_c @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_38_comp__assoc,axiom,
    ! [F2: a > a,G2: c > a,H: c > c] :
      ( ( comp_c_a_c @ ( comp_a_a_c @ F2 @ G2 ) @ H )
      = ( comp_a_a_c @ F2 @ ( comp_c_a_c @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_39_comp__assoc,axiom,
    ! [F2: a > b,G2: b > a,H: c > b] :
      ( ( comp_b_b_c @ ( comp_a_b_b @ F2 @ G2 ) @ H )
      = ( comp_a_b_c @ F2 @ ( comp_b_a_c @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_40_comp__eq__dest,axiom,
    ! [A: b > a,B: c > b,C: b > a,D: c > b,V: c] :
      ( ( ( comp_b_a_c @ A @ B )
        = ( comp_b_a_c @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_41_comp__eq__dest,axiom,
    ! [A: b > a,B: c > b,C: c > a,D: c > c,V: c] :
      ( ( ( comp_b_a_c @ A @ B )
        = ( comp_c_a_c @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_42_comp__eq__dest,axiom,
    ! [A: b > a,B: c > b,C: a > a,D: c > a,V: c] :
      ( ( ( comp_b_a_c @ A @ B )
        = ( comp_a_a_c @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_43_comp__eq__dest,axiom,
    ! [A: c > a,B: c > c,C: b > a,D: c > b,V: c] :
      ( ( ( comp_c_a_c @ A @ B )
        = ( comp_b_a_c @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_44_comp__eq__dest,axiom,
    ! [A: c > a,B: c > c,C: c > a,D: c > c,V: c] :
      ( ( ( comp_c_a_c @ A @ B )
        = ( comp_c_a_c @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_45_comp__eq__dest,axiom,
    ! [A: c > a,B: c > c,C: a > a,D: c > a,V: c] :
      ( ( ( comp_c_a_c @ A @ B )
        = ( comp_a_a_c @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_46_comp__eq__dest,axiom,
    ! [A: b > b,B: c > b,C: b > b,D: c > b,V: c] :
      ( ( ( comp_b_b_c @ A @ B )
        = ( comp_b_b_c @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_47_comp__eq__dest,axiom,
    ! [A: b > a,B: b > b,C: b > a,D: b > b,V: b] :
      ( ( ( comp_b_a_b @ A @ B )
        = ( comp_b_a_b @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_48_comp__eq__dest,axiom,
    ! [A: b > a,B: b > b,C: a > a,D: b > a,V: b] :
      ( ( ( comp_b_a_b @ A @ B )
        = ( comp_a_a_b @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_49_comp__eq__dest,axiom,
    ! [A: a > a,B: c > a,C: b > a,D: c > b,V: c] :
      ( ( ( comp_a_a_c @ A @ B )
        = ( comp_b_a_c @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_50_comp__eq__elim,axiom,
    ! [A: b > a,B: c > b,C: b > a,D: c > b] :
      ( ( ( comp_b_a_c @ A @ B )
        = ( comp_b_a_c @ C @ D ) )
     => ! [V2: c] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_51_comp__eq__elim,axiom,
    ! [A: b > a,B: c > b,C: c > a,D: c > c] :
      ( ( ( comp_b_a_c @ A @ B )
        = ( comp_c_a_c @ C @ D ) )
     => ! [V2: c] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_52_comp__eq__elim,axiom,
    ! [A: b > a,B: c > b,C: a > a,D: c > a] :
      ( ( ( comp_b_a_c @ A @ B )
        = ( comp_a_a_c @ C @ D ) )
     => ! [V2: c] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_53_comp__eq__elim,axiom,
    ! [A: c > a,B: c > c,C: b > a,D: c > b] :
      ( ( ( comp_c_a_c @ A @ B )
        = ( comp_b_a_c @ C @ D ) )
     => ! [V2: c] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_54_comp__eq__elim,axiom,
    ! [A: c > a,B: c > c,C: c > a,D: c > c] :
      ( ( ( comp_c_a_c @ A @ B )
        = ( comp_c_a_c @ C @ D ) )
     => ! [V2: c] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_55_comp__eq__elim,axiom,
    ! [A: c > a,B: c > c,C: a > a,D: c > a] :
      ( ( ( comp_c_a_c @ A @ B )
        = ( comp_a_a_c @ C @ D ) )
     => ! [V2: c] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_56_comp__eq__elim,axiom,
    ! [A: b > b,B: c > b,C: b > b,D: c > b] :
      ( ( ( comp_b_b_c @ A @ B )
        = ( comp_b_b_c @ C @ D ) )
     => ! [V2: c] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_57_comp__eq__elim,axiom,
    ! [A: b > a,B: b > b,C: b > a,D: b > b] :
      ( ( ( comp_b_a_b @ A @ B )
        = ( comp_b_a_b @ C @ D ) )
     => ! [V2: b] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_58_comp__eq__elim,axiom,
    ! [A: b > a,B: b > b,C: a > a,D: b > a] :
      ( ( ( comp_b_a_b @ A @ B )
        = ( comp_a_a_b @ C @ D ) )
     => ! [V2: b] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_59_comp__eq__elim,axiom,
    ! [A: a > a,B: c > a,C: b > a,D: c > b] :
      ( ( ( comp_a_a_c @ A @ B )
        = ( comp_b_a_c @ C @ D ) )
     => ! [V2: c] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_60_comp__cong,axiom,
    ! [F2: b > a,G2: c > b,X2: c,F3: b > a,G3: c > b,X3: c] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( F3 @ ( G3 @ X3 ) ) )
     => ( ( comp_b_a_c @ F2 @ G2 @ X2 )
        = ( comp_b_a_c @ F3 @ G3 @ X3 ) ) ) ).

% comp_cong
thf(fact_61_comp__cong,axiom,
    ! [F2: b > a,G2: c > b,X2: c,F3: c > a,G3: c > c,X3: c] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( F3 @ ( G3 @ X3 ) ) )
     => ( ( comp_b_a_c @ F2 @ G2 @ X2 )
        = ( comp_c_a_c @ F3 @ G3 @ X3 ) ) ) ).

% comp_cong
thf(fact_62_comp__cong,axiom,
    ! [F2: b > a,G2: c > b,X2: c,F3: b > a,G3: b > b,X3: b] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( F3 @ ( G3 @ X3 ) ) )
     => ( ( comp_b_a_c @ F2 @ G2 @ X2 )
        = ( comp_b_a_b @ F3 @ G3 @ X3 ) ) ) ).

% comp_cong
thf(fact_63_comp__cong,axiom,
    ! [F2: b > a,G2: c > b,X2: c,F3: a > a,G3: c > a,X3: c] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( F3 @ ( G3 @ X3 ) ) )
     => ( ( comp_b_a_c @ F2 @ G2 @ X2 )
        = ( comp_a_a_c @ F3 @ G3 @ X3 ) ) ) ).

% comp_cong
thf(fact_64_comp__cong,axiom,
    ! [F2: b > a,G2: c > b,X2: c,F3: a > a,G3: b > a,X3: b] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( F3 @ ( G3 @ X3 ) ) )
     => ( ( comp_b_a_c @ F2 @ G2 @ X2 )
        = ( comp_a_a_b @ F3 @ G3 @ X3 ) ) ) ).

% comp_cong
thf(fact_65_comp__cong,axiom,
    ! [F2: c > a,G2: c > c,X2: c,F3: b > a,G3: c > b,X3: c] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( F3 @ ( G3 @ X3 ) ) )
     => ( ( comp_c_a_c @ F2 @ G2 @ X2 )
        = ( comp_b_a_c @ F3 @ G3 @ X3 ) ) ) ).

% comp_cong
thf(fact_66_comp__cong,axiom,
    ! [F2: c > a,G2: c > c,X2: c,F3: c > a,G3: c > c,X3: c] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( F3 @ ( G3 @ X3 ) ) )
     => ( ( comp_c_a_c @ F2 @ G2 @ X2 )
        = ( comp_c_a_c @ F3 @ G3 @ X3 ) ) ) ).

% comp_cong
thf(fact_67_comp__cong,axiom,
    ! [F2: c > a,G2: c > c,X2: c,F3: b > a,G3: b > b,X3: b] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( F3 @ ( G3 @ X3 ) ) )
     => ( ( comp_c_a_c @ F2 @ G2 @ X2 )
        = ( comp_b_a_b @ F3 @ G3 @ X3 ) ) ) ).

% comp_cong
thf(fact_68_comp__cong,axiom,
    ! [F2: c > a,G2: c > c,X2: c,F3: a > a,G3: c > a,X3: c] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( F3 @ ( G3 @ X3 ) ) )
     => ( ( comp_c_a_c @ F2 @ G2 @ X2 )
        = ( comp_a_a_c @ F3 @ G3 @ X3 ) ) ) ).

% comp_cong
thf(fact_69_comp__cong,axiom,
    ! [F2: c > a,G2: c > c,X2: c,F3: a > a,G3: b > a,X3: b] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( F3 @ ( G3 @ X3 ) ) )
     => ( ( comp_c_a_c @ F2 @ G2 @ X2 )
        = ( comp_a_a_b @ F3 @ G3 @ X3 ) ) ) ).

% comp_cong
thf(fact_70_comp__eq__dest__lhs,axiom,
    ! [A: b > a,B: c > b,C: c > a,V: c] :
      ( ( ( comp_b_a_c @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_71_comp__eq__dest__lhs,axiom,
    ! [A: c > a,B: c > c,C: c > a,V: c] :
      ( ( ( comp_c_a_c @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_72_comp__eq__dest__lhs,axiom,
    ! [A: b > b,B: c > b,C: c > b,V: c] :
      ( ( ( comp_b_b_c @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_73_comp__eq__dest__lhs,axiom,
    ! [A: b > a,B: b > b,C: b > a,V: b] :
      ( ( ( comp_b_a_b @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_74_comp__eq__dest__lhs,axiom,
    ! [A: a > a,B: c > a,C: c > a,V: c] :
      ( ( ( comp_a_a_c @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_75_comp__eq__dest__lhs,axiom,
    ! [A: a > a,B: b > a,C: b > a,V: b] :
      ( ( ( comp_a_a_b @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_76_comp__eq__dest__lhs,axiom,
    ! [A: a > b,B: b > a,C: b > b,V: b] :
      ( ( ( comp_a_b_b @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_77_comp__eq__dest__lhs,axiom,
    ! [A: a > b,B: a > a,C: a > b,V: a] :
      ( ( ( comp_a_b_a @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_78_comp__eq__dest__lhs,axiom,
    ! [A: a > a,B: rat > a,C: rat > a,V: rat] :
      ( ( ( comp_a_a_rat @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_79_comp__eq__dest__lhs,axiom,
    ! [A: a > a,B: nat > a,C: nat > a,V: nat] :
      ( ( ( comp_a_a_nat @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_80_comp__apply__eq,axiom,
    ! [F2: b > a,G2: c > b,X2: c,H: b > a,K: c > b] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_b_a_c @ F2 @ G2 @ X2 )
        = ( comp_b_a_c @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_81_comp__apply__eq,axiom,
    ! [F2: b > a,G2: c > b,X2: c,H: c > a,K: c > c] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_b_a_c @ F2 @ G2 @ X2 )
        = ( comp_c_a_c @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_82_comp__apply__eq,axiom,
    ! [F2: b > a,G2: c > b,X2: c,H: a > a,K: c > a] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_b_a_c @ F2 @ G2 @ X2 )
        = ( comp_a_a_c @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_83_comp__apply__eq,axiom,
    ! [F2: c > a,G2: c > c,X2: c,H: b > a,K: c > b] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_c_a_c @ F2 @ G2 @ X2 )
        = ( comp_b_a_c @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_84_comp__apply__eq,axiom,
    ! [F2: c > a,G2: c > c,X2: c,H: c > a,K: c > c] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_c_a_c @ F2 @ G2 @ X2 )
        = ( comp_c_a_c @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_85_comp__apply__eq,axiom,
    ! [F2: c > a,G2: c > c,X2: c,H: a > a,K: c > a] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_c_a_c @ F2 @ G2 @ X2 )
        = ( comp_a_a_c @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_86_comp__apply__eq,axiom,
    ! [F2: b > b,G2: c > b,X2: c,H: b > b,K: c > b] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_b_b_c @ F2 @ G2 @ X2 )
        = ( comp_b_b_c @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_87_comp__apply__eq,axiom,
    ! [F2: b > a,G2: b > b,X2: b,H: b > a,K: b > b] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_b_a_b @ F2 @ G2 @ X2 )
        = ( comp_b_a_b @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_88_comp__apply__eq,axiom,
    ! [F2: b > a,G2: b > b,X2: b,H: a > a,K: b > a] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_b_a_b @ F2 @ G2 @ X2 )
        = ( comp_a_a_b @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_89_comp__apply__eq,axiom,
    ! [F2: a > a,G2: c > a,X2: c,H: b > a,K: c > b] :
      ( ( ( F2 @ ( G2 @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_a_a_c @ F2 @ G2 @ X2 )
        = ( comp_b_a_c @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_90_function__factors__left,axiom,
    ! [G2: c > b,F2: c > a] :
      ( ( ! [X: c,Y: c] :
            ( ( ( G2 @ X )
              = ( G2 @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H2: b > a] :
            ( F2
            = ( comp_b_a_c @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_91_function__factors__left,axiom,
    ! [G2: c > c,F2: c > a] :
      ( ( ! [X: c,Y: c] :
            ( ( ( G2 @ X )
              = ( G2 @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H2: c > a] :
            ( F2
            = ( comp_c_a_c @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_92_function__factors__left,axiom,
    ! [G2: c > b,F2: c > b] :
      ( ( ! [X: c,Y: c] :
            ( ( ( G2 @ X )
              = ( G2 @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H2: b > b] :
            ( F2
            = ( comp_b_b_c @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_93_function__factors__left,axiom,
    ! [G2: b > b,F2: b > a] :
      ( ( ! [X: b,Y: b] :
            ( ( ( G2 @ X )
              = ( G2 @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H2: b > a] :
            ( F2
            = ( comp_b_a_b @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_94_function__factors__left,axiom,
    ! [G2: c > a,F2: c > a] :
      ( ( ! [X: c,Y: c] :
            ( ( ( G2 @ X )
              = ( G2 @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H2: a > a] :
            ( F2
            = ( comp_a_a_c @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_95_function__factors__left,axiom,
    ! [G2: b > a,F2: b > a] :
      ( ( ! [X: b,Y: b] :
            ( ( ( G2 @ X )
              = ( G2 @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H2: a > a] :
            ( F2
            = ( comp_a_a_b @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_96_function__factors__left,axiom,
    ! [G2: b > a,F2: b > b] :
      ( ( ! [X: b,Y: b] :
            ( ( ( G2 @ X )
              = ( G2 @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H2: a > b] :
            ( F2
            = ( comp_a_b_b @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_97_function__factors__left,axiom,
    ! [G2: a > a,F2: a > b] :
      ( ( ! [X: a,Y: a] :
            ( ( ( G2 @ X )
              = ( G2 @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H2: a > b] :
            ( F2
            = ( comp_a_b_a @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_98_function__factors__left,axiom,
    ! [G2: rat > a,F2: rat > a] :
      ( ( ! [X: rat,Y: rat] :
            ( ( ( G2 @ X )
              = ( G2 @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H2: a > a] :
            ( F2
            = ( comp_a_a_rat @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_99_function__factors__left,axiom,
    ! [G2: nat > a,F2: nat > a] :
      ( ( ! [X: nat,Y: nat] :
            ( ( ( G2 @ X )
              = ( G2 @ Y ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) )
      = ( ? [H2: a > a] :
            ( F2
            = ( comp_a_a_nat @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_100_function__factors__right,axiom,
    ! [G2: b > a,F2: c > a] :
      ( ( ! [X: c] :
          ? [Y: b] :
            ( ( G2 @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H2: c > b] :
            ( F2
            = ( comp_b_a_c @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_101_function__factors__right,axiom,
    ! [G2: c > a,F2: c > a] :
      ( ( ! [X: c] :
          ? [Y: c] :
            ( ( G2 @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H2: c > c] :
            ( F2
            = ( comp_c_a_c @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_102_function__factors__right,axiom,
    ! [G2: b > b,F2: c > b] :
      ( ( ! [X: c] :
          ? [Y: b] :
            ( ( G2 @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H2: c > b] :
            ( F2
            = ( comp_b_b_c @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_103_function__factors__right,axiom,
    ! [G2: b > a,F2: b > a] :
      ( ( ! [X: b] :
          ? [Y: b] :
            ( ( G2 @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H2: b > b] :
            ( F2
            = ( comp_b_a_b @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_104_function__factors__right,axiom,
    ! [G2: a > a,F2: c > a] :
      ( ( ! [X: c] :
          ? [Y: a] :
            ( ( G2 @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H2: c > a] :
            ( F2
            = ( comp_a_a_c @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_105_function__factors__right,axiom,
    ! [G2: a > a,F2: b > a] :
      ( ( ! [X: b] :
          ? [Y: a] :
            ( ( G2 @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H2: b > a] :
            ( F2
            = ( comp_a_a_b @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_106_function__factors__right,axiom,
    ! [G2: a > b,F2: b > b] :
      ( ( ! [X: b] :
          ? [Y: a] :
            ( ( G2 @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H2: b > a] :
            ( F2
            = ( comp_a_b_b @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_107_function__factors__right,axiom,
    ! [G2: a > b,F2: a > b] :
      ( ( ! [X: a] :
          ? [Y: a] :
            ( ( G2 @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H2: a > a] :
            ( F2
            = ( comp_a_b_a @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_108_function__factors__right,axiom,
    ! [G2: a > a,F2: rat > a] :
      ( ( ! [X: rat] :
          ? [Y: a] :
            ( ( G2 @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H2: rat > a] :
            ( F2
            = ( comp_a_a_rat @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_109_function__factors__right,axiom,
    ! [G2: a > a,F2: nat > a] :
      ( ( ! [X: nat] :
          ? [Y: a] :
            ( ( G2 @ Y )
            = ( F2 @ X ) ) )
      = ( ? [H2: nat > a] :
            ( F2
            = ( comp_a_a_nat @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_110_type__copy__map__cong0,axiom,
    ! [M: c > b,G2: b > c,X2: b,N: b > b,H: b > b,F2: b > a] :
      ( ( ( M @ ( G2 @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_c_a_b @ ( comp_b_a_c @ F2 @ M ) @ G2 @ X2 )
        = ( comp_b_a_b @ ( comp_b_a_b @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_111_type__copy__map__cong0,axiom,
    ! [M: c > b,G2: b > c,X2: b,N: a > b,H: b > a,F2: b > a] :
      ( ( ( M @ ( G2 @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_c_a_b @ ( comp_b_a_c @ F2 @ M ) @ G2 @ X2 )
        = ( comp_a_a_b @ ( comp_b_a_a @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_112_type__copy__map__cong0,axiom,
    ! [M: c > c,G2: b > c,X2: b,N: b > c,H: b > b,F2: c > a] :
      ( ( ( M @ ( G2 @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_c_a_b @ ( comp_c_a_c @ F2 @ M ) @ G2 @ X2 )
        = ( comp_b_a_b @ ( comp_c_a_b @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_113_type__copy__map__cong0,axiom,
    ! [M: c > c,G2: b > c,X2: b,N: a > c,H: b > a,F2: c > a] :
      ( ( ( M @ ( G2 @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_c_a_b @ ( comp_c_a_c @ F2 @ M ) @ G2 @ X2 )
        = ( comp_a_a_b @ ( comp_c_a_a @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_114_type__copy__map__cong0,axiom,
    ! [M: c > b,G2: c > c,X2: c,N: b > b,H: c > b,F2: b > b] :
      ( ( ( M @ ( G2 @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_c_b_c @ ( comp_b_b_c @ F2 @ M ) @ G2 @ X2 )
        = ( comp_b_b_c @ ( comp_b_b_b @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_115_type__copy__map__cong0,axiom,
    ! [M: c > a,G2: b > c,X2: b,N: b > a,H: b > b,F2: a > a] :
      ( ( ( M @ ( G2 @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_c_a_b @ ( comp_a_a_c @ F2 @ M ) @ G2 @ X2 )
        = ( comp_b_a_b @ ( comp_a_a_b @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_116_type__copy__map__cong0,axiom,
    ! [M: c > a,G2: b > c,X2: b,N: a > a,H: b > a,F2: a > a] :
      ( ( ( M @ ( G2 @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_c_a_b @ ( comp_a_a_c @ F2 @ M ) @ G2 @ X2 )
        = ( comp_a_a_b @ ( comp_a_a_a @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_117_type__copy__map__cong0,axiom,
    ! [M: b > c,G2: c > b,X2: c,N: c > c,H: c > c,F2: c > a] :
      ( ( ( M @ ( G2 @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_b_a_c @ ( comp_c_a_b @ F2 @ M ) @ G2 @ X2 )
        = ( comp_c_a_c @ ( comp_c_a_c @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_118_type__copy__map__cong0,axiom,
    ! [M: b > b,G2: c > b,X2: c,N: b > b,H: c > b,F2: b > a] :
      ( ( ( M @ ( G2 @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_b_a_c @ ( comp_b_a_b @ F2 @ M ) @ G2 @ X2 )
        = ( comp_b_a_c @ ( comp_b_a_b @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_119_type__copy__map__cong0,axiom,
    ! [M: b > b,G2: c > b,X2: c,N: c > b,H: c > c,F2: b > a] :
      ( ( ( M @ ( G2 @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_b_a_c @ ( comp_b_a_b @ F2 @ M ) @ G2 @ X2 )
        = ( comp_c_a_c @ ( comp_b_a_c @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_120_rewriteR__comp__comp2,axiom,
    ! [G2: c > b,H: b > c,R1: a > b,R2: b > a,F2: b > a,L: a > a] :
      ( ( ( comp_c_b_b @ G2 @ H )
        = ( comp_a_b_b @ R1 @ R2 ) )
     => ( ( ( comp_b_a_a @ F2 @ R1 )
          = L )
       => ( ( comp_c_a_b @ ( comp_b_a_c @ F2 @ G2 ) @ H )
          = ( comp_a_a_b @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_121_rewriteR__comp__comp2,axiom,
    ! [G2: c > c,H: b > c,R1: b > c,R2: b > b,F2: c > a,L: b > a] :
      ( ( ( comp_c_c_b @ G2 @ H )
        = ( comp_b_c_b @ R1 @ R2 ) )
     => ( ( ( comp_c_a_b @ F2 @ R1 )
          = L )
       => ( ( comp_c_a_b @ ( comp_c_a_c @ F2 @ G2 ) @ H )
          = ( comp_b_a_b @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_122_rewriteR__comp__comp2,axiom,
    ! [G2: c > c,H: b > c,R1: a > c,R2: b > a,F2: c > a,L: a > a] :
      ( ( ( comp_c_c_b @ G2 @ H )
        = ( comp_a_c_b @ R1 @ R2 ) )
     => ( ( ( comp_c_a_a @ F2 @ R1 )
          = L )
       => ( ( comp_c_a_b @ ( comp_c_a_c @ F2 @ G2 ) @ H )
          = ( comp_a_a_b @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_123_rewriteR__comp__comp2,axiom,
    ! [G2: c > b,H: c > c,R1: a > b,R2: c > a,F2: b > a,L: a > a] :
      ( ( ( comp_c_b_c @ G2 @ H )
        = ( comp_a_b_c @ R1 @ R2 ) )
     => ( ( ( comp_b_a_a @ F2 @ R1 )
          = L )
       => ( ( comp_c_a_c @ ( comp_b_a_c @ F2 @ G2 ) @ H )
          = ( comp_a_a_c @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_124_rewriteR__comp__comp2,axiom,
    ! [G2: c > c,H: c > c,R1: b > c,R2: c > b,F2: c > a,L: b > a] :
      ( ( ( comp_c_c_c @ G2 @ H )
        = ( comp_b_c_c @ R1 @ R2 ) )
     => ( ( ( comp_c_a_b @ F2 @ R1 )
          = L )
       => ( ( comp_c_a_c @ ( comp_c_a_c @ F2 @ G2 ) @ H )
          = ( comp_b_a_c @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_125_rewriteR__comp__comp2,axiom,
    ! [G2: c > c,H: c > c,R1: a > c,R2: c > a,F2: c > a,L: a > a] :
      ( ( ( comp_c_c_c @ G2 @ H )
        = ( comp_a_c_c @ R1 @ R2 ) )
     => ( ( ( comp_c_a_a @ F2 @ R1 )
          = L )
       => ( ( comp_c_a_c @ ( comp_c_a_c @ F2 @ G2 ) @ H )
          = ( comp_a_a_c @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_126_rewriteR__comp__comp2,axiom,
    ! [G2: b > b,H: b > b,R1: a > b,R2: b > a,F2: b > a,L: a > a] :
      ( ( ( comp_b_b_b @ G2 @ H )
        = ( comp_a_b_b @ R1 @ R2 ) )
     => ( ( ( comp_b_a_a @ F2 @ R1 )
          = L )
       => ( ( comp_b_a_b @ ( comp_b_a_b @ F2 @ G2 ) @ H )
          = ( comp_a_a_b @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_127_rewriteR__comp__comp2,axiom,
    ! [G2: c > b,H: c > c,R1: c > b,R2: c > c,F2: b > a,L: c > a] :
      ( ( ( comp_c_b_c @ G2 @ H )
        = ( comp_c_b_c @ R1 @ R2 ) )
     => ( ( ( comp_b_a_c @ F2 @ R1 )
          = L )
       => ( ( comp_c_a_c @ ( comp_b_a_c @ F2 @ G2 ) @ H )
          = ( comp_c_a_c @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_128_rewriteR__comp__comp2,axiom,
    ! [G2: b > b,H: b > b,R1: c > b,R2: b > c,F2: b > a,L: c > a] :
      ( ( ( comp_b_b_b @ G2 @ H )
        = ( comp_c_b_b @ R1 @ R2 ) )
     => ( ( ( comp_b_a_c @ F2 @ R1 )
          = L )
       => ( ( comp_b_a_b @ ( comp_b_a_b @ F2 @ G2 ) @ H )
          = ( comp_c_a_b @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_129_rewriteR__comp__comp2,axiom,
    ! [G2: a > b,H: c > a,R1: c > b,R2: c > c,F2: b > a,L: c > a] :
      ( ( ( comp_a_b_c @ G2 @ H )
        = ( comp_c_b_c @ R1 @ R2 ) )
     => ( ( ( comp_b_a_c @ F2 @ R1 )
          = L )
       => ( ( comp_a_a_c @ ( comp_b_a_a @ F2 @ G2 ) @ H )
          = ( comp_c_a_c @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_130_rewriteL__comp__comp2,axiom,
    ! [F2: a > b,G2: a > a,L1: b > b,L2: a > b,H: c > a,R: c > b] :
      ( ( ( comp_a_b_a @ F2 @ G2 )
        = ( comp_b_b_a @ L1 @ L2 ) )
     => ( ( ( comp_a_b_c @ L2 @ H )
          = R )
       => ( ( comp_a_b_c @ F2 @ ( comp_a_a_c @ G2 @ H ) )
          = ( comp_b_b_c @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_131_rewriteL__comp__comp2,axiom,
    ! [F2: a > a,G2: a > a,L1: b > a,L2: a > b,H: c > a,R: c > b] :
      ( ( ( comp_a_a_a @ F2 @ G2 )
        = ( comp_b_a_a @ L1 @ L2 ) )
     => ( ( ( comp_a_b_c @ L2 @ H )
          = R )
       => ( ( comp_a_a_c @ F2 @ ( comp_a_a_c @ G2 @ H ) )
          = ( comp_b_a_c @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_132_rewriteL__comp__comp2,axiom,
    ! [F2: a > a,G2: a > a,L1: c > a,L2: a > c,H: c > a,R: c > c] :
      ( ( ( comp_a_a_a @ F2 @ G2 )
        = ( comp_c_a_a @ L1 @ L2 ) )
     => ( ( ( comp_a_c_c @ L2 @ H )
          = R )
       => ( ( comp_a_a_c @ F2 @ ( comp_a_a_c @ G2 @ H ) )
          = ( comp_c_a_c @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_133_rewriteL__comp__comp2,axiom,
    ! [F2: a > a,G2: a > a,L1: b > a,L2: a > b,H: b > a,R: b > b] :
      ( ( ( comp_a_a_a @ F2 @ G2 )
        = ( comp_b_a_a @ L1 @ L2 ) )
     => ( ( ( comp_a_b_b @ L2 @ H )
          = R )
       => ( ( comp_a_a_b @ F2 @ ( comp_a_a_b @ G2 @ H ) )
          = ( comp_b_a_b @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_134_rewriteL__comp__comp2,axiom,
    ! [F2: b > b,G2: b > b,L1: a > b,L2: b > a,H: c > b,R: c > a] :
      ( ( ( comp_b_b_b @ F2 @ G2 )
        = ( comp_a_b_b @ L1 @ L2 ) )
     => ( ( ( comp_b_a_c @ L2 @ H )
          = R )
       => ( ( comp_b_b_c @ F2 @ ( comp_b_b_c @ G2 @ H ) )
          = ( comp_a_b_c @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_135_rewriteL__comp__comp2,axiom,
    ! [F2: a > b,G2: b > a,L1: b > b,L2: b > b,H: c > b,R: c > b] :
      ( ( ( comp_a_b_b @ F2 @ G2 )
        = ( comp_b_b_b @ L1 @ L2 ) )
     => ( ( ( comp_b_b_c @ L2 @ H )
          = R )
       => ( ( comp_a_b_c @ F2 @ ( comp_b_a_c @ G2 @ H ) )
          = ( comp_b_b_c @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_136_rewriteL__comp__comp2,axiom,
    ! [F2: b > b,G2: b > b,L1: b > b,L2: b > b,H: c > b,R: c > b] :
      ( ( ( comp_b_b_b @ F2 @ G2 )
        = ( comp_b_b_b @ L1 @ L2 ) )
     => ( ( ( comp_b_b_c @ L2 @ H )
          = R )
       => ( ( comp_b_b_c @ F2 @ ( comp_b_b_c @ G2 @ H ) )
          = ( comp_b_b_c @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_137_rewriteL__comp__comp2,axiom,
    ! [F2: b > a,G2: a > b,L1: a > a,L2: a > a,H: c > a,R: c > a] :
      ( ( ( comp_b_a_a @ F2 @ G2 )
        = ( comp_a_a_a @ L1 @ L2 ) )
     => ( ( ( comp_a_a_c @ L2 @ H )
          = R )
       => ( ( comp_b_a_c @ F2 @ ( comp_a_b_c @ G2 @ H ) )
          = ( comp_a_a_c @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_138_rewriteL__comp__comp2,axiom,
    ! [F2: c > a,G2: a > c,L1: a > a,L2: a > a,H: c > a,R: c > a] :
      ( ( ( comp_c_a_a @ F2 @ G2 )
        = ( comp_a_a_a @ L1 @ L2 ) )
     => ( ( ( comp_a_a_c @ L2 @ H )
          = R )
       => ( ( comp_c_a_c @ F2 @ ( comp_a_c_c @ G2 @ H ) )
          = ( comp_a_a_c @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_139_rewriteL__comp__comp2,axiom,
    ! [F2: b > b,G2: a > b,L1: a > b,L2: a > a,H: c > a,R: c > a] :
      ( ( ( comp_b_b_a @ F2 @ G2 )
        = ( comp_a_b_a @ L1 @ L2 ) )
     => ( ( ( comp_a_a_c @ L2 @ H )
          = R )
       => ( ( comp_b_b_c @ F2 @ ( comp_a_b_c @ G2 @ H ) )
          = ( comp_a_b_c @ L1 @ R ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_140_rewriteR__comp__comp,axiom,
    ! [G2: c > b,H: b > c,R: b > b,F2: b > a] :
      ( ( ( comp_c_b_b @ G2 @ H )
        = R )
     => ( ( comp_c_a_b @ ( comp_b_a_c @ F2 @ G2 ) @ H )
        = ( comp_b_a_b @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_141_rewriteR__comp__comp,axiom,
    ! [G2: c > b,H: c > c,R: c > b,F2: b > b] :
      ( ( ( comp_c_b_c @ G2 @ H )
        = R )
     => ( ( comp_c_b_c @ ( comp_b_b_c @ F2 @ G2 ) @ H )
        = ( comp_b_b_c @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_142_rewriteR__comp__comp,axiom,
    ! [G2: c > a,H: b > c,R: b > a,F2: a > a] :
      ( ( ( comp_c_a_b @ G2 @ H )
        = R )
     => ( ( comp_c_a_b @ ( comp_a_a_c @ F2 @ G2 ) @ H )
        = ( comp_a_a_b @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_143_rewriteR__comp__comp,axiom,
    ! [G2: b > c,H: c > b,R: c > c,F2: c > a] :
      ( ( ( comp_b_c_c @ G2 @ H )
        = R )
     => ( ( comp_b_a_c @ ( comp_c_a_b @ F2 @ G2 ) @ H )
        = ( comp_c_a_c @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_144_rewriteR__comp__comp,axiom,
    ! [G2: c > b,H: c > c,R: c > b,F2: b > a] :
      ( ( ( comp_c_b_c @ G2 @ H )
        = R )
     => ( ( comp_c_a_c @ ( comp_b_a_c @ F2 @ G2 ) @ H )
        = ( comp_b_a_c @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_145_rewriteR__comp__comp,axiom,
    ! [G2: c > c,H: c > c,R: c > c,F2: c > a] :
      ( ( ( comp_c_c_c @ G2 @ H )
        = R )
     => ( ( comp_c_a_c @ ( comp_c_a_c @ F2 @ G2 ) @ H )
        = ( comp_c_a_c @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_146_rewriteR__comp__comp,axiom,
    ! [G2: b > b,H: b > b,R: b > b,F2: b > a] :
      ( ( ( comp_b_b_b @ G2 @ H )
        = R )
     => ( ( comp_b_a_b @ ( comp_b_a_b @ F2 @ G2 ) @ H )
        = ( comp_b_a_b @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_147_rewriteR__comp__comp,axiom,
    ! [G2: a > b,H: c > a,R: c > b,F2: b > a] :
      ( ( ( comp_a_b_c @ G2 @ H )
        = R )
     => ( ( comp_a_a_c @ ( comp_b_a_a @ F2 @ G2 ) @ H )
        = ( comp_b_a_c @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_148_rewriteR__comp__comp,axiom,
    ! [G2: a > c,H: c > a,R: c > c,F2: c > a] :
      ( ( ( comp_a_c_c @ G2 @ H )
        = R )
     => ( ( comp_a_a_c @ ( comp_c_a_a @ F2 @ G2 ) @ H )
        = ( comp_c_a_c @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_149_rewriteR__comp__comp,axiom,
    ! [G2: a > b,H: b > a,R: b > b,F2: b > a] :
      ( ( ( comp_a_b_b @ G2 @ H )
        = R )
     => ( ( comp_a_a_b @ ( comp_b_a_a @ F2 @ G2 ) @ H )
        = ( comp_b_a_b @ F2 @ R ) ) ) ).

% rewriteR_comp_comp
thf(fact_150_rewriteL__comp__comp,axiom,
    ! [F2: a > b,G2: b > a,L: b > b,H: c > b] :
      ( ( ( comp_a_b_b @ F2 @ G2 )
        = L )
     => ( ( comp_a_b_c @ F2 @ ( comp_b_a_c @ G2 @ H ) )
        = ( comp_b_b_c @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_151_rewriteL__comp__comp,axiom,
    ! [F2: b > a,G2: a > b,L: a > a,H: c > a] :
      ( ( ( comp_b_a_a @ F2 @ G2 )
        = L )
     => ( ( comp_b_a_c @ F2 @ ( comp_a_b_c @ G2 @ H ) )
        = ( comp_a_a_c @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_152_rewriteL__comp__comp,axiom,
    ! [F2: c > a,G2: b > c,L: b > a,H: c > b] :
      ( ( ( comp_c_a_b @ F2 @ G2 )
        = L )
     => ( ( comp_c_a_c @ F2 @ ( comp_b_c_c @ G2 @ H ) )
        = ( comp_b_a_c @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_153_rewriteL__comp__comp,axiom,
    ! [F2: c > a,G2: a > c,L: a > a,H: c > a] :
      ( ( ( comp_c_a_a @ F2 @ G2 )
        = L )
     => ( ( comp_c_a_c @ F2 @ ( comp_a_c_c @ G2 @ H ) )
        = ( comp_a_a_c @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_154_rewriteL__comp__comp,axiom,
    ! [F2: b > b,G2: b > b,L: b > b,H: c > b] :
      ( ( ( comp_b_b_b @ F2 @ G2 )
        = L )
     => ( ( comp_b_b_c @ F2 @ ( comp_b_b_c @ G2 @ H ) )
        = ( comp_b_b_c @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_155_rewriteL__comp__comp,axiom,
    ! [F2: b > a,G2: a > b,L: a > a,H: b > a] :
      ( ( ( comp_b_a_a @ F2 @ G2 )
        = L )
     => ( ( comp_b_a_b @ F2 @ ( comp_a_b_b @ G2 @ H ) )
        = ( comp_a_a_b @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_156_rewriteL__comp__comp,axiom,
    ! [F2: a > a,G2: a > a,L: a > a,H: c > a] :
      ( ( ( comp_a_a_a @ F2 @ G2 )
        = L )
     => ( ( comp_a_a_c @ F2 @ ( comp_a_a_c @ G2 @ H ) )
        = ( comp_a_a_c @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_157_rewriteL__comp__comp,axiom,
    ! [F2: a > a,G2: a > a,L: a > a,H: b > a] :
      ( ( ( comp_a_a_a @ F2 @ G2 )
        = L )
     => ( ( comp_a_a_b @ F2 @ ( comp_a_a_b @ G2 @ H ) )
        = ( comp_a_a_b @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_158_rewriteL__comp__comp,axiom,
    ! [F2: b > a,G2: c > b,L: c > a,H: c > c] :
      ( ( ( comp_b_a_c @ F2 @ G2 )
        = L )
     => ( ( comp_b_a_c @ F2 @ ( comp_c_b_c @ G2 @ H ) )
        = ( comp_c_a_c @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_159_rewriteL__comp__comp,axiom,
    ! [F2: b > a,G2: c > b,L: c > a,H: b > c] :
      ( ( ( comp_b_a_c @ F2 @ G2 )
        = L )
     => ( ( comp_b_a_b @ F2 @ ( comp_c_b_b @ G2 @ H ) )
        = ( comp_c_a_b @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_160_fcomp__comp,axiom,
    ( fcomp_c_b_a
    = ( ^ [F: c > b,G: b > a] : ( comp_b_a_c @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_161_fcomp__comp,axiom,
    ( fcomp_c_c_a
    = ( ^ [F: c > c,G: c > a] : ( comp_c_a_c @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_162_fcomp__comp,axiom,
    ( fcomp_c_b_b
    = ( ^ [F: c > b,G: b > b] : ( comp_b_b_c @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_163_fcomp__comp,axiom,
    ( fcomp_b_b_a
    = ( ^ [F: b > b,G: b > a] : ( comp_b_a_b @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_164_fcomp__comp,axiom,
    ( fcomp_c_a_a
    = ( ^ [F: c > a,G: a > a] : ( comp_a_a_c @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_165_fcomp__comp,axiom,
    ( fcomp_b_a_a
    = ( ^ [F: b > a,G: a > a] : ( comp_a_a_b @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_166_fcomp__comp,axiom,
    ( fcomp_b_a_b
    = ( ^ [F: b > a,G: a > b] : ( comp_a_b_b @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_167_fcomp__comp,axiom,
    ( fcomp_a_a_b
    = ( ^ [F: a > a,G: a > b] : ( comp_a_b_a @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_168_fcomp__comp,axiom,
    ( fcomp_rat_a_a
    = ( ^ [F: rat > a,G: a > a] : ( comp_a_a_rat @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_169_fcomp__comp,axiom,
    ( fcomp_nat_a_a
    = ( ^ [F: nat > a,G: a > a] : ( comp_a_a_nat @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_170_vec__contains__img,axiom,
    ! [A: b,V: vec_b,F2: b > a] :
      ( ( member_b @ A @ ( vec_set_b @ V ) )
     => ( member_a @ ( F2 @ A ) @ ( vec_set_a @ ( map_vec_b_a @ F2 @ V ) ) ) ) ).

% vec_contains_img
thf(fact_171_vec__contains__img,axiom,
    ! [A: c,V: vec_c,F2: c > b] :
      ( ( member_c @ A @ ( vec_set_c @ V ) )
     => ( member_b @ ( F2 @ A ) @ ( vec_set_b @ ( map_vec_c_b @ F2 @ V ) ) ) ) ).

% vec_contains_img
thf(fact_172_vec__contains__img,axiom,
    ! [A: c,V: vec_c,F2: c > a] :
      ( ( member_c @ A @ ( vec_set_c @ V ) )
     => ( member_a @ ( F2 @ A ) @ ( vec_set_a @ ( map_vec_c_a @ F2 @ V ) ) ) ) ).

% vec_contains_img
thf(fact_173_vec__contains__img,axiom,
    ! [A: c,V: vec_c,F2: c > c] :
      ( ( member_c @ A @ ( vec_set_c @ V ) )
     => ( member_c @ ( F2 @ A ) @ ( vec_set_c @ ( map_vec_c_c @ F2 @ V ) ) ) ) ).

% vec_contains_img
thf(fact_174_vec__contains__img,axiom,
    ! [A: c,V: vec_c,F2: c > rat] :
      ( ( member_c @ A @ ( vec_set_c @ V ) )
     => ( member_rat @ ( F2 @ A ) @ ( vec_set_rat @ ( map_vec_c_rat @ F2 @ V ) ) ) ) ).

% vec_contains_img
thf(fact_175_vec__contains__img,axiom,
    ! [A: c,V: vec_c,F2: c > nat] :
      ( ( member_c @ A @ ( vec_set_c @ V ) )
     => ( member_nat @ ( F2 @ A ) @ ( vec_set_nat @ ( map_vec_c_nat @ F2 @ V ) ) ) ) ).

% vec_contains_img
thf(fact_176_vec__contains__img,axiom,
    ! [A: a,V: vec_a,F2: a > c] :
      ( ( member_a @ A @ ( vec_set_a @ V ) )
     => ( member_c @ ( F2 @ A ) @ ( vec_set_c @ ( map_vec_a_c @ F2 @ V ) ) ) ) ).

% vec_contains_img
thf(fact_177_vec__contains__img,axiom,
    ! [A: a,V: vec_a,F2: a > a] :
      ( ( member_a @ A @ ( vec_set_a @ V ) )
     => ( member_a @ ( F2 @ A ) @ ( vec_set_a @ ( map_vec_a_a @ F2 @ V ) ) ) ) ).

% vec_contains_img
thf(fact_178_vec__contains__img,axiom,
    ! [A: a,V: vec_a,F2: a > b] :
      ( ( member_a @ A @ ( vec_set_a @ V ) )
     => ( member_b @ ( F2 @ A ) @ ( vec_set_b @ ( map_vec_a_b @ F2 @ V ) ) ) ) ).

% vec_contains_img
thf(fact_179_vec__contains__img,axiom,
    ! [A: a,V: vec_a,F2: a > rat] :
      ( ( member_a @ A @ ( vec_set_a @ V ) )
     => ( member_rat @ ( F2 @ A ) @ ( vec_set_rat @ ( map_vec_a_rat @ F2 @ V ) ) ) ) ).

% vec_contains_img
thf(fact_180_map__fun__def,axiom,
    ( map_fun_c_b_b_a
    = ( ^ [F: c > b,G: b > a,H2: b > b] : ( comp_b_a_c @ ( comp_b_a_b @ G @ H2 ) @ F ) ) ) ).

% map_fun_def
thf(fact_181_map__fun__def,axiom,
    ( map_fun_c_b_a_a
    = ( ^ [F: c > b,G: a > a,H2: b > a] : ( comp_b_a_c @ ( comp_a_a_b @ G @ H2 ) @ F ) ) ) ).

% map_fun_def
thf(fact_182_map__fun__def,axiom,
    ( map_fun_c_c_b_a
    = ( ^ [F: c > c,G: b > a,H2: c > b] : ( comp_c_a_c @ ( comp_b_a_c @ G @ H2 ) @ F ) ) ) ).

% map_fun_def
thf(fact_183_map__fun__def,axiom,
    ( map_fun_c_c_c_a
    = ( ^ [F: c > c,G: c > a,H2: c > c] : ( comp_c_a_c @ ( comp_c_a_c @ G @ H2 ) @ F ) ) ) ).

% map_fun_def
thf(fact_184_map__fun__def,axiom,
    ( map_fun_c_c_a_a
    = ( ^ [F: c > c,G: a > a,H2: c > a] : ( comp_c_a_c @ ( comp_a_a_c @ G @ H2 ) @ F ) ) ) ).

% map_fun_def
thf(fact_185_map__fun__def,axiom,
    ( map_fun_b_b_b_a
    = ( ^ [F: b > b,G: b > a,H2: b > b] : ( comp_b_a_b @ ( comp_b_a_b @ G @ H2 ) @ F ) ) ) ).

% map_fun_def
thf(fact_186_map__fun__def,axiom,
    ( map_fun_b_b_a_a
    = ( ^ [F: b > b,G: a > a,H2: b > a] : ( comp_b_a_b @ ( comp_a_a_b @ G @ H2 ) @ F ) ) ) ).

% map_fun_def
thf(fact_187_map__fun__def,axiom,
    ( map_fun_c_b_a_b
    = ( ^ [F: c > b,G: a > b,H2: b > a] : ( comp_b_b_c @ ( comp_a_b_b @ G @ H2 ) @ F ) ) ) ).

% map_fun_def
thf(fact_188_map__fun__def,axiom,
    ( map_fun_c_a_a_a
    = ( ^ [F: c > a,G: a > a,H2: a > a] : ( comp_a_a_c @ ( comp_a_a_a @ G @ H2 ) @ F ) ) ) ).

% map_fun_def
thf(fact_189_map__fun__def,axiom,
    ( map_fun_b_a_a_a
    = ( ^ [F: b > a,G: a > a,H2: a > a] : ( comp_a_a_b @ ( comp_a_a_a @ G @ H2 ) @ F ) ) ) ).

% map_fun_def
thf(fact_190_map__fun_Ocompositionality,axiom,
    ! [F2: c > b,G2: b > a,H: b > a,I: c > b,Fun: a > c] :
      ( ( map_fun_c_b_b_a @ F2 @ G2 @ ( map_fun_b_a_c_b @ H @ I @ Fun ) )
      = ( map_fun_c_a_c_a @ ( comp_b_a_c @ H @ F2 ) @ ( comp_b_a_c @ G2 @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_191_map__fun_Ocompositionality,axiom,
    ! [F2: c > b,G2: c > a,H: b > a,I: c > c,Fun: a > c] :
      ( ( map_fun_c_b_c_a @ F2 @ G2 @ ( map_fun_b_a_c_c @ H @ I @ Fun ) )
      = ( map_fun_c_a_c_a @ ( comp_b_a_c @ H @ F2 ) @ ( comp_c_a_c @ G2 @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_192_map__fun_Ocompositionality,axiom,
    ! [F2: c > b,G2: b > b,H: b > a,I: c > b,Fun: a > c] :
      ( ( map_fun_c_b_b_b @ F2 @ G2 @ ( map_fun_b_a_c_b @ H @ I @ Fun ) )
      = ( map_fun_c_a_c_b @ ( comp_b_a_c @ H @ F2 ) @ ( comp_b_b_c @ G2 @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_193_map__fun_Ocompositionality,axiom,
    ! [F2: c > b,G2: b > a,H: b > a,I: b > b,Fun: a > b] :
      ( ( map_fun_c_b_b_a @ F2 @ G2 @ ( map_fun_b_a_b_b @ H @ I @ Fun ) )
      = ( map_fun_c_a_b_a @ ( comp_b_a_c @ H @ F2 ) @ ( comp_b_a_b @ G2 @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_194_map__fun_Ocompositionality,axiom,
    ! [F2: c > b,G2: a > a,H: b > a,I: c > a,Fun: a > c] :
      ( ( map_fun_c_b_a_a @ F2 @ G2 @ ( map_fun_b_a_c_a @ H @ I @ Fun ) )
      = ( map_fun_c_a_c_a @ ( comp_b_a_c @ H @ F2 ) @ ( comp_a_a_c @ G2 @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_195_map__fun_Ocompositionality,axiom,
    ! [F2: c > b,G2: a > a,H: b > a,I: b > a,Fun: a > b] :
      ( ( map_fun_c_b_a_a @ F2 @ G2 @ ( map_fun_b_a_b_a @ H @ I @ Fun ) )
      = ( map_fun_c_a_b_a @ ( comp_b_a_c @ H @ F2 ) @ ( comp_a_a_b @ G2 @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_196_map__fun_Ocompositionality,axiom,
    ! [F2: c > c,G2: b > a,H: c > a,I: c > b,Fun: a > c] :
      ( ( map_fun_c_c_b_a @ F2 @ G2 @ ( map_fun_c_a_c_b @ H @ I @ Fun ) )
      = ( map_fun_c_a_c_a @ ( comp_c_a_c @ H @ F2 ) @ ( comp_b_a_c @ G2 @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_197_map__fun_Ocompositionality,axiom,
    ! [F2: c > c,G2: c > a,H: c > a,I: c > c,Fun: a > c] :
      ( ( map_fun_c_c_c_a @ F2 @ G2 @ ( map_fun_c_a_c_c @ H @ I @ Fun ) )
      = ( map_fun_c_a_c_a @ ( comp_c_a_c @ H @ F2 ) @ ( comp_c_a_c @ G2 @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_198_map__fun_Ocompositionality,axiom,
    ! [F2: c > c,G2: b > b,H: c > a,I: c > b,Fun: a > c] :
      ( ( map_fun_c_c_b_b @ F2 @ G2 @ ( map_fun_c_a_c_b @ H @ I @ Fun ) )
      = ( map_fun_c_a_c_b @ ( comp_c_a_c @ H @ F2 ) @ ( comp_b_b_c @ G2 @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_199_map__fun_Ocompositionality,axiom,
    ! [F2: c > c,G2: b > a,H: c > a,I: b > b,Fun: a > b] :
      ( ( map_fun_c_c_b_a @ F2 @ G2 @ ( map_fun_c_a_b_b @ H @ I @ Fun ) )
      = ( map_fun_c_a_b_a @ ( comp_c_a_c @ H @ F2 ) @ ( comp_b_a_b @ G2 @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_200_map__fun_Ocomp,axiom,
    ! [F2: c > b,G2: b > a,H: b > a,I: c > b] :
      ( ( comp_b_b_c_a_a_c @ ( map_fun_c_b_b_a @ F2 @ G2 ) @ ( map_fun_b_a_c_b @ H @ I ) )
      = ( map_fun_c_a_c_a @ ( comp_b_a_c @ H @ F2 ) @ ( comp_b_a_c @ G2 @ I ) ) ) ).

% map_fun.comp
thf(fact_201_map__fun_Ocomp,axiom,
    ! [F2: c > b,G2: c > a,H: b > a,I: c > c] :
      ( ( comp_b_c_c_a_a_c @ ( map_fun_c_b_c_a @ F2 @ G2 ) @ ( map_fun_b_a_c_c @ H @ I ) )
      = ( map_fun_c_a_c_a @ ( comp_b_a_c @ H @ F2 ) @ ( comp_c_a_c @ G2 @ I ) ) ) ).

% map_fun.comp
thf(fact_202_map__fun_Ocomp,axiom,
    ! [F2: c > b,G2: b > b,H: b > a,I: c > b] :
      ( ( comp_b_b_c_b_a_c @ ( map_fun_c_b_b_b @ F2 @ G2 ) @ ( map_fun_b_a_c_b @ H @ I ) )
      = ( map_fun_c_a_c_b @ ( comp_b_a_c @ H @ F2 ) @ ( comp_b_b_c @ G2 @ I ) ) ) ).

% map_fun.comp
thf(fact_203_map__fun_Ocomp,axiom,
    ! [F2: c > b,G2: b > a,H: b > a,I: b > b] :
      ( ( comp_b_b_c_a_a_b @ ( map_fun_c_b_b_a @ F2 @ G2 ) @ ( map_fun_b_a_b_b @ H @ I ) )
      = ( map_fun_c_a_b_a @ ( comp_b_a_c @ H @ F2 ) @ ( comp_b_a_b @ G2 @ I ) ) ) ).

% map_fun.comp
thf(fact_204_map__fun_Ocomp,axiom,
    ! [F2: c > b,G2: a > a,H: b > a,I: c > a] :
      ( ( comp_b_a_c_a_a_c @ ( map_fun_c_b_a_a @ F2 @ G2 ) @ ( map_fun_b_a_c_a @ H @ I ) )
      = ( map_fun_c_a_c_a @ ( comp_b_a_c @ H @ F2 ) @ ( comp_a_a_c @ G2 @ I ) ) ) ).

% map_fun.comp
thf(fact_205_map__fun_Ocomp,axiom,
    ! [F2: c > b,G2: a > a,H: b > a,I: b > a] :
      ( ( comp_b_a_c_a_a_b @ ( map_fun_c_b_a_a @ F2 @ G2 ) @ ( map_fun_b_a_b_a @ H @ I ) )
      = ( map_fun_c_a_b_a @ ( comp_b_a_c @ H @ F2 ) @ ( comp_a_a_b @ G2 @ I ) ) ) ).

% map_fun.comp
thf(fact_206_map__fun_Ocomp,axiom,
    ! [F2: c > c,G2: b > a,H: c > a,I: c > b] :
      ( ( comp_c_b_c_a_a_c @ ( map_fun_c_c_b_a @ F2 @ G2 ) @ ( map_fun_c_a_c_b @ H @ I ) )
      = ( map_fun_c_a_c_a @ ( comp_c_a_c @ H @ F2 ) @ ( comp_b_a_c @ G2 @ I ) ) ) ).

% map_fun.comp
thf(fact_207_map__fun_Ocomp,axiom,
    ! [F2: c > c,G2: c > a,H: c > a,I: c > c] :
      ( ( comp_c_c_c_a_a_c @ ( map_fun_c_c_c_a @ F2 @ G2 ) @ ( map_fun_c_a_c_c @ H @ I ) )
      = ( map_fun_c_a_c_a @ ( comp_c_a_c @ H @ F2 ) @ ( comp_c_a_c @ G2 @ I ) ) ) ).

% map_fun.comp
thf(fact_208_map__fun_Ocomp,axiom,
    ! [F2: c > c,G2: b > b,H: c > a,I: c > b] :
      ( ( comp_c_b_c_b_a_c @ ( map_fun_c_c_b_b @ F2 @ G2 ) @ ( map_fun_c_a_c_b @ H @ I ) )
      = ( map_fun_c_a_c_b @ ( comp_c_a_c @ H @ F2 ) @ ( comp_b_b_c @ G2 @ I ) ) ) ).

% map_fun.comp
thf(fact_209_map__fun_Ocomp,axiom,
    ! [F2: c > c,G2: b > a,H: c > a,I: b > b] :
      ( ( comp_c_b_c_a_a_b @ ( map_fun_c_c_b_a @ F2 @ G2 ) @ ( map_fun_c_a_b_b @ H @ I ) )
      = ( map_fun_c_a_b_a @ ( comp_c_a_c @ H @ F2 ) @ ( comp_b_a_b @ G2 @ I ) ) ) ).

% map_fun.comp
thf(fact_210_id__comp,axiom,
    ! [G2: rat > a] :
      ( ( comp_a_a_rat @ id_a @ G2 )
      = G2 ) ).

% id_comp
thf(fact_211_id__comp,axiom,
    ! [G2: nat > a] :
      ( ( comp_a_a_nat @ id_a @ G2 )
      = G2 ) ).

% id_comp
thf(fact_212_id__comp,axiom,
    ! [G2: a > a] :
      ( ( comp_a_a_a @ id_a @ G2 )
      = G2 ) ).

% id_comp
thf(fact_213_id__comp,axiom,
    ! [G2: c > b] :
      ( ( comp_b_b_c @ id_b @ G2 )
      = G2 ) ).

% id_comp
thf(fact_214_id__comp,axiom,
    ! [G2: c > a] :
      ( ( comp_a_a_c @ id_a @ G2 )
      = G2 ) ).

% id_comp
thf(fact_215_id__comp,axiom,
    ! [G2: b > a] :
      ( ( comp_a_a_b @ id_a @ G2 )
      = G2 ) ).

% id_comp
thf(fact_216_comp__id,axiom,
    ! [F2: a > b] :
      ( ( comp_a_b_a @ F2 @ id_a )
      = F2 ) ).

% comp_id
thf(fact_217_comp__id,axiom,
    ! [F2: a > a] :
      ( ( comp_a_a_a @ F2 @ id_a )
      = F2 ) ).

% comp_id
thf(fact_218_comp__id,axiom,
    ! [F2: c > a] :
      ( ( comp_c_a_c @ F2 @ id_c )
      = F2 ) ).

% comp_id
thf(fact_219_comp__id,axiom,
    ! [F2: b > a] :
      ( ( comp_b_a_b @ F2 @ id_b )
      = F2 ) ).

% comp_id
thf(fact_220_id__apply,axiom,
    ( id_c_a
    = ( ^ [X: c > a] : X ) ) ).

% id_apply
thf(fact_221_id__apply,axiom,
    ( id_b_a
    = ( ^ [X: b > a] : X ) ) ).

% id_apply
thf(fact_222_id__apply,axiom,
    ( id_c
    = ( ^ [X: c] : X ) ) ).

% id_apply
thf(fact_223_id__apply,axiom,
    ( id_b
    = ( ^ [X: b] : X ) ) ).

% id_apply
thf(fact_224_id__apply,axiom,
    ( id_a
    = ( ^ [X: a] : X ) ) ).

% id_apply
thf(fact_225_map__fun__apply,axiom,
    ( map_fun_a_a_c_b
    = ( ^ [F: a > a,G: c > b,H2: a > c,X: a] : ( G @ ( H2 @ ( F @ X ) ) ) ) ) ).

% map_fun_apply
thf(fact_226_map__fun__apply,axiom,
    ( map_fun_a_a_c_a
    = ( ^ [F: a > a,G: c > a,H2: a > c,X: a] : ( G @ ( H2 @ ( F @ X ) ) ) ) ) ).

% map_fun_apply
thf(fact_227_map__fun__apply,axiom,
    ( map_fun_a_a_b_b
    = ( ^ [F: a > a,G: b > b,H2: a > b,X: a] : ( G @ ( H2 @ ( F @ X ) ) ) ) ) ).

% map_fun_apply
thf(fact_228_map__fun__apply,axiom,
    ( map_fun_a_a_b_a
    = ( ^ [F: a > a,G: b > a,H2: a > b,X: a] : ( G @ ( H2 @ ( F @ X ) ) ) ) ) ).

% map_fun_apply
thf(fact_229_map__fun__apply,axiom,
    ( map_fun_a_a_a_a
    = ( ^ [F: a > a,G: a > a,H2: a > a,X: a] : ( G @ ( H2 @ ( F @ X ) ) ) ) ) ).

% map_fun_apply
thf(fact_230_fcomp__apply,axiom,
    ( fcomp_c_b_b
    = ( ^ [F: c > b,G: b > b,X: c] : ( G @ ( F @ X ) ) ) ) ).

% fcomp_apply
thf(fact_231_fcomp__apply,axiom,
    ( fcomp_c_b_a
    = ( ^ [F: c > b,G: b > a,X: c] : ( G @ ( F @ X ) ) ) ) ).

% fcomp_apply
thf(fact_232_fcomp__apply,axiom,
    ( fcomp_c_a_a
    = ( ^ [F: c > a,G: a > a,X: c] : ( G @ ( F @ X ) ) ) ) ).

% fcomp_apply
thf(fact_233_fcomp__apply,axiom,
    ( fcomp_b_b_a
    = ( ^ [F: b > b,G: b > a,X: b] : ( G @ ( F @ X ) ) ) ) ).

% fcomp_apply
thf(fact_234_fcomp__apply,axiom,
    ( fcomp_b_a_a
    = ( ^ [F: b > a,G: a > a,X: b] : ( G @ ( F @ X ) ) ) ) ).

% fcomp_apply
thf(fact_235_fun_Omap__id,axiom,
    ! [T: rat > a] :
      ( ( comp_a_a_rat @ id_a @ T )
      = T ) ).

% fun.map_id
thf(fact_236_fun_Omap__id,axiom,
    ! [T: nat > a] :
      ( ( comp_a_a_nat @ id_a @ T )
      = T ) ).

% fun.map_id
thf(fact_237_fun_Omap__id,axiom,
    ! [T: a > a] :
      ( ( comp_a_a_a @ id_a @ T )
      = T ) ).

% fun.map_id
thf(fact_238_fun_Omap__id,axiom,
    ! [T: c > b] :
      ( ( comp_b_b_c @ id_b @ T )
      = T ) ).

% fun.map_id
thf(fact_239_fun_Omap__id,axiom,
    ! [T: c > a] :
      ( ( comp_a_a_c @ id_a @ T )
      = T ) ).

% fun.map_id
thf(fact_240_fun_Omap__id,axiom,
    ! [T: b > a] :
      ( ( comp_a_a_b @ id_a @ T )
      = T ) ).

% fun.map_id
thf(fact_241_fcomp__id,axiom,
    ! [F2: c > b] :
      ( ( fcomp_c_b_b @ F2 @ id_b )
      = F2 ) ).

% fcomp_id
thf(fact_242_fcomp__id,axiom,
    ! [F2: c > a] :
      ( ( fcomp_c_a_a @ F2 @ id_a )
      = F2 ) ).

% fcomp_id
thf(fact_243_fcomp__id,axiom,
    ! [F2: b > a] :
      ( ( fcomp_b_a_a @ F2 @ id_a )
      = F2 ) ).

% fcomp_id
thf(fact_244_id__fcomp,axiom,
    ! [G2: b > a] :
      ( ( fcomp_b_b_a @ id_b @ G2 )
      = G2 ) ).

% id_fcomp
thf(fact_245_id__def,axiom,
    ( id_c_a
    = ( ^ [X: c > a] : X ) ) ).

% id_def
thf(fact_246_id__def,axiom,
    ( id_b_a
    = ( ^ [X: b > a] : X ) ) ).

% id_def
thf(fact_247_id__def,axiom,
    ( id_c
    = ( ^ [X: c] : X ) ) ).

% id_def
thf(fact_248_id__def,axiom,
    ( id_b
    = ( ^ [X: b] : X ) ) ).

% id_def
thf(fact_249_id__def,axiom,
    ( id_a
    = ( ^ [X: a] : X ) ) ).

% id_def
thf(fact_250_eq__id__iff,axiom,
    ! [F2: ( c > a ) > c > a] :
      ( ( ! [X: c > a] :
            ( ( F2 @ X )
            = X ) )
      = ( F2 = id_c_a ) ) ).

% eq_id_iff
thf(fact_251_eq__id__iff,axiom,
    ! [F2: ( b > a ) > b > a] :
      ( ( ! [X: b > a] :
            ( ( F2 @ X )
            = X ) )
      = ( F2 = id_b_a ) ) ).

% eq_id_iff
thf(fact_252_eq__id__iff,axiom,
    ! [F2: c > c] :
      ( ( ! [X: c] :
            ( ( F2 @ X )
            = X ) )
      = ( F2 = id_c ) ) ).

% eq_id_iff
thf(fact_253_eq__id__iff,axiom,
    ! [F2: b > b] :
      ( ( ! [X: b] :
            ( ( F2 @ X )
            = X ) )
      = ( F2 = id_b ) ) ).

% eq_id_iff
thf(fact_254_eq__id__iff,axiom,
    ! [F2: a > a] :
      ( ( ! [X: a] :
            ( ( F2 @ X )
            = X ) )
      = ( F2 = id_a ) ) ).

% eq_id_iff
thf(fact_255_fcomp__assoc,axiom,
    ! [F2: c > b,G2: b > a,H: a > b] :
      ( ( fcomp_c_a_b @ ( fcomp_c_b_a @ F2 @ G2 ) @ H )
      = ( fcomp_c_b_b @ F2 @ ( fcomp_b_a_b @ G2 @ H ) ) ) ).

% fcomp_assoc
thf(fact_256_fcomp__assoc,axiom,
    ! [F2: c > c,G2: c > b,H: b > b] :
      ( ( fcomp_c_b_b @ ( fcomp_c_c_b @ F2 @ G2 ) @ H )
      = ( fcomp_c_c_b @ F2 @ ( fcomp_c_b_b @ G2 @ H ) ) ) ).

% fcomp_assoc
thf(fact_257_fcomp__assoc,axiom,
    ! [F2: c > b,G2: b > b,H: b > b] :
      ( ( fcomp_c_b_b @ ( fcomp_c_b_b @ F2 @ G2 ) @ H )
      = ( fcomp_c_b_b @ F2 @ ( fcomp_b_b_b @ G2 @ H ) ) ) ).

% fcomp_assoc
thf(fact_258_fcomp__assoc,axiom,
    ! [F2: c > c,G2: c > b,H: b > a] :
      ( ( fcomp_c_b_a @ ( fcomp_c_c_b @ F2 @ G2 ) @ H )
      = ( fcomp_c_c_a @ F2 @ ( fcomp_c_b_a @ G2 @ H ) ) ) ).

% fcomp_assoc
thf(fact_259_fcomp__assoc,axiom,
    ! [F2: c > a,G2: a > b,H: b > a] :
      ( ( fcomp_c_b_a @ ( fcomp_c_a_b @ F2 @ G2 ) @ H )
      = ( fcomp_c_a_a @ F2 @ ( fcomp_a_b_a @ G2 @ H ) ) ) ).

% fcomp_assoc
thf(fact_260_fcomp__assoc,axiom,
    ! [F2: c > b,G2: b > b,H: b > a] :
      ( ( fcomp_c_b_a @ ( fcomp_c_b_b @ F2 @ G2 ) @ H )
      = ( fcomp_c_b_a @ F2 @ ( fcomp_b_b_a @ G2 @ H ) ) ) ).

% fcomp_assoc
thf(fact_261_fcomp__assoc,axiom,
    ! [F2: c > c,G2: c > a,H: a > a] :
      ( ( fcomp_c_a_a @ ( fcomp_c_c_a @ F2 @ G2 ) @ H )
      = ( fcomp_c_c_a @ F2 @ ( fcomp_c_a_a @ G2 @ H ) ) ) ).

% fcomp_assoc
thf(fact_262_fcomp__assoc,axiom,
    ! [F2: c > b,G2: b > a,H: a > a] :
      ( ( fcomp_c_a_a @ ( fcomp_c_b_a @ F2 @ G2 ) @ H )
      = ( fcomp_c_b_a @ F2 @ ( fcomp_b_a_a @ G2 @ H ) ) ) ).

% fcomp_assoc
thf(fact_263_fcomp__assoc,axiom,
    ! [F2: c > a,G2: a > a,H: a > a] :
      ( ( fcomp_c_a_a @ ( fcomp_c_a_a @ F2 @ G2 ) @ H )
      = ( fcomp_c_a_a @ F2 @ ( fcomp_a_a_a @ G2 @ H ) ) ) ).

% fcomp_assoc
thf(fact_264_fcomp__assoc,axiom,
    ! [F2: b > c,G2: c > b,H: b > a] :
      ( ( fcomp_b_b_a @ ( fcomp_b_c_b @ F2 @ G2 ) @ H )
      = ( fcomp_b_c_a @ F2 @ ( fcomp_c_b_a @ G2 @ H ) ) ) ).

% fcomp_assoc
thf(fact_265_fcomp__def,axiom,
    ( fcomp_c_b_b
    = ( ^ [F: c > b,G: b > b,X: c] : ( G @ ( F @ X ) ) ) ) ).

% fcomp_def
thf(fact_266_fcomp__def,axiom,
    ( fcomp_c_b_a
    = ( ^ [F: c > b,G: b > a,X: c] : ( G @ ( F @ X ) ) ) ) ).

% fcomp_def
thf(fact_267_fcomp__def,axiom,
    ( fcomp_c_a_a
    = ( ^ [F: c > a,G: a > a,X: c] : ( G @ ( F @ X ) ) ) ) ).

% fcomp_def
thf(fact_268_fcomp__def,axiom,
    ( fcomp_b_b_a
    = ( ^ [F: b > b,G: b > a,X: b] : ( G @ ( F @ X ) ) ) ) ).

% fcomp_def
thf(fact_269_fcomp__def,axiom,
    ( fcomp_b_a_a
    = ( ^ [F: b > a,G: a > a,X: b] : ( G @ ( F @ X ) ) ) ) ).

% fcomp_def
thf(fact_270_fun_Omap__id0,axiom,
    ( ( comp_a_a_rat @ id_a )
    = id_rat_a ) ).

% fun.map_id0
thf(fact_271_fun_Omap__id0,axiom,
    ( ( comp_a_a_nat @ id_a )
    = id_nat_a ) ).

% fun.map_id0
thf(fact_272_fun_Omap__id0,axiom,
    ( ( comp_a_a_a @ id_a )
    = id_a_a ) ).

% fun.map_id0
thf(fact_273_fun_Omap__id0,axiom,
    ( ( comp_b_b_c @ id_b )
    = id_c_b ) ).

% fun.map_id0
thf(fact_274_fun_Omap__id0,axiom,
    ( ( comp_a_a_c @ id_a )
    = id_c_a ) ).

% fun.map_id0
thf(fact_275_fun_Omap__id0,axiom,
    ( ( comp_a_a_b @ id_a )
    = id_b_a ) ).

% fun.map_id0
thf(fact_276_pointfree__idE,axiom,
    ! [F2: a > b,G2: b > a,X2: b] :
      ( ( ( comp_a_b_b @ F2 @ G2 )
        = id_b )
     => ( ( F2 @ ( G2 @ X2 ) )
        = X2 ) ) ).

% pointfree_idE
thf(fact_277_pointfree__idE,axiom,
    ! [F2: a > a,G2: a > a,X2: a] :
      ( ( ( comp_a_a_a @ F2 @ G2 )
        = id_a )
     => ( ( F2 @ ( G2 @ X2 ) )
        = X2 ) ) ).

% pointfree_idE
thf(fact_278_comp__eq__id__dest,axiom,
    ! [A: b > a,B: c > b,C: c > a,V: c] :
      ( ( ( comp_b_a_c @ A @ B )
        = ( comp_a_a_c @ id_a @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_279_comp__eq__id__dest,axiom,
    ! [A: c > a,B: c > c,C: c > a,V: c] :
      ( ( ( comp_c_a_c @ A @ B )
        = ( comp_a_a_c @ id_a @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_280_comp__eq__id__dest,axiom,
    ! [A: b > b,B: c > b,C: c > b,V: c] :
      ( ( ( comp_b_b_c @ A @ B )
        = ( comp_b_b_c @ id_b @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_281_comp__eq__id__dest,axiom,
    ! [A: b > a,B: b > b,C: b > a,V: b] :
      ( ( ( comp_b_a_b @ A @ B )
        = ( comp_a_a_b @ id_a @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_282_comp__eq__id__dest,axiom,
    ! [A: a > a,B: c > a,C: c > a,V: c] :
      ( ( ( comp_a_a_c @ A @ B )
        = ( comp_a_a_c @ id_a @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_283_comp__eq__id__dest,axiom,
    ! [A: a > a,B: b > a,C: b > a,V: b] :
      ( ( ( comp_a_a_b @ A @ B )
        = ( comp_a_a_b @ id_a @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_284_comp__eq__id__dest,axiom,
    ! [A: a > b,B: b > a,C: b > b,V: b] :
      ( ( ( comp_a_b_b @ A @ B )
        = ( comp_b_b_b @ id_b @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_285_comp__eq__id__dest,axiom,
    ! [A: a > b,B: a > a,C: a > b,V: a] :
      ( ( ( comp_a_b_a @ A @ B )
        = ( comp_b_b_a @ id_b @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_286_comp__eq__id__dest,axiom,
    ! [A: a > a,B: rat > a,C: rat > a,V: rat] :
      ( ( ( comp_a_a_rat @ A @ B )
        = ( comp_a_a_rat @ id_a @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_287_comp__eq__id__dest,axiom,
    ! [A: a > a,B: nat > a,C: nat > a,V: nat] :
      ( ( ( comp_a_a_nat @ A @ B )
        = ( comp_a_a_nat @ id_a @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_288_map__fun__id,axiom,
    ( ( map_fun_c_c_c_c @ id_c @ id_c )
    = id_c_c ) ).

% map_fun_id
thf(fact_289_map__fun__id,axiom,
    ( ( map_fun_c_c_b_b @ id_c @ id_b )
    = id_c_b ) ).

% map_fun_id
thf(fact_290_map__fun__id,axiom,
    ( ( map_fun_c_c_a_a @ id_c @ id_a )
    = id_c_a ) ).

% map_fun_id
thf(fact_291_map__fun__id,axiom,
    ( ( map_fun_b_b_c_c @ id_b @ id_c )
    = id_b_c ) ).

% map_fun_id
thf(fact_292_map__fun__id,axiom,
    ( ( map_fun_b_b_b_b @ id_b @ id_b )
    = id_b_b ) ).

% map_fun_id
thf(fact_293_map__fun__id,axiom,
    ( ( map_fun_b_b_a_a @ id_b @ id_a )
    = id_b_a ) ).

% map_fun_id
thf(fact_294_map__fun__id,axiom,
    ( ( map_fun_a_a_c_c @ id_a @ id_c )
    = id_a_c ) ).

% map_fun_id
thf(fact_295_map__fun__id,axiom,
    ( ( map_fun_a_a_b_b @ id_a @ id_b )
    = id_a_b ) ).

% map_fun_id
thf(fact_296_map__fun__id,axiom,
    ( ( map_fun_a_a_a_a @ id_a @ id_a )
    = id_a_a ) ).

% map_fun_id
thf(fact_297_map__fun__id,axiom,
    ( ( map_fun_c_a_c_a_c_c @ id_c_a @ id_c )
    = id_c_a_c ) ).

% map_fun_id
thf(fact_298_isomorphism__expand,axiom,
    ! [F2: c > c,G2: c > c] :
      ( ( ( ( comp_c_c_c @ F2 @ G2 )
          = id_c )
        & ( ( comp_c_c_c @ G2 @ F2 )
          = id_c ) )
      = ( ! [X: c] :
            ( ( F2 @ ( G2 @ X ) )
            = X )
        & ! [X: c] :
            ( ( G2 @ ( F2 @ X ) )
            = X ) ) ) ).

% isomorphism_expand
thf(fact_299_isomorphism__expand,axiom,
    ! [F2: b > c,G2: c > b] :
      ( ( ( ( comp_b_c_c @ F2 @ G2 )
          = id_c )
        & ( ( comp_c_b_b @ G2 @ F2 )
          = id_b ) )
      = ( ! [X: c] :
            ( ( F2 @ ( G2 @ X ) )
            = X )
        & ! [X: b] :
            ( ( G2 @ ( F2 @ X ) )
            = X ) ) ) ).

% isomorphism_expand
thf(fact_300_isomorphism__expand,axiom,
    ! [F2: a > c,G2: c > a] :
      ( ( ( ( comp_a_c_c @ F2 @ G2 )
          = id_c )
        & ( ( comp_c_a_a @ G2 @ F2 )
          = id_a ) )
      = ( ! [X: c] :
            ( ( F2 @ ( G2 @ X ) )
            = X )
        & ! [X: a] :
            ( ( G2 @ ( F2 @ X ) )
            = X ) ) ) ).

% isomorphism_expand
thf(fact_301_isomorphism__expand,axiom,
    ! [F2: c > b,G2: b > c] :
      ( ( ( ( comp_c_b_b @ F2 @ G2 )
          = id_b )
        & ( ( comp_b_c_c @ G2 @ F2 )
          = id_c ) )
      = ( ! [X: b] :
            ( ( F2 @ ( G2 @ X ) )
            = X )
        & ! [X: c] :
            ( ( G2 @ ( F2 @ X ) )
            = X ) ) ) ).

% isomorphism_expand
thf(fact_302_isomorphism__expand,axiom,
    ! [F2: b > b,G2: b > b] :
      ( ( ( ( comp_b_b_b @ F2 @ G2 )
          = id_b )
        & ( ( comp_b_b_b @ G2 @ F2 )
          = id_b ) )
      = ( ! [X: b] :
            ( ( F2 @ ( G2 @ X ) )
            = X )
        & ! [X: b] :
            ( ( G2 @ ( F2 @ X ) )
            = X ) ) ) ).

% isomorphism_expand
thf(fact_303_isomorphism__expand,axiom,
    ! [F2: c > a,G2: a > c] :
      ( ( ( ( comp_c_a_a @ F2 @ G2 )
          = id_a )
        & ( ( comp_a_c_c @ G2 @ F2 )
          = id_c ) )
      = ( ! [X: a] :
            ( ( F2 @ ( G2 @ X ) )
            = X )
        & ! [X: c] :
            ( ( G2 @ ( F2 @ X ) )
            = X ) ) ) ).

% isomorphism_expand
thf(fact_304_isomorphism__expand,axiom,
    ! [F2: b > a,G2: a > b] :
      ( ( ( ( comp_b_a_a @ F2 @ G2 )
          = id_a )
        & ( ( comp_a_b_b @ G2 @ F2 )
          = id_b ) )
      = ( ! [X: a] :
            ( ( F2 @ ( G2 @ X ) )
            = X )
        & ! [X: b] :
            ( ( G2 @ ( F2 @ X ) )
            = X ) ) ) ).

% isomorphism_expand
thf(fact_305_isomorphism__expand,axiom,
    ! [F2: a > b,G2: b > a] :
      ( ( ( ( comp_a_b_b @ F2 @ G2 )
          = id_b )
        & ( ( comp_b_a_a @ G2 @ F2 )
          = id_a ) )
      = ( ! [X: b] :
            ( ( F2 @ ( G2 @ X ) )
            = X )
        & ! [X: a] :
            ( ( G2 @ ( F2 @ X ) )
            = X ) ) ) ).

% isomorphism_expand
thf(fact_306_isomorphism__expand,axiom,
    ! [F2: a > a,G2: a > a] :
      ( ( ( ( comp_a_a_a @ F2 @ G2 )
          = id_a )
        & ( ( comp_a_a_a @ G2 @ F2 )
          = id_a ) )
      = ( ! [X: a] :
            ( ( F2 @ ( G2 @ X ) )
            = X )
        & ! [X: a] :
            ( ( G2 @ ( F2 @ X ) )
            = X ) ) ) ).

% isomorphism_expand
thf(fact_307_isomorphism__expand,axiom,
    ! [F2: c > c > a,G2: ( c > a ) > c] :
      ( ( ( ( comp_c_c_a_c_a @ F2 @ G2 )
          = id_c_a )
        & ( ( comp_c_a_c_c @ G2 @ F2 )
          = id_c ) )
      = ( ! [X: c > a] :
            ( ( F2 @ ( G2 @ X ) )
            = X )
        & ! [X: c] :
            ( ( G2 @ ( F2 @ X ) )
            = X ) ) ) ).

% isomorphism_expand
thf(fact_308_left__right__inverse__eq,axiom,
    ! [F2: c > c,G2: c > c,H: c > c] :
      ( ( ( comp_c_c_c @ F2 @ G2 )
        = id_c )
     => ( ( ( comp_c_c_c @ G2 @ H )
          = id_c )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_309_left__right__inverse__eq,axiom,
    ! [F2: b > c,G2: c > b,H: b > c] :
      ( ( ( comp_b_c_c @ F2 @ G2 )
        = id_c )
     => ( ( ( comp_c_b_b @ G2 @ H )
          = id_b )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_310_left__right__inverse__eq,axiom,
    ! [F2: a > c,G2: c > a,H: a > c] :
      ( ( ( comp_a_c_c @ F2 @ G2 )
        = id_c )
     => ( ( ( comp_c_a_a @ G2 @ H )
          = id_a )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_311_left__right__inverse__eq,axiom,
    ! [F2: c > b,G2: b > c,H: c > b] :
      ( ( ( comp_c_b_b @ F2 @ G2 )
        = id_b )
     => ( ( ( comp_b_c_c @ G2 @ H )
          = id_c )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_312_left__right__inverse__eq,axiom,
    ! [F2: b > b,G2: b > b,H: b > b] :
      ( ( ( comp_b_b_b @ F2 @ G2 )
        = id_b )
     => ( ( ( comp_b_b_b @ G2 @ H )
          = id_b )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_313_left__right__inverse__eq,axiom,
    ! [F2: c > a,G2: a > c,H: c > a] :
      ( ( ( comp_c_a_a @ F2 @ G2 )
        = id_a )
     => ( ( ( comp_a_c_c @ G2 @ H )
          = id_c )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_314_left__right__inverse__eq,axiom,
    ! [F2: b > a,G2: a > b,H: b > a] :
      ( ( ( comp_b_a_a @ F2 @ G2 )
        = id_a )
     => ( ( ( comp_a_b_b @ G2 @ H )
          = id_b )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_315_left__right__inverse__eq,axiom,
    ! [F2: a > b,G2: b > a,H: a > b] :
      ( ( ( comp_a_b_b @ F2 @ G2 )
        = id_b )
     => ( ( ( comp_b_a_a @ G2 @ H )
          = id_a )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_316_left__right__inverse__eq,axiom,
    ! [F2: a > a,G2: a > a,H: a > a] :
      ( ( ( comp_a_a_a @ F2 @ G2 )
        = id_a )
     => ( ( ( comp_a_a_a @ G2 @ H )
          = id_a )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_317_left__right__inverse__eq,axiom,
    ! [F2: c > c > a,G2: ( c > a ) > c,H: c > c > a] :
      ( ( ( comp_c_c_a_c_a @ F2 @ G2 )
        = id_c_a )
     => ( ( ( comp_c_a_c_c @ G2 @ H )
          = id_c )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_318_boolean__algebra__class_Ominus__comp__minus,axiom,
    ( ( comp_s7380310864032827023et_rat @ uminus2201863774496077783et_rat @ uminus2201863774496077783et_rat )
    = id_set_rat ) ).

% boolean_algebra_class.minus_comp_minus
thf(fact_319_boolean__algebra__class_Ominus__comp__minus,axiom,
    ( ( comp_s8964582002068861047et_nat @ uminus5710092332889474511et_nat @ uminus5710092332889474511et_nat )
    = id_set_nat ) ).

% boolean_algebra_class.minus_comp_minus
thf(fact_320_group__add__class_Ominus__comp__minus,axiom,
    ( ( comp_rat_rat_rat @ uminus_uminus_rat @ uminus_uminus_rat )
    = id_rat ) ).

% group_add_class.minus_comp_minus
thf(fact_321_o__prs_I2_J,axiom,
    ! [R12: c > c > $o,Abs1: c > c,Rep1: c > c,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: b > b > $o,Abs3: b > b,Rep3: b > b] :
      ( ( quotient3_c_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_b_b @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu5567477910418985390_b_c_a @ id_b_a @ ( map_fu3580789755375985064_a_c_a @ ( map_fun_c_c_b_b @ Abs1 @ id_b ) @ ( map_fun_c_c_a_a @ Rep1 @ id_a ) ) @ comp_b_a_c )
            = comp_b_a_c ) ) ) ) ).

% o_prs(2)
thf(fact_322_o__prs_I2_J,axiom,
    ! [R12: c > c > $o,Abs1: c > b,Rep1: b > c,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: b > b > $o,Abs3: b > b,Rep3: b > b] :
      ( ( quotient3_c_b @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_b_b @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu2001909935855630000_b_b_a @ id_b_a @ ( map_fu1196190707875695402_a_b_a @ ( map_fun_c_b_b_b @ Abs1 @ id_b ) @ ( map_fun_b_c_a_a @ Rep1 @ id_a ) ) @ comp_b_a_c )
            = comp_b_a_b ) ) ) ) ).

% o_prs(2)
thf(fact_323_o__prs_I2_J,axiom,
    ! [R12: c > c > $o,Abs1: c > c,Rep1: c > c,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: b > b > $o,Abs3: b > b,Rep3: b > b] :
      ( ( quotient3_c_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_b_b @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu2764701474353874862_c_c_a @ id_c_a @ ( map_fu6618264312679260842_a_c_a @ ( map_fun_c_c_c_c @ Abs1 @ id_c ) @ ( map_fun_c_c_a_a @ Rep1 @ id_a ) ) @ comp_c_a_c )
            = comp_c_a_c ) ) ) ) ).

% o_prs(2)
thf(fact_324_o__prs_I2_J,axiom,
    ! [R12: c > c > $o,Abs1: c > c,Rep1: c > c,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: b > b > $o,Abs3: b > b,Rep3: b > b] :
      ( ( quotient3_c_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_b_b @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu6441608359614649650_b_c_b @ id_b_b @ ( map_fu5259323019911985834_b_c_b @ ( map_fun_c_c_b_b @ Abs1 @ id_b ) @ ( map_fun_c_c_b_b @ Rep1 @ id_b ) ) @ comp_b_b_c )
            = comp_b_b_c ) ) ) ) ).

% o_prs(2)
thf(fact_325_o__prs_I2_J,axiom,
    ! [R12: b > b > $o,Abs1: b > c,Rep1: c > b,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: b > b > $o,Abs3: b > b,Rep3: b > b] :
      ( ( quotient3_b_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_b_b @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu593469440952318896_b_c_a @ id_b_a @ ( map_fu2703170745860999978_a_c_a @ ( map_fun_b_c_b_b @ Abs1 @ id_b ) @ ( map_fun_c_b_a_a @ Rep1 @ id_a ) ) @ comp_b_a_b )
            = comp_b_a_c ) ) ) ) ).

% o_prs(2)
thf(fact_326_o__prs_I2_J,axiom,
    ! [R12: c > c > $o,Abs1: c > c,Rep1: c > c,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: b > b > $o,Abs3: b > b,Rep3: b > b] :
      ( ( quotient3_c_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_b_b @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu8370254346484095918_a_c_a @ id_a_a @ ( map_fu543315198072709286_a_c_a @ ( map_fun_c_c_a_a @ Abs1 @ id_a ) @ ( map_fun_c_c_a_a @ Rep1 @ id_a ) ) @ comp_a_a_c )
            = comp_a_a_c ) ) ) ) ).

% o_prs(2)
thf(fact_327_o__prs_I2_J,axiom,
    ! [R12: c > c > $o,Abs1: c > b,Rep1: b > c,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: b > b > $o,Abs3: b > b,Rep3: b > b] :
      ( ( quotient3_c_b @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_b_b @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu4804686371920740528_a_b_a @ id_a_a @ ( map_fu7382088187427195432_a_b_a @ ( map_fun_c_b_a_a @ Abs1 @ id_a ) @ ( map_fun_b_c_a_a @ Rep1 @ id_a ) ) @ comp_a_a_c )
            = comp_a_a_b ) ) ) ) ).

% o_prs(2)
thf(fact_328_o__prs_I2_J,axiom,
    ! [R12: c > c > $o,Abs1: c > rat,Rep1: rat > c,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: b > b > $o,Abs3: b > b,Rep3: b > b] :
      ( ( quotient3_c_rat @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_b_b @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu1803736159731858670_rat_a @ id_a_a @ ( map_fu5905973340530010510_rat_a @ ( map_fun_c_rat_a_a @ Abs1 @ id_a ) @ ( map_fun_rat_c_a_a @ Rep1 @ id_a ) ) @ comp_a_a_c )
            = comp_a_a_rat ) ) ) ) ).

% o_prs(2)
thf(fact_329_o__prs_I2_J,axiom,
    ! [R12: c > c > $o,Abs1: c > nat,Rep1: nat > c,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: b > b > $o,Abs3: b > b,Rep3: b > b] :
      ( ( quotient3_c_nat @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_b_b @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu6764128878074885870_nat_a @ id_a_a @ ( map_fu8093432620143282062_nat_a @ ( map_fun_c_nat_a_a @ Abs1 @ id_a ) @ ( map_fun_nat_c_a_a @ Rep1 @ id_a ) ) @ comp_a_a_c )
            = comp_a_a_nat ) ) ) ) ).

% o_prs(2)
thf(fact_330_o__prs_I2_J,axiom,
    ! [R12: c > c > $o,Abs1: c > a,Rep1: a > c,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: b > b > $o,Abs3: b > b,Rep3: b > b] :
      ( ( quotient3_c_a @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_b_b @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu1239118397357385138_a_a_a @ id_a_a @ ( map_fu4997489139926905770_a_a_a @ ( map_fun_c_a_a_a @ Abs1 @ id_a ) @ ( map_fun_a_c_a_a @ Rep1 @ id_a ) ) @ comp_a_a_c )
            = comp_a_a_a ) ) ) ) ).

% o_prs(2)
thf(fact_331_mem__Collect__eq,axiom,
    ! [A: c,P: c > $o] :
      ( ( member_c @ A @ ( collect_c @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_332_mem__Collect__eq,axiom,
    ! [A: b,P: b > $o] :
      ( ( member_b @ A @ ( collect_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_333_mem__Collect__eq,axiom,
    ! [A: a,P: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_334_mem__Collect__eq,axiom,
    ! [A: rat,P: rat > $o] :
      ( ( member_rat @ A @ ( collect_rat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_335_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_336_Collect__mem__eq,axiom,
    ! [A2: set_c] :
      ( ( collect_c
        @ ^ [X: c] : ( member_c @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_337_Collect__mem__eq,axiom,
    ! [A2: set_b] :
      ( ( collect_b
        @ ^ [X: b] : ( member_b @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_338_Collect__mem__eq,axiom,
    ! [A2: set_a] :
      ( ( collect_a
        @ ^ [X: a] : ( member_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_339_Collect__mem__eq,axiom,
    ! [A2: set_rat] :
      ( ( collect_rat
        @ ^ [X: rat] : ( member_rat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_340_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_341_Collect__cong,axiom,
    ! [P: rat > $o,Q: rat > $o] :
      ( ! [X4: rat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_rat @ P )
        = ( collect_rat @ Q ) ) ) ).

% Collect_cong
thf(fact_342_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_343_DEADID_Opred__map,axiom,
    ! [X2: c > a] :
      ( ( bNF_pred_DEADID_c_a @ ( id_c_a @ X2 ) )
      = ( bNF_pred_DEADID_c_a @ X2 ) ) ).

% DEADID.pred_map
thf(fact_344_DEADID_Opred__map,axiom,
    ! [X2: b > a] :
      ( ( bNF_pred_DEADID_b_a @ ( id_b_a @ X2 ) )
      = ( bNF_pred_DEADID_b_a @ X2 ) ) ).

% DEADID.pred_map
thf(fact_345_DEADID_Opred__map,axiom,
    ! [X2: c] :
      ( ( bNF_pred_DEADID_c @ ( id_c @ X2 ) )
      = ( bNF_pred_DEADID_c @ X2 ) ) ).

% DEADID.pred_map
thf(fact_346_DEADID_Opred__map,axiom,
    ! [X2: b] :
      ( ( bNF_pred_DEADID_b @ ( id_b @ X2 ) )
      = ( bNF_pred_DEADID_b @ X2 ) ) ).

% DEADID.pred_map
thf(fact_347_DEADID_Opred__map,axiom,
    ! [X2: a] :
      ( ( bNF_pred_DEADID_a @ ( id_a @ X2 ) )
      = ( bNF_pred_DEADID_a @ X2 ) ) ).

% DEADID.pred_map
thf(fact_348_DEADID_Omap__cong__pred,axiom,
    ! [X2: c > a,Ya: c > a] :
      ( ( X2 = Ya )
     => ( ( bNF_pred_DEADID_c_a @ Ya )
       => ( ( id_c_a @ X2 )
          = ( id_c_a @ Ya ) ) ) ) ).

% DEADID.map_cong_pred
thf(fact_349_DEADID_Omap__cong__pred,axiom,
    ! [X2: b > a,Ya: b > a] :
      ( ( X2 = Ya )
     => ( ( bNF_pred_DEADID_b_a @ Ya )
       => ( ( id_b_a @ X2 )
          = ( id_b_a @ Ya ) ) ) ) ).

% DEADID.map_cong_pred
thf(fact_350_DEADID_Omap__cong__pred,axiom,
    ! [X2: c,Ya: c] :
      ( ( X2 = Ya )
     => ( ( bNF_pred_DEADID_c @ Ya )
       => ( ( id_c @ X2 )
          = ( id_c @ Ya ) ) ) ) ).

% DEADID.map_cong_pred
thf(fact_351_DEADID_Omap__cong__pred,axiom,
    ! [X2: b,Ya: b] :
      ( ( X2 = Ya )
     => ( ( bNF_pred_DEADID_b @ Ya )
       => ( ( id_b @ X2 )
          = ( id_b @ Ya ) ) ) ) ).

% DEADID.map_cong_pred
thf(fact_352_DEADID_Omap__cong__pred,axiom,
    ! [X2: a,Ya: a] :
      ( ( X2 = Ya )
     => ( ( bNF_pred_DEADID_a @ Ya )
       => ( ( id_a @ X2 )
          = ( id_a @ Ya ) ) ) ) ).

% DEADID.map_cong_pred
thf(fact_353_vec__set__map,axiom,
    ! [F2: nat > rat,V: vec_nat] :
      ( ( vec_set_rat @ ( map_vec_nat_rat @ F2 @ V ) )
      = ( image_nat_rat @ F2 @ ( vec_set_nat @ V ) ) ) ).

% vec_set_map
thf(fact_354_vec__set__map,axiom,
    ! [F2: b > a,V: vec_b] :
      ( ( vec_set_a @ ( map_vec_b_a @ F2 @ V ) )
      = ( image_b_a @ F2 @ ( vec_set_b @ V ) ) ) ).

% vec_set_map
thf(fact_355_vec__set__map,axiom,
    ! [F2: c > b,V: vec_c] :
      ( ( vec_set_b @ ( map_vec_c_b @ F2 @ V ) )
      = ( image_c_b @ F2 @ ( vec_set_c @ V ) ) ) ).

% vec_set_map
thf(fact_356_vec__set__map,axiom,
    ! [F2: c > a,V: vec_c] :
      ( ( vec_set_a @ ( map_vec_c_a @ F2 @ V ) )
      = ( image_c_a @ F2 @ ( vec_set_c @ V ) ) ) ).

% vec_set_map
thf(fact_357_vec__set__map,axiom,
    ! [F2: c > c,V: vec_c] :
      ( ( vec_set_c @ ( map_vec_c_c @ F2 @ V ) )
      = ( image_c_c @ F2 @ ( vec_set_c @ V ) ) ) ).

% vec_set_map
thf(fact_358_vec__set__map,axiom,
    ! [F2: a > c,V: vec_a] :
      ( ( vec_set_c @ ( map_vec_a_c @ F2 @ V ) )
      = ( image_a_c @ F2 @ ( vec_set_a @ V ) ) ) ).

% vec_set_map
thf(fact_359_vec__set__map,axiom,
    ! [F2: b > c,V: vec_b] :
      ( ( vec_set_c @ ( map_vec_b_c @ F2 @ V ) )
      = ( image_b_c @ F2 @ ( vec_set_b @ V ) ) ) ).

% vec_set_map
thf(fact_360_vec__set__map,axiom,
    ! [F2: rat > c,V: vec_rat] :
      ( ( vec_set_c @ ( map_vec_rat_c @ F2 @ V ) )
      = ( image_rat_c @ F2 @ ( vec_set_rat @ V ) ) ) ).

% vec_set_map
thf(fact_361_vec__set__map,axiom,
    ! [F2: nat > c,V: vec_nat] :
      ( ( vec_set_c @ ( map_vec_nat_c @ F2 @ V ) )
      = ( image_nat_c @ F2 @ ( vec_set_nat @ V ) ) ) ).

% vec_set_map
thf(fact_362_vec__set__map,axiom,
    ! [F2: a > a,V: vec_a] :
      ( ( vec_set_a @ ( map_vec_a_a @ F2 @ V ) )
      = ( image_a_a @ F2 @ ( vec_set_a @ V ) ) ) ).

% vec_set_map
thf(fact_363_image__id,axiom,
    ( ( image_c_a_c_a @ id_c_a )
    = id_set_c_a ) ).

% image_id
thf(fact_364_image__id,axiom,
    ( ( image_b_a_b_a @ id_b_a )
    = id_set_b_a ) ).

% image_id
thf(fact_365_image__id,axiom,
    ( ( image_c_c @ id_c )
    = id_set_c ) ).

% image_id
thf(fact_366_image__id,axiom,
    ( ( image_b_b @ id_b )
    = id_set_b ) ).

% image_id
thf(fact_367_image__id,axiom,
    ( ( image_a_a @ id_a )
    = id_set_a ) ).

% image_id
thf(fact_368_cond__prs,axiom,
    ! [R4: b > b > $o,Absf: b > b,Repf: b > b,A: $o,B: b,C: b] :
      ( ( quotient3_b_b @ R4 @ Absf @ Repf )
     => ( ( A
         => ( ( Absf @ ( if_b @ A @ ( Repf @ B ) @ ( Repf @ C ) ) )
            = B ) )
        & ( ~ A
         => ( ( Absf @ ( if_b @ A @ ( Repf @ B ) @ ( Repf @ C ) ) )
            = C ) ) ) ) ).

% cond_prs
thf(fact_369_cond__prs,axiom,
    ! [R4: b > b > $o,Absf: b > a,Repf: a > b,A: $o,B: a,C: a] :
      ( ( quotient3_b_a @ R4 @ Absf @ Repf )
     => ( ( A
         => ( ( Absf @ ( if_b @ A @ ( Repf @ B ) @ ( Repf @ C ) ) )
            = B ) )
        & ( ~ A
         => ( ( Absf @ ( if_b @ A @ ( Repf @ B ) @ ( Repf @ C ) ) )
            = C ) ) ) ) ).

% cond_prs
thf(fact_370_cond__prs,axiom,
    ! [R4: a > a > $o,Absf: a > c,Repf: c > a,A: $o,B: c,C: c] :
      ( ( quotient3_a_c @ R4 @ Absf @ Repf )
     => ( ( A
         => ( ( Absf @ ( if_a @ A @ ( Repf @ B ) @ ( Repf @ C ) ) )
            = B ) )
        & ( ~ A
         => ( ( Absf @ ( if_a @ A @ ( Repf @ B ) @ ( Repf @ C ) ) )
            = C ) ) ) ) ).

% cond_prs
thf(fact_371_cond__prs,axiom,
    ! [R4: a > a > $o,Absf: a > b,Repf: b > a,A: $o,B: b,C: b] :
      ( ( quotient3_a_b @ R4 @ Absf @ Repf )
     => ( ( A
         => ( ( Absf @ ( if_a @ A @ ( Repf @ B ) @ ( Repf @ C ) ) )
            = B ) )
        & ( ~ A
         => ( ( Absf @ ( if_a @ A @ ( Repf @ B ) @ ( Repf @ C ) ) )
            = C ) ) ) ) ).

% cond_prs
thf(fact_372_cond__prs,axiom,
    ! [R4: a > a > $o,Absf: a > a,Repf: a > a,A: $o,B: a,C: a] :
      ( ( quotient3_a_a @ R4 @ Absf @ Repf )
     => ( ( A
         => ( ( Absf @ ( if_a @ A @ ( Repf @ B ) @ ( Repf @ C ) ) )
            = B ) )
        & ( ~ A
         => ( ( Absf @ ( if_a @ A @ ( Repf @ B ) @ ( Repf @ C ) ) )
            = C ) ) ) ) ).

% cond_prs
thf(fact_373_Quotient3I,axiom,
    ! [Abs: b > b,Rep: b > b,R4: b > b > $o] :
      ( ! [A3: b] :
          ( ( Abs @ ( Rep @ A3 ) )
          = A3 )
     => ( ! [A3: b] : ( R4 @ ( Rep @ A3 ) @ ( Rep @ A3 ) )
       => ( ! [R5: b,S: b] :
              ( ( R4 @ R5 @ S )
              = ( ( R4 @ R5 @ R5 )
                & ( R4 @ S @ S )
                & ( ( Abs @ R5 )
                  = ( Abs @ S ) ) ) )
         => ( quotient3_b_b @ R4 @ Abs @ Rep ) ) ) ) ).

% Quotient3I
thf(fact_374_Quotient3I,axiom,
    ! [Abs: b > a,Rep: a > b,R4: b > b > $o] :
      ( ! [A3: a] :
          ( ( Abs @ ( Rep @ A3 ) )
          = A3 )
     => ( ! [A3: a] : ( R4 @ ( Rep @ A3 ) @ ( Rep @ A3 ) )
       => ( ! [R5: b,S: b] :
              ( ( R4 @ R5 @ S )
              = ( ( R4 @ R5 @ R5 )
                & ( R4 @ S @ S )
                & ( ( Abs @ R5 )
                  = ( Abs @ S ) ) ) )
         => ( quotient3_b_a @ R4 @ Abs @ Rep ) ) ) ) ).

% Quotient3I
thf(fact_375_Quotient3I,axiom,
    ! [Abs: a > c,Rep: c > a,R4: a > a > $o] :
      ( ! [A3: c] :
          ( ( Abs @ ( Rep @ A3 ) )
          = A3 )
     => ( ! [A3: c] : ( R4 @ ( Rep @ A3 ) @ ( Rep @ A3 ) )
       => ( ! [R5: a,S: a] :
              ( ( R4 @ R5 @ S )
              = ( ( R4 @ R5 @ R5 )
                & ( R4 @ S @ S )
                & ( ( Abs @ R5 )
                  = ( Abs @ S ) ) ) )
         => ( quotient3_a_c @ R4 @ Abs @ Rep ) ) ) ) ).

% Quotient3I
thf(fact_376_Quotient3I,axiom,
    ! [Abs: a > b,Rep: b > a,R4: a > a > $o] :
      ( ! [A3: b] :
          ( ( Abs @ ( Rep @ A3 ) )
          = A3 )
     => ( ! [A3: b] : ( R4 @ ( Rep @ A3 ) @ ( Rep @ A3 ) )
       => ( ! [R5: a,S: a] :
              ( ( R4 @ R5 @ S )
              = ( ( R4 @ R5 @ R5 )
                & ( R4 @ S @ S )
                & ( ( Abs @ R5 )
                  = ( Abs @ S ) ) ) )
         => ( quotient3_a_b @ R4 @ Abs @ Rep ) ) ) ) ).

% Quotient3I
thf(fact_377_Quotient3I,axiom,
    ! [Abs: a > a,Rep: a > a,R4: a > a > $o] :
      ( ! [A3: a] :
          ( ( Abs @ ( Rep @ A3 ) )
          = A3 )
     => ( ! [A3: a] : ( R4 @ ( Rep @ A3 ) @ ( Rep @ A3 ) )
       => ( ! [R5: a,S: a] :
              ( ( R4 @ R5 @ S )
              = ( ( R4 @ R5 @ R5 )
                & ( R4 @ S @ S )
                & ( ( Abs @ R5 )
                  = ( Abs @ S ) ) ) )
         => ( quotient3_a_a @ R4 @ Abs @ Rep ) ) ) ) ).

% Quotient3I
thf(fact_378_equals__rsp,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b,Xa: b,Xb: b,Ya: b,Yb: b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ Xa @ Xb )
       => ( ( R4 @ Ya @ Yb )
         => ( ( R4 @ Xa @ Ya )
            = ( R4 @ Xb @ Yb ) ) ) ) ) ).

% equals_rsp
thf(fact_379_equals__rsp,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b,Xa: b,Xb: b,Ya: b,Yb: b] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ Xa @ Xb )
       => ( ( R4 @ Ya @ Yb )
         => ( ( R4 @ Xa @ Ya )
            = ( R4 @ Xb @ Yb ) ) ) ) ) ).

% equals_rsp
thf(fact_380_equals__rsp,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a,Xa: a,Xb: a,Ya: a,Yb: a] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( R4 @ Xa @ Xb )
       => ( ( R4 @ Ya @ Yb )
         => ( ( R4 @ Xa @ Ya )
            = ( R4 @ Xb @ Yb ) ) ) ) ) ).

% equals_rsp
thf(fact_381_equals__rsp,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a,Xa: a,Xb: a,Ya: a,Yb: a] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ Xa @ Xb )
       => ( ( R4 @ Ya @ Yb )
         => ( ( R4 @ Xa @ Ya )
            = ( R4 @ Xb @ Yb ) ) ) ) ) ).

% equals_rsp
thf(fact_382_equals__rsp,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a,Xa: a,Xb: a,Ya: a,Yb: a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ Xa @ Xb )
       => ( ( R4 @ Ya @ Yb )
         => ( ( R4 @ Xa @ Ya )
            = ( R4 @ Xb @ Yb ) ) ) ) ) ).

% equals_rsp
thf(fact_383_rep__abs__rsp,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b,X1: b,X22: b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ X1 @ X22 )
       => ( R4 @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).

% rep_abs_rsp
thf(fact_384_rep__abs__rsp,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b,X1: b,X22: b] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ X1 @ X22 )
       => ( R4 @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).

% rep_abs_rsp
thf(fact_385_rep__abs__rsp,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a,X1: a,X22: a] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( R4 @ X1 @ X22 )
       => ( R4 @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).

% rep_abs_rsp
thf(fact_386_rep__abs__rsp,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a,X1: a,X22: a] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ X1 @ X22 )
       => ( R4 @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).

% rep_abs_rsp
thf(fact_387_rep__abs__rsp,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a,X1: a,X22: a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ X1 @ X22 )
       => ( R4 @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).

% rep_abs_rsp
thf(fact_388_Quotient3__def,axiom,
    ( quotient3_b_b
    = ( ^ [R6: b > b > $o,Abs4: b > b,Rep4: b > b] :
          ( ! [A4: b] :
              ( ( Abs4 @ ( Rep4 @ A4 ) )
              = A4 )
          & ! [A4: b] : ( R6 @ ( Rep4 @ A4 ) @ ( Rep4 @ A4 ) )
          & ! [R7: b,S2: b] :
              ( ( R6 @ R7 @ S2 )
              = ( ( R6 @ R7 @ R7 )
                & ( R6 @ S2 @ S2 )
                & ( ( Abs4 @ R7 )
                  = ( Abs4 @ S2 ) ) ) ) ) ) ) ).

% Quotient3_def
thf(fact_389_Quotient3__def,axiom,
    ( quotient3_b_a
    = ( ^ [R6: b > b > $o,Abs4: b > a,Rep4: a > b] :
          ( ! [A4: a] :
              ( ( Abs4 @ ( Rep4 @ A4 ) )
              = A4 )
          & ! [A4: a] : ( R6 @ ( Rep4 @ A4 ) @ ( Rep4 @ A4 ) )
          & ! [R7: b,S2: b] :
              ( ( R6 @ R7 @ S2 )
              = ( ( R6 @ R7 @ R7 )
                & ( R6 @ S2 @ S2 )
                & ( ( Abs4 @ R7 )
                  = ( Abs4 @ S2 ) ) ) ) ) ) ) ).

% Quotient3_def
thf(fact_390_Quotient3__def,axiom,
    ( quotient3_a_c
    = ( ^ [R6: a > a > $o,Abs4: a > c,Rep4: c > a] :
          ( ! [A4: c] :
              ( ( Abs4 @ ( Rep4 @ A4 ) )
              = A4 )
          & ! [A4: c] : ( R6 @ ( Rep4 @ A4 ) @ ( Rep4 @ A4 ) )
          & ! [R7: a,S2: a] :
              ( ( R6 @ R7 @ S2 )
              = ( ( R6 @ R7 @ R7 )
                & ( R6 @ S2 @ S2 )
                & ( ( Abs4 @ R7 )
                  = ( Abs4 @ S2 ) ) ) ) ) ) ) ).

% Quotient3_def
thf(fact_391_Quotient3__def,axiom,
    ( quotient3_a_b
    = ( ^ [R6: a > a > $o,Abs4: a > b,Rep4: b > a] :
          ( ! [A4: b] :
              ( ( Abs4 @ ( Rep4 @ A4 ) )
              = A4 )
          & ! [A4: b] : ( R6 @ ( Rep4 @ A4 ) @ ( Rep4 @ A4 ) )
          & ! [R7: a,S2: a] :
              ( ( R6 @ R7 @ S2 )
              = ( ( R6 @ R7 @ R7 )
                & ( R6 @ S2 @ S2 )
                & ( ( Abs4 @ R7 )
                  = ( Abs4 @ S2 ) ) ) ) ) ) ) ).

% Quotient3_def
thf(fact_392_Quotient3__def,axiom,
    ( quotient3_a_a
    = ( ^ [R6: a > a > $o,Abs4: a > a,Rep4: a > a] :
          ( ! [A4: a] :
              ( ( Abs4 @ ( Rep4 @ A4 ) )
              = A4 )
          & ! [A4: a] : ( R6 @ ( Rep4 @ A4 ) @ ( Rep4 @ A4 ) )
          & ! [R7: a,S2: a] :
              ( ( R6 @ R7 @ S2 )
              = ( ( R6 @ R7 @ R7 )
                & ( R6 @ S2 @ S2 )
                & ( ( Abs4 @ R7 )
                  = ( Abs4 @ S2 ) ) ) ) ) ) ) ).

% Quotient3_def
thf(fact_393_Quotient3__rel,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b,R: b,S3: b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( ( R4 @ R @ R )
          & ( R4 @ S3 @ S3 )
          & ( ( Abs @ R )
            = ( Abs @ S3 ) ) )
        = ( R4 @ R @ S3 ) ) ) ).

% Quotient3_rel
thf(fact_394_Quotient3__rel,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b,R: b,S3: b] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( ( R4 @ R @ R )
          & ( R4 @ S3 @ S3 )
          & ( ( Abs @ R )
            = ( Abs @ S3 ) ) )
        = ( R4 @ R @ S3 ) ) ) ).

% Quotient3_rel
thf(fact_395_Quotient3__rel,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a,R: a,S3: a] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( ( R4 @ R @ R )
          & ( R4 @ S3 @ S3 )
          & ( ( Abs @ R )
            = ( Abs @ S3 ) ) )
        = ( R4 @ R @ S3 ) ) ) ).

% Quotient3_rel
thf(fact_396_Quotient3__rel,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a,R: a,S3: a] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( ( R4 @ R @ R )
          & ( R4 @ S3 @ S3 )
          & ( ( Abs @ R )
            = ( Abs @ S3 ) ) )
        = ( R4 @ R @ S3 ) ) ) ).

% Quotient3_rel
thf(fact_397_Quotient3__rel,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a,R: a,S3: a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( ( R4 @ R @ R )
          & ( R4 @ S3 @ S3 )
          & ( ( Abs @ R )
            = ( Abs @ S3 ) ) )
        = ( R4 @ R @ S3 ) ) ) ).

% Quotient3_rel
thf(fact_398_Quotient3__refl1,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b,R: b,S3: b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( R4 @ R @ R ) ) ) ).

% Quotient3_refl1
thf(fact_399_Quotient3__refl1,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b,R: b,S3: b] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( R4 @ R @ R ) ) ) ).

% Quotient3_refl1
thf(fact_400_Quotient3__refl1,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a,R: a,S3: a] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( R4 @ R @ R ) ) ) ).

% Quotient3_refl1
thf(fact_401_Quotient3__refl1,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a,R: a,S3: a] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( R4 @ R @ R ) ) ) ).

% Quotient3_refl1
thf(fact_402_Quotient3__refl1,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a,R: a,S3: a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( R4 @ R @ R ) ) ) ).

% Quotient3_refl1
thf(fact_403_Quotient3__refl2,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b,R: b,S3: b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( R4 @ S3 @ S3 ) ) ) ).

% Quotient3_refl2
thf(fact_404_Quotient3__refl2,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b,R: b,S3: b] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( R4 @ S3 @ S3 ) ) ) ).

% Quotient3_refl2
thf(fact_405_Quotient3__refl2,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a,R: a,S3: a] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( R4 @ S3 @ S3 ) ) ) ).

% Quotient3_refl2
thf(fact_406_Quotient3__refl2,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a,R: a,S3: a] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( R4 @ S3 @ S3 ) ) ) ).

% Quotient3_refl2
thf(fact_407_Quotient3__refl2,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a,R: a,S3: a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( R4 @ S3 @ S3 ) ) ) ).

% Quotient3_refl2
thf(fact_408_rep__abs__rsp__left,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b,X1: b,X22: b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ X1 @ X22 )
       => ( R4 @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).

% rep_abs_rsp_left
thf(fact_409_rep__abs__rsp__left,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b,X1: b,X22: b] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ X1 @ X22 )
       => ( R4 @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).

% rep_abs_rsp_left
thf(fact_410_rep__abs__rsp__left,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a,X1: a,X22: a] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( R4 @ X1 @ X22 )
       => ( R4 @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).

% rep_abs_rsp_left
thf(fact_411_rep__abs__rsp__left,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a,X1: a,X22: a] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ X1 @ X22 )
       => ( R4 @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).

% rep_abs_rsp_left
thf(fact_412_rep__abs__rsp__left,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a,X1: a,X22: a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ X1 @ X22 )
       => ( R4 @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).

% rep_abs_rsp_left
thf(fact_413_Quotient3__abs__rep,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b,A: b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( Abs @ ( Rep @ A ) )
        = A ) ) ).

% Quotient3_abs_rep
thf(fact_414_Quotient3__abs__rep,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b,A: a] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( Abs @ ( Rep @ A ) )
        = A ) ) ).

% Quotient3_abs_rep
thf(fact_415_Quotient3__abs__rep,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a,A: c] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( Abs @ ( Rep @ A ) )
        = A ) ) ).

% Quotient3_abs_rep
thf(fact_416_Quotient3__abs__rep,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a,A: b] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( Abs @ ( Rep @ A ) )
        = A ) ) ).

% Quotient3_abs_rep
thf(fact_417_Quotient3__abs__rep,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a,A: a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( Abs @ ( Rep @ A ) )
        = A ) ) ).

% Quotient3_abs_rep
thf(fact_418_Quotient3__rel__abs,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b,R: b,S3: b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( ( Abs @ R )
          = ( Abs @ S3 ) ) ) ) ).

% Quotient3_rel_abs
thf(fact_419_Quotient3__rel__abs,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b,R: b,S3: b] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( ( Abs @ R )
          = ( Abs @ S3 ) ) ) ) ).

% Quotient3_rel_abs
thf(fact_420_Quotient3__rel__abs,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a,R: a,S3: a] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( ( Abs @ R )
          = ( Abs @ S3 ) ) ) ) ).

% Quotient3_rel_abs
thf(fact_421_Quotient3__rel__abs,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a,R: a,S3: a] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( ( Abs @ R )
          = ( Abs @ S3 ) ) ) ) ).

% Quotient3_rel_abs
thf(fact_422_Quotient3__rel__abs,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a,R: a,S3: a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ S3 )
       => ( ( Abs @ R )
          = ( Abs @ S3 ) ) ) ) ).

% Quotient3_rel_abs
thf(fact_423_Quotient3__rel__rep,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b,A: b,B: b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ ( Rep @ A ) @ ( Rep @ B ) )
        = ( A = B ) ) ) ).

% Quotient3_rel_rep
thf(fact_424_Quotient3__rel__rep,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b,A: a,B: a] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ ( Rep @ A ) @ ( Rep @ B ) )
        = ( A = B ) ) ) ).

% Quotient3_rel_rep
thf(fact_425_Quotient3__rel__rep,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a,A: c,B: c] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( R4 @ ( Rep @ A ) @ ( Rep @ B ) )
        = ( A = B ) ) ) ).

% Quotient3_rel_rep
thf(fact_426_Quotient3__rel__rep,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a,A: b,B: b] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ ( Rep @ A ) @ ( Rep @ B ) )
        = ( A = B ) ) ) ).

% Quotient3_rel_rep
thf(fact_427_Quotient3__rel__rep,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a,A: a,B: a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ ( Rep @ A ) @ ( Rep @ B ) )
        = ( A = B ) ) ) ).

% Quotient3_rel_rep
thf(fact_428_Quotient3__rep__abs,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b,R: b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ R )
       => ( R4 @ ( Rep @ ( Abs @ R ) ) @ R ) ) ) ).

% Quotient3_rep_abs
thf(fact_429_Quotient3__rep__abs,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b,R: b] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ R )
       => ( R4 @ ( Rep @ ( Abs @ R ) ) @ R ) ) ) ).

% Quotient3_rep_abs
thf(fact_430_Quotient3__rep__abs,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a,R: a] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ R )
       => ( R4 @ ( Rep @ ( Abs @ R ) ) @ R ) ) ) ).

% Quotient3_rep_abs
thf(fact_431_Quotient3__rep__abs,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a,R: a] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ R )
       => ( R4 @ ( Rep @ ( Abs @ R ) ) @ R ) ) ) ).

% Quotient3_rep_abs
thf(fact_432_Quotient3__rep__abs,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a,R: a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( R4 @ R @ R )
       => ( R4 @ ( Rep @ ( Abs @ R ) ) @ R ) ) ) ).

% Quotient3_rep_abs
thf(fact_433_Quotient3__rep__reflp,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b,A: b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( R4 @ ( Rep @ A ) @ ( Rep @ A ) ) ) ).

% Quotient3_rep_reflp
thf(fact_434_Quotient3__rep__reflp,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b,A: a] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( R4 @ ( Rep @ A ) @ ( Rep @ A ) ) ) ).

% Quotient3_rep_reflp
thf(fact_435_Quotient3__rep__reflp,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a,A: c] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( R4 @ ( Rep @ A ) @ ( Rep @ A ) ) ) ).

% Quotient3_rep_reflp
thf(fact_436_Quotient3__rep__reflp,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a,A: b] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( R4 @ ( Rep @ A ) @ ( Rep @ A ) ) ) ).

% Quotient3_rep_reflp
thf(fact_437_Quotient3__rep__reflp,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a,A: a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( R4 @ ( Rep @ A ) @ ( Rep @ A ) ) ) ).

% Quotient3_rep_reflp
thf(fact_438_identity__quotient3,axiom,
    ( quotient3_c_a_c_a
    @ ^ [Y2: c > a,Z: c > a] : ( Y2 = Z )
    @ id_c_a
    @ id_c_a ) ).

% identity_quotient3
thf(fact_439_identity__quotient3,axiom,
    ( quotient3_b_a_b_a
    @ ^ [Y2: b > a,Z: b > a] : ( Y2 = Z )
    @ id_b_a
    @ id_b_a ) ).

% identity_quotient3
thf(fact_440_identity__quotient3,axiom,
    ( quotient3_c_c
    @ ^ [Y2: c,Z: c] : ( Y2 = Z )
    @ id_c
    @ id_c ) ).

% identity_quotient3
thf(fact_441_identity__quotient3,axiom,
    ( quotient3_b_b
    @ ^ [Y2: b,Z: b] : ( Y2 = Z )
    @ id_b
    @ id_b ) ).

% identity_quotient3
thf(fact_442_identity__quotient3,axiom,
    ( quotient3_a_a
    @ ^ [Y2: a,Z: a] : ( Y2 = Z )
    @ id_a
    @ id_a ) ).

% identity_quotient3
thf(fact_443_if__prs,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( map_fu7515637755774237654_b_b_b @ id_o @ ( map_fun_b_b_b_b_b_b2 @ Rep @ ( map_fun_b_b_b_b @ Rep @ Abs ) ) @ if_b )
        = if_b ) ) ).

% if_prs
thf(fact_444_if__prs,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( map_fu2226394946436255765_a_a_a @ id_o @ ( map_fun_a_b_b_b_a_a @ Rep @ ( map_fun_a_b_b_a @ Rep @ Abs ) ) @ if_b )
        = if_a ) ) ).

% if_prs
thf(fact_445_if__prs,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( map_fu2836974390496659864_c_c_c @ id_o @ ( map_fun_c_a_a_a_c_c @ Rep @ ( map_fun_c_a_a_c @ Rep @ Abs ) ) @ if_a )
        = if_c ) ) ).

% if_prs
thf(fact_446_if__prs,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( map_fu6771103618013453783_b_b_b @ id_o @ ( map_fun_b_a_a_a_b_b @ Rep @ ( map_fun_b_a_a_b @ Rep @ Abs ) ) @ if_a )
        = if_b ) ) ).

% if_prs
thf(fact_447_if__prs,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( map_fu1481860808675471894_a_a_a @ id_o @ ( map_fun_a_a_a_a_a_a @ Rep @ ( map_fun_a_a_a_a @ Rep @ Abs ) ) @ if_a )
        = if_a ) ) ).

% if_prs
thf(fact_448_let__prs,axiom,
    ! [R12: b > b > $o,Abs1: b > c,Rep1: c > b,R22: a > a > $o,Abs2: a > a,Rep2: a > a] :
      ( ( quotient3_b_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_a_a @ R22 @ Abs2 @ Rep2 )
       => ( ( map_fu7466819186754988118_a_c_c @ Rep2 @ ( map_fun_a_c_a_b_b_c @ ( map_fun_a_a_c_b @ Abs2 @ Rep1 ) @ Abs1 )
            @ ^ [S2: a,F: a > b] : ( F @ S2 ) )
          = ( ^ [S2: a,F: a > c] : ( F @ S2 ) ) ) ) ) ).

% let_prs
thf(fact_449_let__prs,axiom,
    ! [R12: b > b > $o,Abs1: b > b,Rep1: b > b,R22: b > b > $o,Abs2: b > b,Rep2: b > b] :
      ( ( quotient3_b_b @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( map_fu4555376265042084374_b_b_b @ Rep2 @ ( map_fun_b_b_b_b_b_b @ ( map_fun_b_b_b_b @ Abs2 @ Rep1 ) @ Abs1 )
            @ ^ [S2: b,F: b > b] : ( F @ S2 ) )
          = ( ^ [S2: b,F: b > b] : ( F @ S2 ) ) ) ) ) ).

% let_prs
thf(fact_450_let__prs,axiom,
    ! [R12: b > b > $o,Abs1: b > b,Rep1: b > b,R22: b > b > $o,Abs2: b > a,Rep2: a > b] :
      ( ( quotient3_b_b @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_a @ R22 @ Abs2 @ Rep2 )
       => ( ( map_fu4204360803803164886_a_b_b @ Rep2 @ ( map_fun_a_b_b_b_b_b @ ( map_fun_b_a_b_b @ Abs2 @ Rep1 ) @ Abs1 )
            @ ^ [S2: b,F: b > b] : ( F @ S2 ) )
          = ( ^ [S2: a,F: a > b] : ( F @ S2 ) ) ) ) ) ).

% let_prs
thf(fact_451_let__prs,axiom,
    ! [R12: b > b > $o,Abs1: b > b,Rep1: b > b,R22: a > a > $o,Abs2: a > c,Rep2: c > a] :
      ( ( quotient3_b_b @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_a_c @ R22 @ Abs2 @ Rep2 )
       => ( ( map_fu8804499105534584982_c_b_b @ Rep2 @ ( map_fun_c_b_a_b_b_b @ ( map_fun_a_c_b_b @ Abs2 @ Rep1 ) @ Abs1 )
            @ ^ [S2: a,F: a > b] : ( F @ S2 ) )
          = ( ^ [S2: c,F: c > b] : ( F @ S2 ) ) ) ) ) ).

% let_prs
thf(fact_452_let__prs,axiom,
    ! [R12: b > b > $o,Abs1: b > b,Rep1: b > b,R22: a > a > $o,Abs2: a > b,Rep2: b > a] :
      ( ( quotient3_b_b @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_a_b @ R22 @ Abs2 @ Rep2 )
       => ( ( map_fu8453483644295665494_b_b_b @ Rep2 @ ( map_fun_b_b_a_b_b_b @ ( map_fun_a_b_b_b @ Abs2 @ Rep1 ) @ Abs1 )
            @ ^ [S2: a,F: a > b] : ( F @ S2 ) )
          = ( ^ [S2: b,F: b > b] : ( F @ S2 ) ) ) ) ) ).

% let_prs
thf(fact_453_let__prs,axiom,
    ! [R12: b > b > $o,Abs1: b > b,Rep1: b > b,R22: a > a > $o,Abs2: a > a,Rep2: a > a] :
      ( ( quotient3_b_b @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_a_a @ R22 @ Abs2 @ Rep2 )
       => ( ( map_fu8102468183056746006_a_b_b @ Rep2 @ ( map_fun_a_b_a_b_b_b @ ( map_fun_a_a_b_b @ Abs2 @ Rep1 ) @ Abs1 )
            @ ^ [S2: a,F: a > b] : ( F @ S2 ) )
          = ( ^ [S2: a,F: a > b] : ( F @ S2 ) ) ) ) ) ).

% let_prs
thf(fact_454_let__prs,axiom,
    ! [R12: b > b > $o,Abs1: b > a,Rep1: a > b,R22: b > b > $o,Abs2: b > b,Rep2: b > b] :
      ( ( quotient3_b_a @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( map_fu5191025261343842262_b_a_a @ Rep2 @ ( map_fun_b_a_b_b_b_a @ ( map_fun_b_b_a_b @ Abs2 @ Rep1 ) @ Abs1 )
            @ ^ [S2: b,F: b > b] : ( F @ S2 ) )
          = ( ^ [S2: b,F: b > a] : ( F @ S2 ) ) ) ) ) ).

% let_prs
thf(fact_455_let__prs,axiom,
    ! [R12: b > b > $o,Abs1: b > a,Rep1: a > b,R22: b > b > $o,Abs2: b > a,Rep2: a > b] :
      ( ( quotient3_b_a @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_a @ R22 @ Abs2 @ Rep2 )
       => ( ( map_fu4840009800104922774_a_a_a @ Rep2 @ ( map_fun_a_a_b_b_b_a @ ( map_fun_b_a_a_b @ Abs2 @ Rep1 ) @ Abs1 )
            @ ^ [S2: b,F: b > b] : ( F @ S2 ) )
          = ( ^ [S2: a,F: a > a] : ( F @ S2 ) ) ) ) ) ).

% let_prs
thf(fact_456_let__prs,axiom,
    ! [R12: b > b > $o,Abs1: b > a,Rep1: a > b,R22: a > a > $o,Abs2: a > c,Rep2: c > a] :
      ( ( quotient3_b_a @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_a_c @ R22 @ Abs2 @ Rep2 )
       => ( ( map_fu216776064981567062_c_a_a @ Rep2 @ ( map_fun_c_a_a_b_b_a @ ( map_fun_a_c_a_b @ Abs2 @ Rep1 ) @ Abs1 )
            @ ^ [S2: a,F: a > b] : ( F @ S2 ) )
          = ( ^ [S2: c,F: c > a] : ( F @ S2 ) ) ) ) ) ).

% let_prs
thf(fact_457_let__prs,axiom,
    ! [R12: b > b > $o,Abs1: b > a,Rep1: a > b,R22: a > a > $o,Abs2: a > b,Rep2: b > a] :
      ( ( quotient3_b_a @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_a_b @ R22 @ Abs2 @ Rep2 )
       => ( ( map_fu9089132640597423382_b_a_a @ Rep2 @ ( map_fun_b_a_a_b_b_a @ ( map_fun_a_b_a_b @ Abs2 @ Rep1 ) @ Abs1 )
            @ ^ [S2: a,F: a > b] : ( F @ S2 ) )
          = ( ^ [S2: b,F: b > a] : ( F @ S2 ) ) ) ) ) ).

% let_prs
thf(fact_458_image__eq__imp__comp,axiom,
    ! [F2: c > b,A2: set_c,G2: c > b,B2: set_c,H: b > a] :
      ( ( ( image_c_b @ F2 @ A2 )
        = ( image_c_b @ G2 @ B2 ) )
     => ( ( image_c_a @ ( comp_b_a_c @ H @ F2 ) @ A2 )
        = ( image_c_a @ ( comp_b_a_c @ H @ G2 ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_459_image__eq__imp__comp,axiom,
    ! [F2: c > b,A2: set_c,G2: b > b,B2: set_b,H: b > a] :
      ( ( ( image_c_b @ F2 @ A2 )
        = ( image_b_b @ G2 @ B2 ) )
     => ( ( image_c_a @ ( comp_b_a_c @ H @ F2 ) @ A2 )
        = ( image_b_a @ ( comp_b_a_b @ H @ G2 ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_460_image__eq__imp__comp,axiom,
    ! [F2: c > c,A2: set_c,G2: c > c,B2: set_c,H: c > a] :
      ( ( ( image_c_c @ F2 @ A2 )
        = ( image_c_c @ G2 @ B2 ) )
     => ( ( image_c_a @ ( comp_c_a_c @ H @ F2 ) @ A2 )
        = ( image_c_a @ ( comp_c_a_c @ H @ G2 ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_461_image__eq__imp__comp,axiom,
    ! [F2: c > b,A2: set_c,G2: c > b,B2: set_c,H: b > b] :
      ( ( ( image_c_b @ F2 @ A2 )
        = ( image_c_b @ G2 @ B2 ) )
     => ( ( image_c_b @ ( comp_b_b_c @ H @ F2 ) @ A2 )
        = ( image_c_b @ ( comp_b_b_c @ H @ G2 ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_462_image__eq__imp__comp,axiom,
    ! [F2: b > b,A2: set_b,G2: c > b,B2: set_c,H: b > a] :
      ( ( ( image_b_b @ F2 @ A2 )
        = ( image_c_b @ G2 @ B2 ) )
     => ( ( image_b_a @ ( comp_b_a_b @ H @ F2 ) @ A2 )
        = ( image_c_a @ ( comp_b_a_c @ H @ G2 ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_463_image__eq__imp__comp,axiom,
    ! [F2: b > b,A2: set_b,G2: b > b,B2: set_b,H: b > a] :
      ( ( ( image_b_b @ F2 @ A2 )
        = ( image_b_b @ G2 @ B2 ) )
     => ( ( image_b_a @ ( comp_b_a_b @ H @ F2 ) @ A2 )
        = ( image_b_a @ ( comp_b_a_b @ H @ G2 ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_464_image__eq__imp__comp,axiom,
    ! [F2: c > a,A2: set_c,G2: c > a,B2: set_c,H: a > a] :
      ( ( ( image_c_a @ F2 @ A2 )
        = ( image_c_a @ G2 @ B2 ) )
     => ( ( image_c_a @ ( comp_a_a_c @ H @ F2 ) @ A2 )
        = ( image_c_a @ ( comp_a_a_c @ H @ G2 ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_465_image__eq__imp__comp,axiom,
    ! [F2: c > a,A2: set_c,G2: b > a,B2: set_b,H: a > a] :
      ( ( ( image_c_a @ F2 @ A2 )
        = ( image_b_a @ G2 @ B2 ) )
     => ( ( image_c_a @ ( comp_a_a_c @ H @ F2 ) @ A2 )
        = ( image_b_a @ ( comp_a_a_b @ H @ G2 ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_466_image__eq__imp__comp,axiom,
    ! [F2: b > a,A2: set_b,G2: c > a,B2: set_c,H: a > a] :
      ( ( ( image_b_a @ F2 @ A2 )
        = ( image_c_a @ G2 @ B2 ) )
     => ( ( image_b_a @ ( comp_a_a_b @ H @ F2 ) @ A2 )
        = ( image_c_a @ ( comp_a_a_c @ H @ G2 ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_467_image__eq__imp__comp,axiom,
    ! [F2: b > a,A2: set_b,G2: b > a,B2: set_b,H: a > a] :
      ( ( ( image_b_a @ F2 @ A2 )
        = ( image_b_a @ G2 @ B2 ) )
     => ( ( image_b_a @ ( comp_a_a_b @ H @ F2 ) @ A2 )
        = ( image_b_a @ ( comp_a_a_b @ H @ G2 ) @ B2 ) ) ) ).

% image_eq_imp_comp
thf(fact_468_image__comp,axiom,
    ! [F2: b > a,G2: c > b,R: set_c] :
      ( ( image_b_a @ F2 @ ( image_c_b @ G2 @ R ) )
      = ( image_c_a @ ( comp_b_a_c @ F2 @ G2 ) @ R ) ) ).

% image_comp
thf(fact_469_image__comp,axiom,
    ! [F2: c > a,G2: c > c,R: set_c] :
      ( ( image_c_a @ F2 @ ( image_c_c @ G2 @ R ) )
      = ( image_c_a @ ( comp_c_a_c @ F2 @ G2 ) @ R ) ) ).

% image_comp
thf(fact_470_image__comp,axiom,
    ! [F2: b > b,G2: c > b,R: set_c] :
      ( ( image_b_b @ F2 @ ( image_c_b @ G2 @ R ) )
      = ( image_c_b @ ( comp_b_b_c @ F2 @ G2 ) @ R ) ) ).

% image_comp
thf(fact_471_image__comp,axiom,
    ! [F2: b > a,G2: b > b,R: set_b] :
      ( ( image_b_a @ F2 @ ( image_b_b @ G2 @ R ) )
      = ( image_b_a @ ( comp_b_a_b @ F2 @ G2 ) @ R ) ) ).

% image_comp
thf(fact_472_image__comp,axiom,
    ! [F2: a > a,G2: c > a,R: set_c] :
      ( ( image_a_a @ F2 @ ( image_c_a @ G2 @ R ) )
      = ( image_c_a @ ( comp_a_a_c @ F2 @ G2 ) @ R ) ) ).

% image_comp
thf(fact_473_image__comp,axiom,
    ! [F2: a > a,G2: b > a,R: set_b] :
      ( ( image_a_a @ F2 @ ( image_b_a @ G2 @ R ) )
      = ( image_b_a @ ( comp_a_a_b @ F2 @ G2 ) @ R ) ) ).

% image_comp
thf(fact_474_image__comp,axiom,
    ! [F2: rat > rat,G2: nat > rat,R: set_nat] :
      ( ( image_rat_rat @ F2 @ ( image_nat_rat @ G2 @ R ) )
      = ( image_nat_rat @ ( comp_rat_rat_nat @ F2 @ G2 ) @ R ) ) ).

% image_comp
thf(fact_475_image__comp,axiom,
    ! [F2: nat > rat,G2: nat > nat,R: set_nat] :
      ( ( image_nat_rat @ F2 @ ( image_nat_nat @ G2 @ R ) )
      = ( image_nat_rat @ ( comp_nat_rat_nat @ F2 @ G2 ) @ R ) ) ).

% image_comp
thf(fact_476_image__comp,axiom,
    ! [F2: c > c,G2: a > c,R: set_a] :
      ( ( image_c_c @ F2 @ ( image_a_c @ G2 @ R ) )
      = ( image_a_c @ ( comp_c_c_a @ F2 @ G2 ) @ R ) ) ).

% image_comp
thf(fact_477_image__comp,axiom,
    ! [F2: c > b,G2: a > c,R: set_a] :
      ( ( image_c_b @ F2 @ ( image_a_c @ G2 @ R ) )
      = ( image_a_b @ ( comp_c_b_a @ F2 @ G2 ) @ R ) ) ).

% image_comp
thf(fact_478_abs__o__rep,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( comp_b_b_b @ Abs @ Rep )
        = id_b ) ) ).

% abs_o_rep
thf(fact_479_abs__o__rep,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( comp_b_a_a @ Abs @ Rep )
        = id_a ) ) ).

% abs_o_rep
thf(fact_480_abs__o__rep,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( comp_a_c_c @ Abs @ Rep )
        = id_c ) ) ).

% abs_o_rep
thf(fact_481_abs__o__rep,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( comp_a_b_b @ Abs @ Rep )
        = id_b ) ) ).

% abs_o_rep
thf(fact_482_abs__o__rep,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( comp_a_a_a @ Abs @ Rep )
        = id_a ) ) ).

% abs_o_rep
thf(fact_483_o__prs_I1_J,axiom,
    ! [R12: c > c > $o,Abs1: c > c,Rep1: c > c,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: a > a > $o,Abs3: a > a,Rep3: a > a] :
      ( ( quotient3_c_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_a_a @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu5567477910418985390_b_c_a @ ( map_fun_b_b_a_a @ Abs2 @ Rep3 ) @ ( map_fu3580789755375985064_a_c_a @ ( map_fun_c_c_b_b @ Abs1 @ Rep2 ) @ ( map_fun_c_c_a_a @ Rep1 @ Abs3 ) ) @ comp_b_a_c )
            = comp_b_a_c ) ) ) ) ).

% o_prs(1)
thf(fact_484_o__prs_I1_J,axiom,
    ! [R12: c > c > $o,Abs1: c > c,Rep1: c > c,R22: b > b > $o,Abs2: b > c,Rep2: c > b,R3: a > a > $o,Abs3: a > a,Rep3: a > a] :
      ( ( quotient3_c_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_c @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_a_a @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu3776588507943935598_c_c_a @ ( map_fun_b_c_a_a @ Abs2 @ Rep3 ) @ ( map_fu6953506126080630569_a_c_a @ ( map_fun_c_c_c_b @ Abs1 @ Rep2 ) @ ( map_fun_c_c_a_a @ Rep1 @ Abs3 ) ) @ comp_b_a_c )
            = comp_c_a_c ) ) ) ) ).

% o_prs(1)
thf(fact_485_o__prs_I1_J,axiom,
    ! [R12: c > c > $o,Abs1: c > c,Rep1: c > c,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: a > a > $o,Abs3: a > b,Rep3: b > a] :
      ( ( quotient3_c_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_a_b @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu5478692276376878896_b_c_b @ ( map_fun_b_b_b_a @ Abs2 @ Rep3 ) @ ( map_fu3580789759679213865_a_c_b @ ( map_fun_c_c_b_b @ Abs1 @ Rep2 ) @ ( map_fun_c_c_a_b @ Rep1 @ Abs3 ) ) @ comp_b_a_c )
            = comp_b_b_c ) ) ) ) ).

% o_prs(1)
thf(fact_486_o__prs_I1_J,axiom,
    ! [R12: c > c > $o,Abs1: c > b,Rep1: b > c,R22: b > b > $o,Abs2: b > b,Rep2: b > b,R3: a > a > $o,Abs3: a > a,Rep3: a > a] :
      ( ( quotient3_c_b @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_a_a @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu2001909935855630000_b_b_a @ ( map_fun_b_b_a_a @ Abs2 @ Rep3 ) @ ( map_fu1196190707875695402_a_b_a @ ( map_fun_c_b_b_b @ Abs1 @ Rep2 ) @ ( map_fun_b_c_a_a @ Rep1 @ Abs3 ) ) @ comp_b_a_c )
            = comp_b_a_b ) ) ) ) ).

% o_prs(1)
thf(fact_487_o__prs_I1_J,axiom,
    ! [R12: c > c > $o,Abs1: c > c,Rep1: c > c,R22: b > b > $o,Abs2: b > a,Rep2: a > b,R3: a > a > $o,Abs3: a > a,Rep3: a > a] :
      ( ( quotient3_c_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_a @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_a_a @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu7358367312894035182_a_c_a @ ( map_fun_b_a_a_a @ Abs2 @ Rep3 ) @ ( map_fu208073384671339559_a_c_a @ ( map_fun_c_c_a_b @ Abs1 @ Rep2 ) @ ( map_fun_c_c_a_a @ Rep1 @ Abs3 ) ) @ comp_b_a_c )
            = comp_a_a_c ) ) ) ) ).

% o_prs(1)
thf(fact_488_o__prs_I1_J,axiom,
    ! [R12: c > c > $o,Abs1: c > b,Rep1: b > c,R22: b > b > $o,Abs2: b > a,Rep2: a > b,R3: a > a > $o,Abs3: a > a,Rep3: a > a] :
      ( ( quotient3_c_b @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_b_a @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_a_a @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu3792799338330679792_a_b_a @ ( map_fun_b_a_a_a @ Abs2 @ Rep3 ) @ ( map_fu7046846374025825705_a_b_a @ ( map_fun_c_b_a_b @ Abs1 @ Rep2 ) @ ( map_fun_b_c_a_a @ Rep1 @ Abs3 ) ) @ comp_b_a_c )
            = comp_a_a_b ) ) ) ) ).

% o_prs(1)
thf(fact_489_o__prs_I1_J,axiom,
    ! [R12: c > c > $o,Abs1: c > c,Rep1: c > c,R22: c > c > $o,Abs2: c > b,Rep2: b > c,R3: a > a > $o,Abs3: a > a,Rep3: a > a] :
      ( ( quotient3_c_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_c_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_a_a @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu4555590876828924654_b_c_a @ ( map_fun_c_b_a_a @ Abs2 @ Rep3 ) @ ( map_fu3245547941974615337_a_c_a @ ( map_fun_c_c_b_c @ Abs1 @ Rep2 ) @ ( map_fun_c_c_a_a @ Rep1 @ Abs3 ) ) @ comp_c_a_c )
            = comp_b_a_c ) ) ) ) ).

% o_prs(1)
thf(fact_490_o__prs_I1_J,axiom,
    ! [R12: c > c > $o,Abs1: c > c,Rep1: c > c,R22: c > c > $o,Abs2: c > c,Rep2: c > c,R3: a > a > $o,Abs3: a > a,Rep3: a > a] :
      ( ( quotient3_c_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_c_c @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_a_a @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu2764701474353874862_c_c_a @ ( map_fun_c_c_a_a @ Abs2 @ Rep3 ) @ ( map_fu6618264312679260842_a_c_a @ ( map_fun_c_c_c_c @ Abs1 @ Rep2 ) @ ( map_fun_c_c_a_a @ Rep1 @ Abs3 ) ) @ comp_c_a_c )
            = comp_c_a_c ) ) ) ) ).

% o_prs(1)
thf(fact_491_o__prs_I1_J,axiom,
    ! [R12: c > c > $o,Abs1: c > c,Rep1: c > c,R22: c > c > $o,Abs2: c > b,Rep2: b > c,R3: a > a > $o,Abs3: a > b,Rep3: b > a] :
      ( ( quotient3_c_c @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_c_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_a_b @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu4466805242786818160_b_c_b @ ( map_fun_c_b_b_a @ Abs2 @ Rep3 ) @ ( map_fu3245547946277844138_a_c_b @ ( map_fun_c_c_b_c @ Abs1 @ Rep2 ) @ ( map_fun_c_c_a_b @ Rep1 @ Abs3 ) ) @ comp_c_a_c )
            = comp_b_b_c ) ) ) ) ).

% o_prs(1)
thf(fact_492_o__prs_I1_J,axiom,
    ! [R12: c > c > $o,Abs1: c > b,Rep1: b > c,R22: c > c > $o,Abs2: c > b,Rep2: b > c,R3: a > a > $o,Abs3: a > a,Rep3: a > a] :
      ( ( quotient3_c_b @ R12 @ Abs1 @ Rep1 )
     => ( ( quotient3_c_b @ R22 @ Abs2 @ Rep2 )
       => ( ( quotient3_a_a @ R3 @ Abs3 @ Rep3 )
         => ( ( map_fu990022902265569264_b_b_a @ ( map_fun_c_b_a_a @ Abs2 @ Rep3 ) @ ( map_fu860948894474325675_a_b_a @ ( map_fun_c_b_b_c @ Abs1 @ Rep2 ) @ ( map_fun_b_c_a_a @ Rep1 @ Abs3 ) ) @ comp_c_a_c )
            = comp_b_a_b ) ) ) ) ).

% o_prs(1)
thf(fact_493_id__prs,axiom,
    ! [R4: c > c > $o,Abs: c > c,Rep: c > c] :
      ( ( quotient3_c_c @ R4 @ Abs @ Rep )
     => ( ( map_fun_c_c_c_c @ Rep @ Abs @ id_c )
        = id_c ) ) ).

% id_prs
thf(fact_494_id__prs,axiom,
    ! [R4: c > c > $o,Abs: c > b,Rep: b > c] :
      ( ( quotient3_c_b @ R4 @ Abs @ Rep )
     => ( ( map_fun_b_c_c_b @ Rep @ Abs @ id_c )
        = id_b ) ) ).

% id_prs
thf(fact_495_id__prs,axiom,
    ! [R4: c > c > $o,Abs: c > a,Rep: a > c] :
      ( ( quotient3_c_a @ R4 @ Abs @ Rep )
     => ( ( map_fun_a_c_c_a @ Rep @ Abs @ id_c )
        = id_a ) ) ).

% id_prs
thf(fact_496_id__prs,axiom,
    ! [R4: b > b > $o,Abs: b > c,Rep: c > b] :
      ( ( quotient3_b_c @ R4 @ Abs @ Rep )
     => ( ( map_fun_c_b_b_c @ Rep @ Abs @ id_b )
        = id_c ) ) ).

% id_prs
thf(fact_497_id__prs,axiom,
    ! [R4: b > b > $o,Abs: b > b,Rep: b > b] :
      ( ( quotient3_b_b @ R4 @ Abs @ Rep )
     => ( ( map_fun_b_b_b_b @ Rep @ Abs @ id_b )
        = id_b ) ) ).

% id_prs
thf(fact_498_id__prs,axiom,
    ! [R4: b > b > $o,Abs: b > a,Rep: a > b] :
      ( ( quotient3_b_a @ R4 @ Abs @ Rep )
     => ( ( map_fun_a_b_b_a @ Rep @ Abs @ id_b )
        = id_a ) ) ).

% id_prs
thf(fact_499_id__prs,axiom,
    ! [R4: a > a > $o,Abs: a > c,Rep: c > a] :
      ( ( quotient3_a_c @ R4 @ Abs @ Rep )
     => ( ( map_fun_c_a_a_c @ Rep @ Abs @ id_a )
        = id_c ) ) ).

% id_prs
thf(fact_500_id__prs,axiom,
    ! [R4: a > a > $o,Abs: a > b,Rep: b > a] :
      ( ( quotient3_a_b @ R4 @ Abs @ Rep )
     => ( ( map_fun_b_a_a_b @ Rep @ Abs @ id_a )
        = id_b ) ) ).

% id_prs
thf(fact_501_id__prs,axiom,
    ! [R4: a > a > $o,Abs: a > a,Rep: a > a] :
      ( ( quotient3_a_a @ R4 @ Abs @ Rep )
     => ( ( map_fun_a_a_a_a @ Rep @ Abs @ id_a )
        = id_a ) ) ).

% id_prs
thf(fact_502_id__prs,axiom,
    ! [R4: ( c > a ) > ( c > a ) > $o,Abs: ( c > a ) > c,Rep: c > c > a] :
      ( ( quotient3_c_a_c @ R4 @ Abs @ Rep )
     => ( ( map_fun_c_c_a_c_a_c @ Rep @ Abs @ id_c_a )
        = id_c ) ) ).

% id_prs
thf(fact_503_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_504_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_505_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_506_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [X2: set_rat] :
      ( ( uminus2201863774496077783et_rat @ ( uminus2201863774496077783et_rat @ X2 ) )
      = X2 ) ).

% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_507_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [X2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( uminus5710092332889474511et_nat @ X2 ) )
      = X2 ) ).

% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_508_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [X2: set_rat,Y3: set_rat] :
      ( ( ( uminus2201863774496077783et_rat @ X2 )
        = ( uminus2201863774496077783et_rat @ Y3 ) )
      = ( X2 = Y3 ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_509_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( ( uminus5710092332889474511et_nat @ X2 )
        = ( uminus5710092332889474511et_nat @ Y3 ) )
      = ( X2 = Y3 ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_510_image__eqI,axiom,
    ! [B: rat,F2: nat > rat,X2: nat,A2: set_nat] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( member_nat @ X2 @ A2 )
       => ( member_rat @ B @ ( image_nat_rat @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_511_image__eqI,axiom,
    ! [B: rat,F2: rat > rat,X2: rat,A2: set_rat] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( member_rat @ X2 @ A2 )
       => ( member_rat @ B @ ( image_rat_rat @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_512_image__eqI,axiom,
    ! [B: nat,F2: rat > nat,X2: rat,A2: set_rat] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( member_rat @ X2 @ A2 )
       => ( member_nat @ B @ ( image_rat_nat @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_513_image__eqI,axiom,
    ! [B: c,F2: rat > c,X2: rat,A2: set_rat] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( member_rat @ X2 @ A2 )
       => ( member_c @ B @ ( image_rat_c @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_514_image__eqI,axiom,
    ! [B: b,F2: rat > b,X2: rat,A2: set_rat] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( member_rat @ X2 @ A2 )
       => ( member_b @ B @ ( image_rat_b @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_515_image__eqI,axiom,
    ! [B: a,F2: rat > a,X2: rat,A2: set_rat] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( member_rat @ X2 @ A2 )
       => ( member_a @ B @ ( image_rat_a @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_516_image__eqI,axiom,
    ! [B: nat,F2: nat > nat,X2: nat,A2: set_nat] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( member_nat @ X2 @ A2 )
       => ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_517_image__eqI,axiom,
    ! [B: c,F2: nat > c,X2: nat,A2: set_nat] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( member_nat @ X2 @ A2 )
       => ( member_c @ B @ ( image_nat_c @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_518_image__eqI,axiom,
    ! [B: b,F2: nat > b,X2: nat,A2: set_nat] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( member_nat @ X2 @ A2 )
       => ( member_b @ B @ ( image_nat_b @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_519_image__eqI,axiom,
    ! [B: a,F2: nat > a,X2: nat,A2: set_nat] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( member_nat @ X2 @ A2 )
       => ( member_a @ B @ ( image_nat_a @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_520_Inf_OINF__id__eq,axiom,
    ! [Inf: set_c_a > c > a,A2: set_c_a] :
      ( ( Inf @ ( image_c_a_c_a @ id_c_a @ A2 ) )
      = ( Inf @ A2 ) ) ).

% Inf.INF_id_eq
thf(fact_521_Inf_OINF__id__eq,axiom,
    ! [Inf: set_b_a > b > a,A2: set_b_a] :
      ( ( Inf @ ( image_b_a_b_a @ id_b_a @ A2 ) )
      = ( Inf @ A2 ) ) ).

% Inf.INF_id_eq
thf(fact_522_Inf_OINF__id__eq,axiom,
    ! [Inf: set_c > c,A2: set_c] :
      ( ( Inf @ ( image_c_c @ id_c @ A2 ) )
      = ( Inf @ A2 ) ) ).

% Inf.INF_id_eq
thf(fact_523_Inf_OINF__id__eq,axiom,
    ! [Inf: set_b > b,A2: set_b] :
      ( ( Inf @ ( image_b_b @ id_b @ A2 ) )
      = ( Inf @ A2 ) ) ).

% Inf.INF_id_eq
thf(fact_524_Inf_OINF__id__eq,axiom,
    ! [Inf: set_a > a,A2: set_a] :
      ( ( Inf @ ( image_a_a @ id_a @ A2 ) )
      = ( Inf @ A2 ) ) ).

% Inf.INF_id_eq
thf(fact_525_Sup_OSUP__id__eq,axiom,
    ! [Sup: set_c_a > c > a,A2: set_c_a] :
      ( ( Sup @ ( image_c_a_c_a @ id_c_a @ A2 ) )
      = ( Sup @ A2 ) ) ).

% Sup.SUP_id_eq
thf(fact_526_Sup_OSUP__id__eq,axiom,
    ! [Sup: set_b_a > b > a,A2: set_b_a] :
      ( ( Sup @ ( image_b_a_b_a @ id_b_a @ A2 ) )
      = ( Sup @ A2 ) ) ).

% Sup.SUP_id_eq
thf(fact_527_Sup_OSUP__id__eq,axiom,
    ! [Sup: set_c > c,A2: set_c] :
      ( ( Sup @ ( image_c_c @ id_c @ A2 ) )
      = ( Sup @ A2 ) ) ).

% Sup.SUP_id_eq
thf(fact_528_Sup_OSUP__id__eq,axiom,
    ! [Sup: set_b > b,A2: set_b] :
      ( ( Sup @ ( image_b_b @ id_b @ A2 ) )
      = ( Sup @ A2 ) ) ).

% Sup.SUP_id_eq
thf(fact_529_Sup_OSUP__id__eq,axiom,
    ! [Sup: set_a > a,A2: set_a] :
      ( ( Sup @ ( image_a_a @ id_a @ A2 ) )
      = ( Sup @ A2 ) ) ).

% Sup.SUP_id_eq
thf(fact_530_comp__image,axiom,
    ! [F2: b > a,G2: c > b] :
      ( ( comp_s7019688007442371646_set_c @ ( image_b_a @ F2 ) @ ( image_c_b @ G2 ) )
      = ( image_c_a @ ( comp_b_a_c @ F2 @ G2 ) ) ) ).

% comp_image
thf(fact_531_comp__image,axiom,
    ! [F2: c > a,G2: c > c] :
      ( ( comp_s4903259949451896639_set_c @ ( image_c_a @ F2 ) @ ( image_c_c @ G2 ) )
      = ( image_c_a @ ( comp_c_a_c @ F2 @ G2 ) ) ) ).

% comp_image
thf(fact_532_comp__image,axiom,
    ! [F2: b > b,G2: c > b] :
      ( ( comp_s8507650044737170173_set_c @ ( image_b_b @ F2 ) @ ( image_c_b @ G2 ) )
      = ( image_c_b @ ( comp_b_b_c @ F2 @ G2 ) ) ) ).

% comp_image
thf(fact_533_comp__image,axiom,
    ! [F2: b > a,G2: b > b] :
      ( ( comp_s7019688003139142845_set_b @ ( image_b_a @ F2 ) @ ( image_b_b @ G2 ) )
      = ( image_b_a @ ( comp_b_a_b @ F2 @ G2 ) ) ) ).

% comp_image
thf(fact_534_comp__image,axiom,
    ! [F2: a > a,G2: c > a] :
      ( ( comp_s9136116065432846653_set_c @ ( image_a_a @ F2 ) @ ( image_c_a @ G2 ) )
      = ( image_c_a @ ( comp_a_a_c @ F2 @ G2 ) ) ) ).

% comp_image
thf(fact_535_comp__image,axiom,
    ! [F2: a > a,G2: b > a] :
      ( ( comp_s9136116061129617852_set_b @ ( image_a_a @ F2 ) @ ( image_b_a @ G2 ) )
      = ( image_b_a @ ( comp_a_a_b @ F2 @ G2 ) ) ) ).

% comp_image
thf(fact_536_comp__image,axiom,
    ! [F2: rat > rat,G2: nat > rat] :
      ( ( comp_s1665167385571447943et_nat @ ( image_rat_rat @ F2 ) @ ( image_nat_rat @ G2 ) )
      = ( image_nat_rat @ ( comp_rat_rat_nat @ F2 @ G2 ) ) ) ).

% comp_image
thf(fact_537_comp__image,axiom,
    ! [F2: nat > rat,G2: nat > nat] :
      ( ( comp_s8495322428307219071et_nat @ ( image_nat_rat @ F2 ) @ ( image_nat_nat @ G2 ) )
      = ( image_nat_rat @ ( comp_nat_rat_nat @ F2 @ G2 ) ) ) ).

% comp_image
thf(fact_538_comp__image,axiom,
    ! [F2: c > c,G2: a > c] :
      ( ( comp_s7879184015435036091_set_a @ ( image_c_c @ F2 ) @ ( image_a_c @ G2 ) )
      = ( image_a_c @ ( comp_c_c_a @ F2 @ G2 ) ) ) ).

% comp_image
thf(fact_539_comp__image,axiom,
    ! [F2: c > b,G2: a > c] :
      ( ( comp_s6391221978140237564_set_a @ ( image_c_b @ F2 ) @ ( image_a_c @ G2 ) )
      = ( image_a_b @ ( comp_c_b_a @ F2 @ G2 ) ) ) ).

% comp_image
thf(fact_540_Sup_OSUP__image,axiom,
    ! [Sup: set_a > a,G2: b > a,F2: c > b,A2: set_c] :
      ( ( Sup @ ( image_b_a @ G2 @ ( image_c_b @ F2 @ A2 ) ) )
      = ( Sup @ ( image_c_a @ ( comp_b_a_c @ G2 @ F2 ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_541_Sup_OSUP__image,axiom,
    ! [Sup: set_a > a,G2: c > a,F2: c > c,A2: set_c] :
      ( ( Sup @ ( image_c_a @ G2 @ ( image_c_c @ F2 @ A2 ) ) )
      = ( Sup @ ( image_c_a @ ( comp_c_a_c @ G2 @ F2 ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_542_Sup_OSUP__image,axiom,
    ! [Sup: set_b > b,G2: b > b,F2: c > b,A2: set_c] :
      ( ( Sup @ ( image_b_b @ G2 @ ( image_c_b @ F2 @ A2 ) ) )
      = ( Sup @ ( image_c_b @ ( comp_b_b_c @ G2 @ F2 ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_543_Sup_OSUP__image,axiom,
    ! [Sup: set_a > a,G2: b > a,F2: b > b,A2: set_b] :
      ( ( Sup @ ( image_b_a @ G2 @ ( image_b_b @ F2 @ A2 ) ) )
      = ( Sup @ ( image_b_a @ ( comp_b_a_b @ G2 @ F2 ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_544_Sup_OSUP__image,axiom,
    ! [Sup: set_a > a,G2: a > a,F2: c > a,A2: set_c] :
      ( ( Sup @ ( image_a_a @ G2 @ ( image_c_a @ F2 @ A2 ) ) )
      = ( Sup @ ( image_c_a @ ( comp_a_a_c @ G2 @ F2 ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_545_Sup_OSUP__image,axiom,
    ! [Sup: set_a > a,G2: a > a,F2: b > a,A2: set_b] :
      ( ( Sup @ ( image_a_a @ G2 @ ( image_b_a @ F2 @ A2 ) ) )
      = ( Sup @ ( image_b_a @ ( comp_a_a_b @ G2 @ F2 ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_546_Sup_OSUP__image,axiom,
    ! [Sup: set_rat > rat,G2: rat > rat,F2: nat > rat,A2: set_nat] :
      ( ( Sup @ ( image_rat_rat @ G2 @ ( image_nat_rat @ F2 @ A2 ) ) )
      = ( Sup @ ( image_nat_rat @ ( comp_rat_rat_nat @ G2 @ F2 ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_547_Sup_OSUP__image,axiom,
    ! [Sup: set_rat > rat,G2: nat > rat,F2: nat > nat,A2: set_nat] :
      ( ( Sup @ ( image_nat_rat @ G2 @ ( image_nat_nat @ F2 @ A2 ) ) )
      = ( Sup @ ( image_nat_rat @ ( comp_nat_rat_nat @ G2 @ F2 ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_548_Sup_OSUP__image,axiom,
    ! [Sup: set_c > c,G2: c > c,F2: a > c,A2: set_a] :
      ( ( Sup @ ( image_c_c @ G2 @ ( image_a_c @ F2 @ A2 ) ) )
      = ( Sup @ ( image_a_c @ ( comp_c_c_a @ G2 @ F2 ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_549_Sup_OSUP__image,axiom,
    ! [Sup: set_b > b,G2: c > b,F2: a > c,A2: set_a] :
      ( ( Sup @ ( image_c_b @ G2 @ ( image_a_c @ F2 @ A2 ) ) )
      = ( Sup @ ( image_a_b @ ( comp_c_b_a @ G2 @ F2 ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_550_rev__image__eqI,axiom,
    ! [X2: nat,A2: set_nat,B: rat,F2: nat > rat] :
      ( ( member_nat @ X2 @ A2 )
     => ( ( B
          = ( F2 @ X2 ) )
       => ( member_rat @ B @ ( image_nat_rat @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_551_rev__image__eqI,axiom,
    ! [X2: rat,A2: set_rat,B: rat,F2: rat > rat] :
      ( ( member_rat @ X2 @ A2 )
     => ( ( B
          = ( F2 @ X2 ) )
       => ( member_rat @ B @ ( image_rat_rat @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_552_rev__image__eqI,axiom,
    ! [X2: rat,A2: set_rat,B: nat,F2: rat > nat] :
      ( ( member_rat @ X2 @ A2 )
     => ( ( B
          = ( F2 @ X2 ) )
       => ( member_nat @ B @ ( image_rat_nat @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_553_rev__image__eqI,axiom,
    ! [X2: rat,A2: set_rat,B: c,F2: rat > c] :
      ( ( member_rat @ X2 @ A2 )
     => ( ( B
          = ( F2 @ X2 ) )
       => ( member_c @ B @ ( image_rat_c @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_554_rev__image__eqI,axiom,
    ! [X2: rat,A2: set_rat,B: b,F2: rat > b] :
      ( ( member_rat @ X2 @ A2 )
     => ( ( B
          = ( F2 @ X2 ) )
       => ( member_b @ B @ ( image_rat_b @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_555_rev__image__eqI,axiom,
    ! [X2: rat,A2: set_rat,B: a,F2: rat > a] :
      ( ( member_rat @ X2 @ A2 )
     => ( ( B
          = ( F2 @ X2 ) )
       => ( member_a @ B @ ( image_rat_a @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_556_rev__image__eqI,axiom,
    ! [X2: nat,A2: set_nat,B: nat,F2: nat > nat] :
      ( ( member_nat @ X2 @ A2 )
     => ( ( B
          = ( F2 @ X2 ) )
       => ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_557_rev__image__eqI,axiom,
    ! [X2: nat,A2: set_nat,B: c,F2: nat > c] :
      ( ( member_nat @ X2 @ A2 )
     => ( ( B
          = ( F2 @ X2 ) )
       => ( member_c @ B @ ( image_nat_c @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_558_rev__image__eqI,axiom,
    ! [X2: nat,A2: set_nat,B: b,F2: nat > b] :
      ( ( member_nat @ X2 @ A2 )
     => ( ( B
          = ( F2 @ X2 ) )
       => ( member_b @ B @ ( image_nat_b @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_559_rev__image__eqI,axiom,
    ! [X2: nat,A2: set_nat,B: a,F2: nat > a] :
      ( ( member_nat @ X2 @ A2 )
     => ( ( B
          = ( F2 @ X2 ) )
       => ( member_a @ B @ ( image_nat_a @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_560_ball__imageD,axiom,
    ! [F2: b > b,A2: set_b,P: b > $o] :
      ( ! [X4: b] :
          ( ( member_b @ X4 @ ( image_b_b @ F2 @ A2 ) )
         => ( P @ X4 ) )
     => ! [X5: b] :
          ( ( member_b @ X5 @ A2 )
         => ( P @ ( F2 @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_561_ball__imageD,axiom,
    ! [F2: b > a,A2: set_b,P: a > $o] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ ( image_b_a @ F2 @ A2 ) )
         => ( P @ X4 ) )
     => ! [X5: b] :
          ( ( member_b @ X5 @ A2 )
         => ( P @ ( F2 @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_562_ball__imageD,axiom,
    ! [F2: a > c,A2: set_a,P: c > $o] :
      ( ! [X4: c] :
          ( ( member_c @ X4 @ ( image_a_c @ F2 @ A2 ) )
         => ( P @ X4 ) )
     => ! [X5: a] :
          ( ( member_a @ X5 @ A2 )
         => ( P @ ( F2 @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_563_ball__imageD,axiom,
    ! [F2: a > b,A2: set_a,P: b > $o] :
      ( ! [X4: b] :
          ( ( member_b @ X4 @ ( image_a_b @ F2 @ A2 ) )
         => ( P @ X4 ) )
     => ! [X5: a] :
          ( ( member_a @ X5 @ A2 )
         => ( P @ ( F2 @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_564_ball__imageD,axiom,
    ! [F2: a > a,A2: set_a,P: a > $o] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ ( image_a_a @ F2 @ A2 ) )
         => ( P @ X4 ) )
     => ! [X5: a] :
          ( ( member_a @ X5 @ A2 )
         => ( P @ ( F2 @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_565_ball__imageD,axiom,
    ! [F2: nat > rat,A2: set_nat,P: rat > $o] :
      ( ! [X4: rat] :
          ( ( member_rat @ X4 @ ( image_nat_rat @ F2 @ A2 ) )
         => ( P @ X4 ) )
     => ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( P @ ( F2 @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_566_image__cong,axiom,
    ! [M: set_b,N: set_b,F2: b > b,G2: b > b] :
      ( ( M = N )
     => ( ! [X4: b] :
            ( ( member_b @ X4 @ N )
           => ( ( F2 @ X4 )
              = ( G2 @ X4 ) ) )
       => ( ( image_b_b @ F2 @ M )
          = ( image_b_b @ G2 @ N ) ) ) ) ).

% image_cong
thf(fact_567_image__cong,axiom,
    ! [M: set_b,N: set_b,F2: b > a,G2: b > a] :
      ( ( M = N )
     => ( ! [X4: b] :
            ( ( member_b @ X4 @ N )
           => ( ( F2 @ X4 )
              = ( G2 @ X4 ) ) )
       => ( ( image_b_a @ F2 @ M )
          = ( image_b_a @ G2 @ N ) ) ) ) ).

% image_cong
thf(fact_568_image__cong,axiom,
    ! [M: set_a,N: set_a,F2: a > c,G2: a > c] :
      ( ( M = N )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ N )
           => ( ( F2 @ X4 )
              = ( G2 @ X4 ) ) )
       => ( ( image_a_c @ F2 @ M )
          = ( image_a_c @ G2 @ N ) ) ) ) ).

% image_cong
thf(fact_569_image__cong,axiom,
    ! [M: set_a,N: set_a,F2: a > b,G2: a > b] :
      ( ( M = N )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ N )
           => ( ( F2 @ X4 )
              = ( G2 @ X4 ) ) )
       => ( ( image_a_b @ F2 @ M )
          = ( image_a_b @ G2 @ N ) ) ) ) ).

% image_cong
thf(fact_570_image__cong,axiom,
    ! [M: set_a,N: set_a,F2: a > a,G2: a > a] :
      ( ( M = N )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ N )
           => ( ( F2 @ X4 )
              = ( G2 @ X4 ) ) )
       => ( ( image_a_a @ F2 @ M )
          = ( image_a_a @ G2 @ N ) ) ) ) ).

% image_cong
thf(fact_571_image__cong,axiom,
    ! [M: set_nat,N: set_nat,F2: nat > rat,G2: nat > rat] :
      ( ( M = N )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ N )
           => ( ( F2 @ X4 )
              = ( G2 @ X4 ) ) )
       => ( ( image_nat_rat @ F2 @ M )
          = ( image_nat_rat @ G2 @ N ) ) ) ) ).

% image_cong
thf(fact_572_bex__imageD,axiom,
    ! [F2: b > b,A2: set_b,P: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( image_b_b @ F2 @ A2 ) )
          & ( P @ X5 ) )
     => ? [X4: b] :
          ( ( member_b @ X4 @ A2 )
          & ( P @ ( F2 @ X4 ) ) ) ) ).

% bex_imageD
thf(fact_573_bex__imageD,axiom,
    ! [F2: b > a,A2: set_b,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( image_b_a @ F2 @ A2 ) )
          & ( P @ X5 ) )
     => ? [X4: b] :
          ( ( member_b @ X4 @ A2 )
          & ( P @ ( F2 @ X4 ) ) ) ) ).

% bex_imageD
thf(fact_574_bex__imageD,axiom,
    ! [F2: a > c,A2: set_a,P: c > $o] :
      ( ? [X5: c] :
          ( ( member_c @ X5 @ ( image_a_c @ F2 @ A2 ) )
          & ( P @ X5 ) )
     => ? [X4: a] :
          ( ( member_a @ X4 @ A2 )
          & ( P @ ( F2 @ X4 ) ) ) ) ).

% bex_imageD
thf(fact_575_bex__imageD,axiom,
    ! [F2: a > b,A2: set_a,P: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( image_a_b @ F2 @ A2 ) )
          & ( P @ X5 ) )
     => ? [X4: a] :
          ( ( member_a @ X4 @ A2 )
          & ( P @ ( F2 @ X4 ) ) ) ) ).

% bex_imageD
thf(fact_576_bex__imageD,axiom,
    ! [F2: a > a,A2: set_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( image_a_a @ F2 @ A2 ) )
          & ( P @ X5 ) )
     => ? [X4: a] :
          ( ( member_a @ X4 @ A2 )
          & ( P @ ( F2 @ X4 ) ) ) ) ).

% bex_imageD
thf(fact_577_bex__imageD,axiom,
    ! [F2: nat > rat,A2: set_nat,P: rat > $o] :
      ( ? [X5: rat] :
          ( ( member_rat @ X5 @ ( image_nat_rat @ F2 @ A2 ) )
          & ( P @ X5 ) )
     => ? [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
          & ( P @ ( F2 @ X4 ) ) ) ) ).

% bex_imageD
thf(fact_578_image__iff,axiom,
    ! [Z2: c,F2: a > c,A2: set_a] :
      ( ( member_c @ Z2 @ ( image_a_c @ F2 @ A2 ) )
      = ( ? [X: a] :
            ( ( member_a @ X @ A2 )
            & ( Z2
              = ( F2 @ X ) ) ) ) ) ).

% image_iff
thf(fact_579_image__iff,axiom,
    ! [Z2: b,F2: b > b,A2: set_b] :
      ( ( member_b @ Z2 @ ( image_b_b @ F2 @ A2 ) )
      = ( ? [X: b] :
            ( ( member_b @ X @ A2 )
            & ( Z2
              = ( F2 @ X ) ) ) ) ) ).

% image_iff
thf(fact_580_image__iff,axiom,
    ! [Z2: b,F2: a > b,A2: set_a] :
      ( ( member_b @ Z2 @ ( image_a_b @ F2 @ A2 ) )
      = ( ? [X: a] :
            ( ( member_a @ X @ A2 )
            & ( Z2
              = ( F2 @ X ) ) ) ) ) ).

% image_iff
thf(fact_581_image__iff,axiom,
    ! [Z2: a,F2: b > a,A2: set_b] :
      ( ( member_a @ Z2 @ ( image_b_a @ F2 @ A2 ) )
      = ( ? [X: b] :
            ( ( member_b @ X @ A2 )
            & ( Z2
              = ( F2 @ X ) ) ) ) ) ).

% image_iff
thf(fact_582_image__iff,axiom,
    ! [Z2: a,F2: a > a,A2: set_a] :
      ( ( member_a @ Z2 @ ( image_a_a @ F2 @ A2 ) )
      = ( ? [X: a] :
            ( ( member_a @ X @ A2 )
            & ( Z2
              = ( F2 @ X ) ) ) ) ) ).

% image_iff
thf(fact_583_image__iff,axiom,
    ! [Z2: rat,F2: nat > rat,A2: set_nat] :
      ( ( member_rat @ Z2 @ ( image_nat_rat @ F2 @ A2 ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ A2 )
            & ( Z2
              = ( F2 @ X ) ) ) ) ) ).

% image_iff
thf(fact_584_imageI,axiom,
    ! [X2: nat,A2: set_nat,F2: nat > rat] :
      ( ( member_nat @ X2 @ A2 )
     => ( member_rat @ ( F2 @ X2 ) @ ( image_nat_rat @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_585_imageI,axiom,
    ! [X2: rat,A2: set_rat,F2: rat > rat] :
      ( ( member_rat @ X2 @ A2 )
     => ( member_rat @ ( F2 @ X2 ) @ ( image_rat_rat @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_586_imageI,axiom,
    ! [X2: rat,A2: set_rat,F2: rat > nat] :
      ( ( member_rat @ X2 @ A2 )
     => ( member_nat @ ( F2 @ X2 ) @ ( image_rat_nat @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_587_imageI,axiom,
    ! [X2: rat,A2: set_rat,F2: rat > c] :
      ( ( member_rat @ X2 @ A2 )
     => ( member_c @ ( F2 @ X2 ) @ ( image_rat_c @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_588_imageI,axiom,
    ! [X2: rat,A2: set_rat,F2: rat > b] :
      ( ( member_rat @ X2 @ A2 )
     => ( member_b @ ( F2 @ X2 ) @ ( image_rat_b @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_589_imageI,axiom,
    ! [X2: rat,A2: set_rat,F2: rat > a] :
      ( ( member_rat @ X2 @ A2 )
     => ( member_a @ ( F2 @ X2 ) @ ( image_rat_a @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_590_imageI,axiom,
    ! [X2: nat,A2: set_nat,F2: nat > nat] :
      ( ( member_nat @ X2 @ A2 )
     => ( member_nat @ ( F2 @ X2 ) @ ( image_nat_nat @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_591_imageI,axiom,
    ! [X2: nat,A2: set_nat,F2: nat > c] :
      ( ( member_nat @ X2 @ A2 )
     => ( member_c @ ( F2 @ X2 ) @ ( image_nat_c @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_592_imageI,axiom,
    ! [X2: nat,A2: set_nat,F2: nat > b] :
      ( ( member_nat @ X2 @ A2 )
     => ( member_b @ ( F2 @ X2 ) @ ( image_nat_b @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_593_imageI,axiom,
    ! [X2: nat,A2: set_nat,F2: nat > a] :
      ( ( member_nat @ X2 @ A2 )
     => ( member_a @ ( F2 @ X2 ) @ ( image_nat_a @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_594_Sup_OSUP__cong,axiom,
    ! [A2: set_b,B2: set_b,C2: b > b,D2: b > b,Sup: set_b > b] :
      ( ( A2 = B2 )
     => ( ! [X4: b] :
            ( ( member_b @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( Sup @ ( image_b_b @ C2 @ A2 ) )
          = ( Sup @ ( image_b_b @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_595_Sup_OSUP__cong,axiom,
    ! [A2: set_b,B2: set_b,C2: b > a,D2: b > a,Sup: set_a > a] :
      ( ( A2 = B2 )
     => ( ! [X4: b] :
            ( ( member_b @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( Sup @ ( image_b_a @ C2 @ A2 ) )
          = ( Sup @ ( image_b_a @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_596_Sup_OSUP__cong,axiom,
    ! [A2: set_a,B2: set_a,C2: a > c,D2: a > c,Sup: set_c > c] :
      ( ( A2 = B2 )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( Sup @ ( image_a_c @ C2 @ A2 ) )
          = ( Sup @ ( image_a_c @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_597_Sup_OSUP__cong,axiom,
    ! [A2: set_a,B2: set_a,C2: a > b,D2: a > b,Sup: set_b > b] :
      ( ( A2 = B2 )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( Sup @ ( image_a_b @ C2 @ A2 ) )
          = ( Sup @ ( image_a_b @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_598_Sup_OSUP__cong,axiom,
    ! [A2: set_a,B2: set_a,C2: a > a,D2: a > a,Sup: set_a > a] :
      ( ( A2 = B2 )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( Sup @ ( image_a_a @ C2 @ A2 ) )
          = ( Sup @ ( image_a_a @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_599_Sup_OSUP__cong,axiom,
    ! [A2: set_nat,B2: set_nat,C2: nat > rat,D2: nat > rat,Sup: set_rat > rat] :
      ( ( A2 = B2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( Sup @ ( image_nat_rat @ C2 @ A2 ) )
          = ( Sup @ ( image_nat_rat @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_600_Inf_OINF__cong,axiom,
    ! [A2: set_b,B2: set_b,C2: b > b,D2: b > b,Inf: set_b > b] :
      ( ( A2 = B2 )
     => ( ! [X4: b] :
            ( ( member_b @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( Inf @ ( image_b_b @ C2 @ A2 ) )
          = ( Inf @ ( image_b_b @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_601_Inf_OINF__cong,axiom,
    ! [A2: set_b,B2: set_b,C2: b > a,D2: b > a,Inf: set_a > a] :
      ( ( A2 = B2 )
     => ( ! [X4: b] :
            ( ( member_b @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( Inf @ ( image_b_a @ C2 @ A2 ) )
          = ( Inf @ ( image_b_a @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_602_Inf_OINF__cong,axiom,
    ! [A2: set_a,B2: set_a,C2: a > c,D2: a > c,Inf: set_c > c] :
      ( ( A2 = B2 )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( Inf @ ( image_a_c @ C2 @ A2 ) )
          = ( Inf @ ( image_a_c @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_603_Inf_OINF__cong,axiom,
    ! [A2: set_a,B2: set_a,C2: a > b,D2: a > b,Inf: set_b > b] :
      ( ( A2 = B2 )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( Inf @ ( image_a_b @ C2 @ A2 ) )
          = ( Inf @ ( image_a_b @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_604_Inf_OINF__cong,axiom,
    ! [A2: set_a,B2: set_a,C2: a > a,D2: a > a,Inf: set_a > a] :
      ( ( A2 = B2 )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( Inf @ ( image_a_a @ C2 @ A2 ) )
          = ( Inf @ ( image_a_a @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_605_Inf_OINF__cong,axiom,
    ! [A2: set_nat,B2: set_nat,C2: nat > rat,D2: nat > rat,Inf: set_rat > rat] :
      ( ( A2 = B2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( Inf @ ( image_nat_rat @ C2 @ A2 ) )
          = ( Inf @ ( image_nat_rat @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_606_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_607_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_608_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_609_Inf_OINF__image,axiom,
    ! [Inf: set_a > a,G2: b > a,F2: c > b,A2: set_c] :
      ( ( Inf @ ( image_b_a @ G2 @ ( image_c_b @ F2 @ A2 ) ) )
      = ( Inf @ ( image_c_a @ ( comp_b_a_c @ G2 @ F2 ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_610_Inf_OINF__image,axiom,
    ! [Inf: set_a > a,G2: c > a,F2: c > c,A2: set_c] :
      ( ( Inf @ ( image_c_a @ G2 @ ( image_c_c @ F2 @ A2 ) ) )
      = ( Inf @ ( image_c_a @ ( comp_c_a_c @ G2 @ F2 ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_611_Inf_OINF__image,axiom,
    ! [Inf: set_b > b,G2: b > b,F2: c > b,A2: set_c] :
      ( ( Inf @ ( image_b_b @ G2 @ ( image_c_b @ F2 @ A2 ) ) )
      = ( Inf @ ( image_c_b @ ( comp_b_b_c @ G2 @ F2 ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_612_Inf_OINF__image,axiom,
    ! [Inf: set_a > a,G2: b > a,F2: b > b,A2: set_b] :
      ( ( Inf @ ( image_b_a @ G2 @ ( image_b_b @ F2 @ A2 ) ) )
      = ( Inf @ ( image_b_a @ ( comp_b_a_b @ G2 @ F2 ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_613_Inf_OINF__image,axiom,
    ! [Inf: set_a > a,G2: a > a,F2: c > a,A2: set_c] :
      ( ( Inf @ ( image_a_a @ G2 @ ( image_c_a @ F2 @ A2 ) ) )
      = ( Inf @ ( image_c_a @ ( comp_a_a_c @ G2 @ F2 ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_614_Inf_OINF__image,axiom,
    ! [Inf: set_a > a,G2: a > a,F2: b > a,A2: set_b] :
      ( ( Inf @ ( image_a_a @ G2 @ ( image_b_a @ F2 @ A2 ) ) )
      = ( Inf @ ( image_b_a @ ( comp_a_a_b @ G2 @ F2 ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_615_Inf_OINF__image,axiom,
    ! [Inf: set_rat > rat,G2: rat > rat,F2: nat > rat,A2: set_nat] :
      ( ( Inf @ ( image_rat_rat @ G2 @ ( image_nat_rat @ F2 @ A2 ) ) )
      = ( Inf @ ( image_nat_rat @ ( comp_rat_rat_nat @ G2 @ F2 ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_616_Inf_OINF__image,axiom,
    ! [Inf: set_rat > rat,G2: nat > rat,F2: nat > nat,A2: set_nat] :
      ( ( Inf @ ( image_nat_rat @ G2 @ ( image_nat_nat @ F2 @ A2 ) ) )
      = ( Inf @ ( image_nat_rat @ ( comp_nat_rat_nat @ G2 @ F2 ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_617_Inf_OINF__image,axiom,
    ! [Inf: set_c > c,G2: c > c,F2: a > c,A2: set_a] :
      ( ( Inf @ ( image_c_c @ G2 @ ( image_a_c @ F2 @ A2 ) ) )
      = ( Inf @ ( image_a_c @ ( comp_c_c_a @ G2 @ F2 ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_618_Inf_OINF__image,axiom,
    ! [Inf: set_b > b,G2: c > b,F2: a > c,A2: set_a] :
      ( ( Inf @ ( image_c_b @ G2 @ ( image_a_c @ F2 @ A2 ) ) )
      = ( Inf @ ( image_a_b @ ( comp_c_b_a @ G2 @ F2 ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_619_image__bind,axiom,
    ! [F2: nat > nat,A2: set_nat,G2: nat > set_nat] :
      ( ( image_nat_nat @ F2 @ ( bind_nat_nat @ A2 @ G2 ) )
      = ( bind_nat_nat @ A2 @ ( comp_s3433241188411525313at_nat @ ( image_nat_nat @ F2 ) @ G2 ) ) ) ).

% image_bind
thf(fact_620_image__bind,axiom,
    ! [F2: nat > rat,A2: set_nat,G2: nat > set_nat] :
      ( ( image_nat_rat @ F2 @ ( bind_nat_nat @ A2 @ G2 ) )
      = ( bind_nat_rat @ A2 @ ( comp_s8255929034757298889at_nat @ ( image_nat_rat @ F2 ) @ G2 ) ) ) ).

% image_bind
thf(fact_621_bind__image,axiom,
    ! [F2: nat > nat,A2: set_nat,G2: nat > set_nat] :
      ( ( bind_nat_nat @ ( image_nat_nat @ F2 @ A2 ) @ G2 )
      = ( bind_nat_nat @ A2 @ ( comp_nat_set_nat_nat @ G2 @ F2 ) ) ) ).

% bind_image
thf(fact_622_bind__image,axiom,
    ! [F2: nat > rat,A2: set_nat,G2: rat > set_nat] :
      ( ( bind_rat_nat @ ( image_nat_rat @ F2 @ A2 ) @ G2 )
      = ( bind_nat_nat @ A2 @ ( comp_rat_set_nat_nat @ G2 @ F2 ) ) ) ).

% bind_image
thf(fact_623_surj__uminus,axiom,
    ( ( image_rat_rat @ uminus_uminus_rat @ top_top_set_rat )
    = top_top_set_rat ) ).

% surj_uminus
thf(fact_624_vimage__id,axiom,
    ( ( vimage_c_a_c_a @ id_c_a )
    = id_set_c_a ) ).

% vimage_id
thf(fact_625_vimage__id,axiom,
    ( ( vimage_b_a_b_a @ id_b_a )
    = id_set_b_a ) ).

% vimage_id
thf(fact_626_vimage__id,axiom,
    ( ( vimage_c_c @ id_c )
    = id_set_c ) ).

% vimage_id
thf(fact_627_vimage__id,axiom,
    ( ( vimage_b_b @ id_b )
    = id_set_b ) ).

% vimage_id
thf(fact_628_vimage__id,axiom,
    ( ( vimage_a_a @ id_a )
    = id_set_a ) ).

% vimage_id
thf(fact_629_SUP__id__eq,axiom,
    ! [A2: set_set_rat] :
      ( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ id_set_rat @ A2 ) )
      = ( comple3890839924845867745et_rat @ A2 ) ) ).

% SUP_id_eq
thf(fact_630_SUP__id__eq,axiom,
    ! [A2: set_set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ id_set_nat @ A2 ) )
      = ( comple7399068483239264473et_nat @ A2 ) ) ).

% SUP_id_eq
thf(fact_631_SUP__id__eq,axiom,
    ! [A2: set_nat] :
      ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ id_nat @ A2 ) )
      = ( complete_Sup_Sup_nat @ A2 ) ) ).

% SUP_id_eq
thf(fact_632_INF__id__eq,axiom,
    ! [A2: set_set_rat] :
      ( ( comple4298007329820168263et_rat @ ( image_3939399684171694371et_rat @ id_set_rat @ A2 ) )
      = ( comple4298007329820168263et_rat @ A2 ) ) ).

% INF_id_eq
thf(fact_633_INF__id__eq,axiom,
    ! [A2: set_set_nat] :
      ( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ id_set_nat @ A2 ) )
      = ( comple7806235888213564991et_nat @ A2 ) ) ).

% INF_id_eq
thf(fact_634_INF__id__eq,axiom,
    ! [A2: set_nat] :
      ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ id_nat @ A2 ) )
      = ( complete_Inf_Inf_nat @ A2 ) ) ).

% INF_id_eq
thf(fact_635_set_Ocomp,axiom,
    ! [F2: c > b,G2: b > a] :
      ( ( comp_s772240036570735290_set_a @ ( vimage_c_b @ F2 ) @ ( vimage_b_a @ G2 ) )
      = ( vimage_c_a @ ( comp_b_a_c @ G2 @ F2 ) ) ) ).

% set.comp
thf(fact_636_set_Ocomp,axiom,
    ! [F2: c > c,G2: c > a] :
      ( ( comp_s7879184015435036091_set_a @ ( vimage_c_c @ F2 ) @ ( vimage_c_a @ G2 ) )
      = ( vimage_c_a @ ( comp_c_a_c @ G2 @ F2 ) ) ) ).

% set.comp
thf(fact_637_set_Ocomp,axiom,
    ! [F2: c > b,G2: b > b] :
      ( ( comp_s772240040873964091_set_b @ ( vimage_c_b @ F2 ) @ ( vimage_b_b @ G2 ) )
      = ( vimage_c_b @ ( comp_b_b_c @ G2 @ F2 ) ) ) ).

% set.comp
thf(fact_638_set_Ocomp,axiom,
    ! [F2: b > b,G2: b > a] :
      ( ( comp_s8507650036130712571_set_a @ ( vimage_b_b @ F2 ) @ ( vimage_b_a @ G2 ) )
      = ( vimage_b_a @ ( comp_b_a_b @ G2 @ F2 ) ) ) ).

% set.comp
thf(fact_639_set_Ocomp,axiom,
    ! [F2: c > a,G2: a > a] :
      ( ( comp_s2888668094561210297_set_a @ ( vimage_c_a @ F2 ) @ ( vimage_a_a @ G2 ) )
      = ( vimage_c_a @ ( comp_a_a_c @ G2 @ F2 ) ) ) ).

% set.comp
thf(fact_640_set_Ocomp,axiom,
    ! [F2: b > a,G2: a > a] :
      ( ( comp_s1400706057266411770_set_a @ ( vimage_b_a @ F2 ) @ ( vimage_a_a @ G2 ) )
      = ( vimage_b_a @ ( comp_a_a_b @ G2 @ F2 ) ) ) ).

% set.comp
thf(fact_641_set_Ocomp,axiom,
    ! [F2: c > c,G2: c > b] :
      ( ( comp_s7879184019738264892_set_b @ ( vimage_c_c @ F2 ) @ ( vimage_c_b @ G2 ) )
      = ( vimage_c_b @ ( comp_c_b_c @ G2 @ F2 ) ) ) ).

% set.comp
thf(fact_642_set_Ocomp,axiom,
    ! [F2: b > c,G2: c > b] :
      ( ( comp_s6391221982443466365_set_b @ ( vimage_b_c @ F2 ) @ ( vimage_c_b @ G2 ) )
      = ( vimage_b_b @ ( comp_c_b_b @ G2 @ F2 ) ) ) ).

% set.comp
thf(fact_643_set_Ocomp,axiom,
    ! [F2: b > c,G2: c > a] :
      ( ( comp_s6391221978140237564_set_a @ ( vimage_b_c @ F2 ) @ ( vimage_c_a @ G2 ) )
      = ( vimage_b_a @ ( comp_c_a_b @ G2 @ F2 ) ) ) ).

% set.comp
thf(fact_644_set_Ocomp,axiom,
    ! [F2: a > c,G2: c > a] :
      ( ( comp_s4903259940845439037_set_a @ ( vimage_a_c @ F2 ) @ ( vimage_c_a @ G2 ) )
      = ( vimage_a_a @ ( comp_c_a_a @ G2 @ F2 ) ) ) ).

% set.comp
thf(fact_645_surj__id,axiom,
    ( ( image_c_a_c_a @ id_c_a @ top_top_set_c_a )
    = top_top_set_c_a ) ).

% surj_id
thf(fact_646_surj__id,axiom,
    ( ( image_b_a_b_a @ id_b_a @ top_top_set_b_a )
    = top_top_set_b_a ) ).

% surj_id
thf(fact_647_surj__id,axiom,
    ( ( image_c_c @ id_c @ top_top_set_c )
    = top_top_set_c ) ).

% surj_id
thf(fact_648_surj__id,axiom,
    ( ( image_b_b @ id_b @ top_top_set_b )
    = top_top_set_b ) ).

% surj_id
thf(fact_649_surj__id,axiom,
    ( ( image_a_a @ id_a @ top_top_set_a )
    = top_top_set_a ) ).

% surj_id
thf(fact_650_surj__id,axiom,
    ( ( image_nat_nat @ id_nat @ top_top_set_nat )
    = top_top_set_nat ) ).

% surj_id
thf(fact_651_surj__id,axiom,
    ( ( image_rat_rat @ id_rat @ top_top_set_rat )
    = top_top_set_rat ) ).

% surj_id
thf(fact_652_comp__surj,axiom,
    ! [F2: c > b,G2: b > a] :
      ( ( ( image_c_b @ F2 @ top_top_set_c )
        = top_top_set_b )
     => ( ( ( image_b_a @ G2 @ top_top_set_b )
          = top_top_set_a )
       => ( ( image_c_a @ ( comp_b_a_c @ G2 @ F2 ) @ top_top_set_c )
          = top_top_set_a ) ) ) ).

% comp_surj
thf(fact_653_comp__surj,axiom,
    ! [F2: c > c,G2: c > a] :
      ( ( ( image_c_c @ F2 @ top_top_set_c )
        = top_top_set_c )
     => ( ( ( image_c_a @ G2 @ top_top_set_c )
          = top_top_set_a )
       => ( ( image_c_a @ ( comp_c_a_c @ G2 @ F2 ) @ top_top_set_c )
          = top_top_set_a ) ) ) ).

% comp_surj
thf(fact_654_comp__surj,axiom,
    ! [F2: c > b,G2: b > b] :
      ( ( ( image_c_b @ F2 @ top_top_set_c )
        = top_top_set_b )
     => ( ( ( image_b_b @ G2 @ top_top_set_b )
          = top_top_set_b )
       => ( ( image_c_b @ ( comp_b_b_c @ G2 @ F2 ) @ top_top_set_c )
          = top_top_set_b ) ) ) ).

% comp_surj
thf(fact_655_comp__surj,axiom,
    ! [F2: b > b,G2: b > a] :
      ( ( ( image_b_b @ F2 @ top_top_set_b )
        = top_top_set_b )
     => ( ( ( image_b_a @ G2 @ top_top_set_b )
          = top_top_set_a )
       => ( ( image_b_a @ ( comp_b_a_b @ G2 @ F2 ) @ top_top_set_b )
          = top_top_set_a ) ) ) ).

% comp_surj
thf(fact_656_comp__surj,axiom,
    ! [F2: c > a,G2: a > a] :
      ( ( ( image_c_a @ F2 @ top_top_set_c )
        = top_top_set_a )
     => ( ( ( image_a_a @ G2 @ top_top_set_a )
          = top_top_set_a )
       => ( ( image_c_a @ ( comp_a_a_c @ G2 @ F2 ) @ top_top_set_c )
          = top_top_set_a ) ) ) ).

% comp_surj
thf(fact_657_comp__surj,axiom,
    ! [F2: b > a,G2: a > a] :
      ( ( ( image_b_a @ F2 @ top_top_set_b )
        = top_top_set_a )
     => ( ( ( image_a_a @ G2 @ top_top_set_a )
          = top_top_set_a )
       => ( ( image_b_a @ ( comp_a_a_b @ G2 @ F2 ) @ top_top_set_b )
          = top_top_set_a ) ) ) ).

% comp_surj
thf(fact_658_comp__surj,axiom,
    ! [F2: nat > nat,G2: nat > nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_nat @ G2 @ top_top_set_nat )
          = top_top_set_nat )
       => ( ( image_nat_nat @ ( comp_nat_nat_nat @ G2 @ F2 ) @ top_top_set_nat )
          = top_top_set_nat ) ) ) ).

% comp_surj
thf(fact_659_comp__surj,axiom,
    ! [F2: nat > nat,G2: nat > rat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_rat @ G2 @ top_top_set_nat )
          = top_top_set_rat )
       => ( ( image_nat_rat @ ( comp_nat_rat_nat @ G2 @ F2 ) @ top_top_set_nat )
          = top_top_set_rat ) ) ) ).

% comp_surj
thf(fact_660_comp__surj,axiom,
    ! [F2: nat > rat,G2: rat > nat] :
      ( ( ( image_nat_rat @ F2 @ top_top_set_nat )
        = top_top_set_rat )
     => ( ( ( image_rat_nat @ G2 @ top_top_set_rat )
          = top_top_set_nat )
       => ( ( image_nat_nat @ ( comp_rat_nat_nat @ G2 @ F2 ) @ top_top_set_nat )
          = top_top_set_nat ) ) ) ).

% comp_surj
thf(fact_661_comp__surj,axiom,
    ! [F2: nat > rat,G2: rat > rat] :
      ( ( ( image_nat_rat @ F2 @ top_top_set_nat )
        = top_top_set_rat )
     => ( ( ( image_rat_rat @ G2 @ top_top_set_rat )
          = top_top_set_rat )
       => ( ( image_nat_rat @ ( comp_rat_rat_nat @ G2 @ F2 ) @ top_top_set_nat )
          = top_top_set_rat ) ) ) ).

% comp_surj
thf(fact_662_fun_Omap__ident__strong,axiom,
    ! [T: c > b,F2: b > b] :
      ( ! [Z3: b] :
          ( ( member_b @ Z3 @ ( image_c_b @ T @ top_top_set_c ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( comp_b_b_c @ F2 @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_663_fun_Omap__ident__strong,axiom,
    ! [T: c > a,F2: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( image_c_a @ T @ top_top_set_c ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( comp_a_a_c @ F2 @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_664_fun_Omap__ident__strong,axiom,
    ! [T: b > a,F2: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( image_b_a @ T @ top_top_set_b ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( comp_a_a_b @ F2 @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_665_fun_Omap__ident__strong,axiom,
    ! [T: nat > rat,F2: rat > rat] :
      ( ! [Z3: rat] :
          ( ( member_rat @ Z3 @ ( image_nat_rat @ T @ top_top_set_nat ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( comp_rat_rat_nat @ F2 @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_666_fun_Omap__ident__strong,axiom,
    ! [T: nat > nat,F2: nat > nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( image_nat_nat @ T @ top_top_set_nat ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( comp_nat_nat_nat @ F2 @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_667_fun_Omap__ident__strong,axiom,
    ! [T: nat > c,F2: c > c] :
      ( ! [Z3: c] :
          ( ( member_c @ Z3 @ ( image_nat_c @ T @ top_top_set_nat ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( comp_c_c_nat @ F2 @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_668_fun_Omap__ident__strong,axiom,
    ! [T: nat > b,F2: b > b] :
      ( ! [Z3: b] :
          ( ( member_b @ Z3 @ ( image_nat_b @ T @ top_top_set_nat ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( comp_b_b_nat @ F2 @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_669_fun_Omap__ident__strong,axiom,
    ! [T: nat > a,F2: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( image_nat_a @ T @ top_top_set_nat ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( comp_a_a_nat @ F2 @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_670_fun_Omap__ident__strong,axiom,
    ! [T: rat > rat,F2: rat > rat] :
      ( ! [Z3: rat] :
          ( ( member_rat @ Z3 @ ( image_rat_rat @ T @ top_top_set_rat ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( comp_rat_rat_rat @ F2 @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_671_fun_Omap__ident__strong,axiom,
    ! [T: rat > nat,F2: nat > nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( image_rat_nat @ T @ top_top_set_rat ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( comp_nat_nat_rat @ F2 @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_672_UNIV__I,axiom,
    ! [X2: c] : ( member_c @ X2 @ top_top_set_c ) ).

% UNIV_I
thf(fact_673_UNIV__I,axiom,
    ! [X2: b] : ( member_b @ X2 @ top_top_set_b ) ).

% UNIV_I
thf(fact_674_UNIV__I,axiom,
    ! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).

% UNIV_I
thf(fact_675_UNIV__I,axiom,
    ! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_676_UNIV__I,axiom,
    ! [X2: rat] : ( member_rat @ X2 @ top_top_set_rat ) ).

% UNIV_I
thf(fact_677_Inter__UNIV__conv_I1_J,axiom,
    ! [A2: set_set_c] :
      ( ( ( comple6135023387286571239_set_c @ A2 )
        = top_top_set_c )
      = ( ! [X: set_c] :
            ( ( member_set_c @ X @ A2 )
           => ( X = top_top_set_c ) ) ) ) ).

% Inter_UNIV_conv(1)
thf(fact_678_Inter__UNIV__conv_I1_J,axiom,
    ! [A2: set_set_b] :
      ( ( ( comple6135023382983342438_set_b @ A2 )
        = top_top_set_b )
      = ( ! [X: set_b] :
            ( ( member_set_b @ X @ A2 )
           => ( X = top_top_set_b ) ) ) ) ).

% Inter_UNIV_conv(1)
thf(fact_679_Inter__UNIV__conv_I1_J,axiom,
    ! [A2: set_set_a] :
      ( ( ( comple6135023378680113637_set_a @ A2 )
        = top_top_set_a )
      = ( ! [X: set_a] :
            ( ( member_set_a @ X @ A2 )
           => ( X = top_top_set_a ) ) ) ) ).

% Inter_UNIV_conv(1)
thf(fact_680_Inter__UNIV__conv_I1_J,axiom,
    ! [A2: set_set_nat] :
      ( ( ( comple7806235888213564991et_nat @ A2 )
        = top_top_set_nat )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A2 )
           => ( X = top_top_set_nat ) ) ) ) ).

% Inter_UNIV_conv(1)
thf(fact_681_Inter__UNIV__conv_I1_J,axiom,
    ! [A2: set_set_rat] :
      ( ( ( comple4298007329820168263et_rat @ A2 )
        = top_top_set_rat )
      = ( ! [X: set_rat] :
            ( ( member_set_rat @ X @ A2 )
           => ( X = top_top_set_rat ) ) ) ) ).

% Inter_UNIV_conv(1)
thf(fact_682_Inter__UNIV__conv_I2_J,axiom,
    ! [A2: set_set_c] :
      ( ( top_top_set_c
        = ( comple6135023387286571239_set_c @ A2 ) )
      = ( ! [X: set_c] :
            ( ( member_set_c @ X @ A2 )
           => ( X = top_top_set_c ) ) ) ) ).

% Inter_UNIV_conv(2)
thf(fact_683_Inter__UNIV__conv_I2_J,axiom,
    ! [A2: set_set_b] :
      ( ( top_top_set_b
        = ( comple6135023382983342438_set_b @ A2 ) )
      = ( ! [X: set_b] :
            ( ( member_set_b @ X @ A2 )
           => ( X = top_top_set_b ) ) ) ) ).

% Inter_UNIV_conv(2)
thf(fact_684_Inter__UNIV__conv_I2_J,axiom,
    ! [A2: set_set_a] :
      ( ( top_top_set_a
        = ( comple6135023378680113637_set_a @ A2 ) )
      = ( ! [X: set_a] :
            ( ( member_set_a @ X @ A2 )
           => ( X = top_top_set_a ) ) ) ) ).

% Inter_UNIV_conv(2)
thf(fact_685_Inter__UNIV__conv_I2_J,axiom,
    ! [A2: set_set_nat] :
      ( ( top_top_set_nat
        = ( comple7806235888213564991et_nat @ A2 ) )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A2 )
           => ( X = top_top_set_nat ) ) ) ) ).

% Inter_UNIV_conv(2)
thf(fact_686_Inter__UNIV__conv_I2_J,axiom,
    ! [A2: set_set_rat] :
      ( ( top_top_set_rat
        = ( comple4298007329820168263et_rat @ A2 ) )
      = ( ! [X: set_rat] :
            ( ( member_set_rat @ X @ A2 )
           => ( X = top_top_set_rat ) ) ) ) ).

% Inter_UNIV_conv(2)
thf(fact_687_vimage__eq,axiom,
    ! [A: rat,F2: rat > rat,B2: set_rat] :
      ( ( member_rat @ A @ ( vimage_rat_rat @ F2 @ B2 ) )
      = ( member_rat @ ( F2 @ A ) @ B2 ) ) ).

% vimage_eq
thf(fact_688_vimage__eq,axiom,
    ! [A: rat,F2: rat > nat,B2: set_nat] :
      ( ( member_rat @ A @ ( vimage_rat_nat @ F2 @ B2 ) )
      = ( member_nat @ ( F2 @ A ) @ B2 ) ) ).

% vimage_eq
thf(fact_689_vimage__eq,axiom,
    ! [A: rat,F2: rat > c,B2: set_c] :
      ( ( member_rat @ A @ ( vimage_rat_c @ F2 @ B2 ) )
      = ( member_c @ ( F2 @ A ) @ B2 ) ) ).

% vimage_eq
thf(fact_690_vimage__eq,axiom,
    ! [A: rat,F2: rat > b,B2: set_b] :
      ( ( member_rat @ A @ ( vimage_rat_b @ F2 @ B2 ) )
      = ( member_b @ ( F2 @ A ) @ B2 ) ) ).

% vimage_eq
thf(fact_691_vimage__eq,axiom,
    ! [A: rat,F2: rat > a,B2: set_a] :
      ( ( member_rat @ A @ ( vimage_rat_a @ F2 @ B2 ) )
      = ( member_a @ ( F2 @ A ) @ B2 ) ) ).

% vimage_eq
thf(fact_692_vimage__eq,axiom,
    ! [A: nat,F2: nat > rat,B2: set_rat] :
      ( ( member_nat @ A @ ( vimage_nat_rat @ F2 @ B2 ) )
      = ( member_rat @ ( F2 @ A ) @ B2 ) ) ).

% vimage_eq
thf(fact_693_vimage__eq,axiom,
    ! [A: nat,F2: nat > nat,B2: set_nat] :
      ( ( member_nat @ A @ ( vimage_nat_nat @ F2 @ B2 ) )
      = ( member_nat @ ( F2 @ A ) @ B2 ) ) ).

% vimage_eq
thf(fact_694_vimage__eq,axiom,
    ! [A: nat,F2: nat > c,B2: set_c] :
      ( ( member_nat @ A @ ( vimage_nat_c @ F2 @ B2 ) )
      = ( member_c @ ( F2 @ A ) @ B2 ) ) ).

% vimage_eq
thf(fact_695_vimage__eq,axiom,
    ! [A: nat,F2: nat > b,B2: set_b] :
      ( ( member_nat @ A @ ( vimage_nat_b @ F2 @ B2 ) )
      = ( member_b @ ( F2 @ A ) @ B2 ) ) ).

% vimage_eq
thf(fact_696_vimage__eq,axiom,
    ! [A: nat,F2: nat > a,B2: set_a] :
      ( ( member_nat @ A @ ( vimage_nat_a @ F2 @ B2 ) )
      = ( member_a @ ( F2 @ A ) @ B2 ) ) ).

% vimage_eq
thf(fact_697_vimageI,axiom,
    ! [F2: rat > rat,A: rat,B: rat,B2: set_rat] :
      ( ( ( F2 @ A )
        = B )
     => ( ( member_rat @ B @ B2 )
       => ( member_rat @ A @ ( vimage_rat_rat @ F2 @ B2 ) ) ) ) ).

% vimageI
thf(fact_698_vimageI,axiom,
    ! [F2: nat > rat,A: nat,B: rat,B2: set_rat] :
      ( ( ( F2 @ A )
        = B )
     => ( ( member_rat @ B @ B2 )
       => ( member_nat @ A @ ( vimage_nat_rat @ F2 @ B2 ) ) ) ) ).

% vimageI
thf(fact_699_vimageI,axiom,
    ! [F2: c > rat,A: c,B: rat,B2: set_rat] :
      ( ( ( F2 @ A )
        = B )
     => ( ( member_rat @ B @ B2 )
       => ( member_c @ A @ ( vimage_c_rat @ F2 @ B2 ) ) ) ) ).

% vimageI
thf(fact_700_vimageI,axiom,
    ! [F2: b > rat,A: b,B: rat,B2: set_rat] :
      ( ( ( F2 @ A )
        = B )
     => ( ( member_rat @ B @ B2 )
       => ( member_b @ A @ ( vimage_b_rat @ F2 @ B2 ) ) ) ) ).

% vimageI
thf(fact_701_vimageI,axiom,
    ! [F2: a > rat,A: a,B: rat,B2: set_rat] :
      ( ( ( F2 @ A )
        = B )
     => ( ( member_rat @ B @ B2 )
       => ( member_a @ A @ ( vimage_a_rat @ F2 @ B2 ) ) ) ) ).

% vimageI
thf(fact_702_vimageI,axiom,
    ! [F2: rat > nat,A: rat,B: nat,B2: set_nat] :
      ( ( ( F2 @ A )
        = B )
     => ( ( member_nat @ B @ B2 )
       => ( member_rat @ A @ ( vimage_rat_nat @ F2 @ B2 ) ) ) ) ).

% vimageI
thf(fact_703_vimageI,axiom,
    ! [F2: nat > nat,A: nat,B: nat,B2: set_nat] :
      ( ( ( F2 @ A )
        = B )
     => ( ( member_nat @ B @ B2 )
       => ( member_nat @ A @ ( vimage_nat_nat @ F2 @ B2 ) ) ) ) ).

% vimageI
thf(fact_704_vimageI,axiom,
    ! [F2: c > nat,A: c,B: nat,B2: set_nat] :
      ( ( ( F2 @ A )
        = B )
     => ( ( member_nat @ B @ B2 )
       => ( member_c @ A @ ( vimage_c_nat @ F2 @ B2 ) ) ) ) ).

% vimageI
thf(fact_705_vimageI,axiom,
    ! [F2: b > nat,A: b,B: nat,B2: set_nat] :
      ( ( ( F2 @ A )
        = B )
     => ( ( member_nat @ B @ B2 )
       => ( member_b @ A @ ( vimage_b_nat @ F2 @ B2 ) ) ) ) ).

% vimageI
thf(fact_706_vimageI,axiom,
    ! [F2: a > nat,A: a,B: nat,B2: set_nat] :
      ( ( ( F2 @ A )
        = B )
     => ( ( member_nat @ B @ B2 )
       => ( member_a @ A @ ( vimage_a_nat @ F2 @ B2 ) ) ) ) ).

% vimageI
thf(fact_707_member__bind,axiom,
    ! [X2: nat,A2: set_nat,F2: nat > set_nat] :
      ( ( member_nat @ X2 @ ( bind_nat_nat @ A2 @ F2 ) )
      = ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ) ).

% member_bind
thf(fact_708_Inf__top__conv_I1_J,axiom,
    ! [A2: set_set_c] :
      ( ( ( comple6135023387286571239_set_c @ A2 )
        = top_top_set_c )
      = ( ! [X: set_c] :
            ( ( member_set_c @ X @ A2 )
           => ( X = top_top_set_c ) ) ) ) ).

% Inf_top_conv(1)
thf(fact_709_Inf__top__conv_I1_J,axiom,
    ! [A2: set_set_b] :
      ( ( ( comple6135023382983342438_set_b @ A2 )
        = top_top_set_b )
      = ( ! [X: set_b] :
            ( ( member_set_b @ X @ A2 )
           => ( X = top_top_set_b ) ) ) ) ).

% Inf_top_conv(1)
thf(fact_710_Inf__top__conv_I1_J,axiom,
    ! [A2: set_set_a] :
      ( ( ( comple6135023378680113637_set_a @ A2 )
        = top_top_set_a )
      = ( ! [X: set_a] :
            ( ( member_set_a @ X @ A2 )
           => ( X = top_top_set_a ) ) ) ) ).

% Inf_top_conv(1)
thf(fact_711_Inf__top__conv_I1_J,axiom,
    ! [A2: set_rat_o] :
      ( ( ( comple2477142665972227838_rat_o @ A2 )
        = top_top_rat_o )
      = ( ! [X: rat > $o] :
            ( ( member_rat_o @ X @ A2 )
           => ( X = top_top_rat_o ) ) ) ) ).

% Inf_top_conv(1)
thf(fact_712_Inf__top__conv_I1_J,axiom,
    ! [A2: set_nat_o] :
      ( ( ( comple6214475593288795910_nat_o @ A2 )
        = top_top_nat_o )
      = ( ! [X: nat > $o] :
            ( ( member_nat_o @ X @ A2 )
           => ( X = top_top_nat_o ) ) ) ) ).

% Inf_top_conv(1)
thf(fact_713_Inf__top__conv_I1_J,axiom,
    ! [A2: set_set_nat] :
      ( ( ( comple7806235888213564991et_nat @ A2 )
        = top_top_set_nat )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A2 )
           => ( X = top_top_set_nat ) ) ) ) ).

% Inf_top_conv(1)
thf(fact_714_Inf__top__conv_I1_J,axiom,
    ! [A2: set_set_rat] :
      ( ( ( comple4298007329820168263et_rat @ A2 )
        = top_top_set_rat )
      = ( ! [X: set_rat] :
            ( ( member_set_rat @ X @ A2 )
           => ( X = top_top_set_rat ) ) ) ) ).

% Inf_top_conv(1)
thf(fact_715_Inf__top__conv_I2_J,axiom,
    ! [A2: set_set_c] :
      ( ( top_top_set_c
        = ( comple6135023387286571239_set_c @ A2 ) )
      = ( ! [X: set_c] :
            ( ( member_set_c @ X @ A2 )
           => ( X = top_top_set_c ) ) ) ) ).

% Inf_top_conv(2)
thf(fact_716_Inf__top__conv_I2_J,axiom,
    ! [A2: set_set_b] :
      ( ( top_top_set_b
        = ( comple6135023382983342438_set_b @ A2 ) )
      = ( ! [X: set_b] :
            ( ( member_set_b @ X @ A2 )
           => ( X = top_top_set_b ) ) ) ) ).

% Inf_top_conv(2)
thf(fact_717_Inf__top__conv_I2_J,axiom,
    ! [A2: set_set_a] :
      ( ( top_top_set_a
        = ( comple6135023378680113637_set_a @ A2 ) )
      = ( ! [X: set_a] :
            ( ( member_set_a @ X @ A2 )
           => ( X = top_top_set_a ) ) ) ) ).

% Inf_top_conv(2)
thf(fact_718_Inf__top__conv_I2_J,axiom,
    ! [A2: set_rat_o] :
      ( ( top_top_rat_o
        = ( comple2477142665972227838_rat_o @ A2 ) )
      = ( ! [X: rat > $o] :
            ( ( member_rat_o @ X @ A2 )
           => ( X = top_top_rat_o ) ) ) ) ).

% Inf_top_conv(2)
thf(fact_719_Inf__top__conv_I2_J,axiom,
    ! [A2: set_nat_o] :
      ( ( top_top_nat_o
        = ( comple6214475593288795910_nat_o @ A2 ) )
      = ( ! [X: nat > $o] :
            ( ( member_nat_o @ X @ A2 )
           => ( X = top_top_nat_o ) ) ) ) ).

% Inf_top_conv(2)
thf(fact_720_Inf__top__conv_I2_J,axiom,
    ! [A2: set_set_nat] :
      ( ( top_top_set_nat
        = ( comple7806235888213564991et_nat @ A2 ) )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A2 )
           => ( X = top_top_set_nat ) ) ) ) ).

% Inf_top_conv(2)
thf(fact_721_Inf__top__conv_I2_J,axiom,
    ! [A2: set_set_rat] :
      ( ( top_top_set_rat
        = ( comple4298007329820168263et_rat @ A2 ) )
      = ( ! [X: set_rat] :
            ( ( member_set_rat @ X @ A2 )
           => ( X = top_top_set_rat ) ) ) ) ).

% Inf_top_conv(2)
thf(fact_722_vimage__UNIV,axiom,
    ! [F2: nat > nat] :
      ( ( vimage_nat_nat @ F2 @ top_top_set_nat )
      = top_top_set_nat ) ).

% vimage_UNIV
thf(fact_723_vimage__UNIV,axiom,
    ! [F2: rat > nat] :
      ( ( vimage_rat_nat @ F2 @ top_top_set_nat )
      = top_top_set_rat ) ).

% vimage_UNIV
thf(fact_724_vimage__UNIV,axiom,
    ! [F2: nat > rat] :
      ( ( vimage_nat_rat @ F2 @ top_top_set_rat )
      = top_top_set_nat ) ).

% vimage_UNIV
thf(fact_725_vimage__UNIV,axiom,
    ! [F2: rat > rat] :
      ( ( vimage_rat_rat @ F2 @ top_top_set_rat )
      = top_top_set_rat ) ).

% vimage_UNIV
thf(fact_726_vimage__UNIV,axiom,
    ! [F2: c > nat] :
      ( ( vimage_c_nat @ F2 @ top_top_set_nat )
      = top_top_set_c ) ).

% vimage_UNIV
thf(fact_727_vimage__UNIV,axiom,
    ! [F2: b > nat] :
      ( ( vimage_b_nat @ F2 @ top_top_set_nat )
      = top_top_set_b ) ).

% vimage_UNIV
thf(fact_728_vimage__UNIV,axiom,
    ! [F2: a > nat] :
      ( ( vimage_a_nat @ F2 @ top_top_set_nat )
      = top_top_set_a ) ).

% vimage_UNIV
thf(fact_729_vimage__UNIV,axiom,
    ! [F2: c > rat] :
      ( ( vimage_c_rat @ F2 @ top_top_set_rat )
      = top_top_set_c ) ).

% vimage_UNIV
thf(fact_730_vimage__UNIV,axiom,
    ! [F2: b > rat] :
      ( ( vimage_b_rat @ F2 @ top_top_set_rat )
      = top_top_set_b ) ).

% vimage_UNIV
thf(fact_731_vimage__UNIV,axiom,
    ! [F2: a > rat] :
      ( ( vimage_a_rat @ F2 @ top_top_set_rat )
      = top_top_set_a ) ).

% vimage_UNIV
thf(fact_732_Sup__UNIV,axiom,
    ( ( comple2307003618534512845_set_c @ top_top_set_set_c )
    = top_top_set_c ) ).

% Sup_UNIV
thf(fact_733_Sup__UNIV,axiom,
    ( ( comple2307003614231284044_set_b @ top_top_set_set_b )
    = top_top_set_b ) ).

% Sup_UNIV
thf(fact_734_Sup__UNIV,axiom,
    ( ( comple2307003609928055243_set_a @ top_top_set_set_a )
    = top_top_set_a ) ).

% Sup_UNIV
thf(fact_735_Sup__UNIV,axiom,
    ( ( comple4580332206425622756_rat_o @ top_top_set_rat_o )
    = top_top_rat_o ) ).

% Sup_UNIV
thf(fact_736_Sup__UNIV,axiom,
    ( ( comple8317665133742190828_nat_o @ top_top_set_nat_o )
    = top_top_nat_o ) ).

% Sup_UNIV
thf(fact_737_Sup__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
    = top_top_set_nat ) ).

% Sup_UNIV
thf(fact_738_Sup__UNIV,axiom,
    ( ( comple3890839924845867745et_rat @ top_top_set_set_rat )
    = top_top_set_rat ) ).

% Sup_UNIV
thf(fact_739_vimage__Compl,axiom,
    ! [F2: c > b,A2: set_b] :
      ( ( vimage_c_b @ F2 @ ( uminus_uminus_set_b @ A2 ) )
      = ( uminus_uminus_set_c @ ( vimage_c_b @ F2 @ A2 ) ) ) ).

% vimage_Compl
thf(fact_740_vimage__Compl,axiom,
    ! [F2: c > a,A2: set_a] :
      ( ( vimage_c_a @ F2 @ ( uminus_uminus_set_a @ A2 ) )
      = ( uminus_uminus_set_c @ ( vimage_c_a @ F2 @ A2 ) ) ) ).

% vimage_Compl
thf(fact_741_vimage__Compl,axiom,
    ! [F2: b > b,A2: set_b] :
      ( ( vimage_b_b @ F2 @ ( uminus_uminus_set_b @ A2 ) )
      = ( uminus_uminus_set_b @ ( vimage_b_b @ F2 @ A2 ) ) ) ).

% vimage_Compl
thf(fact_742_vimage__Compl,axiom,
    ! [F2: b > a,A2: set_a] :
      ( ( vimage_b_a @ F2 @ ( uminus_uminus_set_a @ A2 ) )
      = ( uminus_uminus_set_b @ ( vimage_b_a @ F2 @ A2 ) ) ) ).

% vimage_Compl
thf(fact_743_vimage__Compl,axiom,
    ! [F2: a > a,A2: set_a] :
      ( ( vimage_a_a @ F2 @ ( uminus_uminus_set_a @ A2 ) )
      = ( uminus_uminus_set_a @ ( vimage_a_a @ F2 @ A2 ) ) ) ).

% vimage_Compl
thf(fact_744_vimage__Compl,axiom,
    ! [F2: rat > rat,A2: set_rat] :
      ( ( vimage_rat_rat @ F2 @ ( uminus2201863774496077783et_rat @ A2 ) )
      = ( uminus2201863774496077783et_rat @ ( vimage_rat_rat @ F2 @ A2 ) ) ) ).

% vimage_Compl
thf(fact_745_vimage__Compl,axiom,
    ! [F2: nat > rat,A2: set_rat] :
      ( ( vimage_nat_rat @ F2 @ ( uminus2201863774496077783et_rat @ A2 ) )
      = ( uminus5710092332889474511et_nat @ ( vimage_nat_rat @ F2 @ A2 ) ) ) ).

% vimage_Compl
thf(fact_746_vimage__Compl,axiom,
    ! [F2: rat > nat,A2: set_nat] :
      ( ( vimage_rat_nat @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( uminus2201863774496077783et_rat @ ( vimage_rat_nat @ F2 @ A2 ) ) ) ).

% vimage_Compl
thf(fact_747_vimage__Compl,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( vimage_nat_nat @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( uminus5710092332889474511et_nat @ ( vimage_nat_nat @ F2 @ A2 ) ) ) ).

% vimage_Compl
thf(fact_748_bind__UNION,axiom,
    ( bind_nat_nat
    = ( ^ [A5: set_nat,F: nat > set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A5 ) ) ) ) ).

% bind_UNION
thf(fact_749_Union__UNIV,axiom,
    ( ( comple2307003618534512845_set_c @ top_top_set_set_c )
    = top_top_set_c ) ).

% Union_UNIV
thf(fact_750_Union__UNIV,axiom,
    ( ( comple2307003614231284044_set_b @ top_top_set_set_b )
    = top_top_set_b ) ).

% Union_UNIV
thf(fact_751_Union__UNIV,axiom,
    ( ( comple2307003609928055243_set_a @ top_top_set_set_a )
    = top_top_set_a ) ).

% Union_UNIV
thf(fact_752_Union__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
    = top_top_set_nat ) ).

% Union_UNIV
thf(fact_753_Union__UNIV,axiom,
    ( ( comple3890839924845867745et_rat @ top_top_set_set_rat )
    = top_top_set_rat ) ).

% Union_UNIV
thf(fact_754_vimage__Collect,axiom,
    ! [P: b > $o,F2: c > b,Q: c > $o] :
      ( ! [X4: c] :
          ( ( P @ ( F2 @ X4 ) )
          = ( Q @ X4 ) )
     => ( ( vimage_c_b @ F2 @ ( collect_b @ P ) )
        = ( collect_c @ Q ) ) ) ).

% vimage_Collect
thf(fact_755_vimage__Collect,axiom,
    ! [P: a > $o,F2: c > a,Q: c > $o] :
      ( ! [X4: c] :
          ( ( P @ ( F2 @ X4 ) )
          = ( Q @ X4 ) )
     => ( ( vimage_c_a @ F2 @ ( collect_a @ P ) )
        = ( collect_c @ Q ) ) ) ).

% vimage_Collect
thf(fact_756_vimage__Collect,axiom,
    ! [P: b > $o,F2: b > b,Q: b > $o] :
      ( ! [X4: b] :
          ( ( P @ ( F2 @ X4 ) )
          = ( Q @ X4 ) )
     => ( ( vimage_b_b @ F2 @ ( collect_b @ P ) )
        = ( collect_b @ Q ) ) ) ).

% vimage_Collect
thf(fact_757_vimage__Collect,axiom,
    ! [P: a > $o,F2: b > a,Q: b > $o] :
      ( ! [X4: b] :
          ( ( P @ ( F2 @ X4 ) )
          = ( Q @ X4 ) )
     => ( ( vimage_b_a @ F2 @ ( collect_a @ P ) )
        = ( collect_b @ Q ) ) ) ).

% vimage_Collect
thf(fact_758_vimage__Collect,axiom,
    ! [P: a > $o,F2: a > a,Q: a > $o] :
      ( ! [X4: a] :
          ( ( P @ ( F2 @ X4 ) )
          = ( Q @ X4 ) )
     => ( ( vimage_a_a @ F2 @ ( collect_a @ P ) )
        = ( collect_a @ Q ) ) ) ).

% vimage_Collect
thf(fact_759_vimage__Collect,axiom,
    ! [P: rat > $o,F2: rat > rat,Q: rat > $o] :
      ( ! [X4: rat] :
          ( ( P @ ( F2 @ X4 ) )
          = ( Q @ X4 ) )
     => ( ( vimage_rat_rat @ F2 @ ( collect_rat @ P ) )
        = ( collect_rat @ Q ) ) ) ).

% vimage_Collect
thf(fact_760_vimage__Collect,axiom,
    ! [P: rat > $o,F2: nat > rat,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ ( F2 @ X4 ) )
          = ( Q @ X4 ) )
     => ( ( vimage_nat_rat @ F2 @ ( collect_rat @ P ) )
        = ( collect_nat @ Q ) ) ) ).

% vimage_Collect
thf(fact_761_vimage__Collect,axiom,
    ! [P: nat > $o,F2: rat > nat,Q: rat > $o] :
      ( ! [X4: rat] :
          ( ( P @ ( F2 @ X4 ) )
          = ( Q @ X4 ) )
     => ( ( vimage_rat_nat @ F2 @ ( collect_nat @ P ) )
        = ( collect_rat @ Q ) ) ) ).

% vimage_Collect
thf(fact_762_vimage__Collect,axiom,
    ! [P: nat > $o,F2: nat > nat,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ ( F2 @ X4 ) )
          = ( Q @ X4 ) )
     => ( ( vimage_nat_nat @ F2 @ ( collect_nat @ P ) )
        = ( collect_nat @ Q ) ) ) ).

% vimage_Collect
thf(fact_763_UNIV__witness,axiom,
    ? [X4: c] : ( member_c @ X4 @ top_top_set_c ) ).

% UNIV_witness
thf(fact_764_UNIV__witness,axiom,
    ? [X4: b] : ( member_b @ X4 @ top_top_set_b ) ).

% UNIV_witness
thf(fact_765_UNIV__witness,axiom,
    ? [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).

% UNIV_witness
thf(fact_766_UNIV__witness,axiom,
    ? [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_767_UNIV__witness,axiom,
    ? [X4: rat] : ( member_rat @ X4 @ top_top_set_rat ) ).

% UNIV_witness
thf(fact_768_UNIV__eq__I,axiom,
    ! [A2: set_c] :
      ( ! [X4: c] : ( member_c @ X4 @ A2 )
     => ( top_top_set_c = A2 ) ) ).

% UNIV_eq_I
thf(fact_769_UNIV__eq__I,axiom,
    ! [A2: set_b] :
      ( ! [X4: b] : ( member_b @ X4 @ A2 )
     => ( top_top_set_b = A2 ) ) ).

% UNIV_eq_I
thf(fact_770_UNIV__eq__I,axiom,
    ! [A2: set_a] :
      ( ! [X4: a] : ( member_a @ X4 @ A2 )
     => ( top_top_set_a = A2 ) ) ).

% UNIV_eq_I
thf(fact_771_UNIV__eq__I,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] : ( member_nat @ X4 @ A2 )
     => ( top_top_set_nat = A2 ) ) ).

% UNIV_eq_I
thf(fact_772_UNIV__eq__I,axiom,
    ! [A2: set_rat] :
      ( ! [X4: rat] : ( member_rat @ X4 @ A2 )
     => ( top_top_set_rat = A2 ) ) ).

% UNIV_eq_I
thf(fact_773_vimageI2,axiom,
    ! [F2: rat > rat,A: rat,A2: set_rat] :
      ( ( member_rat @ ( F2 @ A ) @ A2 )
     => ( member_rat @ A @ ( vimage_rat_rat @ F2 @ A2 ) ) ) ).

% vimageI2
thf(fact_774_vimageI2,axiom,
    ! [F2: nat > rat,A: nat,A2: set_rat] :
      ( ( member_rat @ ( F2 @ A ) @ A2 )
     => ( member_nat @ A @ ( vimage_nat_rat @ F2 @ A2 ) ) ) ).

% vimageI2
thf(fact_775_vimageI2,axiom,
    ! [F2: c > rat,A: c,A2: set_rat] :
      ( ( member_rat @ ( F2 @ A ) @ A2 )
     => ( member_c @ A @ ( vimage_c_rat @ F2 @ A2 ) ) ) ).

% vimageI2
thf(fact_776_vimageI2,axiom,
    ! [F2: b > rat,A: b,A2: set_rat] :
      ( ( member_rat @ ( F2 @ A ) @ A2 )
     => ( member_b @ A @ ( vimage_b_rat @ F2 @ A2 ) ) ) ).

% vimageI2
thf(fact_777_vimageI2,axiom,
    ! [F2: a > rat,A: a,A2: set_rat] :
      ( ( member_rat @ ( F2 @ A ) @ A2 )
     => ( member_a @ A @ ( vimage_a_rat @ F2 @ A2 ) ) ) ).

% vimageI2
thf(fact_778_vimageI2,axiom,
    ! [F2: rat > nat,A: rat,A2: set_nat] :
      ( ( member_nat @ ( F2 @ A ) @ A2 )
     => ( member_rat @ A @ ( vimage_rat_nat @ F2 @ A2 ) ) ) ).

% vimageI2
thf(fact_779_vimageI2,axiom,
    ! [F2: nat > nat,A: nat,A2: set_nat] :
      ( ( member_nat @ ( F2 @ A ) @ A2 )
     => ( member_nat @ A @ ( vimage_nat_nat @ F2 @ A2 ) ) ) ).

% vimageI2
thf(fact_780_vimageI2,axiom,
    ! [F2: c > nat,A: c,A2: set_nat] :
      ( ( member_nat @ ( F2 @ A ) @ A2 )
     => ( member_c @ A @ ( vimage_c_nat @ F2 @ A2 ) ) ) ).

% vimageI2
thf(fact_781_vimageI2,axiom,
    ! [F2: b > nat,A: b,A2: set_nat] :
      ( ( member_nat @ ( F2 @ A ) @ A2 )
     => ( member_b @ A @ ( vimage_b_nat @ F2 @ A2 ) ) ) ).

% vimageI2
thf(fact_782_vimageI2,axiom,
    ! [F2: a > nat,A: a,A2: set_nat] :
      ( ( member_nat @ ( F2 @ A ) @ A2 )
     => ( member_a @ A @ ( vimage_a_nat @ F2 @ A2 ) ) ) ).

% vimageI2
thf(fact_783_vimageE,axiom,
    ! [A: rat,F2: rat > rat,B2: set_rat] :
      ( ( member_rat @ A @ ( vimage_rat_rat @ F2 @ B2 ) )
     => ( member_rat @ ( F2 @ A ) @ B2 ) ) ).

% vimageE
thf(fact_784_vimageE,axiom,
    ! [A: rat,F2: rat > nat,B2: set_nat] :
      ( ( member_rat @ A @ ( vimage_rat_nat @ F2 @ B2 ) )
     => ( member_nat @ ( F2 @ A ) @ B2 ) ) ).

% vimageE
thf(fact_785_vimageE,axiom,
    ! [A: rat,F2: rat > c,B2: set_c] :
      ( ( member_rat @ A @ ( vimage_rat_c @ F2 @ B2 ) )
     => ( member_c @ ( F2 @ A ) @ B2 ) ) ).

% vimageE
thf(fact_786_vimageE,axiom,
    ! [A: rat,F2: rat > b,B2: set_b] :
      ( ( member_rat @ A @ ( vimage_rat_b @ F2 @ B2 ) )
     => ( member_b @ ( F2 @ A ) @ B2 ) ) ).

% vimageE
thf(fact_787_vimageE,axiom,
    ! [A: rat,F2: rat > a,B2: set_a] :
      ( ( member_rat @ A @ ( vimage_rat_a @ F2 @ B2 ) )
     => ( member_a @ ( F2 @ A ) @ B2 ) ) ).

% vimageE
thf(fact_788_vimageE,axiom,
    ! [A: nat,F2: nat > rat,B2: set_rat] :
      ( ( member_nat @ A @ ( vimage_nat_rat @ F2 @ B2 ) )
     => ( member_rat @ ( F2 @ A ) @ B2 ) ) ).

% vimageE
thf(fact_789_vimageE,axiom,
    ! [A: nat,F2: nat > nat,B2: set_nat] :
      ( ( member_nat @ A @ ( vimage_nat_nat @ F2 @ B2 ) )
     => ( member_nat @ ( F2 @ A ) @ B2 ) ) ).

% vimageE
thf(fact_790_vimageE,axiom,
    ! [A: nat,F2: nat > c,B2: set_c] :
      ( ( member_nat @ A @ ( vimage_nat_c @ F2 @ B2 ) )
     => ( member_c @ ( F2 @ A ) @ B2 ) ) ).

% vimageE
thf(fact_791_vimageE,axiom,
    ! [A: nat,F2: nat > b,B2: set_b] :
      ( ( member_nat @ A @ ( vimage_nat_b @ F2 @ B2 ) )
     => ( member_b @ ( F2 @ A ) @ B2 ) ) ).

% vimageE
thf(fact_792_vimageE,axiom,
    ! [A: nat,F2: nat > a,B2: set_a] :
      ( ( member_nat @ A @ ( vimage_nat_a @ F2 @ B2 ) )
     => ( member_a @ ( F2 @ A ) @ B2 ) ) ).

% vimageE
thf(fact_793_vimageD,axiom,
    ! [A: rat,F2: rat > rat,A2: set_rat] :
      ( ( member_rat @ A @ ( vimage_rat_rat @ F2 @ A2 ) )
     => ( member_rat @ ( F2 @ A ) @ A2 ) ) ).

% vimageD
thf(fact_794_vimageD,axiom,
    ! [A: rat,F2: rat > nat,A2: set_nat] :
      ( ( member_rat @ A @ ( vimage_rat_nat @ F2 @ A2 ) )
     => ( member_nat @ ( F2 @ A ) @ A2 ) ) ).

% vimageD
thf(fact_795_vimageD,axiom,
    ! [A: rat,F2: rat > c,A2: set_c] :
      ( ( member_rat @ A @ ( vimage_rat_c @ F2 @ A2 ) )
     => ( member_c @ ( F2 @ A ) @ A2 ) ) ).

% vimageD
thf(fact_796_vimageD,axiom,
    ! [A: rat,F2: rat > b,A2: set_b] :
      ( ( member_rat @ A @ ( vimage_rat_b @ F2 @ A2 ) )
     => ( member_b @ ( F2 @ A ) @ A2 ) ) ).

% vimageD
thf(fact_797_vimageD,axiom,
    ! [A: rat,F2: rat > a,A2: set_a] :
      ( ( member_rat @ A @ ( vimage_rat_a @ F2 @ A2 ) )
     => ( member_a @ ( F2 @ A ) @ A2 ) ) ).

% vimageD
thf(fact_798_vimageD,axiom,
    ! [A: nat,F2: nat > rat,A2: set_rat] :
      ( ( member_nat @ A @ ( vimage_nat_rat @ F2 @ A2 ) )
     => ( member_rat @ ( F2 @ A ) @ A2 ) ) ).

% vimageD
thf(fact_799_vimageD,axiom,
    ! [A: nat,F2: nat > nat,A2: set_nat] :
      ( ( member_nat @ A @ ( vimage_nat_nat @ F2 @ A2 ) )
     => ( member_nat @ ( F2 @ A ) @ A2 ) ) ).

% vimageD
thf(fact_800_vimageD,axiom,
    ! [A: nat,F2: nat > c,A2: set_c] :
      ( ( member_nat @ A @ ( vimage_nat_c @ F2 @ A2 ) )
     => ( member_c @ ( F2 @ A ) @ A2 ) ) ).

% vimageD
thf(fact_801_vimageD,axiom,
    ! [A: nat,F2: nat > b,A2: set_b] :
      ( ( member_nat @ A @ ( vimage_nat_b @ F2 @ A2 ) )
     => ( member_b @ ( F2 @ A ) @ A2 ) ) ).

% vimageD
thf(fact_802_vimageD,axiom,
    ! [A: nat,F2: nat > a,A2: set_a] :
      ( ( member_nat @ A @ ( vimage_nat_a @ F2 @ A2 ) )
     => ( member_a @ ( F2 @ A ) @ A2 ) ) ).

% vimageD
thf(fact_803_surj__image__vimage__eq,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( image_nat_nat @ F2 @ ( vimage_nat_nat @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_804_surj__image__vimage__eq,axiom,
    ! [F2: nat > rat,A2: set_rat] :
      ( ( ( image_nat_rat @ F2 @ top_top_set_nat )
        = top_top_set_rat )
     => ( ( image_nat_rat @ F2 @ ( vimage_nat_rat @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_805_surj__image__vimage__eq,axiom,
    ! [F2: rat > nat,A2: set_nat] :
      ( ( ( image_rat_nat @ F2 @ top_top_set_rat )
        = top_top_set_nat )
     => ( ( image_rat_nat @ F2 @ ( vimage_rat_nat @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_806_surj__image__vimage__eq,axiom,
    ! [F2: rat > rat,A2: set_rat] :
      ( ( ( image_rat_rat @ F2 @ top_top_set_rat )
        = top_top_set_rat )
     => ( ( image_rat_rat @ F2 @ ( vimage_rat_rat @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_807_surj__image__vimage__eq,axiom,
    ! [F2: nat > c,A2: set_c] :
      ( ( ( image_nat_c @ F2 @ top_top_set_nat )
        = top_top_set_c )
     => ( ( image_nat_c @ F2 @ ( vimage_nat_c @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_808_surj__image__vimage__eq,axiom,
    ! [F2: nat > b,A2: set_b] :
      ( ( ( image_nat_b @ F2 @ top_top_set_nat )
        = top_top_set_b )
     => ( ( image_nat_b @ F2 @ ( vimage_nat_b @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_809_surj__image__vimage__eq,axiom,
    ! [F2: nat > a,A2: set_a] :
      ( ( ( image_nat_a @ F2 @ top_top_set_nat )
        = top_top_set_a )
     => ( ( image_nat_a @ F2 @ ( vimage_nat_a @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_810_surj__image__vimage__eq,axiom,
    ! [F2: rat > c,A2: set_c] :
      ( ( ( image_rat_c @ F2 @ top_top_set_rat )
        = top_top_set_c )
     => ( ( image_rat_c @ F2 @ ( vimage_rat_c @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_811_surj__image__vimage__eq,axiom,
    ! [F2: rat > b,A2: set_b] :
      ( ( ( image_rat_b @ F2 @ top_top_set_rat )
        = top_top_set_b )
     => ( ( image_rat_b @ F2 @ ( vimage_rat_b @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_812_surj__image__vimage__eq,axiom,
    ! [F2: rat > a,A2: set_a] :
      ( ( ( image_rat_a @ F2 @ top_top_set_rat )
        = top_top_set_a )
     => ( ( image_rat_a @ F2 @ ( vimage_rat_a @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_813_uminus__Inf,axiom,
    ! [A2: set_set_rat] :
      ( ( uminus2201863774496077783et_rat @ ( comple4298007329820168263et_rat @ A2 ) )
      = ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ uminus2201863774496077783et_rat @ A2 ) ) ) ).

% uminus_Inf
thf(fact_814_uminus__Inf,axiom,
    ! [A2: set_set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( comple7806235888213564991et_nat @ A2 ) )
      = ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ uminus5710092332889474511et_nat @ A2 ) ) ) ).

% uminus_Inf
thf(fact_815_uminus__Sup,axiom,
    ! [A2: set_set_rat] :
      ( ( uminus2201863774496077783et_rat @ ( comple3890839924845867745et_rat @ A2 ) )
      = ( comple4298007329820168263et_rat @ ( image_3939399684171694371et_rat @ uminus2201863774496077783et_rat @ A2 ) ) ) ).

% uminus_Sup
thf(fact_816_uminus__Sup,axiom,
    ! [A2: set_set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( comple7399068483239264473et_nat @ A2 ) )
      = ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ uminus5710092332889474511et_nat @ A2 ) ) ) ).

% uminus_Sup
thf(fact_817_vimage__comp,axiom,
    ! [F2: c > b,G2: b > a,X2: set_a] :
      ( ( vimage_c_b @ F2 @ ( vimage_b_a @ G2 @ X2 ) )
      = ( vimage_c_a @ ( comp_b_a_c @ G2 @ F2 ) @ X2 ) ) ).

% vimage_comp
thf(fact_818_vimage__comp,axiom,
    ! [F2: c > c,G2: c > a,X2: set_a] :
      ( ( vimage_c_c @ F2 @ ( vimage_c_a @ G2 @ X2 ) )
      = ( vimage_c_a @ ( comp_c_a_c @ G2 @ F2 ) @ X2 ) ) ).

% vimage_comp
thf(fact_819_vimage__comp,axiom,
    ! [F2: c > b,G2: b > b,X2: set_b] :
      ( ( vimage_c_b @ F2 @ ( vimage_b_b @ G2 @ X2 ) )
      = ( vimage_c_b @ ( comp_b_b_c @ G2 @ F2 ) @ X2 ) ) ).

% vimage_comp
thf(fact_820_vimage__comp,axiom,
    ! [F2: b > b,G2: b > a,X2: set_a] :
      ( ( vimage_b_b @ F2 @ ( vimage_b_a @ G2 @ X2 ) )
      = ( vimage_b_a @ ( comp_b_a_b @ G2 @ F2 ) @ X2 ) ) ).

% vimage_comp
thf(fact_821_vimage__comp,axiom,
    ! [F2: c > a,G2: a > a,X2: set_a] :
      ( ( vimage_c_a @ F2 @ ( vimage_a_a @ G2 @ X2 ) )
      = ( vimage_c_a @ ( comp_a_a_c @ G2 @ F2 ) @ X2 ) ) ).

% vimage_comp
thf(fact_822_vimage__comp,axiom,
    ! [F2: b > a,G2: a > a,X2: set_a] :
      ( ( vimage_b_a @ F2 @ ( vimage_a_a @ G2 @ X2 ) )
      = ( vimage_b_a @ ( comp_a_a_b @ G2 @ F2 ) @ X2 ) ) ).

% vimage_comp
thf(fact_823_vimage__comp,axiom,
    ! [F2: c > c,G2: c > b,X2: set_b] :
      ( ( vimage_c_c @ F2 @ ( vimage_c_b @ G2 @ X2 ) )
      = ( vimage_c_b @ ( comp_c_b_c @ G2 @ F2 ) @ X2 ) ) ).

% vimage_comp
thf(fact_824_vimage__comp,axiom,
    ! [F2: b > c,G2: c > b,X2: set_b] :
      ( ( vimage_b_c @ F2 @ ( vimage_c_b @ G2 @ X2 ) )
      = ( vimage_b_b @ ( comp_c_b_b @ G2 @ F2 ) @ X2 ) ) ).

% vimage_comp
thf(fact_825_vimage__comp,axiom,
    ! [F2: b > c,G2: c > a,X2: set_a] :
      ( ( vimage_b_c @ F2 @ ( vimage_c_a @ G2 @ X2 ) )
      = ( vimage_b_a @ ( comp_c_a_b @ G2 @ F2 ) @ X2 ) ) ).

% vimage_comp
thf(fact_826_vimage__comp,axiom,
    ! [F2: a > c,G2: c > a,X2: set_a] :
      ( ( vimage_a_c @ F2 @ ( vimage_c_a @ G2 @ X2 ) )
      = ( vimage_a_a @ ( comp_c_a_a @ G2 @ F2 ) @ X2 ) ) ).

% vimage_comp
thf(fact_827_set_Ocompositionality,axiom,
    ! [F2: c > b,G2: b > a,Set: set_a] :
      ( ( vimage_c_b @ F2 @ ( vimage_b_a @ G2 @ Set ) )
      = ( vimage_c_a @ ( comp_b_a_c @ G2 @ F2 ) @ Set ) ) ).

% set.compositionality
thf(fact_828_set_Ocompositionality,axiom,
    ! [F2: c > c,G2: c > a,Set: set_a] :
      ( ( vimage_c_c @ F2 @ ( vimage_c_a @ G2 @ Set ) )
      = ( vimage_c_a @ ( comp_c_a_c @ G2 @ F2 ) @ Set ) ) ).

% set.compositionality
thf(fact_829_set_Ocompositionality,axiom,
    ! [F2: c > b,G2: b > b,Set: set_b] :
      ( ( vimage_c_b @ F2 @ ( vimage_b_b @ G2 @ Set ) )
      = ( vimage_c_b @ ( comp_b_b_c @ G2 @ F2 ) @ Set ) ) ).

% set.compositionality
thf(fact_830_set_Ocompositionality,axiom,
    ! [F2: b > b,G2: b > a,Set: set_a] :
      ( ( vimage_b_b @ F2 @ ( vimage_b_a @ G2 @ Set ) )
      = ( vimage_b_a @ ( comp_b_a_b @ G2 @ F2 ) @ Set ) ) ).

% set.compositionality
thf(fact_831_set_Ocompositionality,axiom,
    ! [F2: c > a,G2: a > a,Set: set_a] :
      ( ( vimage_c_a @ F2 @ ( vimage_a_a @ G2 @ Set ) )
      = ( vimage_c_a @ ( comp_a_a_c @ G2 @ F2 ) @ Set ) ) ).

% set.compositionality
thf(fact_832_set_Ocompositionality,axiom,
    ! [F2: b > a,G2: a > a,Set: set_a] :
      ( ( vimage_b_a @ F2 @ ( vimage_a_a @ G2 @ Set ) )
      = ( vimage_b_a @ ( comp_a_a_b @ G2 @ F2 ) @ Set ) ) ).

% set.compositionality
thf(fact_833_set_Ocompositionality,axiom,
    ! [F2: c > c,G2: c > b,Set: set_b] :
      ( ( vimage_c_c @ F2 @ ( vimage_c_b @ G2 @ Set ) )
      = ( vimage_c_b @ ( comp_c_b_c @ G2 @ F2 ) @ Set ) ) ).

% set.compositionality
thf(fact_834_set_Ocompositionality,axiom,
    ! [F2: b > c,G2: c > b,Set: set_b] :
      ( ( vimage_b_c @ F2 @ ( vimage_c_b @ G2 @ Set ) )
      = ( vimage_b_b @ ( comp_c_b_b @ G2 @ F2 ) @ Set ) ) ).

% set.compositionality
thf(fact_835_set_Ocompositionality,axiom,
    ! [F2: b > c,G2: c > a,Set: set_a] :
      ( ( vimage_b_c @ F2 @ ( vimage_c_a @ G2 @ Set ) )
      = ( vimage_b_a @ ( comp_c_a_b @ G2 @ F2 ) @ Set ) ) ).

% set.compositionality
thf(fact_836_set_Ocompositionality,axiom,
    ! [F2: a > c,G2: c > a,Set: set_a] :
      ( ( vimage_a_c @ F2 @ ( vimage_c_a @ G2 @ Set ) )
      = ( vimage_a_a @ ( comp_c_a_a @ G2 @ F2 ) @ Set ) ) ).

% set.compositionality
thf(fact_837_INF__cong,axiom,
    ! [A2: set_rat,B2: set_rat,C2: rat > nat,D2: rat > nat] :
      ( ( A2 = B2 )
     => ( ! [X4: rat] :
            ( ( member_rat @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( complete_Inf_Inf_nat @ ( image_rat_nat @ C2 @ A2 ) )
          = ( complete_Inf_Inf_nat @ ( image_rat_nat @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_838_INF__cong,axiom,
    ! [A2: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat] :
      ( ( A2 = B2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ C2 @ A2 ) )
          = ( complete_Inf_Inf_nat @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_839_INF__cong,axiom,
    ! [A2: set_c,B2: set_c,C2: c > nat,D2: c > nat] :
      ( ( A2 = B2 )
     => ( ! [X4: c] :
            ( ( member_c @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( complete_Inf_Inf_nat @ ( image_c_nat @ C2 @ A2 ) )
          = ( complete_Inf_Inf_nat @ ( image_c_nat @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_840_INF__cong,axiom,
    ! [A2: set_b,B2: set_b,C2: b > nat,D2: b > nat] :
      ( ( A2 = B2 )
     => ( ! [X4: b] :
            ( ( member_b @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( complete_Inf_Inf_nat @ ( image_b_nat @ C2 @ A2 ) )
          = ( complete_Inf_Inf_nat @ ( image_b_nat @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_841_INF__cong,axiom,
    ! [A2: set_a,B2: set_a,C2: a > nat,D2: a > nat] :
      ( ( A2 = B2 )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( complete_Inf_Inf_nat @ ( image_a_nat @ C2 @ A2 ) )
          = ( complete_Inf_Inf_nat @ ( image_a_nat @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_842_INF__cong,axiom,
    ! [A2: set_rat,B2: set_rat,C2: rat > set_rat,D2: rat > set_rat] :
      ( ( A2 = B2 )
     => ( ! [X4: rat] :
            ( ( member_rat @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( comple4298007329820168263et_rat @ ( image_rat_set_rat @ C2 @ A2 ) )
          = ( comple4298007329820168263et_rat @ ( image_rat_set_rat @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_843_INF__cong,axiom,
    ! [A2: set_nat,B2: set_nat,C2: nat > set_rat,D2: nat > set_rat] :
      ( ( A2 = B2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( comple4298007329820168263et_rat @ ( image_nat_set_rat @ C2 @ A2 ) )
          = ( comple4298007329820168263et_rat @ ( image_nat_set_rat @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_844_INF__cong,axiom,
    ! [A2: set_c,B2: set_c,C2: c > set_rat,D2: c > set_rat] :
      ( ( A2 = B2 )
     => ( ! [X4: c] :
            ( ( member_c @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( comple4298007329820168263et_rat @ ( image_c_set_rat @ C2 @ A2 ) )
          = ( comple4298007329820168263et_rat @ ( image_c_set_rat @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_845_INF__cong,axiom,
    ! [A2: set_b,B2: set_b,C2: b > set_rat,D2: b > set_rat] :
      ( ( A2 = B2 )
     => ( ! [X4: b] :
            ( ( member_b @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( comple4298007329820168263et_rat @ ( image_b_set_rat @ C2 @ A2 ) )
          = ( comple4298007329820168263et_rat @ ( image_b_set_rat @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_846_INF__cong,axiom,
    ! [A2: set_a,B2: set_a,C2: a > set_rat,D2: a > set_rat] :
      ( ( A2 = B2 )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( comple4298007329820168263et_rat @ ( image_a_set_rat @ C2 @ A2 ) )
          = ( comple4298007329820168263et_rat @ ( image_a_set_rat @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_847_SUP__cong,axiom,
    ! [A2: set_a,B2: set_a,C2: a > set_nat,D2: a > set_nat] :
      ( ( A2 = B2 )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ C2 @ A2 ) )
          = ( comple7399068483239264473et_nat @ ( image_a_set_nat @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_848_SUP__cong,axiom,
    ! [A2: set_rat,B2: set_rat,C2: rat > nat,D2: rat > nat] :
      ( ( A2 = B2 )
     => ( ! [X4: rat] :
            ( ( member_rat @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( complete_Sup_Sup_nat @ ( image_rat_nat @ C2 @ A2 ) )
          = ( complete_Sup_Sup_nat @ ( image_rat_nat @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_849_SUP__cong,axiom,
    ! [A2: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat] :
      ( ( A2 = B2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C2 @ A2 ) )
          = ( complete_Sup_Sup_nat @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_850_SUP__cong,axiom,
    ! [A2: set_c,B2: set_c,C2: c > nat,D2: c > nat] :
      ( ( A2 = B2 )
     => ( ! [X4: c] :
            ( ( member_c @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( complete_Sup_Sup_nat @ ( image_c_nat @ C2 @ A2 ) )
          = ( complete_Sup_Sup_nat @ ( image_c_nat @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_851_SUP__cong,axiom,
    ! [A2: set_b,B2: set_b,C2: b > nat,D2: b > nat] :
      ( ( A2 = B2 )
     => ( ! [X4: b] :
            ( ( member_b @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( complete_Sup_Sup_nat @ ( image_b_nat @ C2 @ A2 ) )
          = ( complete_Sup_Sup_nat @ ( image_b_nat @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_852_SUP__cong,axiom,
    ! [A2: set_a,B2: set_a,C2: a > nat,D2: a > nat] :
      ( ( A2 = B2 )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ B2 )
           => ( ( C2 @ X4 )
              = ( D2 @ X4 ) ) )
       => ( ( complete_Sup_Sup_nat @ ( image_a_nat @ C2 @ A2 ) )
          = ( complete_Sup_Sup_nat @ ( image_a_nat @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_853_surj__def,axiom,
    ! [F2: nat > nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
      = ( ! [Y: nat] :
          ? [X: nat] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_854_surj__def,axiom,
    ! [F2: nat > rat] :
      ( ( ( image_nat_rat @ F2 @ top_top_set_nat )
        = top_top_set_rat )
      = ( ! [Y: rat] :
          ? [X: nat] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_855_surj__def,axiom,
    ! [F2: rat > nat] :
      ( ( ( image_rat_nat @ F2 @ top_top_set_rat )
        = top_top_set_nat )
      = ( ! [Y: nat] :
          ? [X: rat] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_856_surj__def,axiom,
    ! [F2: rat > rat] :
      ( ( ( image_rat_rat @ F2 @ top_top_set_rat )
        = top_top_set_rat )
      = ( ! [Y: rat] :
          ? [X: rat] :
            ( Y
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_857_surjI,axiom,
    ! [G2: nat > nat,F2: nat > nat] :
      ( ! [X4: nat] :
          ( ( G2 @ ( F2 @ X4 ) )
          = X4 )
     => ( ( image_nat_nat @ G2 @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_858_surjI,axiom,
    ! [G2: nat > rat,F2: rat > nat] :
      ( ! [X4: rat] :
          ( ( G2 @ ( F2 @ X4 ) )
          = X4 )
     => ( ( image_nat_rat @ G2 @ top_top_set_nat )
        = top_top_set_rat ) ) ).

% surjI
thf(fact_859_surjI,axiom,
    ! [G2: rat > nat,F2: nat > rat] :
      ( ! [X4: nat] :
          ( ( G2 @ ( F2 @ X4 ) )
          = X4 )
     => ( ( image_rat_nat @ G2 @ top_top_set_rat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_860_surjI,axiom,
    ! [G2: rat > rat,F2: rat > rat] :
      ( ! [X4: rat] :
          ( ( G2 @ ( F2 @ X4 ) )
          = X4 )
     => ( ( image_rat_rat @ G2 @ top_top_set_rat )
        = top_top_set_rat ) ) ).

% surjI
thf(fact_861_surjE,axiom,
    ! [F2: nat > nat,Y3: nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ~ ! [X4: nat] :
            ( Y3
           != ( F2 @ X4 ) ) ) ).

% surjE
thf(fact_862_surjE,axiom,
    ! [F2: nat > rat,Y3: rat] :
      ( ( ( image_nat_rat @ F2 @ top_top_set_nat )
        = top_top_set_rat )
     => ~ ! [X4: nat] :
            ( Y3
           != ( F2 @ X4 ) ) ) ).

% surjE
thf(fact_863_surjE,axiom,
    ! [F2: rat > nat,Y3: nat] :
      ( ( ( image_rat_nat @ F2 @ top_top_set_rat )
        = top_top_set_nat )
     => ~ ! [X4: rat] :
            ( Y3
           != ( F2 @ X4 ) ) ) ).

% surjE
thf(fact_864_surjE,axiom,
    ! [F2: rat > rat,Y3: rat] :
      ( ( ( image_rat_rat @ F2 @ top_top_set_rat )
        = top_top_set_rat )
     => ~ ! [X4: rat] :
            ( Y3
           != ( F2 @ X4 ) ) ) ).

% surjE
thf(fact_865_surjD,axiom,
    ! [F2: nat > nat,Y3: nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ? [X4: nat] :
          ( Y3
          = ( F2 @ X4 ) ) ) ).

% surjD
thf(fact_866_surjD,axiom,
    ! [F2: nat > rat,Y3: rat] :
      ( ( ( image_nat_rat @ F2 @ top_top_set_nat )
        = top_top_set_rat )
     => ? [X4: nat] :
          ( Y3
          = ( F2 @ X4 ) ) ) ).

% surjD
thf(fact_867_surjD,axiom,
    ! [F2: rat > nat,Y3: nat] :
      ( ( ( image_rat_nat @ F2 @ top_top_set_rat )
        = top_top_set_nat )
     => ? [X4: rat] :
          ( Y3
          = ( F2 @ X4 ) ) ) ).

% surjD
thf(fact_868_surjD,axiom,
    ! [F2: rat > rat,Y3: rat] :
      ( ( ( image_rat_rat @ F2 @ top_top_set_rat )
        = top_top_set_rat )
     => ? [X4: rat] :
          ( Y3
          = ( F2 @ X4 ) ) ) ).

% surjD
thf(fact_869_rangeI,axiom,
    ! [F2: nat > rat,X2: nat] : ( member_rat @ ( F2 @ X2 ) @ ( image_nat_rat @ F2 @ top_top_set_nat ) ) ).

% rangeI
thf(fact_870_range__eqI,axiom,
    ! [B: rat,F2: nat > rat,X2: nat] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( member_rat @ B @ ( image_nat_rat @ F2 @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_871_DEADID_Oin__rel,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [A4: nat,B3: nat] :
        ? [Z4: nat] :
          ( ( member_nat @ Z4 @ top_top_set_nat )
          & ( ( id_nat @ Z4 )
            = A4 )
          & ( ( id_nat @ Z4 )
            = B3 ) ) ) ) ).

% DEADID.in_rel
thf(fact_872_DEADID_Oin__rel,axiom,
    ( ( ^ [Y2: rat,Z: rat] : ( Y2 = Z ) )
    = ( ^ [A4: rat,B3: rat] :
        ? [Z4: rat] :
          ( ( member_rat @ Z4 @ top_top_set_rat )
          & ( ( id_rat @ Z4 )
            = A4 )
          & ( ( id_rat @ Z4 )
            = B3 ) ) ) ) ).

% DEADID.in_rel
thf(fact_873_fun_Oset__map,axiom,
    ! [F2: b > a,V: c > b] :
      ( ( image_c_a @ ( comp_b_a_c @ F2 @ V ) @ top_top_set_c )
      = ( image_b_a @ F2 @ ( image_c_b @ V @ top_top_set_c ) ) ) ).

% fun.set_map
thf(fact_874_fun_Oset__map,axiom,
    ! [F2: c > a,V: c > c] :
      ( ( image_c_a @ ( comp_c_a_c @ F2 @ V ) @ top_top_set_c )
      = ( image_c_a @ F2 @ ( image_c_c @ V @ top_top_set_c ) ) ) ).

% fun.set_map
thf(fact_875_fun_Oset__map,axiom,
    ! [F2: b > b,V: c > b] :
      ( ( image_c_b @ ( comp_b_b_c @ F2 @ V ) @ top_top_set_c )
      = ( image_b_b @ F2 @ ( image_c_b @ V @ top_top_set_c ) ) ) ).

% fun.set_map
thf(fact_876_fun_Oset__map,axiom,
    ! [F2: b > a,V: b > b] :
      ( ( image_b_a @ ( comp_b_a_b @ F2 @ V ) @ top_top_set_b )
      = ( image_b_a @ F2 @ ( image_b_b @ V @ top_top_set_b ) ) ) ).

% fun.set_map
thf(fact_877_fun_Oset__map,axiom,
    ! [F2: a > a,V: c > a] :
      ( ( image_c_a @ ( comp_a_a_c @ F2 @ V ) @ top_top_set_c )
      = ( image_a_a @ F2 @ ( image_c_a @ V @ top_top_set_c ) ) ) ).

% fun.set_map
thf(fact_878_fun_Oset__map,axiom,
    ! [F2: a > a,V: b > a] :
      ( ( image_b_a @ ( comp_a_a_b @ F2 @ V ) @ top_top_set_b )
      = ( image_a_a @ F2 @ ( image_b_a @ V @ top_top_set_b ) ) ) ).

% fun.set_map
thf(fact_879_fun_Oset__map,axiom,
    ! [F2: rat > rat,V: nat > rat] :
      ( ( image_nat_rat @ ( comp_rat_rat_nat @ F2 @ V ) @ top_top_set_nat )
      = ( image_rat_rat @ F2 @ ( image_nat_rat @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_880_fun_Oset__map,axiom,
    ! [F2: nat > rat,V: nat > nat] :
      ( ( image_nat_rat @ ( comp_nat_rat_nat @ F2 @ V ) @ top_top_set_nat )
      = ( image_nat_rat @ F2 @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_881_fun_Oset__map,axiom,
    ! [F2: nat > rat,V: rat > nat] :
      ( ( image_rat_rat @ ( comp_nat_rat_rat @ F2 @ V ) @ top_top_set_rat )
      = ( image_nat_rat @ F2 @ ( image_rat_nat @ V @ top_top_set_rat ) ) ) ).

% fun.set_map
thf(fact_882_fun_Omap__cong,axiom,
    ! [X2: c > b,Ya: c > b,F2: b > a,G2: b > a] :
      ( ( X2 = Ya )
     => ( ! [Z3: b] :
            ( ( member_b @ Z3 @ ( image_c_b @ Ya @ top_top_set_c ) )
           => ( ( F2 @ Z3 )
              = ( G2 @ Z3 ) ) )
       => ( ( comp_b_a_c @ F2 @ X2 )
          = ( comp_b_a_c @ G2 @ Ya ) ) ) ) ).

% fun.map_cong
thf(fact_883_fun_Omap__cong,axiom,
    ! [X2: c > c,Ya: c > c,F2: c > a,G2: c > a] :
      ( ( X2 = Ya )
     => ( ! [Z3: c] :
            ( ( member_c @ Z3 @ ( image_c_c @ Ya @ top_top_set_c ) )
           => ( ( F2 @ Z3 )
              = ( G2 @ Z3 ) ) )
       => ( ( comp_c_a_c @ F2 @ X2 )
          = ( comp_c_a_c @ G2 @ Ya ) ) ) ) ).

% fun.map_cong
thf(fact_884_fun_Omap__cong,axiom,
    ! [X2: c > b,Ya: c > b,F2: b > b,G2: b > b] :
      ( ( X2 = Ya )
     => ( ! [Z3: b] :
            ( ( member_b @ Z3 @ ( image_c_b @ Ya @ top_top_set_c ) )
           => ( ( F2 @ Z3 )
              = ( G2 @ Z3 ) ) )
       => ( ( comp_b_b_c @ F2 @ X2 )
          = ( comp_b_b_c @ G2 @ Ya ) ) ) ) ).

% fun.map_cong
thf(fact_885_fun_Omap__cong,axiom,
    ! [X2: b > b,Ya: b > b,F2: b > a,G2: b > a] :
      ( ( X2 = Ya )
     => ( ! [Z3: b] :
            ( ( member_b @ Z3 @ ( image_b_b @ Ya @ top_top_set_b ) )
           => ( ( F2 @ Z3 )
              = ( G2 @ Z3 ) ) )
       => ( ( comp_b_a_b @ F2 @ X2 )
          = ( comp_b_a_b @ G2 @ Ya ) ) ) ) ).

% fun.map_cong
thf(fact_886_fun_Omap__cong,axiom,
    ! [X2: c > a,Ya: c > a,F2: a > a,G2: a > a] :
      ( ( X2 = Ya )
     => ( ! [Z3: a] :
            ( ( member_a @ Z3 @ ( image_c_a @ Ya @ top_top_set_c ) )
           => ( ( F2 @ Z3 )
              = ( G2 @ Z3 ) ) )
       => ( ( comp_a_a_c @ F2 @ X2 )
          = ( comp_a_a_c @ G2 @ Ya ) ) ) ) ).

% fun.map_cong
thf(fact_887_fun_Omap__cong,axiom,
    ! [X2: b > a,Ya: b > a,F2: a > a,G2: a > a] :
      ( ( X2 = Ya )
     => ( ! [Z3: a] :
            ( ( member_a @ Z3 @ ( image_b_a @ Ya @ top_top_set_b ) )
           => ( ( F2 @ Z3 )
              = ( G2 @ Z3 ) ) )
       => ( ( comp_a_a_b @ F2 @ X2 )
          = ( comp_a_a_b @ G2 @ Ya ) ) ) ) ).

% fun.map_cong
thf(fact_888_fun_Omap__cong0,axiom,
    ! [X2: c > b,F2: b > a,G2: b > a] :
      ( ! [Z3: b] :
          ( ( member_b @ Z3 @ ( image_c_b @ X2 @ top_top_set_c ) )
         => ( ( F2 @ Z3 )
            = ( G2 @ Z3 ) ) )
     => ( ( comp_b_a_c @ F2 @ X2 )
        = ( comp_b_a_c @ G2 @ X2 ) ) ) ).

% fun.map_cong0
thf(fact_889_fun_Omap__cong0,axiom,
    ! [X2: c > c,F2: c > a,G2: c > a] :
      ( ! [Z3: c] :
          ( ( member_c @ Z3 @ ( image_c_c @ X2 @ top_top_set_c ) )
         => ( ( F2 @ Z3 )
            = ( G2 @ Z3 ) ) )
     => ( ( comp_c_a_c @ F2 @ X2 )
        = ( comp_c_a_c @ G2 @ X2 ) ) ) ).

% fun.map_cong0
thf(fact_890_fun_Omap__cong0,axiom,
    ! [X2: c > b,F2: b > b,G2: b > b] :
      ( ! [Z3: b] :
          ( ( member_b @ Z3 @ ( image_c_b @ X2 @ top_top_set_c ) )
         => ( ( F2 @ Z3 )
            = ( G2 @ Z3 ) ) )
     => ( ( comp_b_b_c @ F2 @ X2 )
        = ( comp_b_b_c @ G2 @ X2 ) ) ) ).

% fun.map_cong0
thf(fact_891_fun_Omap__cong0,axiom,
    ! [X2: b > b,F2: b > a,G2: b > a] :
      ( ! [Z3: b] :
          ( ( member_b @ Z3 @ ( image_b_b @ X2 @ top_top_set_b ) )
         => ( ( F2 @ Z3 )
            = ( G2 @ Z3 ) ) )
     => ( ( comp_b_a_b @ F2 @ X2 )
        = ( comp_b_a_b @ G2 @ X2 ) ) ) ).

% fun.map_cong0
thf(fact_892_fun_Omap__cong0,axiom,
    ! [X2: c > a,F2: a > a,G2: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( image_c_a @ X2 @ top_top_set_c ) )
         => ( ( F2 @ Z3 )
            = ( G2 @ Z3 ) ) )
     => ( ( comp_a_a_c @ F2 @ X2 )
        = ( comp_a_a_c @ G2 @ X2 ) ) ) ).

% fun.map_cong0
thf(fact_893_fun_Omap__cong0,axiom,
    ! [X2: b > a,F2: a > a,G2: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( image_b_a @ X2 @ top_top_set_b ) )
         => ( ( F2 @ Z3 )
            = ( G2 @ Z3 ) ) )
     => ( ( comp_a_a_b @ F2 @ X2 )
        = ( comp_a_a_b @ G2 @ X2 ) ) ) ).

% fun.map_cong0
thf(fact_894_fun_Oinj__map__strong,axiom,
    ! [X2: c > b,Xa: c > b,F2: b > a,Fa: b > a] :
      ( ! [Z3: b,Za: b] :
          ( ( member_b @ Z3 @ ( image_c_b @ X2 @ top_top_set_c ) )
         => ( ( member_b @ Za @ ( image_c_b @ Xa @ top_top_set_c ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( comp_b_a_c @ F2 @ X2 )
          = ( comp_b_a_c @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% fun.inj_map_strong
thf(fact_895_fun_Oinj__map__strong,axiom,
    ! [X2: c > c,Xa: c > c,F2: c > a,Fa: c > a] :
      ( ! [Z3: c,Za: c] :
          ( ( member_c @ Z3 @ ( image_c_c @ X2 @ top_top_set_c ) )
         => ( ( member_c @ Za @ ( image_c_c @ Xa @ top_top_set_c ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( comp_c_a_c @ F2 @ X2 )
          = ( comp_c_a_c @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% fun.inj_map_strong
thf(fact_896_fun_Oinj__map__strong,axiom,
    ! [X2: c > b,Xa: c > b,F2: b > b,Fa: b > b] :
      ( ! [Z3: b,Za: b] :
          ( ( member_b @ Z3 @ ( image_c_b @ X2 @ top_top_set_c ) )
         => ( ( member_b @ Za @ ( image_c_b @ Xa @ top_top_set_c ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( comp_b_b_c @ F2 @ X2 )
          = ( comp_b_b_c @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% fun.inj_map_strong
thf(fact_897_fun_Oinj__map__strong,axiom,
    ! [X2: b > b,Xa: b > b,F2: b > a,Fa: b > a] :
      ( ! [Z3: b,Za: b] :
          ( ( member_b @ Z3 @ ( image_b_b @ X2 @ top_top_set_b ) )
         => ( ( member_b @ Za @ ( image_b_b @ Xa @ top_top_set_b ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( comp_b_a_b @ F2 @ X2 )
          = ( comp_b_a_b @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% fun.inj_map_strong
thf(fact_898_fun_Oinj__map__strong,axiom,
    ! [X2: c > a,Xa: c > a,F2: a > a,Fa: a > a] :
      ( ! [Z3: a,Za: a] :
          ( ( member_a @ Z3 @ ( image_c_a @ X2 @ top_top_set_c ) )
         => ( ( member_a @ Za @ ( image_c_a @ Xa @ top_top_set_c ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( comp_a_a_c @ F2 @ X2 )
          = ( comp_a_a_c @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% fun.inj_map_strong
thf(fact_899_fun_Oinj__map__strong,axiom,
    ! [X2: b > a,Xa: b > a,F2: a > a,Fa: a > a] :
      ( ! [Z3: a,Za: a] :
          ( ( member_a @ Z3 @ ( image_b_a @ X2 @ top_top_set_b ) )
         => ( ( member_a @ Za @ ( image_b_a @ Xa @ top_top_set_b ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( comp_a_a_b @ F2 @ X2 )
          = ( comp_a_a_b @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% fun.inj_map_strong
thf(fact_900_surj__fun__eq,axiom,
    ! [F2: c > b,X6: set_c,G1: b > a,G22: b > a] :
      ( ( ( image_c_b @ F2 @ X6 )
        = top_top_set_b )
     => ( ! [X4: c] :
            ( ( member_c @ X4 @ X6 )
           => ( ( comp_b_a_c @ G1 @ F2 @ X4 )
              = ( comp_b_a_c @ G22 @ F2 @ X4 ) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
thf(fact_901_surj__fun__eq,axiom,
    ! [F2: c > c,X6: set_c,G1: c > a,G22: c > a] :
      ( ( ( image_c_c @ F2 @ X6 )
        = top_top_set_c )
     => ( ! [X4: c] :
            ( ( member_c @ X4 @ X6 )
           => ( ( comp_c_a_c @ G1 @ F2 @ X4 )
              = ( comp_c_a_c @ G22 @ F2 @ X4 ) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
thf(fact_902_surj__fun__eq,axiom,
    ! [F2: c > b,X6: set_c,G1: b > b,G22: b > b] :
      ( ( ( image_c_b @ F2 @ X6 )
        = top_top_set_b )
     => ( ! [X4: c] :
            ( ( member_c @ X4 @ X6 )
           => ( ( comp_b_b_c @ G1 @ F2 @ X4 )
              = ( comp_b_b_c @ G22 @ F2 @ X4 ) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
thf(fact_903_surj__fun__eq,axiom,
    ! [F2: b > b,X6: set_b,G1: b > a,G22: b > a] :
      ( ( ( image_b_b @ F2 @ X6 )
        = top_top_set_b )
     => ( ! [X4: b] :
            ( ( member_b @ X4 @ X6 )
           => ( ( comp_b_a_b @ G1 @ F2 @ X4 )
              = ( comp_b_a_b @ G22 @ F2 @ X4 ) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
thf(fact_904_surj__fun__eq,axiom,
    ! [F2: c > a,X6: set_c,G1: a > a,G22: a > a] :
      ( ( ( image_c_a @ F2 @ X6 )
        = top_top_set_a )
     => ( ! [X4: c] :
            ( ( member_c @ X4 @ X6 )
           => ( ( comp_a_a_c @ G1 @ F2 @ X4 )
              = ( comp_a_a_c @ G22 @ F2 @ X4 ) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
thf(fact_905_surj__fun__eq,axiom,
    ! [F2: b > a,X6: set_b,G1: a > a,G22: a > a] :
      ( ( ( image_b_a @ F2 @ X6 )
        = top_top_set_a )
     => ( ! [X4: b] :
            ( ( member_b @ X4 @ X6 )
           => ( ( comp_a_a_b @ G1 @ F2 @ X4 )
              = ( comp_a_a_b @ G22 @ F2 @ X4 ) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
thf(fact_906_iso__tuple__UNIV__I,axiom,
    ! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_907_iso__tuple__UNIV__I,axiom,
    ! [X2: rat] : ( member_rat @ X2 @ top_top_set_rat ) ).

% iso_tuple_UNIV_I
thf(fact_908_type__copy__set__map0,axiom,
    ! [Rep: a > b,Abs: b > a,S4: b > set_rat,M: b > b,F2: nat > rat,S5: b > set_nat,G2: c > b] :
      ( ( type_definition_a_b @ Rep @ Abs @ top_top_set_b )
     => ( ( ( comp_b_set_rat_b @ S4 @ M )
          = ( comp_s1772850473432672326_rat_b @ ( image_nat_rat @ F2 ) @ S5 ) )
       => ( ( comp_a_set_rat_c @ ( comp_b_set_rat_a @ S4 @ Rep ) @ ( comp_b_a_c @ ( comp_b_a_b @ Abs @ M ) @ G2 ) )
          = ( comp_s1772850473432672327_rat_c @ ( image_nat_rat @ F2 ) @ ( comp_b_set_nat_c @ S5 @ G2 ) ) ) ) ) ).

% type_copy_set_map0
thf(fact_909_type__copy__set__map0,axiom,
    ! [Rep: a > a,Abs: a > a,S4: a > set_rat,M: b > a,F2: nat > rat,S5: b > set_nat,G2: c > b] :
      ( ( type_definition_a_a @ Rep @ Abs @ top_top_set_a )
     => ( ( ( comp_a_set_rat_b @ S4 @ M )
          = ( comp_s1772850473432672326_rat_b @ ( image_nat_rat @ F2 ) @ S5 ) )
       => ( ( comp_a_set_rat_c @ ( comp_a_set_rat_a @ S4 @ Rep ) @ ( comp_b_a_c @ ( comp_a_a_b @ Abs @ M ) @ G2 ) )
          = ( comp_s1772850473432672327_rat_c @ ( image_nat_rat @ F2 ) @ ( comp_b_set_nat_c @ S5 @ G2 ) ) ) ) ) ).

% type_copy_set_map0
thf(fact_910_type__copy__set__map0,axiom,
    ! [Rep: a > b,Abs: b > a,S4: b > set_rat,M: c > b,F2: nat > rat,S5: c > set_nat,G2: c > c] :
      ( ( type_definition_a_b @ Rep @ Abs @ top_top_set_b )
     => ( ( ( comp_b_set_rat_c @ S4 @ M )
          = ( comp_s1772850473432672327_rat_c @ ( image_nat_rat @ F2 ) @ S5 ) )
       => ( ( comp_a_set_rat_c @ ( comp_b_set_rat_a @ S4 @ Rep ) @ ( comp_c_a_c @ ( comp_b_a_c @ Abs @ M ) @ G2 ) )
          = ( comp_s1772850473432672327_rat_c @ ( image_nat_rat @ F2 ) @ ( comp_c_set_nat_c @ S5 @ G2 ) ) ) ) ) ).

% type_copy_set_map0
thf(fact_911_type__copy__set__map0,axiom,
    ! [Rep: a > c,Abs: c > a,S4: c > set_rat,M: c > c,F2: nat > rat,S5: c > set_nat,G2: c > c] :
      ( ( type_definition_a_c @ Rep @ Abs @ top_top_set_c )
     => ( ( ( comp_c_set_rat_c @ S4 @ M )
          = ( comp_s1772850473432672327_rat_c @ ( image_nat_rat @ F2 ) @ S5 ) )
       => ( ( comp_a_set_rat_c @ ( comp_c_set_rat_a @ S4 @ Rep ) @ ( comp_c_a_c @ ( comp_c_a_c @ Abs @ M ) @ G2 ) )
          = ( comp_s1772850473432672327_rat_c @ ( image_nat_rat @ F2 ) @ ( comp_c_set_nat_c @ S5 @ G2 ) ) ) ) ) ).

% type_copy_set_map0
thf(fact_912_type__copy__set__map0,axiom,
    ! [Rep: a > a,Abs: a > a,S4: a > set_rat,M: c > a,F2: nat > rat,S5: c > set_nat,G2: c > c] :
      ( ( type_definition_a_a @ Rep @ Abs @ top_top_set_a )
     => ( ( ( comp_a_set_rat_c @ S4 @ M )
          = ( comp_s1772850473432672327_rat_c @ ( image_nat_rat @ F2 ) @ S5 ) )
       => ( ( comp_a_set_rat_c @ ( comp_a_set_rat_a @ S4 @ Rep ) @ ( comp_c_a_c @ ( comp_a_a_c @ Abs @ M ) @ G2 ) )
          = ( comp_s1772850473432672327_rat_c @ ( image_nat_rat @ F2 ) @ ( comp_c_set_nat_c @ S5 @ G2 ) ) ) ) ) ).

% type_copy_set_map0
thf(fact_913_type__copy__set__map0,axiom,
    ! [Rep: a > b,Abs: b > a,S4: b > set_rat,M: b > b,F2: nat > rat,S5: b > set_nat,G2: b > b] :
      ( ( type_definition_a_b @ Rep @ Abs @ top_top_set_b )
     => ( ( ( comp_b_set_rat_b @ S4 @ M )
          = ( comp_s1772850473432672326_rat_b @ ( image_nat_rat @ F2 ) @ S5 ) )
       => ( ( comp_a_set_rat_b @ ( comp_b_set_rat_a @ S4 @ Rep ) @ ( comp_b_a_b @ ( comp_b_a_b @ Abs @ M ) @ G2 ) )
          = ( comp_s1772850473432672326_rat_b @ ( image_nat_rat @ F2 ) @ ( comp_b_set_nat_b @ S5 @ G2 ) ) ) ) ) ).

% type_copy_set_map0
thf(fact_914_type__copy__set__map0,axiom,
    ! [Rep: a > a,Abs: a > a,S4: a > set_rat,M: b > a,F2: nat > rat,S5: b > set_nat,G2: b > b] :
      ( ( type_definition_a_a @ Rep @ Abs @ top_top_set_a )
     => ( ( ( comp_a_set_rat_b @ S4 @ M )
          = ( comp_s1772850473432672326_rat_b @ ( image_nat_rat @ F2 ) @ S5 ) )
       => ( ( comp_a_set_rat_b @ ( comp_a_set_rat_a @ S4 @ Rep ) @ ( comp_b_a_b @ ( comp_a_a_b @ Abs @ M ) @ G2 ) )
          = ( comp_s1772850473432672326_rat_b @ ( image_nat_rat @ F2 ) @ ( comp_b_set_nat_b @ S5 @ G2 ) ) ) ) ) ).

% type_copy_set_map0
thf(fact_915_type__copy__set__map0,axiom,
    ! [Rep: a > nat,Abs: nat > a,S4: nat > set_rat,M: b > nat,F2: nat > rat,S5: b > set_nat,G2: c > b] :
      ( ( type_d7819358849495216162_a_nat @ Rep @ Abs @ top_top_set_nat )
     => ( ( ( comp_nat_set_rat_b @ S4 @ M )
          = ( comp_s1772850473432672326_rat_b @ ( image_nat_rat @ F2 ) @ S5 ) )
       => ( ( comp_a_set_rat_c @ ( comp_nat_set_rat_a @ S4 @ Rep ) @ ( comp_b_a_c @ ( comp_nat_a_b @ Abs @ M ) @ G2 ) )
          = ( comp_s1772850473432672327_rat_c @ ( image_nat_rat @ F2 ) @ ( comp_b_set_nat_c @ S5 @ G2 ) ) ) ) ) ).

% type_copy_set_map0
thf(fact_916_type__copy__set__map0,axiom,
    ! [Rep: a > nat,Abs: nat > a,S4: nat > set_rat,M: c > nat,F2: nat > rat,S5: c > set_nat,G2: c > c] :
      ( ( type_d7819358849495216162_a_nat @ Rep @ Abs @ top_top_set_nat )
     => ( ( ( comp_nat_set_rat_c @ S4 @ M )
          = ( comp_s1772850473432672327_rat_c @ ( image_nat_rat @ F2 ) @ S5 ) )
       => ( ( comp_a_set_rat_c @ ( comp_nat_set_rat_a @ S4 @ Rep ) @ ( comp_c_a_c @ ( comp_nat_a_c @ Abs @ M ) @ G2 ) )
          = ( comp_s1772850473432672327_rat_c @ ( image_nat_rat @ F2 ) @ ( comp_c_set_nat_c @ S5 @ G2 ) ) ) ) ) ).

% type_copy_set_map0
thf(fact_917_type__copy__set__map0,axiom,
    ! [Rep: b > nat,Abs: nat > b,S4: nat > set_rat,M: b > nat,F2: nat > rat,S5: b > set_nat,G2: c > b] :
      ( ( type_d9054803178451610659_b_nat @ Rep @ Abs @ top_top_set_nat )
     => ( ( ( comp_nat_set_rat_b @ S4 @ M )
          = ( comp_s1772850473432672326_rat_b @ ( image_nat_rat @ F2 ) @ S5 ) )
       => ( ( comp_b_set_rat_c @ ( comp_nat_set_rat_b @ S4 @ Rep ) @ ( comp_b_b_c @ ( comp_nat_b_b @ Abs @ M ) @ G2 ) )
          = ( comp_s1772850473432672327_rat_c @ ( image_nat_rat @ F2 ) @ ( comp_b_set_nat_c @ S5 @ G2 ) ) ) ) ) ).

% type_copy_set_map0
thf(fact_918_image__vimage__eq,axiom,
    ! [F2: nat > rat,A2: set_rat] :
      ( ( image_nat_rat @ F2 @ ( vimage_nat_rat @ F2 @ A2 ) )
      = ( inf_inf_set_rat @ A2 @ ( image_nat_rat @ F2 @ top_top_set_nat ) ) ) ).

% image_vimage_eq
thf(fact_919_surj__vimage__empty,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( vimage_nat_nat @ F2 @ A2 )
          = bot_bot_set_nat )
        = ( A2 = bot_bot_set_nat ) ) ) ).

% surj_vimage_empty
thf(fact_920_surj__vimage__empty,axiom,
    ! [F2: nat > rat,A2: set_rat] :
      ( ( ( image_nat_rat @ F2 @ top_top_set_nat )
        = top_top_set_rat )
     => ( ( ( vimage_nat_rat @ F2 @ A2 )
          = bot_bot_set_nat )
        = ( A2 = bot_bot_set_rat ) ) ) ).

% surj_vimage_empty
thf(fact_921_surj__vimage__empty,axiom,
    ! [F2: rat > nat,A2: set_nat] :
      ( ( ( image_rat_nat @ F2 @ top_top_set_rat )
        = top_top_set_nat )
     => ( ( ( vimage_rat_nat @ F2 @ A2 )
          = bot_bot_set_rat )
        = ( A2 = bot_bot_set_nat ) ) ) ).

% surj_vimage_empty
thf(fact_922_surj__vimage__empty,axiom,
    ! [F2: rat > rat,A2: set_rat] :
      ( ( ( image_rat_rat @ F2 @ top_top_set_rat )
        = top_top_set_rat )
     => ( ( ( vimage_rat_rat @ F2 @ A2 )
          = bot_bot_set_rat )
        = ( A2 = bot_bot_set_rat ) ) ) ).

% surj_vimage_empty
thf(fact_923_vimage__subsetD,axiom,
    ! [F2: nat > nat,B2: set_nat,A2: set_nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F2 @ B2 ) @ A2 )
       => ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).

% vimage_subsetD
thf(fact_924_vimage__subsetD,axiom,
    ! [F2: nat > rat,B2: set_rat,A2: set_nat] :
      ( ( ( image_nat_rat @ F2 @ top_top_set_nat )
        = top_top_set_rat )
     => ( ( ord_less_eq_set_nat @ ( vimage_nat_rat @ F2 @ B2 ) @ A2 )
       => ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F2 @ A2 ) ) ) ) ).

% vimage_subsetD
thf(fact_925_vimage__subsetD,axiom,
    ! [F2: rat > nat,B2: set_nat,A2: set_rat] :
      ( ( ( image_rat_nat @ F2 @ top_top_set_rat )
        = top_top_set_nat )
     => ( ( ord_less_eq_set_rat @ ( vimage_rat_nat @ F2 @ B2 ) @ A2 )
       => ( ord_less_eq_set_nat @ B2 @ ( image_rat_nat @ F2 @ A2 ) ) ) ) ).

% vimage_subsetD
thf(fact_926_vimage__subsetD,axiom,
    ! [F2: rat > rat,B2: set_rat,A2: set_rat] :
      ( ( ( image_rat_rat @ F2 @ top_top_set_rat )
        = top_top_set_rat )
     => ( ( ord_less_eq_set_rat @ ( vimage_rat_rat @ F2 @ B2 ) @ A2 )
       => ( ord_less_eq_set_rat @ B2 @ ( image_rat_rat @ F2 @ A2 ) ) ) ) ).

% vimage_subsetD
thf(fact_927_image__is__empty,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( ( image_nat_rat @ F2 @ A2 )
        = bot_bot_set_rat )
      = ( A2 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_928_empty__is__image,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( bot_bot_set_rat
        = ( image_nat_rat @ F2 @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_929_image__empty,axiom,
    ! [F2: nat > rat] :
      ( ( image_nat_rat @ F2 @ bot_bot_set_nat )
      = bot_bot_set_rat ) ).

% image_empty
thf(fact_930_inf__top__left,axiom,
    ! [X2: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ X2 )
      = X2 ) ).

% inf_top_left
thf(fact_931_inf__top__left,axiom,
    ! [X2: set_rat] :
      ( ( inf_inf_set_rat @ top_top_set_rat @ X2 )
      = X2 ) ).

% inf_top_left
thf(fact_932_inf__top__right,axiom,
    ! [X2: set_nat] :
      ( ( inf_inf_set_nat @ X2 @ top_top_set_nat )
      = X2 ) ).

% inf_top_right
thf(fact_933_inf__top__right,axiom,
    ! [X2: set_rat] :
      ( ( inf_inf_set_rat @ X2 @ top_top_set_rat )
      = X2 ) ).

% inf_top_right
thf(fact_934_inf__eq__top__iff,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( ( inf_inf_set_nat @ X2 @ Y3 )
        = top_top_set_nat )
      = ( ( X2 = top_top_set_nat )
        & ( Y3 = top_top_set_nat ) ) ) ).

% inf_eq_top_iff
thf(fact_935_inf__eq__top__iff,axiom,
    ! [X2: set_rat,Y3: set_rat] :
      ( ( ( inf_inf_set_rat @ X2 @ Y3 )
        = top_top_set_rat )
      = ( ( X2 = top_top_set_rat )
        & ( Y3 = top_top_set_rat ) ) ) ).

% inf_eq_top_iff
thf(fact_936_top__eq__inf__iff,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( top_top_set_nat
        = ( inf_inf_set_nat @ X2 @ Y3 ) )
      = ( ( X2 = top_top_set_nat )
        & ( Y3 = top_top_set_nat ) ) ) ).

% top_eq_inf_iff
thf(fact_937_top__eq__inf__iff,axiom,
    ! [X2: set_rat,Y3: set_rat] :
      ( ( top_top_set_rat
        = ( inf_inf_set_rat @ X2 @ Y3 ) )
      = ( ( X2 = top_top_set_rat )
        & ( Y3 = top_top_set_rat ) ) ) ).

% top_eq_inf_iff
thf(fact_938_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B )
        = top_top_set_nat )
      = ( ( A = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_939_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_rat,B: set_rat] :
      ( ( ( inf_inf_set_rat @ A @ B )
        = top_top_set_rat )
      = ( ( A = top_top_set_rat )
        & ( B = top_top_set_rat ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_940_inf__top_Oleft__neutral,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ A )
      = A ) ).

% inf_top.left_neutral
thf(fact_941_inf__top_Oleft__neutral,axiom,
    ! [A: set_rat] :
      ( ( inf_inf_set_rat @ top_top_set_rat @ A )
      = A ) ).

% inf_top.left_neutral
thf(fact_942_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( top_top_set_nat
        = ( inf_inf_set_nat @ A @ B ) )
      = ( ( A = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_943_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_rat,B: set_rat] :
      ( ( top_top_set_rat
        = ( inf_inf_set_rat @ A @ B ) )
      = ( ( A = top_top_set_rat )
        & ( B = top_top_set_rat ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_944_inf__top_Oright__neutral,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ A @ top_top_set_nat )
      = A ) ).

% inf_top.right_neutral
thf(fact_945_inf__top_Oright__neutral,axiom,
    ! [A: set_rat] :
      ( ( inf_inf_set_rat @ A @ top_top_set_rat )
      = A ) ).

% inf_top.right_neutral
thf(fact_946_Int__UNIV,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ A2 @ B2 )
        = top_top_set_nat )
      = ( ( A2 = top_top_set_nat )
        & ( B2 = top_top_set_nat ) ) ) ).

% Int_UNIV
thf(fact_947_Int__UNIV,axiom,
    ! [A2: set_rat,B2: set_rat] :
      ( ( ( inf_inf_set_rat @ A2 @ B2 )
        = top_top_set_rat )
      = ( ( A2 = top_top_set_rat )
        & ( B2 = top_top_set_rat ) ) ) ).

% Int_UNIV
thf(fact_948_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
    = top_top_set_nat ) ).

% boolean_algebra.compl_zero
thf(fact_949_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus2201863774496077783et_rat @ bot_bot_set_rat )
    = top_top_set_rat ) ).

% boolean_algebra.compl_zero
thf(fact_950_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus5710092332889474511et_nat @ top_top_set_nat )
    = bot_bot_set_nat ) ).

% boolean_algebra.compl_one
thf(fact_951_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus2201863774496077783et_rat @ top_top_set_rat )
    = bot_bot_set_rat ) ).

% boolean_algebra.compl_one
thf(fact_952_Inf__empty,axiom,
    ( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
    = top_top_set_nat ) ).

% Inf_empty
thf(fact_953_Inf__empty,axiom,
    ( ( comple4298007329820168263et_rat @ bot_bot_set_set_rat )
    = top_top_set_rat ) ).

% Inf_empty
thf(fact_954_image__Int__subset,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( inf_inf_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( image_nat_rat @ F2 @ B2 ) ) ) ).

% image_Int_subset
thf(fact_955_top_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
     => ( A = top_top_set_nat ) ) ).

% top.extremum_uniqueI
thf(fact_956_top_Oextremum__uniqueI,axiom,
    ! [A: set_rat] :
      ( ( ord_less_eq_set_rat @ top_top_set_rat @ A )
     => ( A = top_top_set_rat ) ) ).

% top.extremum_uniqueI
thf(fact_957_top_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
      = ( A = top_top_set_nat ) ) ).

% top.extremum_unique
thf(fact_958_top_Oextremum__unique,axiom,
    ! [A: set_rat] :
      ( ( ord_less_eq_set_rat @ top_top_set_rat @ A )
      = ( A = top_top_set_rat ) ) ).

% top.extremum_unique
thf(fact_959_top__greatest,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).

% top_greatest
thf(fact_960_top__greatest,axiom,
    ! [A: set_rat] : ( ord_less_eq_set_rat @ A @ top_top_set_rat ) ).

% top_greatest
thf(fact_961_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_962_top__set__def,axiom,
    ( top_top_set_rat
    = ( collect_rat @ top_top_rat_o ) ) ).

% top_set_def
thf(fact_963_boolean__algebra_Oconj__one__right,axiom,
    ! [X2: set_nat] :
      ( ( inf_inf_set_nat @ X2 @ top_top_set_nat )
      = X2 ) ).

% boolean_algebra.conj_one_right
thf(fact_964_boolean__algebra_Oconj__one__right,axiom,
    ! [X2: set_rat] :
      ( ( inf_inf_set_rat @ X2 @ top_top_set_rat )
      = X2 ) ).

% boolean_algebra.conj_one_right
thf(fact_965_Int__UNIV__left,axiom,
    ! [B2: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ B2 )
      = B2 ) ).

% Int_UNIV_left
thf(fact_966_Int__UNIV__left,axiom,
    ! [B2: set_rat] :
      ( ( inf_inf_set_rat @ top_top_set_rat @ B2 )
      = B2 ) ).

% Int_UNIV_left
thf(fact_967_Int__UNIV__right,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
      = A2 ) ).

% Int_UNIV_right
thf(fact_968_Int__UNIV__right,axiom,
    ! [A2: set_rat] :
      ( ( inf_inf_set_rat @ A2 @ top_top_set_rat )
      = A2 ) ).

% Int_UNIV_right
thf(fact_969_subset__UNIV,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_970_subset__UNIV,axiom,
    ! [A2: set_rat] : ( ord_less_eq_set_rat @ A2 @ top_top_set_rat ) ).

% subset_UNIV
thf(fact_971_subset__image__iff,axiom,
    ! [B2: set_rat,F2: nat > rat,A2: set_nat] :
      ( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F2 @ A2 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A2 )
            & ( B2
              = ( image_nat_rat @ F2 @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_972_image__subset__iff,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_rat] :
      ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ B2 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A2 )
           => ( member_rat @ ( F2 @ X ) @ B2 ) ) ) ) ).

% image_subset_iff
thf(fact_973_subset__imageE,axiom,
    ! [B2: set_rat,F2: nat > rat,A2: set_nat] :
      ( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F2 @ A2 ) )
     => ~ ! [C3: set_nat] :
            ( ( ord_less_eq_set_nat @ C3 @ A2 )
           => ( B2
             != ( image_nat_rat @ F2 @ C3 ) ) ) ) ).

% subset_imageE
thf(fact_974_image__subsetI,axiom,
    ! [A2: set_nat,F2: nat > rat,B2: set_rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( member_rat @ ( F2 @ X4 ) @ B2 ) )
     => ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ B2 ) ) ).

% image_subsetI
thf(fact_975_image__mono,axiom,
    ! [A2: set_nat,B2: set_nat,F2: nat > rat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( image_nat_rat @ F2 @ B2 ) ) ) ).

% image_mono
thf(fact_976_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
thf(fact_977_empty__not__UNIV,axiom,
    bot_bot_set_rat != top_top_set_rat ).

% empty_not_UNIV
thf(fact_978_Compl__empty__eq,axiom,
    ( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
    = top_top_set_nat ) ).

% Compl_empty_eq
thf(fact_979_Compl__empty__eq,axiom,
    ( ( uminus2201863774496077783et_rat @ bot_bot_set_rat )
    = top_top_set_rat ) ).

% Compl_empty_eq
thf(fact_980_Compl__UNIV__eq,axiom,
    ( ( uminus5710092332889474511et_nat @ top_top_set_nat )
    = bot_bot_set_nat ) ).

% Compl_UNIV_eq
thf(fact_981_Compl__UNIV__eq,axiom,
    ( ( uminus2201863774496077783et_rat @ top_top_set_rat )
    = bot_bot_set_rat ) ).

% Compl_UNIV_eq
thf(fact_982_Inter__empty,axiom,
    ( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
    = top_top_set_nat ) ).

% Inter_empty
thf(fact_983_Inter__empty,axiom,
    ( ( comple4298007329820168263et_rat @ bot_bot_set_set_rat )
    = top_top_set_rat ) ).

% Inter_empty
thf(fact_984_range__subsetD,axiom,
    ! [F2: nat > rat,B2: set_rat,I: nat] :
      ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ top_top_set_nat ) @ B2 )
     => ( member_rat @ ( F2 @ I ) @ B2 ) ) ).

% range_subsetD
thf(fact_985_surj__Compl__image__subset,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F2 @ A2 ) ) @ ( image_nat_nat @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_986_surj__Compl__image__subset,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( ( image_nat_rat @ F2 @ top_top_set_nat )
        = top_top_set_rat )
     => ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_nat_rat @ F2 @ A2 ) ) @ ( image_nat_rat @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_987_surj__Compl__image__subset,axiom,
    ! [F2: rat > nat,A2: set_rat] :
      ( ( ( image_rat_nat @ F2 @ top_top_set_rat )
        = top_top_set_nat )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_rat_nat @ F2 @ A2 ) ) @ ( image_rat_nat @ F2 @ ( uminus2201863774496077783et_rat @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_988_surj__Compl__image__subset,axiom,
    ! [F2: rat > rat,A2: set_rat] :
      ( ( ( image_rat_rat @ F2 @ top_top_set_rat )
        = top_top_set_rat )
     => ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_rat_rat @ F2 @ A2 ) ) @ ( image_rat_rat @ F2 @ ( uminus2201863774496077783et_rat @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_989_type__copy__map__comp0__undo,axiom,
    ! [Rep: a > c,Abs: c > a,Rep5: b > b,Abs5: b > b,Rep6: b > b,Abs6: b > b,M: b > b,M1: c > b,M2: b > c] :
      ( ( type_definition_a_c @ Rep @ Abs @ top_top_set_c )
     => ( ( type_definition_b_b @ Rep5 @ Abs5 @ top_top_set_b )
       => ( ( type_definition_b_b @ Rep6 @ Abs6 @ top_top_set_b )
         => ( ( ( comp_b_b_b @ ( comp_b_b_b @ Abs5 @ M ) @ Rep6 )
              = ( comp_a_b_b @ ( comp_c_b_a @ ( comp_b_b_c @ Abs5 @ M1 ) @ Rep ) @ ( comp_b_a_b @ ( comp_c_a_b @ Abs @ M2 ) @ Rep6 ) ) )
           => ( ( comp_c_b_b @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_990_type__copy__map__comp0__undo,axiom,
    ! [Rep: a > c,Abs: c > a,Rep5: b > b,Abs5: b > b,Rep6: c > a,Abs6: a > c,M: a > b,M1: c > b,M2: a > c] :
      ( ( type_definition_a_c @ Rep @ Abs @ top_top_set_c )
     => ( ( type_definition_b_b @ Rep5 @ Abs5 @ top_top_set_b )
       => ( ( type_definition_c_a @ Rep6 @ Abs6 @ top_top_set_a )
         => ( ( ( comp_a_b_c @ ( comp_b_b_a @ Abs5 @ M ) @ Rep6 )
              = ( comp_a_b_c @ ( comp_c_b_a @ ( comp_b_b_c @ Abs5 @ M1 ) @ Rep ) @ ( comp_a_a_c @ ( comp_c_a_a @ Abs @ M2 ) @ Rep6 ) ) )
           => ( ( comp_c_b_a @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_991_type__copy__map__comp0__undo,axiom,
    ! [Rep: a > c,Abs: c > a,Rep5: b > b,Abs5: b > b,Rep6: b > a,Abs6: a > b,M: a > b,M1: c > b,M2: a > c] :
      ( ( type_definition_a_c @ Rep @ Abs @ top_top_set_c )
     => ( ( type_definition_b_b @ Rep5 @ Abs5 @ top_top_set_b )
       => ( ( type_definition_b_a @ Rep6 @ Abs6 @ top_top_set_a )
         => ( ( ( comp_a_b_b @ ( comp_b_b_a @ Abs5 @ M ) @ Rep6 )
              = ( comp_a_b_b @ ( comp_c_b_a @ ( comp_b_b_c @ Abs5 @ M1 ) @ Rep ) @ ( comp_a_a_b @ ( comp_c_a_a @ Abs @ M2 ) @ Rep6 ) ) )
           => ( ( comp_c_b_a @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_992_type__copy__map__comp0__undo,axiom,
    ! [Rep: b > c,Abs: c > b,Rep5: b > a,Abs5: a > b,Rep6: c > c,Abs6: c > c,M: c > a,M1: c > a,M2: c > c] :
      ( ( type_definition_b_c @ Rep @ Abs @ top_top_set_c )
     => ( ( type_definition_b_a @ Rep5 @ Abs5 @ top_top_set_a )
       => ( ( type_definition_c_c @ Rep6 @ Abs6 @ top_top_set_c )
         => ( ( ( comp_c_b_c @ ( comp_a_b_c @ Abs5 @ M ) @ Rep6 )
              = ( comp_b_b_c @ ( comp_c_b_b @ ( comp_a_b_c @ Abs5 @ M1 ) @ Rep ) @ ( comp_c_b_c @ ( comp_c_b_c @ Abs @ M2 ) @ Rep6 ) ) )
           => ( ( comp_c_a_c @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_993_type__copy__map__comp0__undo,axiom,
    ! [Rep: b > a,Abs: a > b,Rep5: b > a,Abs5: a > b,Rep6: c > c,Abs6: c > c,M: c > a,M1: a > a,M2: c > a] :
      ( ( type_definition_b_a @ Rep @ Abs @ top_top_set_a )
     => ( ( type_definition_b_a @ Rep5 @ Abs5 @ top_top_set_a )
       => ( ( type_definition_c_c @ Rep6 @ Abs6 @ top_top_set_c )
         => ( ( ( comp_c_b_c @ ( comp_a_b_c @ Abs5 @ M ) @ Rep6 )
              = ( comp_b_b_c @ ( comp_a_b_b @ ( comp_a_b_a @ Abs5 @ M1 ) @ Rep ) @ ( comp_c_b_c @ ( comp_a_b_c @ Abs @ M2 ) @ Rep6 ) ) )
           => ( ( comp_a_a_c @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_994_type__copy__map__comp0__undo,axiom,
    ! [Rep: b > b,Abs: b > b,Rep5: b > a,Abs5: a > b,Rep6: c > c,Abs6: c > c,M: c > a,M1: b > a,M2: c > b] :
      ( ( type_definition_b_b @ Rep @ Abs @ top_top_set_b )
     => ( ( type_definition_b_a @ Rep5 @ Abs5 @ top_top_set_a )
       => ( ( type_definition_c_c @ Rep6 @ Abs6 @ top_top_set_c )
         => ( ( ( comp_c_b_c @ ( comp_a_b_c @ Abs5 @ M ) @ Rep6 )
              = ( comp_b_b_c @ ( comp_b_b_b @ ( comp_a_b_b @ Abs5 @ M1 ) @ Rep ) @ ( comp_c_b_c @ ( comp_b_b_c @ Abs @ M2 ) @ Rep6 ) ) )
           => ( ( comp_b_a_c @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_995_type__copy__map__comp0__undo,axiom,
    ! [Rep: b > c,Abs: c > b,Rep5: a > b,Abs5: b > a,Rep6: b > c,Abs6: c > b,M: c > b,M1: c > b,M2: c > c] :
      ( ( type_definition_b_c @ Rep @ Abs @ top_top_set_c )
     => ( ( type_definition_a_b @ Rep5 @ Abs5 @ top_top_set_b )
       => ( ( type_definition_b_c @ Rep6 @ Abs6 @ top_top_set_c )
         => ( ( ( comp_c_a_b @ ( comp_b_a_c @ Abs5 @ M ) @ Rep6 )
              = ( comp_b_a_b @ ( comp_c_a_b @ ( comp_b_a_c @ Abs5 @ M1 ) @ Rep ) @ ( comp_c_b_b @ ( comp_c_b_c @ Abs @ M2 ) @ Rep6 ) ) )
           => ( ( comp_c_b_c @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_996_type__copy__map__comp0__undo,axiom,
    ! [Rep: b > b,Abs: b > b,Rep5: a > b,Abs5: b > a,Rep6: b > c,Abs6: c > b,M: c > b,M1: b > b,M2: c > b] :
      ( ( type_definition_b_b @ Rep @ Abs @ top_top_set_b )
     => ( ( type_definition_a_b @ Rep5 @ Abs5 @ top_top_set_b )
       => ( ( type_definition_b_c @ Rep6 @ Abs6 @ top_top_set_c )
         => ( ( ( comp_c_a_b @ ( comp_b_a_c @ Abs5 @ M ) @ Rep6 )
              = ( comp_b_a_b @ ( comp_b_a_b @ ( comp_b_a_b @ Abs5 @ M1 ) @ Rep ) @ ( comp_c_b_b @ ( comp_b_b_c @ Abs @ M2 ) @ Rep6 ) ) )
           => ( ( comp_b_b_c @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_997_type__copy__map__comp0__undo,axiom,
    ! [Rep: b > a,Abs: a > b,Rep5: a > b,Abs5: b > a,Rep6: b > c,Abs6: c > b,M: c > b,M1: a > b,M2: c > a] :
      ( ( type_definition_b_a @ Rep @ Abs @ top_top_set_a )
     => ( ( type_definition_a_b @ Rep5 @ Abs5 @ top_top_set_b )
       => ( ( type_definition_b_c @ Rep6 @ Abs6 @ top_top_set_c )
         => ( ( ( comp_c_a_b @ ( comp_b_a_c @ Abs5 @ M ) @ Rep6 )
              = ( comp_b_a_b @ ( comp_a_a_b @ ( comp_b_a_a @ Abs5 @ M1 ) @ Rep ) @ ( comp_c_b_b @ ( comp_a_b_c @ Abs @ M2 ) @ Rep6 ) ) )
           => ( ( comp_a_b_c @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_998_type__copy__map__comp0__undo,axiom,
    ! [Rep: a > a,Abs: a > a,Rep5: a > b,Abs5: b > a,Rep6: b > c,Abs6: c > b,M: c > b,M1: a > b,M2: c > a] :
      ( ( type_definition_a_a @ Rep @ Abs @ top_top_set_a )
     => ( ( type_definition_a_b @ Rep5 @ Abs5 @ top_top_set_b )
       => ( ( type_definition_b_c @ Rep6 @ Abs6 @ top_top_set_c )
         => ( ( ( comp_c_a_b @ ( comp_b_a_c @ Abs5 @ M ) @ Rep6 )
              = ( comp_a_a_b @ ( comp_a_a_a @ ( comp_b_a_a @ Abs5 @ M1 ) @ Rep ) @ ( comp_c_a_b @ ( comp_a_a_c @ Abs @ M2 ) @ Rep6 ) ) )
           => ( ( comp_a_b_c @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_999_type__copy__map__comp0,axiom,
    ! [Rep: a > c,Abs: c > a,M: b > b,M1: c > b,M2: b > c,F2: b > b,G2: b > b] :
      ( ( type_definition_a_c @ Rep @ Abs @ top_top_set_c )
     => ( ( M
          = ( comp_c_b_b @ M1 @ M2 ) )
       => ( ( comp_b_b_b @ ( comp_b_b_b @ F2 @ M ) @ G2 )
          = ( comp_a_b_b @ ( comp_c_b_a @ ( comp_b_b_c @ F2 @ M1 ) @ Rep ) @ ( comp_b_a_b @ ( comp_c_a_b @ Abs @ M2 ) @ G2 ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_1000_type__copy__map__comp0,axiom,
    ! [Rep: a > c,Abs: c > a,M: a > b,M1: c > b,M2: a > c,F2: b > b,G2: c > a] :
      ( ( type_definition_a_c @ Rep @ Abs @ top_top_set_c )
     => ( ( M
          = ( comp_c_b_a @ M1 @ M2 ) )
       => ( ( comp_a_b_c @ ( comp_b_b_a @ F2 @ M ) @ G2 )
          = ( comp_a_b_c @ ( comp_c_b_a @ ( comp_b_b_c @ F2 @ M1 ) @ Rep ) @ ( comp_a_a_c @ ( comp_c_a_a @ Abs @ M2 ) @ G2 ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_1001_type__copy__map__comp0,axiom,
    ! [Rep: a > c,Abs: c > a,M: a > b,M1: c > b,M2: a > c,F2: b > b,G2: b > a] :
      ( ( type_definition_a_c @ Rep @ Abs @ top_top_set_c )
     => ( ( M
          = ( comp_c_b_a @ M1 @ M2 ) )
       => ( ( comp_a_b_b @ ( comp_b_b_a @ F2 @ M ) @ G2 )
          = ( comp_a_b_b @ ( comp_c_b_a @ ( comp_b_b_c @ F2 @ M1 ) @ Rep ) @ ( comp_a_a_b @ ( comp_c_a_a @ Abs @ M2 ) @ G2 ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_1002_type__copy__map__comp0,axiom,
    ! [Rep: b > c,Abs: c > b,M: c > b,M1: c > b,M2: c > c,F2: b > a,G2: b > c] :
      ( ( type_definition_b_c @ Rep @ Abs @ top_top_set_c )
     => ( ( M
          = ( comp_c_b_c @ M1 @ M2 ) )
       => ( ( comp_c_a_b @ ( comp_b_a_c @ F2 @ M ) @ G2 )
          = ( comp_b_a_b @ ( comp_c_a_b @ ( comp_b_a_c @ F2 @ M1 ) @ Rep ) @ ( comp_c_b_b @ ( comp_c_b_c @ Abs @ M2 ) @ G2 ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_1003_type__copy__map__comp0,axiom,
    ! [Rep: b > a,Abs: a > b,M: c > b,M1: a > b,M2: c > a,F2: b > a,G2: b > c] :
      ( ( type_definition_b_a @ Rep @ Abs @ top_top_set_a )
     => ( ( M
          = ( comp_a_b_c @ M1 @ M2 ) )
       => ( ( comp_c_a_b @ ( comp_b_a_c @ F2 @ M ) @ G2 )
          = ( comp_b_a_b @ ( comp_a_a_b @ ( comp_b_a_a @ F2 @ M1 ) @ Rep ) @ ( comp_c_b_b @ ( comp_a_b_c @ Abs @ M2 ) @ G2 ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_1004_type__copy__map__comp0,axiom,
    ! [Rep: a > a,Abs: a > a,M: c > b,M1: a > b,M2: c > a,F2: b > a,G2: b > c] :
      ( ( type_definition_a_a @ Rep @ Abs @ top_top_set_a )
     => ( ( M
          = ( comp_a_b_c @ M1 @ M2 ) )
       => ( ( comp_c_a_b @ ( comp_b_a_c @ F2 @ M ) @ G2 )
          = ( comp_a_a_b @ ( comp_a_a_a @ ( comp_b_a_a @ F2 @ M1 ) @ Rep ) @ ( comp_c_a_b @ ( comp_a_a_c @ Abs @ M2 ) @ G2 ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_1005_type__copy__map__comp0,axiom,
    ! [Rep: a > c,Abs: c > a,M: c > b,M1: c > b,M2: c > c,F2: b > a,G2: b > c] :
      ( ( type_definition_a_c @ Rep @ Abs @ top_top_set_c )
     => ( ( M
          = ( comp_c_b_c @ M1 @ M2 ) )
       => ( ( comp_c_a_b @ ( comp_b_a_c @ F2 @ M ) @ G2 )
          = ( comp_a_a_b @ ( comp_c_a_a @ ( comp_b_a_c @ F2 @ M1 ) @ Rep ) @ ( comp_c_a_b @ ( comp_c_a_c @ Abs @ M2 ) @ G2 ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_1006_type__copy__map__comp0,axiom,
    ! [Rep: b > c,Abs: c > b,M: c > c,M1: c > c,M2: c > c,F2: c > a,G2: b > c] :
      ( ( type_definition_b_c @ Rep @ Abs @ top_top_set_c )
     => ( ( M
          = ( comp_c_c_c @ M1 @ M2 ) )
       => ( ( comp_c_a_b @ ( comp_c_a_c @ F2 @ M ) @ G2 )
          = ( comp_b_a_b @ ( comp_c_a_b @ ( comp_c_a_c @ F2 @ M1 ) @ Rep ) @ ( comp_c_b_b @ ( comp_c_b_c @ Abs @ M2 ) @ G2 ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_1007_type__copy__map__comp0,axiom,
    ! [Rep: b > b,Abs: b > b,M: c > c,M1: b > c,M2: c > b,F2: c > a,G2: b > c] :
      ( ( type_definition_b_b @ Rep @ Abs @ top_top_set_b )
     => ( ( M
          = ( comp_b_c_c @ M1 @ M2 ) )
       => ( ( comp_c_a_b @ ( comp_c_a_c @ F2 @ M ) @ G2 )
          = ( comp_b_a_b @ ( comp_b_a_b @ ( comp_c_a_b @ F2 @ M1 ) @ Rep ) @ ( comp_c_b_b @ ( comp_b_b_c @ Abs @ M2 ) @ G2 ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_1008_type__copy__map__comp0,axiom,
    ! [Rep: b > a,Abs: a > b,M: c > c,M1: a > c,M2: c > a,F2: c > a,G2: b > c] :
      ( ( type_definition_b_a @ Rep @ Abs @ top_top_set_a )
     => ( ( M
          = ( comp_a_c_c @ M1 @ M2 ) )
       => ( ( comp_c_a_b @ ( comp_c_a_c @ F2 @ M ) @ G2 )
          = ( comp_b_a_b @ ( comp_a_a_b @ ( comp_c_a_a @ F2 @ M1 ) @ Rep ) @ ( comp_c_b_b @ ( comp_a_b_c @ Abs @ M2 ) @ G2 ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_1009_image__subset__iff__subset__vimage,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_rat] :
      ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ B2 )
      = ( ord_less_eq_set_nat @ A2 @ ( vimage_nat_rat @ F2 @ B2 ) ) ) ).

% image_subset_iff_subset_vimage
thf(fact_1010_image__vimage__subset,axiom,
    ! [F2: nat > rat,A2: set_rat] : ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ ( vimage_nat_rat @ F2 @ A2 ) ) @ A2 ) ).

% image_vimage_subset
thf(fact_1011_type__copy__map__id0,axiom,
    ! [Rep: a > c,Abs: c > a,M: c > c] :
      ( ( type_definition_a_c @ Rep @ Abs @ top_top_set_c )
     => ( ( M = id_c )
       => ( ( comp_c_a_a @ ( comp_c_a_c @ Abs @ M ) @ Rep )
          = id_a ) ) ) ).

% type_copy_map_id0
thf(fact_1012_type__copy__map__id0,axiom,
    ! [Rep: a > b,Abs: b > a,M: b > b] :
      ( ( type_definition_a_b @ Rep @ Abs @ top_top_set_b )
     => ( ( M = id_b )
       => ( ( comp_b_a_a @ ( comp_b_a_b @ Abs @ M ) @ Rep )
          = id_a ) ) ) ).

% type_copy_map_id0
thf(fact_1013_type__definition_OAbs__image,axiom,
    ! [Rep: rat > nat,Abs: nat > rat,A2: set_nat] :
      ( ( type_d5933939304842882774at_nat @ Rep @ Abs @ A2 )
     => ( ( image_nat_rat @ Abs @ A2 )
        = top_top_set_rat ) ) ).

% type_definition.Abs_image
thf(fact_1014_type__definition_ORep__range,axiom,
    ! [Rep: nat > rat,Abs: rat > nat,A2: set_rat] :
      ( ( type_d5615363888691252950at_rat @ Rep @ Abs @ A2 )
     => ( ( image_nat_rat @ Rep @ top_top_set_nat )
        = A2 ) ) ).

% type_definition.Rep_range
thf(fact_1015_all__subset__image,axiom,
    ! [F2: nat > rat,A2: set_nat,P: set_rat > $o] :
      ( ( ! [B4: set_rat] :
            ( ( ord_less_eq_set_rat @ B4 @ ( image_nat_rat @ F2 @ A2 ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_nat] :
            ( ( ord_less_eq_set_nat @ B4 @ A2 )
           => ( P @ ( image_nat_rat @ F2 @ B4 ) ) ) ) ) ).

% all_subset_image
thf(fact_1016_the__elem__image__unique,axiom,
    ! [A2: set_nat,F2: nat > rat,X2: nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [Y4: nat] :
            ( ( member_nat @ Y4 @ A2 )
           => ( ( F2 @ Y4 )
              = ( F2 @ X2 ) ) )
       => ( ( the_elem_rat @ ( image_nat_rat @ F2 @ A2 ) )
          = ( F2 @ X2 ) ) ) ) ).

% the_elem_image_unique
thf(fact_1017_bdd__below_OI2,axiom,
    ! [A2: set_nat,M: rat,F2: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_rat @ M @ ( F2 @ X4 ) ) )
     => ( condit1103211067700513672ow_rat @ ( image_nat_rat @ F2 @ A2 ) ) ) ).

% bdd_below.I2
thf(fact_1018_bdd__belowI2,axiom,
    ! [A2: set_nat,M3: rat,F2: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_rat @ M3 @ ( F2 @ X4 ) ) )
     => ( condit1103211067700513672ow_rat @ ( image_nat_rat @ F2 @ A2 ) ) ) ).

% bdd_belowI2
thf(fact_1019_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_1020_top__empty__eq,axiom,
    ( top_top_rat_o
    = ( ^ [X: rat] : ( member_rat @ X @ top_top_set_rat ) ) ) ).

% top_empty_eq
thf(fact_1021_image__Fpow__mono,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_rat] :
      ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ B2 )
     => ( ord_le513522071413781156et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F2 ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_rat @ B2 ) ) ) ).

% image_Fpow_mono
thf(fact_1022_insert__image,axiom,
    ! [X2: nat,A2: set_nat,F2: nat > rat] :
      ( ( member_nat @ X2 @ A2 )
     => ( ( insert_rat @ ( F2 @ X2 ) @ ( image_nat_rat @ F2 @ A2 ) )
        = ( image_nat_rat @ F2 @ A2 ) ) ) ).

% insert_image
thf(fact_1023_image__insert,axiom,
    ! [F2: nat > rat,A: nat,B2: set_nat] :
      ( ( image_nat_rat @ F2 @ ( insert_nat @ A @ B2 ) )
      = ( insert_rat @ ( F2 @ A ) @ ( image_nat_rat @ F2 @ B2 ) ) ) ).

% image_insert
thf(fact_1024_sup__top__left,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ top_top_set_nat @ X2 )
      = top_top_set_nat ) ).

% sup_top_left
thf(fact_1025_sup__top__left,axiom,
    ! [X2: set_rat] :
      ( ( sup_sup_set_rat @ top_top_set_rat @ X2 )
      = top_top_set_rat ) ).

% sup_top_left
thf(fact_1026_sup__top__right,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ X2 @ top_top_set_nat )
      = top_top_set_nat ) ).

% sup_top_right
thf(fact_1027_sup__top__right,axiom,
    ! [X2: set_rat] :
      ( ( sup_sup_set_rat @ X2 @ top_top_set_rat )
      = top_top_set_rat ) ).

% sup_top_right
thf(fact_1028_boolean__algebra_Odisj__one__left,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ top_top_set_nat @ X2 )
      = top_top_set_nat ) ).

% boolean_algebra.disj_one_left
thf(fact_1029_boolean__algebra_Odisj__one__left,axiom,
    ! [X2: set_rat] :
      ( ( sup_sup_set_rat @ top_top_set_rat @ X2 )
      = top_top_set_rat ) ).

% boolean_algebra.disj_one_left
thf(fact_1030_boolean__algebra_Odisj__one__right,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ X2 @ top_top_set_nat )
      = top_top_set_nat ) ).

% boolean_algebra.disj_one_right
thf(fact_1031_boolean__algebra_Odisj__one__right,axiom,
    ! [X2: set_rat] :
      ( ( sup_sup_set_rat @ X2 @ top_top_set_rat )
      = top_top_set_rat ) ).

% boolean_algebra.disj_one_right
thf(fact_1032_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ X2 ) )
      = top_top_set_nat ) ).

% boolean_algebra.disj_cancel_right
thf(fact_1033_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X2: set_rat] :
      ( ( sup_sup_set_rat @ X2 @ ( uminus2201863774496077783et_rat @ X2 ) )
      = top_top_set_rat ) ).

% boolean_algebra.disj_cancel_right
thf(fact_1034_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ X2 )
      = top_top_set_nat ) ).

% boolean_algebra.disj_cancel_left
thf(fact_1035_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X2: set_rat] :
      ( ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X2 ) @ X2 )
      = top_top_set_rat ) ).

% boolean_algebra.disj_cancel_left
thf(fact_1036_sup__compl__top__left2,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y3 ) )
      = top_top_set_nat ) ).

% sup_compl_top_left2
thf(fact_1037_sup__compl__top__left2,axiom,
    ! [X2: set_rat,Y3: set_rat] :
      ( ( sup_sup_set_rat @ X2 @ ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X2 ) @ Y3 ) )
      = top_top_set_rat ) ).

% sup_compl_top_left2
thf(fact_1038_sup__compl__top__left1,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( sup_sup_set_nat @ X2 @ Y3 ) )
      = top_top_set_nat ) ).

% sup_compl_top_left1
thf(fact_1039_sup__compl__top__left1,axiom,
    ! [X2: set_rat,Y3: set_rat] :
      ( ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X2 ) @ ( sup_sup_set_rat @ X2 @ Y3 ) )
      = top_top_set_rat ) ).

% sup_compl_top_left1
thf(fact_1040_Un__UNIV__right,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ top_top_set_nat )
      = top_top_set_nat ) ).

% Un_UNIV_right
thf(fact_1041_Un__UNIV__right,axiom,
    ! [A2: set_rat] :
      ( ( sup_sup_set_rat @ A2 @ top_top_set_rat )
      = top_top_set_rat ) ).

% Un_UNIV_right
thf(fact_1042_Un__UNIV__left,axiom,
    ! [B2: set_nat] :
      ( ( sup_sup_set_nat @ top_top_set_nat @ B2 )
      = top_top_set_nat ) ).

% Un_UNIV_left
thf(fact_1043_Un__UNIV__left,axiom,
    ! [B2: set_rat] :
      ( ( sup_sup_set_rat @ top_top_set_rat @ B2 )
      = top_top_set_rat ) ).

% Un_UNIV_left
thf(fact_1044_image__Un,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_nat] :
      ( ( image_nat_rat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( sup_sup_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( image_nat_rat @ F2 @ B2 ) ) ) ).

% image_Un
thf(fact_1045_insert__UNIV,axiom,
    ! [X2: nat] :
      ( ( insert_nat @ X2 @ top_top_set_nat )
      = top_top_set_nat ) ).

% insert_UNIV
thf(fact_1046_insert__UNIV,axiom,
    ! [X2: rat] :
      ( ( insert_rat @ X2 @ top_top_set_rat )
      = top_top_set_rat ) ).

% insert_UNIV
thf(fact_1047_sup__cancel__left2,axiom,
    ! [X2: set_nat,A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ A ) @ ( sup_sup_set_nat @ X2 @ B ) )
      = top_top_set_nat ) ).

% sup_cancel_left2
thf(fact_1048_sup__cancel__left2,axiom,
    ! [X2: set_rat,A: set_rat,B: set_rat] :
      ( ( sup_sup_set_rat @ ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X2 ) @ A ) @ ( sup_sup_set_rat @ X2 @ B ) )
      = top_top_set_rat ) ).

% sup_cancel_left2
thf(fact_1049_sup__cancel__left1,axiom,
    ! [X2: set_nat,A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ A ) @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ B ) )
      = top_top_set_nat ) ).

% sup_cancel_left1
thf(fact_1050_sup__cancel__left1,axiom,
    ! [X2: set_rat,A: set_rat,B: set_rat] :
      ( ( sup_sup_set_rat @ ( sup_sup_set_rat @ X2 @ A ) @ ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X2 ) @ B ) )
      = top_top_set_rat ) ).

% sup_cancel_left1
thf(fact_1051_Inf__sup__eq__top__iff,axiom,
    ! [B2: set_set_nat,A: set_nat] :
      ( ( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ B2 ) @ A )
        = top_top_set_nat )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ B2 )
           => ( ( sup_sup_set_nat @ X @ A )
              = top_top_set_nat ) ) ) ) ).

% Inf_sup_eq_top_iff
thf(fact_1052_Inf__sup__eq__top__iff,axiom,
    ! [B2: set_set_rat,A: set_rat] :
      ( ( ( sup_sup_set_rat @ ( comple4298007329820168263et_rat @ B2 ) @ A )
        = top_top_set_rat )
      = ( ! [X: set_rat] :
            ( ( member_set_rat @ X @ B2 )
           => ( ( sup_sup_set_rat @ X @ A )
              = top_top_set_rat ) ) ) ) ).

% Inf_sup_eq_top_iff
thf(fact_1053_Compl__partition2,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ A2 )
      = top_top_set_nat ) ).

% Compl_partition2
thf(fact_1054_Compl__partition2,axiom,
    ! [A2: set_rat] :
      ( ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ A2 ) @ A2 )
      = top_top_set_rat ) ).

% Compl_partition2
thf(fact_1055_Compl__partition,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
      = top_top_set_nat ) ).

% Compl_partition
thf(fact_1056_Compl__partition,axiom,
    ! [A2: set_rat] :
      ( ( sup_sup_set_rat @ A2 @ ( uminus2201863774496077783et_rat @ A2 ) )
      = top_top_set_rat ) ).

% Compl_partition
thf(fact_1057_sup__shunt,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( ( sup_sup_set_nat @ X2 @ Y3 )
        = top_top_set_nat )
      = ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y3 ) ) ).

% sup_shunt
thf(fact_1058_sup__shunt,axiom,
    ! [X2: set_rat,Y3: set_rat] :
      ( ( ( sup_sup_set_rat @ X2 @ Y3 )
        = top_top_set_rat )
      = ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ X2 ) @ Y3 ) ) ).

% sup_shunt
thf(fact_1059_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_nat,X2: set_nat,Y3: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ X2 )
        = bot_bot_set_nat )
     => ( ( ( sup_sup_set_nat @ A @ X2 )
          = top_top_set_nat )
       => ( ( ( inf_inf_set_nat @ A @ Y3 )
            = bot_bot_set_nat )
         => ( ( ( sup_sup_set_nat @ A @ Y3 )
              = top_top_set_nat )
           => ( X2 = Y3 ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_1060_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_rat,X2: set_rat,Y3: set_rat] :
      ( ( ( inf_inf_set_rat @ A @ X2 )
        = bot_bot_set_rat )
     => ( ( ( sup_sup_set_rat @ A @ X2 )
          = top_top_set_rat )
       => ( ( ( inf_inf_set_rat @ A @ Y3 )
            = bot_bot_set_rat )
         => ( ( ( sup_sup_set_rat @ A @ Y3 )
              = top_top_set_rat )
           => ( X2 = Y3 ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_1061_range__eq__singletonD,axiom,
    ! [F2: nat > rat,A: rat,X2: nat] :
      ( ( ( image_nat_rat @ F2 @ top_top_set_nat )
        = ( insert_rat @ A @ bot_bot_set_rat ) )
     => ( ( F2 @ X2 )
        = A ) ) ).

% range_eq_singletonD
thf(fact_1062_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( ( inf_inf_set_nat @ X2 @ Y3 )
        = bot_bot_set_nat )
     => ( ( ( sup_sup_set_nat @ X2 @ Y3 )
          = top_top_set_nat )
       => ( ( uminus5710092332889474511et_nat @ X2 )
          = Y3 ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1063_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X2: set_rat,Y3: set_rat] :
      ( ( ( inf_inf_set_rat @ X2 @ Y3 )
        = bot_bot_set_rat )
     => ( ( ( sup_sup_set_rat @ X2 @ Y3 )
          = top_top_set_rat )
       => ( ( uminus2201863774496077783et_rat @ X2 )
          = Y3 ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1064_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea778851993438741648et_nat @ inf_inf_set_nat @ sup_sup_set_nat @ uminus5710092332889474511et_nat @ bot_bot_set_nat @ top_top_set_nat ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1065_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea6493995471900120728et_rat @ inf_inf_set_rat @ sup_sup_set_rat @ uminus2201863774496077783et_rat @ bot_bot_set_rat @ top_top_set_rat ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1066_inf__img__fin__dom_H,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( finite_finite_rat @ ( image_nat_rat @ F2 @ A2 ) )
     => ( ~ ( finite_finite_nat @ A2 )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ ( image_nat_rat @ F2 @ A2 ) )
            & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_rat @ F2 @ ( insert_rat @ X4 @ bot_bot_set_rat ) ) @ A2 ) ) ) ) ) ).

% inf_img_fin_dom'
thf(fact_1067_inf__img__fin__dom_H,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A2 ) )
     => ( ~ ( finite_finite_nat @ A2 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ ( image_nat_nat @ F2 @ A2 ) )
            & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ A2 ) ) ) ) ) ).

% inf_img_fin_dom'
thf(fact_1068_inf__img__fin__domE_H,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( finite_finite_rat @ ( image_nat_rat @ F2 @ A2 ) )
     => ( ~ ( finite_finite_nat @ A2 )
       => ~ ! [Y4: rat] :
              ( ( member_rat @ Y4 @ ( image_nat_rat @ F2 @ A2 ) )
             => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_rat @ F2 @ ( insert_rat @ Y4 @ bot_bot_set_rat ) ) @ A2 ) ) ) ) ) ).

% inf_img_fin_domE'
thf(fact_1069_inf__img__fin__domE_H,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A2 ) )
     => ( ~ ( finite_finite_nat @ A2 )
       => ~ ! [Y4: nat] :
              ( ( member_nat @ Y4 @ ( image_nat_nat @ F2 @ A2 ) )
             => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F2 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ A2 ) ) ) ) ) ).

% inf_img_fin_domE'
thf(fact_1070_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_1071_finite__Plus__UNIV__iff,axiom,
    ( ( finite2679478125380364318at_rat @ top_to281834657035230061at_rat )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_rat @ top_top_set_rat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_1072_finite__Plus__UNIV__iff,axiom,
    ( ( finite5871152039838895134at_nat @ top_to2261136804003905389at_nat )
    = ( ( finite_finite_rat @ top_top_set_rat )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_1073_finite__Plus__UNIV__iff,axiom,
    ( ( finite2362923481445498406at_rat @ top_to5104522503381003637at_rat )
    = ( ( finite_finite_rat @ top_top_set_rat )
      & ( finite_finite_rat @ top_top_set_rat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_1074_finite__imageI,axiom,
    ! [F4: set_nat,H: nat > rat] :
      ( ( finite_finite_nat @ F4 )
     => ( finite_finite_rat @ ( image_nat_rat @ H @ F4 ) ) ) ).

% finite_imageI
thf(fact_1075_finite__imageI,axiom,
    ! [F4: set_nat,H: nat > nat] :
      ( ( finite_finite_nat @ F4 )
     => ( finite_finite_nat @ ( image_nat_nat @ H @ F4 ) ) ) ).

% finite_imageI
thf(fact_1076_finite__Union,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ! [M4: set_nat] :
            ( ( member_set_nat @ M4 @ A2 )
           => ( finite_finite_nat @ M4 ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).

% finite_Union
thf(fact_1077_finite__Inter,axiom,
    ! [M: set_set_nat] :
      ( ? [X5: set_nat] :
          ( ( member_set_nat @ X5 @ M )
          & ( finite_finite_nat @ X5 ) )
     => ( finite_finite_nat @ ( comple7806235888213564991et_nat @ M ) ) ) ).

% finite_Inter
thf(fact_1078_finite__UN,axiom,
    ! [A2: set_nat,B2: nat > set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A2 )
             => ( finite_finite_nat @ ( B2 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1079_finite__compl,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ ( uminus5710092332889474511et_nat @ A2 ) )
        = ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_compl
thf(fact_1080_finite__compl,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( finite_finite_rat @ ( uminus2201863774496077783et_rat @ A2 ) )
        = ( finite_finite_rat @ top_top_set_rat ) ) ) ).

% finite_compl
thf(fact_1081_finite__UnionD,axiom,
    ! [A2: set_set_nat] :
      ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A2 ) )
     => ( finite1152437895449049373et_nat @ A2 ) ) ).

% finite_UnionD
thf(fact_1082_finite__bind,axiom,
    ! [S4: set_nat,F2: nat > set_nat] :
      ( ( finite_finite_nat @ S4 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ S4 )
           => ( finite_finite_nat @ ( F2 @ X4 ) ) )
       => ( finite_finite_nat @ ( bind_nat_nat @ S4 @ F2 ) ) ) ) ).

% finite_bind
thf(fact_1083_finite__subset__Union,axiom,
    ! [A2: set_nat,B5: set_set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B5 ) )
       => ~ ! [F5: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F5 )
             => ( ( ord_le6893508408891458716et_nat @ F5 @ B5 )
               => ~ ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ F5 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_1084_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_1085_finite__prod,axiom,
    ( ( finite2668982390342448306at_rat @ top_to7513191607651882425at_rat )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_rat @ top_top_set_rat ) ) ) ).

% finite_prod
thf(fact_1086_finite__prod,axiom,
    ( ( finite5860656304800979122at_nat @ top_to269121717765781945at_nat )
    = ( ( finite_finite_rat @ top_top_set_rat )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_1087_finite__prod,axiom,
    ( ( finite2352427746407582394at_rat @ top_to3112507417142880193at_rat )
    = ( ( finite_finite_rat @ top_top_set_rat )
      & ( finite_finite_rat @ top_top_set_rat ) ) ) ).

% finite_prod
thf(fact_1088_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_1089_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite_finite_rat @ top_top_set_rat )
     => ( finite2668982390342448306at_rat @ top_to7513191607651882425at_rat ) ) ) ).

% finite_Prod_UNIV
thf(fact_1090_finite__Prod__UNIV,axiom,
    ( ( finite_finite_rat @ top_top_set_rat )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite5860656304800979122at_nat @ top_to269121717765781945at_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_1091_finite__Prod__UNIV,axiom,
    ( ( finite_finite_rat @ top_top_set_rat )
   => ( ( finite_finite_rat @ top_top_set_rat )
     => ( finite2352427746407582394at_rat @ top_to3112507417142880193at_rat ) ) ) ).

% finite_Prod_UNIV
thf(fact_1092_infinite__UNIV__char__0,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_char_0
thf(fact_1093_infinite__UNIV__char__0,axiom,
    ~ ( finite_finite_rat @ top_top_set_rat ) ).

% infinite_UNIV_char_0
thf(fact_1094_ex__new__if__finite,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ top_top_set_nat )
     => ( ( finite_finite_nat @ A2 )
       => ? [A3: nat] :
            ~ ( member_nat @ A3 @ A2 ) ) ) ).

% ex_new_if_finite
thf(fact_1095_ex__new__if__finite,axiom,
    ! [A2: set_rat] :
      ( ~ ( finite_finite_rat @ top_top_set_rat )
     => ( ( finite_finite_rat @ A2 )
       => ? [A3: rat] :
            ~ ( member_rat @ A3 @ A2 ) ) ) ).

% ex_new_if_finite
thf(fact_1096_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_1097_Finite__Set_Ofinite__set,axiom,
    ( ( finite6867581373910428453et_rat @ top_top_set_set_rat )
    = ( finite_finite_rat @ top_top_set_rat ) ) ).

% Finite_Set.finite_set
thf(fact_1098_le__cSup__finite,axiom,
    ! [X6: set_nat,X2: nat] :
      ( ( finite_finite_nat @ X6 )
     => ( ( member_nat @ X2 @ X6 )
       => ( ord_less_eq_nat @ X2 @ ( complete_Sup_Sup_nat @ X6 ) ) ) ) ).

% le_cSup_finite
thf(fact_1099_finite__surj,axiom,
    ! [A2: set_nat,B2: set_rat,F2: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F2 @ A2 ) )
       => ( finite_finite_rat @ B2 ) ) ) ).

% finite_surj
thf(fact_1100_finite__surj,axiom,
    ! [A2: set_nat,B2: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A2 ) )
       => ( finite_finite_nat @ B2 ) ) ) ).

% finite_surj
thf(fact_1101_finite__subset__image,axiom,
    ! [B2: set_rat,F2: nat > rat,A2: set_nat] :
      ( ( finite_finite_rat @ B2 )
     => ( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F2 @ A2 ) )
       => ? [C3: set_nat] :
            ( ( ord_less_eq_set_nat @ C3 @ A2 )
            & ( finite_finite_nat @ C3 )
            & ( B2
              = ( image_nat_rat @ F2 @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1102_finite__subset__image,axiom,
    ! [B2: set_nat,F2: nat > nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A2 ) )
       => ? [C3: set_nat] :
            ( ( ord_less_eq_set_nat @ C3 @ A2 )
            & ( finite_finite_nat @ C3 )
            & ( B2
              = ( image_nat_nat @ F2 @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1103_ex__finite__subset__image,axiom,
    ! [F2: nat > rat,A2: set_nat,P: set_rat > $o] :
      ( ( ? [B4: set_rat] :
            ( ( finite_finite_rat @ B4 )
            & ( ord_less_eq_set_rat @ B4 @ ( image_nat_rat @ F2 @ A2 ) )
            & ( P @ B4 ) ) )
      = ( ? [B4: set_nat] :
            ( ( finite_finite_nat @ B4 )
            & ( ord_less_eq_set_nat @ B4 @ A2 )
            & ( P @ ( image_nat_rat @ F2 @ B4 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1104_ex__finite__subset__image,axiom,
    ! [F2: nat > nat,A2: set_nat,P: set_nat > $o] :
      ( ( ? [B4: set_nat] :
            ( ( finite_finite_nat @ B4 )
            & ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F2 @ A2 ) )
            & ( P @ B4 ) ) )
      = ( ? [B4: set_nat] :
            ( ( finite_finite_nat @ B4 )
            & ( ord_less_eq_set_nat @ B4 @ A2 )
            & ( P @ ( image_nat_nat @ F2 @ B4 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1105_all__finite__subset__image,axiom,
    ! [F2: nat > rat,A2: set_nat,P: set_rat > $o] :
      ( ( ! [B4: set_rat] :
            ( ( ( finite_finite_rat @ B4 )
              & ( ord_less_eq_set_rat @ B4 @ ( image_nat_rat @ F2 @ A2 ) ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_nat] :
            ( ( ( finite_finite_nat @ B4 )
              & ( ord_less_eq_set_nat @ B4 @ A2 ) )
           => ( P @ ( image_nat_rat @ F2 @ B4 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1106_all__finite__subset__image,axiom,
    ! [F2: nat > nat,A2: set_nat,P: set_nat > $o] :
      ( ( ! [B4: set_nat] :
            ( ( ( finite_finite_nat @ B4 )
              & ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F2 @ A2 ) ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_nat] :
            ( ( ( finite_finite_nat @ B4 )
              & ( ord_less_eq_set_nat @ B4 @ A2 ) )
           => ( P @ ( image_nat_nat @ F2 @ B4 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1107_cInf__le__finite,axiom,
    ! [X6: set_nat,X2: nat] :
      ( ( finite_finite_nat @ X6 )
     => ( ( member_nat @ X2 @ X6 )
       => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X6 ) @ X2 ) ) ) ).

% cInf_le_finite
thf(fact_1108_finite__vimageD,axiom,
    ! [H: nat > nat,F4: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ H @ F4 ) )
     => ( ( ( image_nat_nat @ H @ top_top_set_nat )
          = top_top_set_nat )
       => ( finite_finite_nat @ F4 ) ) ) ).

% finite_vimageD
thf(fact_1109_finite__vimageD,axiom,
    ! [H: nat > rat,F4: set_rat] :
      ( ( finite_finite_nat @ ( vimage_nat_rat @ H @ F4 ) )
     => ( ( ( image_nat_rat @ H @ top_top_set_nat )
          = top_top_set_rat )
       => ( finite_finite_rat @ F4 ) ) ) ).

% finite_vimageD
thf(fact_1110_finite__vimageD,axiom,
    ! [H: rat > nat,F4: set_nat] :
      ( ( finite_finite_rat @ ( vimage_rat_nat @ H @ F4 ) )
     => ( ( ( image_rat_nat @ H @ top_top_set_rat )
          = top_top_set_nat )
       => ( finite_finite_nat @ F4 ) ) ) ).

% finite_vimageD
thf(fact_1111_finite__vimageD,axiom,
    ! [H: rat > rat,F4: set_rat] :
      ( ( finite_finite_rat @ ( vimage_rat_rat @ H @ F4 ) )
     => ( ( ( image_rat_rat @ H @ top_top_set_rat )
          = top_top_set_rat )
       => ( finite_finite_rat @ F4 ) ) ) ).

% finite_vimageD
thf(fact_1112_finite__vimageD_H,axiom,
    ! [F2: nat > rat,A2: set_rat] :
      ( ( finite_finite_nat @ ( vimage_nat_rat @ F2 @ A2 ) )
     => ( ( ord_less_eq_set_rat @ A2 @ ( image_nat_rat @ F2 @ top_top_set_nat ) )
       => ( finite_finite_rat @ A2 ) ) ) ).

% finite_vimageD'
thf(fact_1113_finite__vimageD_H,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ F2 @ A2 ) )
     => ( ( ord_less_eq_set_nat @ A2 @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_vimageD'
thf(fact_1114_finite__vimageD_H,axiom,
    ! [F2: rat > nat,A2: set_nat] :
      ( ( finite_finite_rat @ ( vimage_rat_nat @ F2 @ A2 ) )
     => ( ( ord_less_eq_set_nat @ A2 @ ( image_rat_nat @ F2 @ top_top_set_rat ) )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_vimageD'
thf(fact_1115_inf__img__fin__domE,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( finite_finite_rat @ ( image_nat_rat @ F2 @ A2 ) )
     => ( ~ ( finite_finite_nat @ A2 )
       => ~ ! [Y4: rat] :
              ( ( member_rat @ Y4 @ ( image_nat_rat @ F2 @ A2 ) )
             => ( finite_finite_nat @ ( vimage_nat_rat @ F2 @ ( insert_rat @ Y4 @ bot_bot_set_rat ) ) ) ) ) ) ).

% inf_img_fin_domE
thf(fact_1116_inf__img__fin__domE,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A2 ) )
     => ( ~ ( finite_finite_nat @ A2 )
       => ~ ! [Y4: nat] :
              ( ( member_nat @ Y4 @ ( image_nat_nat @ F2 @ A2 ) )
             => ( finite_finite_nat @ ( vimage_nat_nat @ F2 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) ) ) ) ) ).

% inf_img_fin_domE
thf(fact_1117_inf__img__fin__dom,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( finite_finite_rat @ ( image_nat_rat @ F2 @ A2 ) )
     => ( ~ ( finite_finite_nat @ A2 )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ ( image_nat_rat @ F2 @ A2 ) )
            & ~ ( finite_finite_nat @ ( vimage_nat_rat @ F2 @ ( insert_rat @ X4 @ bot_bot_set_rat ) ) ) ) ) ) ).

% inf_img_fin_dom
thf(fact_1118_inf__img__fin__dom,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A2 ) )
     => ( ~ ( finite_finite_nat @ A2 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ ( image_nat_nat @ F2 @ A2 ) )
            & ~ ( finite_finite_nat @ ( vimage_nat_nat @ F2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ).

% inf_img_fin_dom
thf(fact_1119_finite__finite__vimage__IntI,axiom,
    ! [F4: set_nat,H: nat > nat,A2: set_nat] :
      ( ( finite_finite_nat @ F4 )
     => ( ! [Y4: nat] :
            ( ( member_nat @ Y4 @ F4 )
           => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ A2 ) ) )
       => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H @ F4 ) @ A2 ) ) ) ) ).

% finite_finite_vimage_IntI
thf(fact_1120_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_1121_finite__option__UNIV,axiom,
    ( ( finite2014924581280026175on_rat @ top_to2540212048668676366on_rat )
    = ( finite_finite_rat @ top_top_set_rat ) ) ).

% finite_option_UNIV
thf(fact_1122_INF__fold__inf,axiom,
    ! [A2: set_nat,F2: nat > set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F2 @ A2 ) )
        = ( finite5529483035118572448et_nat @ ( comp_s2351873599094224870at_nat @ inf_inf_set_nat @ F2 ) @ top_top_set_nat @ A2 ) ) ) ).

% INF_fold_inf
thf(fact_1123_INF__fold__inf,axiom,
    ! [A2: set_nat,F2: nat > set_rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( comple4298007329820168263et_rat @ ( image_nat_set_rat @ F2 @ A2 ) )
        = ( finite2021254476725175720et_rat @ ( comp_s6252137336010071550at_nat @ inf_inf_set_rat @ F2 ) @ top_top_set_rat @ A2 ) ) ) ).

% INF_fold_inf
thf(fact_1124_Inf__fold__inf,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( comple7806235888213564991et_nat @ A2 )
        = ( finite677925301803182934et_nat @ inf_inf_set_nat @ top_top_set_nat @ A2 ) ) ) ).

% Inf_fold_inf
thf(fact_1125_Inf__fold__inf,axiom,
    ! [A2: set_set_rat] :
      ( ( finite6867581373910428453et_rat @ A2 )
     => ( ( comple4298007329820168263et_rat @ A2 )
        = ( finite5923809206502920038et_rat @ inf_inf_set_rat @ top_top_set_rat @ A2 ) ) ) ).

% Inf_fold_inf
thf(fact_1126_bdd__above_OI2,axiom,
    ! [A2: set_nat,F2: nat > rat,M: rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_rat @ ( F2 @ X4 ) @ M ) )
     => ( condit1579696412822616692ve_rat @ ( image_nat_rat @ F2 @ A2 ) ) ) ).

% bdd_above.I2
thf(fact_1127_image__Pow__mono,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_rat] :
      ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ B2 )
     => ( ord_le513522071413781156et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F2 ) @ ( pow_nat @ A2 ) ) @ ( pow_rat @ B2 ) ) ) ).

% image_Pow_mono
thf(fact_1128_Pow__UNIV,axiom,
    ( ( pow_nat @ top_top_set_nat )
    = top_top_set_set_nat ) ).

% Pow_UNIV
thf(fact_1129_Pow__UNIV,axiom,
    ( ( pow_rat @ top_top_set_rat )
    = top_top_set_set_rat ) ).

% Pow_UNIV
thf(fact_1130_image__Pow__surj,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_rat] :
      ( ( ( image_nat_rat @ F2 @ A2 )
        = B2 )
     => ( ( image_4408659257933336347et_rat @ ( image_nat_rat @ F2 ) @ ( pow_nat @ A2 ) )
        = ( pow_rat @ B2 ) ) ) ).

% image_Pow_surj
thf(fact_1131_Diff__UNIV,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ top_top_set_nat )
      = bot_bot_set_nat ) ).

% Diff_UNIV
thf(fact_1132_Diff__UNIV,axiom,
    ! [A2: set_rat] :
      ( ( minus_minus_set_rat @ A2 @ top_top_set_rat )
      = bot_bot_set_rat ) ).

% Diff_UNIV
thf(fact_1133_Compl__eq__Diff__UNIV,axiom,
    ( uminus5710092332889474511et_nat
    = ( minus_minus_set_nat @ top_top_set_nat ) ) ).

% Compl_eq_Diff_UNIV
thf(fact_1134_Compl__eq__Diff__UNIV,axiom,
    ( uminus2201863774496077783et_rat
    = ( minus_minus_set_rat @ top_top_set_rat ) ) ).

% Compl_eq_Diff_UNIV
thf(fact_1135_image__diff__subset,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_rat @ ( minus_minus_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( image_nat_rat @ F2 @ B2 ) ) @ ( image_nat_rat @ F2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).

% image_diff_subset
thf(fact_1136_range__diff,axiom,
    ! [A: rat] :
      ( ( image_rat_rat @ ( minus_minus_rat @ A ) @ top_top_set_rat )
      = top_top_set_rat ) ).

% range_diff
thf(fact_1137_partition__on__inj__image,axiom,
    ! [A2: set_nat,P: set_set_nat,F2: nat > rat] :
      ( ( disjoi4774308525696689793on_nat @ A2 @ P )
     => ( ( inj_on_nat_rat @ F2 @ A2 )
       => ( disjoi4139178465610194057on_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( minus_5007325069933123869et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F2 ) @ P ) @ ( insert_set_rat @ bot_bot_set_rat @ bot_bot_set_set_rat ) ) ) ) ) ).

% partition_on_inj_image
thf(fact_1138_finite__image__iff,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( ( finite_finite_rat @ ( image_nat_rat @ F2 @ A2 ) )
        = ( finite_finite_nat @ A2 ) ) ) ).

% finite_image_iff
thf(fact_1139_finite__image__iff,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A2 )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A2 ) )
        = ( finite_finite_nat @ A2 ) ) ) ).

% finite_image_iff
thf(fact_1140_inj__on__insert,axiom,
    ! [F2: nat > rat,A: nat,A2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ ( insert_nat @ A @ A2 ) )
      = ( ( inj_on_nat_rat @ F2 @ A2 )
        & ~ ( member_rat @ ( F2 @ A ) @ ( image_nat_rat @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_1141_inj__on__image__Pow,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F2 ) @ ( pow_nat @ A2 ) ) ) ).

% inj_on_image_Pow
thf(fact_1142_infinite__iff__countable__subset,axiom,
    ! [S4: set_rat] :
      ( ( ~ ( finite_finite_rat @ S4 ) )
      = ( ? [F: nat > rat] :
            ( ( inj_on_nat_rat @ F @ top_top_set_nat )
            & ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ top_top_set_nat ) @ S4 ) ) ) ) ).

% infinite_iff_countable_subset
thf(fact_1143_infinite__iff__countable__subset,axiom,
    ! [S4: set_nat] :
      ( ( ~ ( finite_finite_nat @ S4 ) )
      = ( ? [F: nat > nat] :
            ( ( inj_on_nat_nat @ F @ top_top_set_nat )
            & ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ S4 ) ) ) ) ).

% infinite_iff_countable_subset
thf(fact_1144_infinite__countable__subset,axiom,
    ! [S4: set_rat] :
      ( ~ ( finite_finite_rat @ S4 )
     => ? [F6: nat > rat] :
          ( ( inj_on_nat_rat @ F6 @ top_top_set_nat )
          & ( ord_less_eq_set_rat @ ( image_nat_rat @ F6 @ top_top_set_nat ) @ S4 ) ) ) ).

% infinite_countable_subset
thf(fact_1145_infinite__countable__subset,axiom,
    ! [S4: set_nat] :
      ( ~ ( finite_finite_nat @ S4 )
     => ? [F6: nat > nat] :
          ( ( inj_on_nat_nat @ F6 @ top_top_set_nat )
          & ( ord_less_eq_set_nat @ ( image_nat_nat @ F6 @ top_top_set_nat ) @ S4 ) ) ) ).

% infinite_countable_subset
thf(fact_1146_finite__imageD,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( finite_finite_rat @ ( image_nat_rat @ F2 @ A2 ) )
     => ( ( inj_on_nat_rat @ F2 @ A2 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_imageD
thf(fact_1147_finite__imageD,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A2 ) )
     => ( ( inj_on_nat_nat @ F2 @ A2 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_imageD
thf(fact_1148_inj__compose,axiom,
    ! [F2: b > a,G2: c > b] :
      ( ( inj_on_b_a @ F2 @ top_top_set_b )
     => ( ( inj_on_c_b @ G2 @ top_top_set_c )
       => ( inj_on_c_a @ ( comp_b_a_c @ F2 @ G2 ) @ top_top_set_c ) ) ) ).

% inj_compose
thf(fact_1149_inj__compose,axiom,
    ! [F2: c > a,G2: c > c] :
      ( ( inj_on_c_a @ F2 @ top_top_set_c )
     => ( ( inj_on_c_c @ G2 @ top_top_set_c )
       => ( inj_on_c_a @ ( comp_c_a_c @ F2 @ G2 ) @ top_top_set_c ) ) ) ).

% inj_compose
thf(fact_1150_inj__compose,axiom,
    ! [F2: b > b,G2: c > b] :
      ( ( inj_on_b_b @ F2 @ top_top_set_b )
     => ( ( inj_on_c_b @ G2 @ top_top_set_c )
       => ( inj_on_c_b @ ( comp_b_b_c @ F2 @ G2 ) @ top_top_set_c ) ) ) ).

% inj_compose
thf(fact_1151_inj__compose,axiom,
    ! [F2: b > a,G2: b > b] :
      ( ( inj_on_b_a @ F2 @ top_top_set_b )
     => ( ( inj_on_b_b @ G2 @ top_top_set_b )
       => ( inj_on_b_a @ ( comp_b_a_b @ F2 @ G2 ) @ top_top_set_b ) ) ) ).

% inj_compose
thf(fact_1152_inj__compose,axiom,
    ! [F2: a > a,G2: c > a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( ( inj_on_c_a @ G2 @ top_top_set_c )
       => ( inj_on_c_a @ ( comp_a_a_c @ F2 @ G2 ) @ top_top_set_c ) ) ) ).

% inj_compose
thf(fact_1153_inj__compose,axiom,
    ! [F2: a > a,G2: b > a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( ( inj_on_b_a @ G2 @ top_top_set_b )
       => ( inj_on_b_a @ ( comp_a_a_b @ F2 @ G2 ) @ top_top_set_b ) ) ) ).

% inj_compose
thf(fact_1154_fun_Oinj__map,axiom,
    ! [F2: b > a] :
      ( ( inj_on_b_a @ F2 @ top_top_set_b )
     => ( inj_on_c_b_c_a @ ( comp_b_a_c @ F2 ) @ top_top_set_c_b ) ) ).

% fun.inj_map
thf(fact_1155_fun_Oinj__map,axiom,
    ! [F2: c > a] :
      ( ( inj_on_c_a @ F2 @ top_top_set_c )
     => ( inj_on_c_c_c_a @ ( comp_c_a_c @ F2 ) @ top_top_set_c_c ) ) ).

% fun.inj_map
thf(fact_1156_fun_Oinj__map,axiom,
    ! [F2: b > b] :
      ( ( inj_on_b_b @ F2 @ top_top_set_b )
     => ( inj_on_c_b_c_b @ ( comp_b_b_c @ F2 ) @ top_top_set_c_b ) ) ).

% fun.inj_map
thf(fact_1157_fun_Oinj__map,axiom,
    ! [F2: b > a] :
      ( ( inj_on_b_a @ F2 @ top_top_set_b )
     => ( inj_on_b_b_b_a @ ( comp_b_a_b @ F2 ) @ top_top_set_b_b ) ) ).

% fun.inj_map
thf(fact_1158_fun_Oinj__map,axiom,
    ! [F2: a > a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( inj_on_c_a_c_a @ ( comp_a_a_c @ F2 ) @ top_top_set_c_a ) ) ).

% fun.inj_map
thf(fact_1159_fun_Oinj__map,axiom,
    ! [F2: a > a] :
      ( ( inj_on_a_a @ F2 @ top_top_set_a )
     => ( inj_on_b_a_b_a @ ( comp_a_a_b @ F2 ) @ top_top_set_b_a ) ) ).

% fun.inj_map
thf(fact_1160_inj__image__mem__iff,axiom,
    ! [F2: nat > rat,A: nat,A2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
     => ( ( member_rat @ ( F2 @ A ) @ ( image_nat_rat @ F2 @ A2 ) )
        = ( member_nat @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_1161_inj__image__eq__iff,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
     => ( ( ( image_nat_rat @ F2 @ A2 )
          = ( image_nat_rat @ F2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% inj_image_eq_iff
thf(fact_1162_range__ex1__eq,axiom,
    ! [F2: nat > rat,B: rat] :
      ( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
     => ( ( member_rat @ B @ ( image_nat_rat @ F2 @ top_top_set_nat ) )
        = ( ? [X: nat] :
              ( ( B
                = ( F2 @ X ) )
              & ! [Y: nat] :
                  ( ( B
                    = ( F2 @ Y ) )
                 => ( Y = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1163_inj__on__image__Fpow,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F2 ) @ ( finite_Fpow_nat @ A2 ) ) ) ).

% inj_on_image_Fpow
thf(fact_1164_inj__on__image,axiom,
    ! [F2: nat > rat,A2: set_set_nat] :
      ( ( inj_on_nat_rat @ F2 @ ( comple7399068483239264473et_nat @ A2 ) )
     => ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F2 ) @ A2 ) ) ).

% inj_on_image
thf(fact_1165_inj__on__imageI2,axiom,
    ! [F3: b > a,F2: c > b,A2: set_c] :
      ( ( inj_on_c_a @ ( comp_b_a_c @ F3 @ F2 ) @ A2 )
     => ( inj_on_c_b @ F2 @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1166_inj__on__imageI2,axiom,
    ! [F3: c > a,F2: c > c,A2: set_c] :
      ( ( inj_on_c_a @ ( comp_c_a_c @ F3 @ F2 ) @ A2 )
     => ( inj_on_c_c @ F2 @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1167_inj__on__imageI2,axiom,
    ! [F3: b > b,F2: c > b,A2: set_c] :
      ( ( inj_on_c_b @ ( comp_b_b_c @ F3 @ F2 ) @ A2 )
     => ( inj_on_c_b @ F2 @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1168_inj__on__imageI2,axiom,
    ! [F3: b > a,F2: b > b,A2: set_b] :
      ( ( inj_on_b_a @ ( comp_b_a_b @ F3 @ F2 ) @ A2 )
     => ( inj_on_b_b @ F2 @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1169_inj__on__imageI2,axiom,
    ! [F3: a > a,F2: c > a,A2: set_c] :
      ( ( inj_on_c_a @ ( comp_a_a_c @ F3 @ F2 ) @ A2 )
     => ( inj_on_c_a @ F2 @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1170_inj__on__imageI2,axiom,
    ! [F3: a > a,F2: b > a,A2: set_b] :
      ( ( inj_on_b_a @ ( comp_a_a_b @ F3 @ F2 ) @ A2 )
     => ( inj_on_b_a @ F2 @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1171_inj__on__Un__image__eq__iff,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
     => ( ( ( image_nat_rat @ F2 @ A2 )
          = ( image_nat_rat @ F2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% inj_on_Un_image_eq_iff
thf(fact_1172_comp__inj__on,axiom,
    ! [F2: c > b,A2: set_c,G2: b > a] :
      ( ( inj_on_c_b @ F2 @ A2 )
     => ( ( inj_on_b_a @ G2 @ ( image_c_b @ F2 @ A2 ) )
       => ( inj_on_c_a @ ( comp_b_a_c @ G2 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1173_comp__inj__on,axiom,
    ! [F2: c > c,A2: set_c,G2: c > a] :
      ( ( inj_on_c_c @ F2 @ A2 )
     => ( ( inj_on_c_a @ G2 @ ( image_c_c @ F2 @ A2 ) )
       => ( inj_on_c_a @ ( comp_c_a_c @ G2 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1174_comp__inj__on,axiom,
    ! [F2: c > b,A2: set_c,G2: b > b] :
      ( ( inj_on_c_b @ F2 @ A2 )
     => ( ( inj_on_b_b @ G2 @ ( image_c_b @ F2 @ A2 ) )
       => ( inj_on_c_b @ ( comp_b_b_c @ G2 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1175_comp__inj__on,axiom,
    ! [F2: b > b,A2: set_b,G2: b > a] :
      ( ( inj_on_b_b @ F2 @ A2 )
     => ( ( inj_on_b_a @ G2 @ ( image_b_b @ F2 @ A2 ) )
       => ( inj_on_b_a @ ( comp_b_a_b @ G2 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1176_comp__inj__on,axiom,
    ! [F2: c > a,A2: set_c,G2: a > a] :
      ( ( inj_on_c_a @ F2 @ A2 )
     => ( ( inj_on_a_a @ G2 @ ( image_c_a @ F2 @ A2 ) )
       => ( inj_on_c_a @ ( comp_a_a_c @ G2 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1177_comp__inj__on,axiom,
    ! [F2: b > a,A2: set_b,G2: a > a] :
      ( ( inj_on_b_a @ F2 @ A2 )
     => ( ( inj_on_a_a @ G2 @ ( image_b_a @ F2 @ A2 ) )
       => ( inj_on_b_a @ ( comp_a_a_b @ G2 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1178_inj__on__imageI,axiom,
    ! [G2: b > a,F2: c > b,A2: set_c] :
      ( ( inj_on_c_a @ ( comp_b_a_c @ G2 @ F2 ) @ A2 )
     => ( inj_on_b_a @ G2 @ ( image_c_b @ F2 @ A2 ) ) ) ).

% inj_on_imageI
thf(fact_1179_inj__on__imageI,axiom,
    ! [G2: c > a,F2: c > c,A2: set_c] :
      ( ( inj_on_c_a @ ( comp_c_a_c @ G2 @ F2 ) @ A2 )
     => ( inj_on_c_a @ G2 @ ( image_c_c @ F2 @ A2 ) ) ) ).

% inj_on_imageI
thf(fact_1180_inj__on__imageI,axiom,
    ! [G2: b > b,F2: c > b,A2: set_c] :
      ( ( inj_on_c_b @ ( comp_b_b_c @ G2 @ F2 ) @ A2 )
     => ( inj_on_b_b @ G2 @ ( image_c_b @ F2 @ A2 ) ) ) ).

% inj_on_imageI
thf(fact_1181_inj__on__imageI,axiom,
    ! [G2: b > a,F2: b > b,A2: set_b] :
      ( ( inj_on_b_a @ ( comp_b_a_b @ G2 @ F2 ) @ A2 )
     => ( inj_on_b_a @ G2 @ ( image_b_b @ F2 @ A2 ) ) ) ).

% inj_on_imageI
thf(fact_1182_inj__on__imageI,axiom,
    ! [G2: a > a,F2: c > a,A2: set_c] :
      ( ( inj_on_c_a @ ( comp_a_a_c @ G2 @ F2 ) @ A2 )
     => ( inj_on_a_a @ G2 @ ( image_c_a @ F2 @ A2 ) ) ) ).

% inj_on_imageI
thf(fact_1183_inj__on__imageI,axiom,
    ! [G2: a > a,F2: b > a,A2: set_b] :
      ( ( inj_on_b_a @ ( comp_a_a_b @ G2 @ F2 ) @ A2 )
     => ( inj_on_a_a @ G2 @ ( image_b_a @ F2 @ A2 ) ) ) ).

% inj_on_imageI
thf(fact_1184_comp__inj__on__iff,axiom,
    ! [F2: c > b,A2: set_c,F3: b > a] :
      ( ( inj_on_c_b @ F2 @ A2 )
     => ( ( inj_on_b_a @ F3 @ ( image_c_b @ F2 @ A2 ) )
        = ( inj_on_c_a @ ( comp_b_a_c @ F3 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on_iff
thf(fact_1185_comp__inj__on__iff,axiom,
    ! [F2: c > c,A2: set_c,F3: c > a] :
      ( ( inj_on_c_c @ F2 @ A2 )
     => ( ( inj_on_c_a @ F3 @ ( image_c_c @ F2 @ A2 ) )
        = ( inj_on_c_a @ ( comp_c_a_c @ F3 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on_iff
thf(fact_1186_comp__inj__on__iff,axiom,
    ! [F2: c > b,A2: set_c,F3: b > b] :
      ( ( inj_on_c_b @ F2 @ A2 )
     => ( ( inj_on_b_b @ F3 @ ( image_c_b @ F2 @ A2 ) )
        = ( inj_on_c_b @ ( comp_b_b_c @ F3 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on_iff
thf(fact_1187_comp__inj__on__iff,axiom,
    ! [F2: b > b,A2: set_b,F3: b > a] :
      ( ( inj_on_b_b @ F2 @ A2 )
     => ( ( inj_on_b_a @ F3 @ ( image_b_b @ F2 @ A2 ) )
        = ( inj_on_b_a @ ( comp_b_a_b @ F3 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on_iff
thf(fact_1188_comp__inj__on__iff,axiom,
    ! [F2: c > a,A2: set_c,F3: a > a] :
      ( ( inj_on_c_a @ F2 @ A2 )
     => ( ( inj_on_a_a @ F3 @ ( image_c_a @ F2 @ A2 ) )
        = ( inj_on_c_a @ ( comp_a_a_c @ F3 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on_iff
thf(fact_1189_comp__inj__on__iff,axiom,
    ! [F2: b > a,A2: set_b,F3: a > a] :
      ( ( inj_on_b_a @ F2 @ A2 )
     => ( ( inj_on_a_a @ F3 @ ( image_b_a @ F2 @ A2 ) )
        = ( inj_on_b_a @ ( comp_a_a_b @ F3 @ F2 ) @ A2 ) ) ) ).

% comp_inj_on_iff
thf(fact_1190_inj__img__insertE,axiom,
    ! [F2: nat > rat,A2: set_nat,X2: rat,B2: set_rat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( ~ ( member_rat @ X2 @ B2 )
       => ( ( ( insert_rat @ X2 @ B2 )
            = ( image_nat_rat @ F2 @ A2 ) )
         => ~ ! [X7: nat,A6: set_nat] :
                ( ~ ( member_nat @ X7 @ A6 )
               => ( ( A2
                    = ( insert_nat @ X7 @ A6 ) )
                 => ( ( X2
                      = ( F2 @ X7 ) )
                   => ( B2
                     != ( image_nat_rat @ F2 @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1191_subset__image__inj,axiom,
    ! [S4: set_rat,F2: nat > rat,T2: set_nat] :
      ( ( ord_less_eq_set_rat @ S4 @ ( image_nat_rat @ F2 @ T2 ) )
      = ( ? [U: set_nat] :
            ( ( ord_less_eq_set_nat @ U @ T2 )
            & ( inj_on_nat_rat @ F2 @ U )
            & ( S4
              = ( image_nat_rat @ F2 @ U ) ) ) ) ) ).

% subset_image_inj
thf(fact_1192_inj__on__image__eq__iff,axiom,
    ! [F2: nat > rat,C2: set_nat,A2: set_nat,B2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ C2 )
     => ( ( ord_less_eq_set_nat @ A2 @ C2 )
       => ( ( ord_less_eq_set_nat @ B2 @ C2 )
         => ( ( ( image_nat_rat @ F2 @ A2 )
              = ( image_nat_rat @ F2 @ B2 ) )
            = ( A2 = B2 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1193_inj__on__image__mem__iff,axiom,
    ! [F2: nat > rat,B2: set_nat,A: nat,A2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ B2 )
     => ( ( member_nat @ A @ B2 )
       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
         => ( ( member_rat @ ( F2 @ A ) @ ( image_nat_rat @ F2 @ A2 ) )
            = ( member_nat @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1194_finite__UNIV__inj__surj,axiom,
    ! [F2: nat > nat] :
      ( ( finite_finite_nat @ top_top_set_nat )
     => ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
       => ( ( image_nat_nat @ F2 @ top_top_set_nat )
          = top_top_set_nat ) ) ) ).

% finite_UNIV_inj_surj
thf(fact_1195_finite__UNIV__inj__surj,axiom,
    ! [F2: rat > rat] :
      ( ( finite_finite_rat @ top_top_set_rat )
     => ( ( inj_on_rat_rat @ F2 @ top_top_set_rat )
       => ( ( image_rat_rat @ F2 @ top_top_set_rat )
          = top_top_set_rat ) ) ) ).

% finite_UNIV_inj_surj
thf(fact_1196_finite__UNIV__surj__inj,axiom,
    ! [F2: nat > nat] :
      ( ( finite_finite_nat @ top_top_set_nat )
     => ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
          = top_top_set_nat )
       => ( inj_on_nat_nat @ F2 @ top_top_set_nat ) ) ) ).

% finite_UNIV_surj_inj
thf(fact_1197_finite__UNIV__surj__inj,axiom,
    ! [F2: rat > rat] :
      ( ( finite_finite_rat @ top_top_set_rat )
     => ( ( ( image_rat_rat @ F2 @ top_top_set_rat )
          = top_top_set_rat )
       => ( inj_on_rat_rat @ F2 @ top_top_set_rat ) ) ) ).

% finite_UNIV_surj_inj
thf(fact_1198_inj__image__subset__iff,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
     => ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( image_nat_rat @ F2 @ B2 ) )
        = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% inj_image_subset_iff
thf(fact_1199_inj__on__iff__surj,axiom,
    ! [A2: set_rat,A7: set_nat] :
      ( ( A2 != bot_bot_set_rat )
     => ( ( ? [F: rat > nat] :
              ( ( inj_on_rat_nat @ F @ A2 )
              & ( ord_less_eq_set_nat @ ( image_rat_nat @ F @ A2 ) @ A7 ) ) )
        = ( ? [G: nat > rat] :
              ( ( image_nat_rat @ G @ A7 )
              = A2 ) ) ) ) ).

% inj_on_iff_surj
thf(fact_1200_inj__on__iff__surj,axiom,
    ! [A2: set_nat,A7: set_rat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ( ? [F: nat > rat] :
              ( ( inj_on_nat_rat @ F @ A2 )
              & ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A2 ) @ A7 ) ) )
        = ( ? [G: rat > nat] :
              ( ( image_rat_nat @ G @ A7 )
              = A2 ) ) ) ) ).

% inj_on_iff_surj
thf(fact_1201_endo__inj__surj,axiom,
    ! [A2: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ A2 )
       => ( ( inj_on_nat_nat @ F2 @ A2 )
         => ( ( image_nat_nat @ F2 @ A2 )
            = A2 ) ) ) ) ).

% endo_inj_surj
thf(fact_1202_Finite__Set_Oinj__on__finite,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_rat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ B2 )
       => ( ( finite_finite_rat @ B2 )
         => ( finite_finite_nat @ A2 ) ) ) ) ).

% Finite_Set.inj_on_finite
thf(fact_1203_Finite__Set_Oinj__on__finite,axiom,
    ! [F2: nat > nat,A2: set_nat,B2: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A2 )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B2 )
       => ( ( finite_finite_nat @ B2 )
         => ( finite_finite_nat @ A2 ) ) ) ) ).

% Finite_Set.inj_on_finite
thf(fact_1204_finite__surj__inj,axiom,
    ! [A2: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( image_nat_nat @ F2 @ A2 ) )
       => ( inj_on_nat_nat @ F2 @ A2 ) ) ) ).

% finite_surj_inj
thf(fact_1205_image__Int,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
     => ( ( image_nat_rat @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
        = ( inf_inf_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( image_nat_rat @ F2 @ B2 ) ) ) ) ).

% image_Int
thf(fact_1206_inj__on__image__Int,axiom,
    ! [F2: nat > rat,C2: set_nat,A2: set_nat,B2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ C2 )
     => ( ( ord_less_eq_set_nat @ A2 @ C2 )
       => ( ( ord_less_eq_set_nat @ B2 @ C2 )
         => ( ( image_nat_rat @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
            = ( inf_inf_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( image_nat_rat @ F2 @ B2 ) ) ) ) ) ) ).

% inj_on_image_Int
thf(fact_1207_image__set__diff,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
     => ( ( image_nat_rat @ F2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
        = ( minus_minus_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( image_nat_rat @ F2 @ B2 ) ) ) ) ).

% image_set_diff
thf(fact_1208_inj__on__image__set__diff,axiom,
    ! [F2: nat > rat,C2: set_nat,A2: set_nat,B2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ C2 )
     => ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C2 )
       => ( ( ord_less_eq_set_nat @ B2 @ C2 )
         => ( ( image_nat_rat @ F2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
            = ( minus_minus_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( image_nat_rat @ F2 @ B2 ) ) ) ) ) ) ).

% inj_on_image_set_diff
thf(fact_1209_inj__vimage__image__eq,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
     => ( ( vimage_nat_rat @ F2 @ ( image_nat_rat @ F2 @ A2 ) )
        = A2 ) ) ).

% inj_vimage_image_eq
thf(fact_1210_finite__vimageI,axiom,
    ! [F4: set_nat,H: nat > nat] :
      ( ( finite_finite_nat @ F4 )
     => ( ( inj_on_nat_nat @ H @ top_top_set_nat )
       => ( finite_finite_nat @ ( vimage_nat_nat @ H @ F4 ) ) ) ) ).

% finite_vimageI
thf(fact_1211_finite__vimageI,axiom,
    ! [F4: set_nat,H: rat > nat] :
      ( ( finite_finite_nat @ F4 )
     => ( ( inj_on_rat_nat @ H @ top_top_set_rat )
       => ( finite_finite_rat @ ( vimage_rat_nat @ H @ F4 ) ) ) ) ).

% finite_vimageI
thf(fact_1212_finite__vimage__IntI,axiom,
    ! [F4: set_nat,H: nat > nat,A2: set_nat] :
      ( ( finite_finite_nat @ F4 )
     => ( ( inj_on_nat_nat @ H @ A2 )
       => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H @ F4 ) @ A2 ) ) ) ) ).

% finite_vimage_IntI
thf(fact_1213_disjnt__inj__on__iff,axiom,
    ! [F2: nat > rat,A8: set_set_nat,X6: set_nat,Y5: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ ( comple7399068483239264473et_nat @ A8 ) )
     => ( ( member_set_nat @ X6 @ A8 )
       => ( ( member_set_nat @ Y5 @ A8 )
         => ( ( disjnt_rat @ ( image_nat_rat @ F2 @ X6 ) @ ( image_nat_rat @ F2 @ Y5 ) )
            = ( disjnt_nat @ X6 @ Y5 ) ) ) ) ) ).

% disjnt_inj_on_iff
thf(fact_1214_partition__on__map,axiom,
    ! [F2: nat > rat,A2: set_nat,P: set_set_nat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( ( disjoi4774308525696689793on_nat @ A2 @ P )
       => ( disjoi4139178465610194057on_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F2 ) @ P ) ) ) ) ).

% partition_on_map
thf(fact_1215_vimage__subsetI,axiom,
    ! [F2: nat > rat,B2: set_rat,A2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
     => ( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F2 @ A2 ) )
       => ( ord_less_eq_set_nat @ ( vimage_nat_rat @ F2 @ B2 ) @ A2 ) ) ) ).

% vimage_subsetI
thf(fact_1216_inj__image__Compl__subset,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
     => ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) ) @ ( uminus2201863774496077783et_rat @ ( image_nat_rat @ F2 @ A2 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_1217_inj__on__Un,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( ( inj_on_nat_rat @ F2 @ A2 )
        & ( inj_on_nat_rat @ F2 @ B2 )
        & ( ( inf_inf_set_rat @ ( image_nat_rat @ F2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( image_nat_rat @ F2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) )
          = bot_bot_set_rat ) ) ) ).

% inj_on_Un
thf(fact_1218_subset__imageE__inj,axiom,
    ! [B2: set_rat,F2: nat > rat,A2: set_nat] :
      ( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F2 @ A2 ) )
     => ~ ! [C3: set_nat] :
            ( ( ord_less_eq_set_nat @ C3 @ A2 )
           => ( ( B2
                = ( image_nat_rat @ F2 @ C3 ) )
             => ~ ( inj_on_nat_rat @ F2 @ C3 ) ) ) ) ).

% subset_imageE_inj
thf(fact_1219_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_1220_range__inj__infinite,axiom,
    ! [F2: nat > rat] :
      ( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
     => ~ ( finite_finite_rat @ ( image_nat_rat @ F2 @ top_top_set_nat ) ) ) ).

% range_inj_infinite
thf(fact_1221_range__inj__infinite,axiom,
    ! [F2: nat > nat] :
      ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
     => ~ ( finite_finite_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ) ).

% range_inj_infinite
thf(fact_1222_inj__on__fun__updI,axiom,
    ! [F2: nat > rat,A2: set_nat,Y3: rat,X2: nat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( ~ ( member_rat @ Y3 @ ( image_nat_rat @ F2 @ A2 ) )
       => ( inj_on_nat_rat @ ( fun_upd_nat_rat @ F2 @ X2 @ Y3 ) @ A2 ) ) ) ).

% inj_on_fun_updI
thf(fact_1223_fun__upd__comp,axiom,
    ! [F2: b > a,G2: c > b,X2: c,Y3: b] :
      ( ( comp_b_a_c @ F2 @ ( fun_upd_c_b @ G2 @ X2 @ Y3 ) )
      = ( fun_upd_c_a @ ( comp_b_a_c @ F2 @ G2 ) @ X2 @ ( F2 @ Y3 ) ) ) ).

% fun_upd_comp
thf(fact_1224_fun__upd__comp,axiom,
    ! [F2: c > a,G2: c > c,X2: c,Y3: c] :
      ( ( comp_c_a_c @ F2 @ ( fun_upd_c_c @ G2 @ X2 @ Y3 ) )
      = ( fun_upd_c_a @ ( comp_c_a_c @ F2 @ G2 ) @ X2 @ ( F2 @ Y3 ) ) ) ).

% fun_upd_comp
thf(fact_1225_fun__upd__comp,axiom,
    ! [F2: b > b,G2: c > b,X2: c,Y3: b] :
      ( ( comp_b_b_c @ F2 @ ( fun_upd_c_b @ G2 @ X2 @ Y3 ) )
      = ( fun_upd_c_b @ ( comp_b_b_c @ F2 @ G2 ) @ X2 @ ( F2 @ Y3 ) ) ) ).

% fun_upd_comp
thf(fact_1226_fun__upd__comp,axiom,
    ! [F2: b > a,G2: b > b,X2: b,Y3: b] :
      ( ( comp_b_a_b @ F2 @ ( fun_upd_b_b @ G2 @ X2 @ Y3 ) )
      = ( fun_upd_b_a @ ( comp_b_a_b @ F2 @ G2 ) @ X2 @ ( F2 @ Y3 ) ) ) ).

% fun_upd_comp
thf(fact_1227_fun__upd__comp,axiom,
    ! [F2: a > a,G2: c > a,X2: c,Y3: a] :
      ( ( comp_a_a_c @ F2 @ ( fun_upd_c_a @ G2 @ X2 @ Y3 ) )
      = ( fun_upd_c_a @ ( comp_a_a_c @ F2 @ G2 ) @ X2 @ ( F2 @ Y3 ) ) ) ).

% fun_upd_comp
thf(fact_1228_fun__upd__comp,axiom,
    ! [F2: a > a,G2: b > a,X2: b,Y3: a] :
      ( ( comp_a_a_b @ F2 @ ( fun_upd_b_a @ G2 @ X2 @ Y3 ) )
      = ( fun_upd_b_a @ ( comp_a_a_b @ F2 @ G2 ) @ X2 @ ( F2 @ Y3 ) ) ) ).

% fun_upd_comp
thf(fact_1229_nat__not__finite,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% nat_not_finite
thf(fact_1230_fun__upd__image,axiom,
    ! [X2: nat,A2: set_nat,F2: nat > rat,Y3: rat] :
      ( ( ( member_nat @ X2 @ A2 )
       => ( ( image_nat_rat @ ( fun_upd_nat_rat @ F2 @ X2 @ Y3 ) @ A2 )
          = ( insert_rat @ Y3 @ ( image_nat_rat @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) )
      & ( ~ ( member_nat @ X2 @ A2 )
       => ( ( image_nat_rat @ ( fun_upd_nat_rat @ F2 @ X2 @ Y3 ) @ A2 )
          = ( image_nat_rat @ F2 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_1231_pairwise__imageI,axiom,
    ! [A2: set_nat,F2: nat > rat,P: rat > rat > $o] :
      ( ! [X4: nat,Y4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( member_nat @ Y4 @ A2 )
           => ( ( X4 != Y4 )
             => ( ( ( F2 @ X4 )
                 != ( F2 @ Y4 ) )
               => ( P @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) ) )
     => ( pairwise_rat @ P @ ( image_nat_rat @ F2 @ A2 ) ) ) ).

% pairwise_imageI
thf(fact_1232_disjoint__image,axiom,
    ! [F2: nat > rat,A2: set_set_nat] :
      ( ( inj_on_nat_rat @ F2 @ ( comple7399068483239264473et_nat @ A2 ) )
     => ( ( pairwise_set_nat @ disjnt_nat @ A2 )
       => ( pairwise_set_rat @ disjnt_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F2 ) @ A2 ) ) ) ) ).

% disjoint_image
thf(fact_1233_infinite__disjoint__family__imp__infinite__UNION,axiom,
    ! [A2: set_nat,F2: nat > set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( F2 @ X4 )
             != bot_bot_set_nat ) )
       => ( ( disjoi6798895846410478970at_nat @ F2 @ A2 )
         => ~ ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ) ) ) ).

% infinite_disjoint_family_imp_infinite_UNION
thf(fact_1234_the__inv__into__onto,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( ( image_rat_nat @ ( the_inv_into_nat_rat @ A2 @ F2 ) @ ( image_nat_rat @ F2 @ A2 ) )
        = A2 ) ) ).

% the_inv_into_onto
thf(fact_1235_the__inv__into__onto,axiom,
    ! [F2: rat > nat,A2: set_rat] :
      ( ( inj_on_rat_nat @ F2 @ A2 )
     => ( ( image_nat_rat @ ( the_inv_into_rat_nat @ A2 @ F2 ) @ ( image_rat_nat @ F2 @ A2 ) )
        = A2 ) ) ).

% the_inv_into_onto
thf(fact_1236_inj__on__the__inv__into,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( inj_on_rat_nat @ ( the_inv_into_nat_rat @ A2 @ F2 ) @ ( image_nat_rat @ F2 @ A2 ) ) ) ).

% inj_on_the_inv_into
thf(fact_1237_f__the__inv__into__f,axiom,
    ! [F2: nat > rat,A2: set_nat,Y3: rat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( ( member_rat @ Y3 @ ( image_nat_rat @ F2 @ A2 ) )
       => ( ( F2 @ ( the_inv_into_nat_rat @ A2 @ F2 @ Y3 ) )
          = Y3 ) ) ) ).

% f_the_inv_into_f
thf(fact_1238_the__inv__into__into,axiom,
    ! [F2: nat > rat,A2: set_nat,X2: rat,B2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( ( member_rat @ X2 @ ( image_nat_rat @ F2 @ A2 ) )
       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
         => ( member_nat @ ( the_inv_into_nat_rat @ A2 @ F2 @ X2 ) @ B2 ) ) ) ) ).

% the_inv_into_into
thf(fact_1239_the__inv__into__comp,axiom,
    ! [F2: b > c,G2: a > b,A2: set_a,X2: c] :
      ( ( inj_on_b_c @ F2 @ ( image_a_b @ G2 @ A2 ) )
     => ( ( inj_on_a_b @ G2 @ A2 )
       => ( ( member_c @ X2 @ ( image_b_c @ F2 @ ( image_a_b @ G2 @ A2 ) ) )
         => ( ( the_inv_into_a_c @ A2 @ ( comp_b_c_a @ F2 @ G2 ) @ X2 )
            = ( comp_b_a_c @ ( the_inv_into_a_b @ A2 @ G2 ) @ ( the_inv_into_b_c @ ( image_a_b @ G2 @ A2 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1240_the__inv__into__comp,axiom,
    ! [F2: c > c,G2: a > c,A2: set_a,X2: c] :
      ( ( inj_on_c_c @ F2 @ ( image_a_c @ G2 @ A2 ) )
     => ( ( inj_on_a_c @ G2 @ A2 )
       => ( ( member_c @ X2 @ ( image_c_c @ F2 @ ( image_a_c @ G2 @ A2 ) ) )
         => ( ( the_inv_into_a_c @ A2 @ ( comp_c_c_a @ F2 @ G2 ) @ X2 )
            = ( comp_c_a_c @ ( the_inv_into_a_c @ A2 @ G2 ) @ ( the_inv_into_c_c @ ( image_a_c @ G2 @ A2 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1241_the__inv__into__comp,axiom,
    ! [F2: b > c,G2: b > b,A2: set_b,X2: c] :
      ( ( inj_on_b_c @ F2 @ ( image_b_b @ G2 @ A2 ) )
     => ( ( inj_on_b_b @ G2 @ A2 )
       => ( ( member_c @ X2 @ ( image_b_c @ F2 @ ( image_b_b @ G2 @ A2 ) ) )
         => ( ( the_inv_into_b_c @ A2 @ ( comp_b_c_b @ F2 @ G2 ) @ X2 )
            = ( comp_b_b_c @ ( the_inv_into_b_b @ A2 @ G2 ) @ ( the_inv_into_b_c @ ( image_b_b @ G2 @ A2 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1242_the__inv__into__comp,axiom,
    ! [F2: b > b,G2: a > b,A2: set_a,X2: b] :
      ( ( inj_on_b_b @ F2 @ ( image_a_b @ G2 @ A2 ) )
     => ( ( inj_on_a_b @ G2 @ A2 )
       => ( ( member_b @ X2 @ ( image_b_b @ F2 @ ( image_a_b @ G2 @ A2 ) ) )
         => ( ( the_inv_into_a_b @ A2 @ ( comp_b_b_a @ F2 @ G2 ) @ X2 )
            = ( comp_b_a_b @ ( the_inv_into_a_b @ A2 @ G2 ) @ ( the_inv_into_b_b @ ( image_a_b @ G2 @ A2 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1243_the__inv__into__comp,axiom,
    ! [F2: a > c,G2: a > a,A2: set_a,X2: c] :
      ( ( inj_on_a_c @ F2 @ ( image_a_a @ G2 @ A2 ) )
     => ( ( inj_on_a_a @ G2 @ A2 )
       => ( ( member_c @ X2 @ ( image_a_c @ F2 @ ( image_a_a @ G2 @ A2 ) ) )
         => ( ( the_inv_into_a_c @ A2 @ ( comp_a_c_a @ F2 @ G2 ) @ X2 )
            = ( comp_a_a_c @ ( the_inv_into_a_a @ A2 @ G2 ) @ ( the_inv_into_a_c @ ( image_a_a @ G2 @ A2 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1244_the__inv__into__comp,axiom,
    ! [F2: a > b,G2: a > a,A2: set_a,X2: b] :
      ( ( inj_on_a_b @ F2 @ ( image_a_a @ G2 @ A2 ) )
     => ( ( inj_on_a_a @ G2 @ A2 )
       => ( ( member_b @ X2 @ ( image_a_b @ F2 @ ( image_a_a @ G2 @ A2 ) ) )
         => ( ( the_inv_into_a_b @ A2 @ ( comp_a_b_a @ F2 @ G2 ) @ X2 )
            = ( comp_a_a_b @ ( the_inv_into_a_a @ A2 @ G2 ) @ ( the_inv_into_a_b @ ( image_a_a @ G2 @ A2 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1245_the__inv__into__comp,axiom,
    ! [F2: b > a,G2: c > b,A2: set_c,X2: a] :
      ( ( inj_on_b_a @ F2 @ ( image_c_b @ G2 @ A2 ) )
     => ( ( inj_on_c_b @ G2 @ A2 )
       => ( ( member_a @ X2 @ ( image_b_a @ F2 @ ( image_c_b @ G2 @ A2 ) ) )
         => ( ( the_inv_into_c_a @ A2 @ ( comp_b_a_c @ F2 @ G2 ) @ X2 )
            = ( comp_b_c_a @ ( the_inv_into_c_b @ A2 @ G2 ) @ ( the_inv_into_b_a @ ( image_c_b @ G2 @ A2 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1246_the__inv__into__comp,axiom,
    ! [F2: c > a,G2: c > c,A2: set_c,X2: a] :
      ( ( inj_on_c_a @ F2 @ ( image_c_c @ G2 @ A2 ) )
     => ( ( inj_on_c_c @ G2 @ A2 )
       => ( ( member_a @ X2 @ ( image_c_a @ F2 @ ( image_c_c @ G2 @ A2 ) ) )
         => ( ( the_inv_into_c_a @ A2 @ ( comp_c_a_c @ F2 @ G2 ) @ X2 )
            = ( comp_c_c_a @ ( the_inv_into_c_c @ A2 @ G2 ) @ ( the_inv_into_c_a @ ( image_c_c @ G2 @ A2 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1247_the__inv__into__comp,axiom,
    ! [F2: b > b,G2: c > b,A2: set_c,X2: b] :
      ( ( inj_on_b_b @ F2 @ ( image_c_b @ G2 @ A2 ) )
     => ( ( inj_on_c_b @ G2 @ A2 )
       => ( ( member_b @ X2 @ ( image_b_b @ F2 @ ( image_c_b @ G2 @ A2 ) ) )
         => ( ( the_inv_into_c_b @ A2 @ ( comp_b_b_c @ F2 @ G2 ) @ X2 )
            = ( comp_b_c_b @ ( the_inv_into_c_b @ A2 @ G2 ) @ ( the_inv_into_b_b @ ( image_c_b @ G2 @ A2 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1248_the__inv__into__comp,axiom,
    ! [F2: b > a,G2: b > b,A2: set_b,X2: a] :
      ( ( inj_on_b_a @ F2 @ ( image_b_b @ G2 @ A2 ) )
     => ( ( inj_on_b_b @ G2 @ A2 )
       => ( ( member_a @ X2 @ ( image_b_a @ F2 @ ( image_b_b @ G2 @ A2 ) ) )
         => ( ( the_inv_into_b_a @ A2 @ ( comp_b_a_b @ F2 @ G2 ) @ X2 )
            = ( comp_b_b_a @ ( the_inv_into_b_b @ A2 @ G2 ) @ ( the_inv_into_b_a @ ( image_b_b @ G2 @ A2 ) @ F2 ) @ X2 ) ) ) ) ) ).

% the_inv_into_comp
thf(fact_1249_card__vimage__inj,axiom,
    ! [F2: nat > rat,A2: set_rat] :
      ( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
     => ( ( ord_less_eq_set_rat @ A2 @ ( image_nat_rat @ F2 @ top_top_set_nat ) )
       => ( ( finite_card_nat @ ( vimage_nat_rat @ F2 @ A2 ) )
          = ( finite_card_rat @ A2 ) ) ) ) ).

% card_vimage_inj
thf(fact_1250_surj__from__nat,axiom,
    ( ( image_nat_nat @ from_nat_nat @ top_top_set_nat )
    = top_top_set_nat ) ).

% surj_from_nat
thf(fact_1251_surj__from__nat,axiom,
    ( ( image_nat_rat @ from_nat_rat @ top_top_set_nat )
    = top_top_set_rat ) ).

% surj_from_nat
thf(fact_1252_ex__inj,axiom,
    ? [To_nat: nat > nat] : ( inj_on_nat_nat @ To_nat @ top_top_set_nat ) ).

% ex_inj
thf(fact_1253_ex__inj,axiom,
    ? [To_nat: rat > nat] : ( inj_on_rat_nat @ To_nat @ top_top_set_rat ) ).

% ex_inj
thf(fact_1254_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ top_top_set_nat )
     => ( ( ( finite_card_nat @ A2 )
          = ( finite_card_nat @ top_top_set_nat ) )
       => ( A2 = top_top_set_nat ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
thf(fact_1255_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ top_top_set_rat )
     => ( ( ( finite_card_rat @ A2 )
          = ( finite_card_rat @ top_top_set_rat ) )
       => ( A2 = top_top_set_rat ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
thf(fact_1256_card__image__le,axiom,
    ! [A2: set_nat,F2: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ord_less_eq_nat @ ( finite_card_rat @ ( image_nat_rat @ F2 @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).

% card_image_le
thf(fact_1257_surj__card__le,axiom,
    ! [A2: set_nat,B2: set_rat,F2: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F2 @ A2 ) )
       => ( ord_less_eq_nat @ ( finite_card_rat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).

% surj_card_le
thf(fact_1258_inj__on__iff__eq__card,axiom,
    ! [A2: set_nat,F2: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( inj_on_nat_rat @ F2 @ A2 )
        = ( ( finite_card_rat @ ( image_nat_rat @ F2 @ A2 ) )
          = ( finite_card_nat @ A2 ) ) ) ) ).

% inj_on_iff_eq_card
thf(fact_1259_eq__card__imp__inj__on,axiom,
    ! [A2: set_nat,F2: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( finite_card_rat @ ( image_nat_rat @ F2 @ A2 ) )
          = ( finite_card_nat @ A2 ) )
       => ( inj_on_nat_rat @ F2 @ A2 ) ) ) ).

% eq_card_imp_inj_on
thf(fact_1260_partition__on__le__set__elements,axiom,
    ! [A2: set_nat,P: set_set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( disjoi4774308525696689793on_nat @ A2 @ P )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ P ) @ ( finite_card_nat @ A2 ) ) ) ) ).

% partition_on_le_set_elements
thf(fact_1261_card__image,axiom,
    ! [F2: nat > rat,A2: set_nat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( ( finite_card_rat @ ( image_nat_rat @ F2 @ A2 ) )
        = ( finite_card_nat @ A2 ) ) ) ).

% card_image
thf(fact_1262_rat__denum,axiom,
    ? [F6: nat > rat] :
      ( ( image_nat_rat @ F6 @ top_top_set_nat )
      = top_top_set_rat ) ).

% rat_denum
thf(fact_1263_surjective__iff__injective__gen,axiom,
    ! [S4: set_nat,T2: set_rat,F2: nat > rat] :
      ( ( finite_finite_nat @ S4 )
     => ( ( finite_finite_rat @ T2 )
       => ( ( ( finite_card_nat @ S4 )
            = ( finite_card_rat @ T2 ) )
         => ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ S4 ) @ T2 )
           => ( ( ! [X: rat] :
                    ( ( member_rat @ X @ T2 )
                   => ? [Y: nat] :
                        ( ( member_nat @ Y @ S4 )
                        & ( ( F2 @ Y )
                          = X ) ) ) )
              = ( inj_on_nat_rat @ F2 @ S4 ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_1264_surjective__iff__injective__gen,axiom,
    ! [S4: set_nat,T2: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ S4 )
     => ( ( finite_finite_nat @ T2 )
       => ( ( ( finite_card_nat @ S4 )
            = ( finite_card_nat @ T2 ) )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S4 ) @ T2 )
           => ( ( ! [X: nat] :
                    ( ( member_nat @ X @ T2 )
                   => ? [Y: nat] :
                        ( ( member_nat @ Y @ S4 )
                        & ( ( F2 @ Y )
                          = X ) ) ) )
              = ( inj_on_nat_nat @ F2 @ S4 ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_1265_inj__on__iff__card__le,axiom,
    ! [A2: set_nat,B2: set_rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_rat @ B2 )
       => ( ( ? [F: nat > rat] :
                ( ( inj_on_nat_rat @ F @ A2 )
                & ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A2 ) @ B2 ) ) )
          = ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_rat @ B2 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_1266_inj__on__iff__card__le,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( ? [F: nat > nat] :
                ( ( inj_on_nat_nat @ F @ A2 )
                & ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) )
          = ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_1267_card__inj__on__le,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_rat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ B2 )
       => ( ( finite_finite_rat @ B2 )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_rat @ B2 ) ) ) ) ) ).

% card_inj_on_le
thf(fact_1268_card__le__inj,axiom,
    ! [A2: set_nat,B2: set_rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_rat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_rat @ B2 ) )
         => ? [F6: nat > rat] :
              ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F6 @ A2 ) @ B2 )
              & ( inj_on_nat_rat @ F6 @ A2 ) ) ) ) ) ).

% card_le_inj
thf(fact_1269_card__le__inj,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
         => ? [F6: nat > nat] :
              ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F6 @ A2 ) @ B2 )
              & ( inj_on_nat_nat @ F6 @ A2 ) ) ) ) ) ).

% card_le_inj
thf(fact_1270_card__bij__eq,axiom,
    ! [F2: rat > nat,A2: set_rat,B2: set_nat,G2: nat > rat] :
      ( ( inj_on_rat_nat @ F2 @ A2 )
     => ( ( ord_less_eq_set_nat @ ( image_rat_nat @ F2 @ A2 ) @ B2 )
       => ( ( inj_on_nat_rat @ G2 @ B2 )
         => ( ( ord_less_eq_set_rat @ ( image_nat_rat @ G2 @ B2 ) @ A2 )
           => ( ( finite_finite_rat @ A2 )
             => ( ( finite_finite_nat @ B2 )
               => ( ( finite_card_rat @ A2 )
                  = ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_1271_card__bij__eq,axiom,
    ! [F2: nat > rat,A2: set_nat,B2: set_rat,G2: rat > nat] :
      ( ( inj_on_nat_rat @ F2 @ A2 )
     => ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ B2 )
       => ( ( inj_on_rat_nat @ G2 @ B2 )
         => ( ( ord_less_eq_set_nat @ ( image_rat_nat @ G2 @ B2 ) @ A2 )
           => ( ( finite_finite_nat @ A2 )
             => ( ( finite_finite_rat @ B2 )
               => ( ( finite_card_nat @ A2 )
                  = ( finite_card_rat @ B2 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_1272_card__bij__eq,axiom,
    ! [F2: nat > nat,A2: set_nat,B2: set_nat,G2: nat > nat] :
      ( ( inj_on_nat_nat @ F2 @ A2 )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B2 )
       => ( ( inj_on_nat_nat @ G2 @ B2 )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G2 @ B2 ) @ A2 )
           => ( ( finite_finite_nat @ A2 )
             => ( ( finite_finite_nat @ B2 )
               => ( ( finite_card_nat @ A2 )
                  = ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_1273_card__partition,axiom,
    ! [C2: set_set_nat,K: nat] :
      ( ( finite1152437895449049373et_nat @ C2 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ C2 ) )
       => ( ! [C4: set_nat] :
              ( ( member_set_nat @ C4 @ C2 )
             => ( ( finite_card_nat @ C4 )
                = K ) )
         => ( ! [C1: set_nat,C22: set_nat] :
                ( ( member_set_nat @ C1 @ C2 )
               => ( ( member_set_nat @ C22 @ C2 )
                 => ( ( C1 != C22 )
                   => ( ( inf_inf_set_nat @ C1 @ C22 )
                      = bot_bot_set_nat ) ) ) )
           => ( ( times_times_nat @ K @ ( finite_card_set_nat @ C2 ) )
              = ( finite_card_nat @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ) ) ) ).

% card_partition
thf(fact_1274_dvd__partition,axiom,
    ! [C2: set_set_nat,K: nat] :
      ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ C2 ) )
     => ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ C2 )
           => ( dvd_dvd_nat @ K @ ( finite_card_nat @ X4 ) ) )
       => ( ! [X4: set_nat] :
              ( ( member_set_nat @ X4 @ C2 )
             => ! [Xa2: set_nat] :
                  ( ( member_set_nat @ Xa2 @ C2 )
                 => ( ( X4 != Xa2 )
                   => ( ( inf_inf_set_nat @ X4 @ Xa2 )
                      = bot_bot_set_nat ) ) ) )
         => ( dvd_dvd_nat @ K @ ( finite_card_nat @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ) ) ).

% dvd_partition
thf(fact_1275_surj__nat__to__rat__surj,axiom,
    ( ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat )
    = top_top_set_rat ) ).

% surj_nat_to_rat_surj
thf(fact_1276_inj__to__nat,axiom,
    inj_on_nat_nat @ to_nat_nat @ top_top_set_nat ).

% inj_to_nat
thf(fact_1277_inj__to__nat,axiom,
    inj_on_rat_nat @ to_nat_rat @ top_top_set_rat ).

% inj_to_nat
thf(fact_1278_Rats__eq__range__nat__to__rat__surj,axiom,
    ( field_6020823756834552118ts_rat
    = ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat ) ) ).

% Rats_eq_range_nat_to_rat_surj

% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
    ! [X2: a,Y3: a] :
      ( ( if_a @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001tf__a_T,axiom,
    ! [X2: a,Y3: a] :
      ( ( if_a @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001tf__b_T,axiom,
    ! [X2: b,Y3: b] :
      ( ( if_b @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001tf__b_T,axiom,
    ! [X2: b,Y3: b] :
      ( ( if_b @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_3_1_If_001tf__c_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001tf__c_T,axiom,
    ! [X2: c,Y3: c] :
      ( ( if_c @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001tf__c_T,axiom,
    ! [X2: c,Y3: c] :
      ( ( if_c @ $true @ X2 @ Y3 )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( map_vec_b_a @ f @ ( map_vec_c_b @ g @ v ) )
    = ( map_vec_c_a @ ( comp_b_a_c @ f @ g ) @ v ) ) ).

%------------------------------------------------------------------------------