TPTP Problem File: SLH0277^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Rewrite_Properties_Reduction/0016_Rewriting/prob_00202_007012__13787410_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1561 ( 381 unt; 285 typ;   0 def)
%            Number of atoms       : 3969 ( 799 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 15921 ( 428   ~;  45   |; 280   &;12931   @)
%                                         (   0 <=>;2237  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   9 avg)
%            Number of types       :   33 (  32 usr)
%            Number of type conns  :  734 ( 734   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  254 ( 253 usr;  16 con; 0-4 aty)
%            Number of variables   : 4433 ( 230   ^;4036   !; 167   ?;4433   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:18:24.460
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
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% Explicit typings (253)
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    relcomp_a_a_term_a_b: set_Product_prod_a_a > set_Pr690693513201511543rm_a_b > set_Pr690693513201511543rm_a_b ).

thf(sy_c_Relation_Orelcomp_001tf__a_001tf__a_001tf__a,type,
    relcomp_a_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).

thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    total_4010809703584253837rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).

thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    total_9110697612923228665_a_nat: set_Pr4934435412358123699_a_nat > set_Pr1811044260758604347_a_nat > $o ).

thf(sy_c_Relation_Ototal__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    total_on_term_a_b: set_term_a_b > set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_Rewriting_Orrstep_001tf__a_001tf__b,type,
    rrstep_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Rewriting_Orstep_001tf__a_001tf__b,type,
    rstep_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Rewriting_Osig__step_001tf__a_001tf__b,type,
    sig_step_a_b: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Rewriting_Osrsteps__with__root__step_001tf__a_001tf__b,type,
    srstep7844470518422762656ep_a_b: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).

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

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    collec1344989105561874898rm_a_b: ( produc4523971788519308903rm_a_b > $o ) > set_Pr2972776593051762503rm_a_b ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    collec925066218191019974_a_nat: ( produc4708774622424448987_a_nat > $o ) > set_Pr1811044260758604347_a_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    collec99177395925924084rm_a_b: ( produc357393685978478089rm_a_b > $o ) > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    collec4464134535221767506_a_nat: ( product_prod_a_nat > $o ) > set_Pr4934435412358123699_a_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    collec3336397797384452498od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).

thf(sy_c_Set_OCollect_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    collect_term_a_b: ( term_a_b > $o ) > set_term_a_b ).

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

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    insert7009541432154983385rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    insert8054603423593749677_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat ).

thf(sy_c_Set_Oinsert_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    insert_term_a_b: term_a_b > set_term_a_b > set_term_a_b ).

thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    is_sin332545968047684349rm_a_b: set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    remove8523906646656265582rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Subterm__and__Context_Octxt__apply__term_001tf__a_001tf__b,type,
    subter2376574525758040790rm_a_b: subterm_and_ctxt_a_b > term_a_b > term_a_b ).

thf(sy_c_Term_Osubst__apply__term_001tf__a_001tf__b_001tf__b,type,
    subst_7999470309526761004_a_b_b: term_a_b > ( b > term_a_b ) > term_a_b ).

thf(sy_c_Term__Context_Ofunas__term_001tf__a_001tf__b,type,
    term_funas_term_a_b: term_a_b > set_Pr4934435412358123699_a_nat ).

thf(sy_c_Terms__Positions_Ofunas__rel_001tf__a_001tf__b_001tf__b,type,
    terms_7988297476397195622_a_b_b: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat ).

thf(sy_c_Terms__Positions_Oterm__to__sig_001tf__a_001tf__b,type,
    terms_8519481630511763164ig_a_b: set_Pr4934435412358123699_a_nat > b > term_a_b > term_a_b ).

thf(sy_c_Transitive__Closure_Oacyclic_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    transi5314701259734769157rm_a_b: set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    transi2615809358984392588rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).

thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    transi2726145917338391738_a_nat: set_Pr1811044260758604347_a_nat > set_Pr1811044260758604347_a_nat ).

thf(sy_c_Transitive__Closure_Ortrancl_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    transi7742714808557438673rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Transitive__Closure_Ortrancl_001tf__a,type,
    transitive_rtrancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
    transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    transi4560671716973800200rm_a_b: set_Pr6848273517546626247rm_a_b > set_Pr6848273517546626247rm_a_b ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    transi2682318148269310460_a_nat: set_Pr8568195457674374599_a_nat > set_Pr8568195457674374599_a_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    transi9211502839322181930rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    transi6805586507763294364_a_nat: set_Pr1811044260758604347_a_nat > set_Pr1811044260758604347_a_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    transi7922773638565587891rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Transitive__Closure_Otrancl_001tf__a,type,
    transitive_trancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).

thf(sy_c_Wellfounded_Olex__prod_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    lex_pr6163557265797435481rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b ).

thf(sy_c_Wellfounded_Omeasure_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    measure_term_a_b: ( term_a_b > nat ) > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    wf_Pro2335863617654816626rm_a_b: set_Pr2972776593051762503rm_a_b > $o ).

thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    wf_Pro8109884346892000340_a_nat: set_Pr1811044260758604347_a_nat > $o ).

thf(sy_c_Wellfounded_Owf_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    wf_term_a_b: set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_Wfrec_Osame__fst_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    same_f7764617898092871340rm_a_b: ( produc357393685978478089rm_a_b > $o ) > ( produc357393685978478089rm_a_b > set_Pr2972776593051762503rm_a_b ) > set_Pr6848273517546626247rm_a_b ).

thf(sy_c_Wfrec_Osame__fst_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    same_f6637753491745052904_a_nat: ( product_prod_a_nat > $o ) > ( product_prod_a_nat > set_Pr1811044260758604347_a_nat ) > set_Pr8568195457674374599_a_nat ).

thf(sy_c_Wfrec_Osame__fst_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    same_f843731131676418198rm_a_b: ( term_a_b > $o ) > ( term_a_b > set_Pr4386577575007340137rm_a_b ) > set_Pr2972776593051762503rm_a_b ).

thf(sy_c_Wfrec_Osame__fst_001tf__a_001t__Nat__Onat,type,
    same_fst_a_nat: ( a > $o ) > ( a > set_Pr1261947904930325089at_nat ) > set_Pr1811044260758604347_a_nat ).

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
    member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    member5793113912742238992rm_a_b: produc2957090337049255143rm_a_b > set_Pr6848273517546626247rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    member3559618881423075344_a_nat: produc1315028609879303911_a_nat > set_Pr8568195457674374599_a_nat > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    member9062615507155100804_a_nat: produc4708774622424448987_a_nat > set_Pr1811044260758604347_a_nat > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

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

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

thf(sy_v__092_060F_062,type,
    f: set_Pr4934435412358123699_a_nat ).

thf(sy_v__092_060R_062,type,
    r: set_Pr4386577575007340137rm_a_b ).

thf(sy_v_s,type,
    s: term_a_b ).

thf(sy_v_t,type,
    t: term_a_b ).

thf(sy_v_ta____,type,
    ta: term_a_b ).

thf(sy_v_v,type,
    v: b ).

% Relevant facts (1275)
thf(fact_0_base_Oprems_I1_J,axiom,
    ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ r ) @ f ).

% base.prems(1)
thf(fact_1_base_Oprems_I2_J,axiom,
    ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ t ) @ f ).

% base.prems(2)
thf(fact_2_base_Ohyps,axiom,
    member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ta @ t ) @ ( rstep_a_b @ r ) ).

% base.hyps
thf(fact_3_assms_I1_J,axiom,
    member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ r ) ) ).

% assms(1)
thf(fact_4_rstep__ruleI,axiom,
    ! [L: term_a_b,R: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R2 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ ( rstep_a_b @ R2 ) ) ) ).

% rstep_ruleI
thf(fact_5_srstepsD,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).

% srstepsD
thf(fact_6_rstep__trancl__sig__step__r,axiom,
    ! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V: b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F @ V @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).

% rstep_trancl_sig_step_r
thf(fact_7_prod_Oinject,axiom,
    ! [X1: produc357393685978478089rm_a_b,X2: produc357393685978478089rm_a_b,Y1: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
      ( ( ( produc1763473618796451543rm_a_b @ X1 @ X2 )
        = ( produc1763473618796451543rm_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_8_prod_Oinject,axiom,
    ! [X1: a,X2: nat,Y1: a,Y2: nat] :
      ( ( ( product_Pair_a_nat @ X1 @ X2 )
        = ( product_Pair_a_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_9_prod_Oinject,axiom,
    ! [X1: product_prod_a_nat,X2: product_prod_a_nat,Y1: product_prod_a_nat,Y2: product_prod_a_nat] :
      ( ( ( produc2026711137822539155_a_nat @ X1 @ X2 )
        = ( produc2026711137822539155_a_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_10_prod_Oinject,axiom,
    ! [X1: term_a_b,X2: term_a_b,Y1: term_a_b,Y2: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ X1 @ X2 )
        = ( produc7020197800436672577rm_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
      ( ( ( produc1763473618796451543rm_a_b @ A @ B )
        = ( produc1763473618796451543rm_a_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,
    ! [A: a,B: nat,A2: a,B2: nat] :
      ( ( ( product_Pair_a_nat @ A @ B )
        = ( product_Pair_a_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
      ( ( ( produc2026711137822539155_a_nat @ A @ B )
        = ( produc2026711137822539155_a_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
    ! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ A @ B )
        = ( produc7020197800436672577rm_a_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_15_sig__stepI,axiom,
    ! [S: term_a_b,F: set_Pr4934435412358123699_a_nat,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
     => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R2 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ R2 ) ) ) ) ) ).

% sig_stepI
thf(fact_16_srstepD,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).

% srstepD
thf(fact_17_trancl_Ocases,axiom,
    ! [A1: product_prod_a_nat,A22: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A1 @ A22 ) @ ( transi6805586507763294364_a_nat @ R ) )
     => ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A1 @ A22 ) @ R )
       => ~ ! [B3: product_prod_a_nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A1 @ B3 ) @ ( transi6805586507763294364_a_nat @ R ) )
             => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B3 @ A22 ) @ R ) ) ) ) ).

% trancl.cases
thf(fact_18_trancl_Ocases,axiom,
    ! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ R )
       => ~ ! [B3: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B3 ) @ ( transi9211502839322181930rm_a_b @ R ) )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ A22 ) @ R ) ) ) ) ).

% trancl.cases
thf(fact_19_trancl_Ocases,axiom,
    ! [A1: a,A22: a,R: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( transitive_trancl_a @ R ) )
     => ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ R )
       => ~ ! [B3: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B3 ) @ ( transitive_trancl_a @ R ) )
             => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A22 ) @ R ) ) ) ) ).

% trancl.cases
thf(fact_20_trancl_Ocases,axiom,
    ! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R )
       => ~ ! [B3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi6264000038957366511cl_nat @ R ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R ) ) ) ) ).

% trancl.cases
thf(fact_21_trancl_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ R )
       => ~ ! [B3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ R ) ) ) ) ).

% trancl.cases
thf(fact_22_trancl_Osimps,axiom,
    ! [A1: product_prod_a_nat,A22: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A1 @ A22 ) @ ( transi6805586507763294364_a_nat @ R ) )
      = ( ? [A3: product_prod_a_nat,B4: product_prod_a_nat] :
            ( ( A1 = A3 )
            & ( A22 = B4 )
            & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ B4 ) @ R ) )
        | ? [A3: product_prod_a_nat,B4: product_prod_a_nat,C: product_prod_a_nat] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ B4 ) @ ( transi6805586507763294364_a_nat @ R ) )
            & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B4 @ C ) @ R ) ) ) ) ).

% trancl.simps
thf(fact_23_trancl_Osimps,axiom,
    ! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( transi9211502839322181930rm_a_b @ R ) )
      = ( ? [A3: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
            ( ( A1 = A3 )
            & ( A22 = B4 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B4 ) @ R ) )
        | ? [A3: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B4 ) @ ( transi9211502839322181930rm_a_b @ R ) )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ C ) @ R ) ) ) ) ).

% trancl.simps
thf(fact_24_trancl_Osimps,axiom,
    ! [A1: a,A22: a,R: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( transitive_trancl_a @ R ) )
      = ( ? [A3: a,B4: a] :
            ( ( A1 = A3 )
            & ( A22 = B4 )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B4 ) @ R ) )
        | ? [A3: a,B4: a,C: a] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B4 ) @ ( transitive_trancl_a @ R ) )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B4 @ C ) @ R ) ) ) ) ).

% trancl.simps
thf(fact_25_trancl_Osimps,axiom,
    ! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R ) )
      = ( ? [A3: nat,B4: nat] :
            ( ( A1 = A3 )
            & ( A22 = B4 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R ) )
        | ? [A3: nat,B4: nat,C: nat] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi6264000038957366511cl_nat @ R ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C ) @ R ) ) ) ) ).

% trancl.simps
thf(fact_26_trancl_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R ) )
      = ( ? [A3: term_a_b,B4: term_a_b] :
            ( ( A1 = A3 )
            & ( A22 = B4 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ R ) )
        | ? [A3: term_a_b,B4: term_a_b,C: term_a_b] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ ( transi7922773638565587891rm_a_b @ R ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C ) @ R ) ) ) ) ).

% trancl.simps
thf(fact_27_trancl_Or__into__trancl,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R )
     => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi6805586507763294364_a_nat @ R ) ) ) ).

% trancl.r_into_trancl
thf(fact_28_trancl_Or__into__trancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R ) ) ) ).

% trancl.r_into_trancl
thf(fact_29_trancl_Or__into__trancl,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R ) ) ) ).

% trancl.r_into_trancl
thf(fact_30_trancl_Or__into__trancl,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ).

% trancl.r_into_trancl
thf(fact_31_trancl_Or__into__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).

% trancl.r_into_trancl
thf(fact_32_trancl__map,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ! [X3: term_a_b,Y3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).

% trancl_map
thf(fact_33_trancl__map,axiom,
    ! [R: set_Product_prod_a_a,F2: a > a,S: set_Product_prod_a_a,X: a,Y: a] :
      ( ! [X3: a,Y3: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transitive_trancl_a @ S ) ) ) ) ).

% trancl_map
thf(fact_34_trancl__map,axiom,
    ! [R: set_Product_prod_a_a,F2: a > nat,S: set_Pr1261947904930325089at_nat,X: a,Y: a] :
      ( ! [X3: a,Y3: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).

% trancl_map
thf(fact_35_trancl__map,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F2: nat > a,S: set_Product_prod_a_a,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transitive_trancl_a @ S ) ) ) ) ).

% trancl_map
thf(fact_36_trancl__map,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F2: nat > nat,S: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).

% trancl_map
thf(fact_37_trancl__map,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,F2: product_prod_a_nat > a,S: set_Product_prod_a_a,X: product_prod_a_nat,Y: product_prod_a_nat] :
      ( ! [X3: product_prod_a_nat,Y3: product_prod_a_nat] :
          ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi6805586507763294364_a_nat @ R ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transitive_trancl_a @ S ) ) ) ) ).

% trancl_map
thf(fact_38_trancl__map,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,F2: product_prod_a_nat > nat,S: set_Pr1261947904930325089at_nat,X: product_prod_a_nat,Y: product_prod_a_nat] :
      ( ! [X3: product_prod_a_nat,Y3: product_prod_a_nat] :
          ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi6805586507763294364_a_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).

% trancl_map
thf(fact_39_trancl__map,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > a,S: set_Product_prod_a_a,X: term_a_b,Y: term_a_b] :
      ( ! [X3: term_a_b,Y3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transitive_trancl_a @ S ) ) ) ) ).

% trancl_map
thf(fact_40_trancl__map,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > nat,S: set_Pr1261947904930325089at_nat,X: term_a_b,Y: term_a_b] :
      ( ! [X3: term_a_b,Y3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).

% trancl_map
thf(fact_41_trancl__map,axiom,
    ! [R: set_Product_prod_a_a,F2: a > product_prod_a_nat,S: set_Pr1811044260758604347_a_nat,X: a,Y: a] :
      ( ! [X3: a,Y3: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
         => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R ) )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi6805586507763294364_a_nat @ S ) ) ) ) ).

% trancl_map
thf(fact_42_tranclE,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi6805586507763294364_a_nat @ R ) )
     => ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R )
       => ~ ! [C2: product_prod_a_nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C2 ) @ ( transi6805586507763294364_a_nat @ R ) )
             => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ C2 @ B ) @ R ) ) ) ) ).

% tranclE
thf(fact_43_tranclE,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
       => ~ ! [C2: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R ) )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ C2 @ B ) @ R ) ) ) ) ).

% tranclE
thf(fact_44_tranclE,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R ) )
     => ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
       => ~ ! [C2: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C2 ) @ ( transitive_trancl_a @ R ) )
             => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ C2 @ B ) @ R ) ) ) ) ).

% tranclE
thf(fact_45_tranclE,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
       => ~ ! [C2: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C2 @ B ) @ R ) ) ) ) ).

% tranclE
thf(fact_46_tranclE,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
       => ~ ! [C2: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ R ) ) ) ) ).

% tranclE
thf(fact_47_trancl__induct2,axiom,
    ! [Ax: produc357393685978478089rm_a_b,Ay: produc357393685978478089rm_a_b,Bx: produc357393685978478089rm_a_b,By: produc357393685978478089rm_a_b,R: set_Pr6848273517546626247rm_a_b,P: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
      ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi4560671716973800200rm_a_b @ R ) )
     => ( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
            ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) ) @ R )
           => ( P @ A4 @ B3 ) )
       => ( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,Aa: produc357393685978478089rm_a_b,Ba: produc357393685978478089rm_a_b] :
              ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) ) @ ( transi4560671716973800200rm_a_b @ R ) )
             => ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) ) @ R )
               => ( ( P @ A4 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_48_trancl__induct2,axiom,
    ! [Ax: product_prod_a_nat,Ay: product_prod_a_nat,Bx: product_prod_a_nat,By: product_prod_a_nat,R: set_Pr8568195457674374599_a_nat,P: product_prod_a_nat > product_prod_a_nat > $o] :
      ( ( member3559618881423075344_a_nat @ ( produc1024945203614457431_a_nat @ ( produc2026711137822539155_a_nat @ Ax @ Ay ) @ ( produc2026711137822539155_a_nat @ Bx @ By ) ) @ ( transi2682318148269310460_a_nat @ R ) )
     => ( ! [A4: product_prod_a_nat,B3: product_prod_a_nat] :
            ( ( member3559618881423075344_a_nat @ ( produc1024945203614457431_a_nat @ ( produc2026711137822539155_a_nat @ Ax @ Ay ) @ ( produc2026711137822539155_a_nat @ A4 @ B3 ) ) @ R )
           => ( P @ A4 @ B3 ) )
       => ( ! [A4: product_prod_a_nat,B3: product_prod_a_nat,Aa: product_prod_a_nat,Ba: product_prod_a_nat] :
              ( ( member3559618881423075344_a_nat @ ( produc1024945203614457431_a_nat @ ( produc2026711137822539155_a_nat @ Ax @ Ay ) @ ( produc2026711137822539155_a_nat @ A4 @ B3 ) ) @ ( transi2682318148269310460_a_nat @ R ) )
             => ( ( member3559618881423075344_a_nat @ ( produc1024945203614457431_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ B3 ) @ ( produc2026711137822539155_a_nat @ Aa @ Ba ) ) @ R )
               => ( ( P @ A4 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_49_trancl__induct2,axiom,
    ! [Ax: a,Ay: nat,Bx: a,By: nat,R: set_Pr1811044260758604347_a_nat,P: a > nat > $o] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi6805586507763294364_a_nat @ R ) )
     => ( ! [A4: a,B3: nat] :
            ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A4 @ B3 ) ) @ R )
           => ( P @ A4 @ B3 ) )
       => ( ! [A4: a,B3: nat,Aa: a,Ba: nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A4 @ B3 ) ) @ ( transi6805586507763294364_a_nat @ R ) )
             => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A4 @ B3 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R )
               => ( ( P @ A4 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_50_trancl__induct2,axiom,
    ! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R: set_Pr2972776593051762503rm_a_b,P: term_a_b > term_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( ! [A4: term_a_b,B3: term_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ R )
           => ( P @ A4 @ B3 ) )
       => ( ! [A4: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ ( transi9211502839322181930rm_a_b @ R ) )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
               => ( ( P @ A4 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_51_trancl__mono,axiom,
    ! [P2: produc4708774622424448987_a_nat,R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ P2 @ ( transi6805586507763294364_a_nat @ R ) )
     => ( ( ord_le1018141841545957851_a_nat @ R @ S )
       => ( member9062615507155100804_a_nat @ P2 @ ( transi6805586507763294364_a_nat @ S ) ) ) ) ).

% trancl_mono
thf(fact_52_trancl__mono,axiom,
    ! [P2: produc4523971788519308903rm_a_b,R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ P2 @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( ( ord_le2593958535671617255rm_a_b @ R @ S )
       => ( member8417600551952982416rm_a_b @ P2 @ ( transi9211502839322181930rm_a_b @ S ) ) ) ) ).

% trancl_mono
thf(fact_53_trancl__mono,axiom,
    ! [P2: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ( ord_le3146513528884898305at_nat @ R @ S )
       => ( member8440522571783428010at_nat @ P2 @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).

% trancl_mono
thf(fact_54_trancl__mono,axiom,
    ! [P2: product_prod_a_a,R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ P2 @ ( transitive_trancl_a @ R ) )
     => ( ( ord_le746702958409616551od_a_a @ R @ S )
       => ( member1426531477525435216od_a_a @ P2 @ ( transitive_trancl_a @ S ) ) ) ) ).

% trancl_mono
thf(fact_55_trancl__mono,axiom,
    ! [P2: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( ( ord_le118470702582115849rm_a_b @ R @ S )
       => ( member5869715511025134514rm_a_b @ P2 @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).

% trancl_mono
thf(fact_56_prod__induct3,axiom,
    ! [P: produc4523971788519308903rm_a_b > $o,X: produc4523971788519308903rm_a_b] :
      ( ! [A4: produc357393685978478089rm_a_b,B3: term_a_b,C2: term_a_b] : ( P @ ( produc1763473618796451543rm_a_b @ A4 @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_57_prod__induct3,axiom,
    ! [P: produc4708774622424448987_a_nat > $o,X: produc4708774622424448987_a_nat] :
      ( ! [A4: product_prod_a_nat,B3: a,C2: nat] : ( P @ ( produc2026711137822539155_a_nat @ A4 @ ( product_Pair_a_nat @ B3 @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_58_prod__cases3,axiom,
    ! [Y: produc4523971788519308903rm_a_b] :
      ~ ! [A4: produc357393685978478089rm_a_b,B3: term_a_b,C2: term_a_b] :
          ( Y
         != ( produc1763473618796451543rm_a_b @ A4 @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_59_prod__cases3,axiom,
    ! [Y: produc4708774622424448987_a_nat] :
      ~ ! [A4: product_prod_a_nat,B3: a,C2: nat] :
          ( Y
         != ( produc2026711137822539155_a_nat @ A4 @ ( product_Pair_a_nat @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_60_Pair__inject,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
      ( ( ( produc1763473618796451543rm_a_b @ A @ B )
        = ( produc1763473618796451543rm_a_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_61_Pair__inject,axiom,
    ! [A: a,B: nat,A2: a,B2: nat] :
      ( ( ( product_Pair_a_nat @ A @ B )
        = ( product_Pair_a_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_62_Pair__inject,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
      ( ( ( produc2026711137822539155_a_nat @ A @ B )
        = ( produc2026711137822539155_a_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_63_Pair__inject,axiom,
    ! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ A @ B )
        = ( produc7020197800436672577rm_a_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_64_prod__cases,axiom,
    ! [P: produc4523971788519308903rm_a_b > $o,P2: produc4523971788519308903rm_a_b] :
      ( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] : ( P @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_65_prod__cases,axiom,
    ! [P: product_prod_a_nat > $o,P2: product_prod_a_nat] :
      ( ! [A4: a,B3: nat] : ( P @ ( product_Pair_a_nat @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_66_prod__cases,axiom,
    ! [P: produc4708774622424448987_a_nat > $o,P2: produc4708774622424448987_a_nat] :
      ( ! [A4: product_prod_a_nat,B3: product_prod_a_nat] : ( P @ ( produc2026711137822539155_a_nat @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_67_prod__cases,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,P2: produc357393685978478089rm_a_b] :
      ( ! [A4: term_a_b,B3: term_a_b] : ( P @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_68_surj__pair,axiom,
    ! [P2: produc4523971788519308903rm_a_b] :
    ? [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
      ( P2
      = ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_69_surj__pair,axiom,
    ! [P2: product_prod_a_nat] :
    ? [X3: a,Y3: nat] :
      ( P2
      = ( product_Pair_a_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_70_surj__pair,axiom,
    ! [P2: produc4708774622424448987_a_nat] :
    ? [X3: product_prod_a_nat,Y3: product_prod_a_nat] :
      ( P2
      = ( produc2026711137822539155_a_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_71_surj__pair,axiom,
    ! [P2: produc357393685978478089rm_a_b] :
    ? [X3: term_a_b,Y3: term_a_b] :
      ( P2
      = ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_72_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4523971788519308903rm_a_b] :
      ~ ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
          ( Y
         != ( produc1763473618796451543rm_a_b @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_73_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_nat] :
      ~ ! [A4: a,B3: nat] :
          ( Y
         != ( product_Pair_a_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_74_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4708774622424448987_a_nat] :
      ~ ! [A4: product_prod_a_nat,B3: product_prod_a_nat] :
          ( Y
         != ( produc2026711137822539155_a_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_75_old_Oprod_Oexhaust,axiom,
    ! [Y: produc357393685978478089rm_a_b] :
      ~ ! [A4: term_a_b,B3: term_a_b] :
          ( Y
         != ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_76_r__into__trancl_H,axiom,
    ! [P2: produc4708774622424448987_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ P2 @ R )
     => ( member9062615507155100804_a_nat @ P2 @ ( transi6805586507763294364_a_nat @ R ) ) ) ).

% r_into_trancl'
thf(fact_77_r__into__trancl_H,axiom,
    ! [P2: produc4523971788519308903rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ P2 @ R )
     => ( member8417600551952982416rm_a_b @ P2 @ ( transi9211502839322181930rm_a_b @ R ) ) ) ).

% r_into_trancl'
thf(fact_78_r__into__trancl_H,axiom,
    ! [P2: product_prod_a_a,R: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ P2 @ R )
     => ( member1426531477525435216od_a_a @ P2 @ ( transitive_trancl_a @ R ) ) ) ).

% r_into_trancl'
thf(fact_79_r__into__trancl_H,axiom,
    ! [P2: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ R )
     => ( member8440522571783428010at_nat @ P2 @ ( transi6264000038957366511cl_nat @ R ) ) ) ).

% r_into_trancl'
thf(fact_80_r__into__trancl_H,axiom,
    ! [P2: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ R )
     => ( member5869715511025134514rm_a_b @ P2 @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).

% r_into_trancl'
thf(fact_81_sig__stepE,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ R2 ) )
     => ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R2 )
         => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
           => ~ ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ) ).

% sig_stepE
thf(fact_82_rstep__term__to__sig__r,axiom,
    ! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V: b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F @ V @ T ) ) @ ( rstep_a_b @ R2 ) ) ) ) ) ).

% rstep_term_to_sig_r
thf(fact_83_rstep__term__to__sig__l,axiom,
    ! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V: b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F @ V @ S ) @ T ) @ ( rstep_a_b @ R2 ) ) ) ) ) ).

% rstep_term_to_sig_l
thf(fact_84_converse__trancl__induct,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,P: product_prod_a_nat > $o] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi6805586507763294364_a_nat @ R ) )
     => ( ! [Y3: product_prod_a_nat] :
            ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ B ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: product_prod_a_nat,Z: product_prod_a_nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ Z ) @ R )
             => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Z @ B ) @ ( transi6805586507763294364_a_nat @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_85_converse__trancl__induct,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( ! [Y3: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ B ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ R )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ B ) @ ( transi9211502839322181930rm_a_b @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_86_converse__trancl__induct,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,P: a > $o] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R ) )
     => ( ! [Y3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ B ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: a,Z: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z ) @ R )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ B ) @ ( transitive_trancl_a @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_87_converse__trancl__induct,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ! [Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ B ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: nat,Z: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_88_converse__trancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( ! [Y3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ B ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: term_a_b,Z: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_89_trancl__trans__induct,axiom,
    ! [X: product_prod_a_nat,Y: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,P: product_prod_a_nat > product_prod_a_nat > $o] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi6805586507763294364_a_nat @ R ) )
     => ( ! [X3: product_prod_a_nat,Y3: product_prod_a_nat] :
            ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
           => ( P @ X3 @ Y3 ) )
       => ( ! [X3: product_prod_a_nat,Y3: product_prod_a_nat,Z: product_prod_a_nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ ( transi6805586507763294364_a_nat @ R ) )
             => ( ( P @ X3 @ Y3 )
               => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ Z ) @ ( transi6805586507763294364_a_nat @ R ) )
                 => ( ( P @ Y3 @ Z )
                   => ( P @ X3 @ Z ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_90_trancl__trans__induct,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
           => ( P @ X3 @ Y3 ) )
       => ( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( transi9211502839322181930rm_a_b @ R ) )
             => ( ( P @ X3 @ Y3 )
               => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ ( transi9211502839322181930rm_a_b @ R ) )
                 => ( ( P @ Y3 @ Z )
                   => ( P @ X3 @ Z ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_91_trancl__trans__induct,axiom,
    ! [X: a,Y: a,R: set_Product_prod_a_a,P: a > a > $o] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R ) )
     => ( ! [X3: a,Y3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
           => ( P @ X3 @ Y3 ) )
       => ( ! [X3: a,Y3: a,Z: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( transitive_trancl_a @ R ) )
             => ( ( P @ X3 @ Y3 )
               => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z ) @ ( transitive_trancl_a @ R ) )
                 => ( ( P @ Y3 @ Z )
                   => ( P @ X3 @ Z ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_92_trancl__trans__induct,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
           => ( P @ X3 @ Y3 ) )
       => ( ! [X3: nat,Y3: nat,Z: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( transi6264000038957366511cl_nat @ R ) )
             => ( ( P @ X3 @ Y3 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ ( transi6264000038957366511cl_nat @ R ) )
                 => ( ( P @ Y3 @ Z )
                   => ( P @ X3 @ Z ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_93_trancl__trans__induct,axiom,
    ! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( ! [X3: term_a_b,Y3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
           => ( P @ X3 @ Y3 ) )
       => ( ! [X3: term_a_b,Y3: term_a_b,Z: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
             => ( ( P @ X3 @ Y3 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ ( transi7922773638565587891rm_a_b @ R ) )
                 => ( ( P @ Y3 @ Z )
                   => ( P @ X3 @ Z ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_94_trancl__into__trancl2,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,C3: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ ( transi6805586507763294364_a_nat @ R ) )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C3 ) @ ( transi6805586507763294364_a_nat @ R ) ) ) ) ).

% trancl_into_trancl2
thf(fact_95_trancl__into__trancl2,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ ( transi9211502839322181930rm_a_b @ R ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( transi9211502839322181930rm_a_b @ R ) ) ) ) ).

% trancl_into_trancl2
thf(fact_96_trancl__into__trancl2,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,C3: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C3 ) @ ( transitive_trancl_a @ R ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C3 ) @ ( transitive_trancl_a @ R ) ) ) ) ).

% trancl_into_trancl2
thf(fact_97_trancl__into__trancl2,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( transi6264000038957366511cl_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).

% trancl_into_trancl2
thf(fact_98_trancl__into__trancl2,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).

% trancl_into_trancl2
thf(fact_99_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,C3: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi6805586507763294364_a_nat @ R ) )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ R )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C3 ) @ ( transi6805586507763294364_a_nat @ R ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_100_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ R )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( transi9211502839322181930rm_a_b @ R ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_101_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,C3: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C3 ) @ R )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C3 ) @ ( transitive_trancl_a @ R ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_102_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_103_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ R )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_104_irrefl__trancl__rD,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat,Y: product_prod_a_nat] :
      ( ! [X3: product_prod_a_nat] :
          ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ X3 ) @ ( transi6805586507763294364_a_nat @ R ) )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ R )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_105_irrefl__trancl__rD,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ X3 ) @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_106_irrefl__trancl__rD,axiom,
    ! [R: set_Product_prod_a_a,X: a,Y: a] :
      ( ! [X3: a] :
          ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ ( transitive_trancl_a @ R ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_107_irrefl__trancl__rD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_108_irrefl__trancl__rD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ! [X3: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_109_converse__tranclE,axiom,
    ! [X: product_prod_a_nat,Z2: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z2 ) @ ( transi6805586507763294364_a_nat @ R ) )
     => ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z2 ) @ R )
       => ~ ! [Y3: product_prod_a_nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y3 ) @ R )
             => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ Z2 ) @ ( transi6805586507763294364_a_nat @ R ) ) ) ) ) ).

% converse_tranclE
thf(fact_110_converse__tranclE,axiom,
    ! [X: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R )
       => ~ ! [Y3: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y3 ) @ R )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R ) ) ) ) ) ).

% converse_tranclE
thf(fact_111_converse__tranclE,axiom,
    ! [X: a,Z2: a,R: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ ( transitive_trancl_a @ R ) )
     => ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ R )
       => ~ ! [Y3: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
             => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z2 ) @ ( transitive_trancl_a @ R ) ) ) ) ) ).

% converse_tranclE
thf(fact_112_converse__tranclE,axiom,
    ! [X: nat,Z2: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R )
       => ~ ! [Y3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ) ).

% converse_tranclE
thf(fact_113_converse__tranclE,axiom,
    ! [X: term_a_b,Z2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R )
       => ~ ! [Y3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ) ).

% converse_tranclE
thf(fact_114_r__r__into__trancl,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R3: set_Pr1811044260758604347_a_nat,C3: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R3 )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ R3 )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C3 ) @ ( transi6805586507763294364_a_nat @ R3 ) ) ) ) ).

% r_r_into_trancl
thf(fact_115_r__r__into__trancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ R3 )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ).

% r_r_into_trancl
thf(fact_116_r__r__into__trancl,axiom,
    ! [A: a,B: a,R3: set_Product_prod_a_a,C3: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R3 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C3 ) @ R3 )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C3 ) @ ( transitive_trancl_a @ R3 ) ) ) ) ).

% r_r_into_trancl
thf(fact_117_r__r__into__trancl,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ R3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).

% r_r_into_trancl
thf(fact_118_r__r__into__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ R3 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).

% r_r_into_trancl
thf(fact_119_mem__Collect__eq,axiom,
    ! [A: produc4523971788519308903rm_a_b,P: produc4523971788519308903rm_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ A @ ( collec1344989105561874898rm_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_120_mem__Collect__eq,axiom,
    ! [A: term_a_b,P: term_a_b > $o] :
      ( ( member_term_a_b @ A @ ( collect_term_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_121_mem__Collect__eq,axiom,
    ! [A: produc4708774622424448987_a_nat,P: produc4708774622424448987_a_nat > $o] :
      ( ( member9062615507155100804_a_nat @ A @ ( collec925066218191019974_a_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_122_mem__Collect__eq,axiom,
    ! [A: product_prod_a_nat,P: product_prod_a_nat > $o] :
      ( ( member5724188588386418708_a_nat @ A @ ( collec4464134535221767506_a_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_123_mem__Collect__eq,axiom,
    ! [A: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_124_Collect__mem__eq,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b] :
      ( ( collec1344989105561874898rm_a_b
        @ ^ [X4: produc4523971788519308903rm_a_b] : ( member8417600551952982416rm_a_b @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_125_Collect__mem__eq,axiom,
    ! [A5: set_term_a_b] :
      ( ( collect_term_a_b
        @ ^ [X4: term_a_b] : ( member_term_a_b @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_126_Collect__mem__eq,axiom,
    ! [A5: set_Pr1811044260758604347_a_nat] :
      ( ( collec925066218191019974_a_nat
        @ ^ [X4: produc4708774622424448987_a_nat] : ( member9062615507155100804_a_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_127_Collect__mem__eq,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat] :
      ( ( collec4464134535221767506_a_nat
        @ ^ [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_128_Collect__mem__eq,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ( collec99177395925924084rm_a_b
        @ ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_129_Collect__cong,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec99177395925924084rm_a_b @ P )
        = ( collec99177395925924084rm_a_b @ Q ) ) ) ).

% Collect_cong
thf(fact_130_Collect__cong,axiom,
    ! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
      ( ! [X3: product_prod_a_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec4464134535221767506_a_nat @ P )
        = ( collec4464134535221767506_a_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_131_trancl__induct,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,P: product_prod_a_nat > $o] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi6805586507763294364_a_nat @ R ) )
     => ( ! [Y3: product_prod_a_nat] :
            ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ Y3 ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: product_prod_a_nat,Z: product_prod_a_nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ Y3 ) @ ( transi6805586507763294364_a_nat @ R ) )
             => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ Z ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_132_trancl__induct,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( ! [Y3: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y3 ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y3 ) @ ( transi9211502839322181930rm_a_b @ R ) )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_133_trancl__induct,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,P: a > $o] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R ) )
     => ( ! [Y3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ Y3 ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: a,Z: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ Y3 ) @ ( transitive_trancl_a @ R ) )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_134_trancl__induct,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ! [Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: nat,Z: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi6264000038957366511cl_nat @ R ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_135_trancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( ! [Y3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: term_a_b,Z: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_136_trancl__trans,axiom,
    ! [X: product_prod_a_nat,Y: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,Z2: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi6805586507763294364_a_nat @ R ) )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ Z2 ) @ ( transi6805586507763294364_a_nat @ R ) )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z2 ) @ ( transi6805586507763294364_a_nat @ R ) ) ) ) ).

% trancl_trans
thf(fact_137_trancl__trans,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,Z2: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R ) ) ) ) ).

% trancl_trans
thf(fact_138_trancl__trans,axiom,
    ! [X: a,Y: a,R: set_Product_prod_a_a,Z2: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ ( transitive_trancl_a @ R ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ ( transitive_trancl_a @ R ) ) ) ) ).

% trancl_trans
thf(fact_139_trancl__trans,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Z2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( transi6264000038957366511cl_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).

% trancl_trans
thf(fact_140_trancl__trans,axiom,
    ! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).

% trancl_trans
thf(fact_141_term__to__sig__id,axiom,
    ! [T: term_a_b,F: set_Pr4934435412358123699_a_nat,V: b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
     => ( ( terms_8519481630511763164ig_a_b @ F @ V @ T )
        = T ) ) ).

% term_to_sig_id
thf(fact_142_fuans__term__term__to__sig,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,V: b,T: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( terms_8519481630511763164ig_a_b @ F @ V @ T ) ) @ F ) ).

% fuans_term_term_to_sig
thf(fact_143_subsetI,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
      ( ! [X3: produc4523971788519308903rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ X3 @ A5 )
         => ( member8417600551952982416rm_a_b @ X3 @ B5 ) )
     => ( ord_le2593958535671617255rm_a_b @ A5 @ B5 ) ) ).

% subsetI
thf(fact_144_subsetI,axiom,
    ! [A5: set_Pr1811044260758604347_a_nat,B5: set_Pr1811044260758604347_a_nat] :
      ( ! [X3: produc4708774622424448987_a_nat] :
          ( ( member9062615507155100804_a_nat @ X3 @ A5 )
         => ( member9062615507155100804_a_nat @ X3 @ B5 ) )
     => ( ord_le1018141841545957851_a_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_145_subsetI,axiom,
    ! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ! [X3: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X3 @ A5 )
         => ( member1426531477525435216od_a_a @ X3 @ B5 ) )
     => ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ).

% subsetI
thf(fact_146_subsetI,axiom,
    ! [A5: set_term_a_b,B5: set_term_a_b] :
      ( ! [X3: term_a_b] :
          ( ( member_term_a_b @ X3 @ A5 )
         => ( member_term_a_b @ X3 @ B5 ) )
     => ( ord_le2705286416250468010rm_a_b @ A5 @ B5 ) ) ).

% subsetI
thf(fact_147_subsetI,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A5 )
         => ( member_nat @ X3 @ B5 ) )
     => ( ord_less_eq_set_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_148_subsetI,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A5 )
         => ( member_a @ X3 @ B5 ) )
     => ( ord_less_eq_set_a @ A5 @ B5 ) ) ).

% subsetI
thf(fact_149_subsetI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
         => ( member5869715511025134514rm_a_b @ X3 @ B5 ) )
     => ( ord_le118470702582115849rm_a_b @ A5 @ B5 ) ) ).

% subsetI
thf(fact_150_subsetI,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ! [X3: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ X3 @ A5 )
         => ( member5724188588386418708_a_nat @ X3 @ B5 ) )
     => ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_151_subset__antisym,axiom,
    ! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
     => ( ( ord_le746702958409616551od_a_a @ B5 @ A5 )
       => ( A5 = B5 ) ) ) ).

% subset_antisym
thf(fact_152_subset__antisym,axiom,
    ! [A5: set_term_a_b,B5: set_term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ A5 @ B5 )
     => ( ( ord_le2705286416250468010rm_a_b @ B5 @ A5 )
       => ( A5 = B5 ) ) ) ).

% subset_antisym
thf(fact_153_subset__antisym,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A5 @ B5 )
     => ( ( ord_less_eq_set_nat @ B5 @ A5 )
       => ( A5 = B5 ) ) ) ).

% subset_antisym
thf(fact_154_subset__antisym,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ( ord_less_eq_set_a @ B5 @ A5 )
       => ( A5 = B5 ) ) ) ).

% subset_antisym
thf(fact_155_subset__antisym,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ A5 )
       => ( A5 = B5 ) ) ) ).

% subset_antisym
thf(fact_156_order__refl,axiom,
    ! [X: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X @ X ) ).

% order_refl
thf(fact_157_order__refl,axiom,
    ! [X: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ X @ X ) ).

% order_refl
thf(fact_158_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_159_order__refl,axiom,
    ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).

% order_refl
thf(fact_160_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_161_order__refl,axiom,
    ! [X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ X ) ).

% order_refl
thf(fact_162_dual__order_Orefl,axiom,
    ! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).

% dual_order.refl
thf(fact_163_dual__order_Orefl,axiom,
    ! [A: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A @ A ) ).

% dual_order.refl
thf(fact_164_dual__order_Orefl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% dual_order.refl
thf(fact_165_dual__order_Orefl,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).

% dual_order.refl
thf(fact_166_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_167_dual__order_Orefl,axiom,
    ! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).

% dual_order.refl
thf(fact_168_trancl__mono__set,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R @ S )
     => ( ord_le2593958535671617255rm_a_b @ ( transi9211502839322181930rm_a_b @ R ) @ ( transi9211502839322181930rm_a_b @ S ) ) ) ).

% trancl_mono_set
thf(fact_169_trancl__mono__set,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ R @ S )
     => ( ord_le3146513528884898305at_nat @ ( transi6264000038957366511cl_nat @ R ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ).

% trancl_mono_set
thf(fact_170_trancl__mono__set,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ R @ S )
     => ( ord_le746702958409616551od_a_a @ ( transitive_trancl_a @ R ) @ ( transitive_trancl_a @ S ) ) ) ).

% trancl_mono_set
thf(fact_171_trancl__mono__set,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R @ S )
     => ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ).

% trancl_mono_set
thf(fact_172_subrelI,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
         => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ S ) )
     => ( ord_le2593958535671617255rm_a_b @ R @ S ) ) ).

% subrelI
thf(fact_173_subrelI,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
      ( ! [X3: product_prod_a_nat,Y3: product_prod_a_nat] :
          ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
         => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le1018141841545957851_a_nat @ R @ S ) ) ).

% subrelI
thf(fact_174_subrelI,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ! [X3: a,Y3: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S ) )
     => ( ord_le746702958409616551od_a_a @ R @ S ) ) ).

% subrelI
thf(fact_175_subrelI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ S ) )
     => ( ord_le118470702582115849rm_a_b @ R @ S ) ) ).

% subrelI
thf(fact_176_subrelI,axiom,
    ! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
      ( ! [X3: a,Y3: nat] :
          ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ R )
         => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le8666007276011122963_a_nat @ R @ S ) ) ).

% subrelI
thf(fact_177_srrstepD,axiom,
    ! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R2 ) ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R2 ) )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).

% srrstepD
thf(fact_178_srsteps__monp,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F @ G )
     => ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R2 ) ) ) ) ) ).

% srsteps_monp
thf(fact_179_rrstep__basicI,axiom,
    ! [L: term_a_b,R: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R2 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ ( rrstep_a_b @ R2 ) ) ) ).

% rrstep_basicI
thf(fact_180_rrstep__rstep__mono,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( rrstep_a_b @ R2 ) @ ( rstep_a_b @ R2 ) ) ).

% rrstep_rstep_mono
thf(fact_181_srrstep__monp,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F @ G )
     => ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R2 ) ) @ ( sig_step_a_b @ G @ ( rrstep_a_b @ R2 ) ) ) ) ).

% srrstep_monp
thf(fact_182_sig__step__mono2,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,L2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le118470702582115849rm_a_b @ R2 @ L2 )
     => ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R2 ) @ ( sig_step_a_b @ F @ L2 ) ) ) ).

% sig_step_mono2
thf(fact_183_sig__step__mono,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F @ G )
     => ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R2 ) @ ( sig_step_a_b @ G @ R2 ) ) ) ).

% sig_step_mono
thf(fact_184_order__antisym__conv,axiom,
    ! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ Y @ X )
     => ( ( ord_le746702958409616551od_a_a @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_185_order__antisym__conv,axiom,
    ! [Y: set_term_a_b,X: set_term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ Y @ X )
     => ( ( ord_le2705286416250468010rm_a_b @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_186_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_187_order__antisym__conv,axiom,
    ! [Y: set_a,X: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X )
     => ( ( ord_less_eq_set_a @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_188_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_189_order__antisym__conv,axiom,
    ! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ Y @ X )
     => ( ( ord_le8666007276011122963_a_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_190_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_191_ord__le__eq__subst,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C3 )
       => ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
              ( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
             => ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le8666007276011122963_a_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_192_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_193_ord__le__eq__subst,axiom,
    ! [A: set_nat,B: set_nat,F2: set_nat > nat,C3: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C3 )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_194_ord__le__eq__subst,axiom,
    ! [A: set_a,B: set_a,F2: set_a > nat,C3: nat] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ( F2 @ B )
          = C3 )
       => ( ! [X3: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_195_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > set_nat,C3: set_nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_196_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > set_a,C3: set_a] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F2 @ A ) @ C3 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_197_ord__le__eq__subst,axiom,
    ! [A: set_nat,B: set_nat,F2: set_nat > set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C3 )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_198_ord__le__eq__subst,axiom,
    ! [A: set_nat,B: set_nat,F2: set_nat > set_a,C3: set_a] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C3 )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F2 @ A ) @ C3 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_199_ord__le__eq__subst,axiom,
    ! [A: set_a,B: set_a,F2: set_a > set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ( F2 @ B )
          = C3 )
       => ( ! [X3: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_200_ord__le__eq__subst,axiom,
    ! [A: set_a,B: set_a,F2: set_a > set_a,C3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ( F2 @ B )
          = C3 )
       => ( ! [X3: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F2 @ A ) @ C3 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_201_ord__eq__le__subst,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
       => ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
              ( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
             => ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_202_ord__eq__le__subst,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C3: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_203_ord__eq__le__subst,axiom,
    ! [A: nat,F2: set_nat > nat,B: set_nat,C3: set_nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C3 )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_204_ord__eq__le__subst,axiom,
    ! [A: nat,F2: set_a > nat,B: set_a,C3: set_a] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_set_a @ B @ C3 )
       => ( ! [X3: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_205_ord__eq__le__subst,axiom,
    ! [A: set_nat,F2: nat > set_nat,B: nat,C3: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_206_ord__eq__le__subst,axiom,
    ! [A: set_a,F2: nat > set_a,B: nat,C3: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F2 @ C3 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_207_ord__eq__le__subst,axiom,
    ! [A: set_nat,F2: set_nat > set_nat,B: set_nat,C3: set_nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C3 )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_208_ord__eq__le__subst,axiom,
    ! [A: set_a,F2: set_nat > set_a,B: set_nat,C3: set_nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C3 )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F2 @ C3 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_209_ord__eq__le__subst,axiom,
    ! [A: set_nat,F2: set_a > set_nat,B: set_a,C3: set_a] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_set_a @ B @ C3 )
       => ( ! [X3: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_210_ord__eq__le__subst,axiom,
    ! [A: set_a,F2: set_a > set_a,B: set_a,C3: set_a] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_set_a @ B @ C3 )
       => ( ! [X3: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F2 @ C3 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_211_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_212_order__eq__refl,axiom,
    ! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
      ( ( X = Y )
     => ( ord_le746702958409616551od_a_a @ X @ Y ) ) ).

% order_eq_refl
thf(fact_213_order__eq__refl,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b] :
      ( ( X = Y )
     => ( ord_le2705286416250468010rm_a_b @ X @ Y ) ) ).

% order_eq_refl
thf(fact_214_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_215_order__eq__refl,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( X = Y )
     => ( ord_less_eq_set_a @ X @ Y ) ) ).

% order_eq_refl
thf(fact_216_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_217_order__eq__refl,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
      ( ( X = Y )
     => ( ord_le8666007276011122963_a_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_218_order__subst2,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ B )
     => ( ( ord_le8666007276011122963_a_nat @ ( F2 @ B ) @ C3 )
       => ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
              ( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
             => ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le8666007276011122963_a_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% order_subst2
thf(fact_219_order__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% order_subst2
thf(fact_220_order__subst2,axiom,
    ! [A: set_nat,B: set_nat,F2: set_nat > nat,C3: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C3 )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% order_subst2
thf(fact_221_order__subst2,axiom,
    ! [A: set_a,B: set_a,F2: set_a > nat,C3: nat] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C3 )
       => ( ! [X3: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% order_subst2
thf(fact_222_order__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > set_nat,C3: set_nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ ( F2 @ B ) @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% order_subst2
thf(fact_223_order__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > set_a,C3: set_a] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_set_a @ ( F2 @ B ) @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F2 @ A ) @ C3 ) ) ) ) ).

% order_subst2
thf(fact_224_order__subst2,axiom,
    ! [A: set_nat,B: set_nat,F2: set_nat > set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ ( F2 @ B ) @ C3 )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% order_subst2
thf(fact_225_order__subst2,axiom,
    ! [A: set_nat,B: set_nat,F2: set_nat > set_a,C3: set_a] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_a @ ( F2 @ B ) @ C3 )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F2 @ A ) @ C3 ) ) ) ) ).

% order_subst2
thf(fact_226_order__subst2,axiom,
    ! [A: set_a,B: set_a,F2: set_a > set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_nat @ ( F2 @ B ) @ C3 )
       => ( ! [X3: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% order_subst2
thf(fact_227_order__subst2,axiom,
    ! [A: set_a,B: set_a,F2: set_a > set_a,C3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ ( F2 @ B ) @ C3 )
       => ( ! [X3: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F2 @ A ) @ C3 ) ) ) ) ).

% order_subst2
thf(fact_228_order__subst1,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ B ) )
     => ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
       => ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
              ( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
             => ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% order_subst1
thf(fact_229_order__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% order_subst1
thf(fact_230_order__subst1,axiom,
    ! [A: set_nat,F2: nat > set_nat,B: nat,C3: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% order_subst1
thf(fact_231_order__subst1,axiom,
    ! [A: set_a,F2: nat > set_a,B: nat,C3: nat] :
      ( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F2 @ C3 ) ) ) ) ) ).

% order_subst1
thf(fact_232_order__subst1,axiom,
    ! [A: nat,F2: set_nat > nat,B: set_nat,C3: set_nat] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C3 )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% order_subst1
thf(fact_233_order__subst1,axiom,
    ! [A: nat,F2: set_a > nat,B: set_a,C3: set_a] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_set_a @ B @ C3 )
       => ( ! [X3: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% order_subst1
thf(fact_234_order__subst1,axiom,
    ! [A: set_nat,F2: set_nat > set_nat,B: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C3 )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% order_subst1
thf(fact_235_order__subst1,axiom,
    ! [A: set_nat,F2: set_a > set_nat,B: set_a,C3: set_a] :
      ( ( ord_less_eq_set_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_set_a @ B @ C3 )
       => ( ! [X3: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% order_subst1
thf(fact_236_order__subst1,axiom,
    ! [A: set_a,F2: set_nat > set_a,B: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C3 )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F2 @ C3 ) ) ) ) ) ).

% order_subst1
thf(fact_237_order__subst1,axiom,
    ! [A: set_a,F2: set_a > set_a,B: set_a,C3: set_a] :
      ( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_set_a @ B @ C3 )
       => ( ! [X3: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F2 @ C3 ) ) ) ) ) ).

% order_subst1
thf(fact_238_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
          ( ( ord_le746702958409616551od_a_a @ A3 @ B4 )
          & ( ord_le746702958409616551od_a_a @ B4 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_239_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_term_a_b,Z3: set_term_a_b] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_term_a_b,B4: set_term_a_b] :
          ( ( ord_le2705286416250468010rm_a_b @ A3 @ B4 )
          & ( ord_le2705286416250468010rm_a_b @ B4 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_240_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z3: set_nat] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B4 )
          & ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_241_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ A3 @ B4 )
          & ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_242_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
    = ( ^ [A3: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A3 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_243_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ A3 @ B4 )
          & ( ord_le8666007276011122963_a_nat @ B4 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_244_antisym,axiom,
    ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B )
     => ( ( ord_le746702958409616551od_a_a @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_245_antisym,axiom,
    ! [A: set_term_a_b,B: set_term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ A @ B )
     => ( ( ord_le2705286416250468010rm_a_b @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_246_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_247_antisym,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_248_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_249_antisym,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ B )
     => ( ( ord_le8666007276011122963_a_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_250_dual__order_Otrans,axiom,
    ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,C3: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ B @ A )
     => ( ( ord_le746702958409616551od_a_a @ C3 @ B )
       => ( ord_le746702958409616551od_a_a @ C3 @ A ) ) ) ).

% dual_order.trans
thf(fact_251_dual__order_Otrans,axiom,
    ! [B: set_term_a_b,A: set_term_a_b,C3: set_term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ B @ A )
     => ( ( ord_le2705286416250468010rm_a_b @ C3 @ B )
       => ( ord_le2705286416250468010rm_a_b @ C3 @ A ) ) ) ).

% dual_order.trans
thf(fact_252_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C3 @ B )
       => ( ord_less_eq_set_nat @ C3 @ A ) ) ) ).

% dual_order.trans
thf(fact_253_dual__order_Otrans,axiom,
    ! [B: set_a,A: set_a,C3: set_a] :
      ( ( ord_less_eq_set_a @ B @ A )
     => ( ( ord_less_eq_set_a @ C3 @ B )
       => ( ord_less_eq_set_a @ C3 @ A ) ) ) ).

% dual_order.trans
thf(fact_254_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C3: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C3 @ B )
       => ( ord_less_eq_nat @ C3 @ A ) ) ) ).

% dual_order.trans
thf(fact_255_dual__order_Otrans,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ B @ A )
     => ( ( ord_le8666007276011122963_a_nat @ C3 @ B )
       => ( ord_le8666007276011122963_a_nat @ C3 @ A ) ) ) ).

% dual_order.trans
thf(fact_256_dual__order_Oantisym,axiom,
    ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ B @ A )
     => ( ( ord_le746702958409616551od_a_a @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_257_dual__order_Oantisym,axiom,
    ! [B: set_term_a_b,A: set_term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ B @ A )
     => ( ( ord_le2705286416250468010rm_a_b @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_258_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_259_dual__order_Oantisym,axiom,
    ! [B: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ A )
     => ( ( ord_less_eq_set_a @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_260_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_261_dual__order_Oantisym,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ B @ A )
     => ( ( ord_le8666007276011122963_a_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_262_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
          ( ( ord_le746702958409616551od_a_a @ B4 @ A3 )
          & ( ord_le746702958409616551od_a_a @ A3 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_263_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_term_a_b,Z3: set_term_a_b] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_term_a_b,B4: set_term_a_b] :
          ( ( ord_le2705286416250468010rm_a_b @ B4 @ A3 )
          & ( ord_le2705286416250468010rm_a_b @ A3 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_264_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z3: set_nat] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ B4 @ A3 )
          & ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_265_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ B4 @ A3 )
          & ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_266_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
    = ( ^ [A3: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_267_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ B4 @ A3 )
          & ( ord_le8666007276011122963_a_nat @ A3 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_268_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: nat,B3: nat] :
            ( ( P @ B3 @ A4 )
           => ( P @ A4 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_269_order__trans,axiom,
    ! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ X @ Y )
     => ( ( ord_le746702958409616551od_a_a @ Y @ Z2 )
       => ( ord_le746702958409616551od_a_a @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_270_order__trans,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b,Z2: set_term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ X @ Y )
     => ( ( ord_le2705286416250468010rm_a_b @ Y @ Z2 )
       => ( ord_le2705286416250468010rm_a_b @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_271_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z2 )
       => ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_272_order__trans,axiom,
    ! [X: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ Y @ Z2 )
       => ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_273_order__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_274_order__trans,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ Y )
     => ( ( ord_le8666007276011122963_a_nat @ Y @ Z2 )
       => ( ord_le8666007276011122963_a_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_275_order_Otrans,axiom,
    ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C3: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B )
     => ( ( ord_le746702958409616551od_a_a @ B @ C3 )
       => ( ord_le746702958409616551od_a_a @ A @ C3 ) ) ) ).

% order.trans
thf(fact_276_order_Otrans,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,C3: set_term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ A @ B )
     => ( ( ord_le2705286416250468010rm_a_b @ B @ C3 )
       => ( ord_le2705286416250468010rm_a_b @ A @ C3 ) ) ) ).

% order.trans
thf(fact_277_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C3 )
       => ( ord_less_eq_set_nat @ A @ C3 ) ) ) ).

% order.trans
thf(fact_278_order_Otrans,axiom,
    ! [A: set_a,B: set_a,C3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ C3 )
       => ( ord_less_eq_set_a @ A @ C3 ) ) ) ).

% order.trans
thf(fact_279_order_Otrans,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ord_less_eq_nat @ A @ C3 ) ) ) ).

% order.trans
thf(fact_280_order_Otrans,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ B )
     => ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
       => ( ord_le8666007276011122963_a_nat @ A @ C3 ) ) ) ).

% order.trans
thf(fact_281_order__antisym,axiom,
    ! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ X @ Y )
     => ( ( ord_le746702958409616551od_a_a @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_282_order__antisym,axiom,
    ! [X: set_term_a_b,Y: set_term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ X @ Y )
     => ( ( ord_le2705286416250468010rm_a_b @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_283_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_284_order__antisym,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_285_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_286_order__antisym,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ Y )
     => ( ( ord_le8666007276011122963_a_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_287_ord__le__eq__trans,axiom,
    ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C3: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B )
     => ( ( B = C3 )
       => ( ord_le746702958409616551od_a_a @ A @ C3 ) ) ) ).

% ord_le_eq_trans
thf(fact_288_ord__le__eq__trans,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,C3: set_term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ A @ B )
     => ( ( B = C3 )
       => ( ord_le2705286416250468010rm_a_b @ A @ C3 ) ) ) ).

% ord_le_eq_trans
thf(fact_289_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( B = C3 )
       => ( ord_less_eq_set_nat @ A @ C3 ) ) ) ).

% ord_le_eq_trans
thf(fact_290_ord__le__eq__trans,axiom,
    ! [A: set_a,B: set_a,C3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( B = C3 )
       => ( ord_less_eq_set_a @ A @ C3 ) ) ) ).

% ord_le_eq_trans
thf(fact_291_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C3 )
       => ( ord_less_eq_nat @ A @ C3 ) ) ) ).

% ord_le_eq_trans
thf(fact_292_ord__le__eq__trans,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ B )
     => ( ( B = C3 )
       => ( ord_le8666007276011122963_a_nat @ A @ C3 ) ) ) ).

% ord_le_eq_trans
thf(fact_293_ord__eq__le__trans,axiom,
    ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C3: set_Product_prod_a_a] :
      ( ( A = B )
     => ( ( ord_le746702958409616551od_a_a @ B @ C3 )
       => ( ord_le746702958409616551od_a_a @ A @ C3 ) ) ) ).

% ord_eq_le_trans
thf(fact_294_ord__eq__le__trans,axiom,
    ! [A: set_term_a_b,B: set_term_a_b,C3: set_term_a_b] :
      ( ( A = B )
     => ( ( ord_le2705286416250468010rm_a_b @ B @ C3 )
       => ( ord_le2705286416250468010rm_a_b @ A @ C3 ) ) ) ).

% ord_eq_le_trans
thf(fact_295_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C3: set_nat] :
      ( ( A = B )
     => ( ( ord_less_eq_set_nat @ B @ C3 )
       => ( ord_less_eq_set_nat @ A @ C3 ) ) ) ).

% ord_eq_le_trans
thf(fact_296_ord__eq__le__trans,axiom,
    ! [A: set_a,B: set_a,C3: set_a] :
      ( ( A = B )
     => ( ( ord_less_eq_set_a @ B @ C3 )
       => ( ord_less_eq_set_a @ A @ C3 ) ) ) ).

% ord_eq_le_trans
thf(fact_297_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ord_less_eq_nat @ A @ C3 ) ) ) ).

% ord_eq_le_trans
thf(fact_298_ord__eq__le__trans,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( A = B )
     => ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
       => ( ord_le8666007276011122963_a_nat @ A @ C3 ) ) ) ).

% ord_eq_le_trans
thf(fact_299_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y4 = Z3 ) )
    = ( ^ [X4: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
          ( ( ord_le746702958409616551od_a_a @ X4 @ Y5 )
          & ( ord_le746702958409616551od_a_a @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_300_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_term_a_b,Z3: set_term_a_b] : ( Y4 = Z3 ) )
    = ( ^ [X4: set_term_a_b,Y5: set_term_a_b] :
          ( ( ord_le2705286416250468010rm_a_b @ X4 @ Y5 )
          & ( ord_le2705286416250468010rm_a_b @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_301_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z3: set_nat] : ( Y4 = Z3 ) )
    = ( ^ [X4: set_nat,Y5: set_nat] :
          ( ( ord_less_eq_set_nat @ X4 @ Y5 )
          & ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_302_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
    = ( ^ [X4: set_a,Y5: set_a] :
          ( ( ord_less_eq_set_a @ X4 @ Y5 )
          & ( ord_less_eq_set_a @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_303_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_304_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] : ( Y4 = Z3 ) )
    = ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ X4 @ Y5 )
          & ( ord_le8666007276011122963_a_nat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_305_le__cases3,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_306_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_307_Collect__mono__iff,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
      ( ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ P ) @ ( collec99177395925924084rm_a_b @ Q ) )
      = ( ! [X4: produc357393685978478089rm_a_b] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_308_Collect__mono__iff,axiom,
    ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
      ( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) )
      = ( ! [X4: product_prod_a_a] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_309_Collect__mono__iff,axiom,
    ! [P: term_a_b > $o,Q: term_a_b > $o] :
      ( ( ord_le2705286416250468010rm_a_b @ ( collect_term_a_b @ P ) @ ( collect_term_a_b @ Q ) )
      = ( ! [X4: term_a_b] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_310_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_311_Collect__mono__iff,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
      = ( ! [X4: a] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_312_Collect__mono__iff,axiom,
    ! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
      ( ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P ) @ ( collec4464134535221767506_a_nat @ Q ) )
      = ( ! [X4: product_prod_a_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_313_set__eq__subset,axiom,
    ( ( ^ [Y4: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y4 = Z3 ) )
    = ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
          ( ( ord_le746702958409616551od_a_a @ A6 @ B6 )
          & ( ord_le746702958409616551od_a_a @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_314_set__eq__subset,axiom,
    ( ( ^ [Y4: set_term_a_b,Z3: set_term_a_b] : ( Y4 = Z3 ) )
    = ( ^ [A6: set_term_a_b,B6: set_term_a_b] :
          ( ( ord_le2705286416250468010rm_a_b @ A6 @ B6 )
          & ( ord_le2705286416250468010rm_a_b @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_315_set__eq__subset,axiom,
    ( ( ^ [Y4: set_nat,Z3: set_nat] : ( Y4 = Z3 ) )
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B6 )
          & ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_316_set__eq__subset,axiom,
    ( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
    = ( ^ [A6: set_a,B6: set_a] :
          ( ( ord_less_eq_set_a @ A6 @ B6 )
          & ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_317_set__eq__subset,axiom,
    ( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] : ( Y4 = Z3 ) )
    = ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ A6 @ B6 )
          & ( ord_le8666007276011122963_a_nat @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_318_subset__trans,axiom,
    ! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,C4: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
     => ( ( ord_le746702958409616551od_a_a @ B5 @ C4 )
       => ( ord_le746702958409616551od_a_a @ A5 @ C4 ) ) ) ).

% subset_trans
thf(fact_319_subset__trans,axiom,
    ! [A5: set_term_a_b,B5: set_term_a_b,C4: set_term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ A5 @ B5 )
     => ( ( ord_le2705286416250468010rm_a_b @ B5 @ C4 )
       => ( ord_le2705286416250468010rm_a_b @ A5 @ C4 ) ) ) ).

% subset_trans
thf(fact_320_subset__trans,axiom,
    ! [A5: set_nat,B5: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A5 @ B5 )
     => ( ( ord_less_eq_set_nat @ B5 @ C4 )
       => ( ord_less_eq_set_nat @ A5 @ C4 ) ) ) ).

% subset_trans
thf(fact_321_subset__trans,axiom,
    ! [A5: set_a,B5: set_a,C4: set_a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ( ord_less_eq_set_a @ B5 @ C4 )
       => ( ord_less_eq_set_a @ A5 @ C4 ) ) ) ).

% subset_trans
thf(fact_322_subset__trans,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ C4 )
       => ( ord_le8666007276011122963_a_nat @ A5 @ C4 ) ) ) ).

% subset_trans
thf(fact_323_Collect__mono,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ P ) @ ( collec99177395925924084rm_a_b @ Q ) ) ) ).

% Collect_mono
thf(fact_324_Collect__mono,axiom,
    ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
      ( ! [X3: product_prod_a_a] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).

% Collect_mono
thf(fact_325_Collect__mono,axiom,
    ! [P: term_a_b > $o,Q: term_a_b > $o] :
      ( ! [X3: term_a_b] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le2705286416250468010rm_a_b @ ( collect_term_a_b @ P ) @ ( collect_term_a_b @ Q ) ) ) ).

% Collect_mono
thf(fact_326_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_327_Collect__mono,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ! [X3: a] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).

% Collect_mono
thf(fact_328_Collect__mono,axiom,
    ! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
      ( ! [X3: product_prod_a_nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P ) @ ( collec4464134535221767506_a_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_329_subset__refl,axiom,
    ! [A5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A5 @ A5 ) ).

% subset_refl
thf(fact_330_subset__refl,axiom,
    ! [A5: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A5 @ A5 ) ).

% subset_refl
thf(fact_331_subset__refl,axiom,
    ! [A5: set_nat] : ( ord_less_eq_set_nat @ A5 @ A5 ) ).

% subset_refl
thf(fact_332_subset__refl,axiom,
    ! [A5: set_a] : ( ord_less_eq_set_a @ A5 @ A5 ) ).

% subset_refl
thf(fact_333_subset__refl,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A5 @ A5 ) ).

% subset_refl
thf(fact_334_subset__iff,axiom,
    ( ord_le2593958535671617255rm_a_b
    = ( ^ [A6: set_Pr2972776593051762503rm_a_b,B6: set_Pr2972776593051762503rm_a_b] :
        ! [T2: produc4523971788519308903rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ T2 @ A6 )
         => ( member8417600551952982416rm_a_b @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_335_subset__iff,axiom,
    ( ord_le1018141841545957851_a_nat
    = ( ^ [A6: set_Pr1811044260758604347_a_nat,B6: set_Pr1811044260758604347_a_nat] :
        ! [T2: produc4708774622424448987_a_nat] :
          ( ( member9062615507155100804_a_nat @ T2 @ A6 )
         => ( member9062615507155100804_a_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_336_subset__iff,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
        ! [T2: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ T2 @ A6 )
         => ( member1426531477525435216od_a_a @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_337_subset__iff,axiom,
    ( ord_le2705286416250468010rm_a_b
    = ( ^ [A6: set_term_a_b,B6: set_term_a_b] :
        ! [T2: term_a_b] :
          ( ( member_term_a_b @ T2 @ A6 )
         => ( member_term_a_b @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_338_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A6 )
         => ( member_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_339_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A6: set_a,B6: set_a] :
        ! [T2: a] :
          ( ( member_a @ T2 @ A6 )
         => ( member_a @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_340_subset__iff,axiom,
    ( ord_le118470702582115849rm_a_b
    = ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
        ! [T2: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ T2 @ A6 )
         => ( member5869715511025134514rm_a_b @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_341_subset__iff,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
        ! [T2: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ T2 @ A6 )
         => ( member5724188588386418708_a_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_342_equalityD2,axiom,
    ! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ( A5 = B5 )
     => ( ord_le746702958409616551od_a_a @ B5 @ A5 ) ) ).

% equalityD2
thf(fact_343_equalityD2,axiom,
    ! [A5: set_term_a_b,B5: set_term_a_b] :
      ( ( A5 = B5 )
     => ( ord_le2705286416250468010rm_a_b @ B5 @ A5 ) ) ).

% equalityD2
thf(fact_344_equalityD2,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( A5 = B5 )
     => ( ord_less_eq_set_nat @ B5 @ A5 ) ) ).

% equalityD2
thf(fact_345_equalityD2,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( A5 = B5 )
     => ( ord_less_eq_set_a @ B5 @ A5 ) ) ).

% equalityD2
thf(fact_346_equalityD2,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( A5 = B5 )
     => ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ).

% equalityD2
thf(fact_347_equalityD1,axiom,
    ! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ( A5 = B5 )
     => ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ).

% equalityD1
thf(fact_348_equalityD1,axiom,
    ! [A5: set_term_a_b,B5: set_term_a_b] :
      ( ( A5 = B5 )
     => ( ord_le2705286416250468010rm_a_b @ A5 @ B5 ) ) ).

% equalityD1
thf(fact_349_equalityD1,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( A5 = B5 )
     => ( ord_less_eq_set_nat @ A5 @ B5 ) ) ).

% equalityD1
thf(fact_350_equalityD1,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( A5 = B5 )
     => ( ord_less_eq_set_a @ A5 @ B5 ) ) ).

% equalityD1
thf(fact_351_equalityD1,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( A5 = B5 )
     => ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ).

% equalityD1
thf(fact_352_subset__eq,axiom,
    ( ord_le2593958535671617255rm_a_b
    = ( ^ [A6: set_Pr2972776593051762503rm_a_b,B6: set_Pr2972776593051762503rm_a_b] :
        ! [X4: produc4523971788519308903rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ X4 @ A6 )
         => ( member8417600551952982416rm_a_b @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_353_subset__eq,axiom,
    ( ord_le1018141841545957851_a_nat
    = ( ^ [A6: set_Pr1811044260758604347_a_nat,B6: set_Pr1811044260758604347_a_nat] :
        ! [X4: produc4708774622424448987_a_nat] :
          ( ( member9062615507155100804_a_nat @ X4 @ A6 )
         => ( member9062615507155100804_a_nat @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_354_subset__eq,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
        ! [X4: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ A6 )
         => ( member1426531477525435216od_a_a @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_355_subset__eq,axiom,
    ( ord_le2705286416250468010rm_a_b
    = ( ^ [A6: set_term_a_b,B6: set_term_a_b] :
        ! [X4: term_a_b] :
          ( ( member_term_a_b @ X4 @ A6 )
         => ( member_term_a_b @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_356_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A6 )
         => ( member_nat @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_357_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A6: set_a,B6: set_a] :
        ! [X4: a] :
          ( ( member_a @ X4 @ A6 )
         => ( member_a @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_358_subset__eq,axiom,
    ( ord_le118470702582115849rm_a_b
    = ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
        ! [X4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X4 @ A6 )
         => ( member5869715511025134514rm_a_b @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_359_subset__eq,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
        ! [X4: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ X4 @ A6 )
         => ( member5724188588386418708_a_nat @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_360_equalityE,axiom,
    ! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ( A5 = B5 )
     => ~ ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
         => ~ ( ord_le746702958409616551od_a_a @ B5 @ A5 ) ) ) ).

% equalityE
thf(fact_361_equalityE,axiom,
    ! [A5: set_term_a_b,B5: set_term_a_b] :
      ( ( A5 = B5 )
     => ~ ( ( ord_le2705286416250468010rm_a_b @ A5 @ B5 )
         => ~ ( ord_le2705286416250468010rm_a_b @ B5 @ A5 ) ) ) ).

% equalityE
thf(fact_362_equalityE,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( A5 = B5 )
     => ~ ( ( ord_less_eq_set_nat @ A5 @ B5 )
         => ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ).

% equalityE
thf(fact_363_equalityE,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( A5 = B5 )
     => ~ ( ( ord_less_eq_set_a @ A5 @ B5 )
         => ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ).

% equalityE
thf(fact_364_equalityE,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( A5 = B5 )
     => ~ ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
         => ~ ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ) ).

% equalityE
thf(fact_365_subsetD,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,C3: produc4523971788519308903rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ A5 @ B5 )
     => ( ( member8417600551952982416rm_a_b @ C3 @ A5 )
       => ( member8417600551952982416rm_a_b @ C3 @ B5 ) ) ) ).

% subsetD
thf(fact_366_subsetD,axiom,
    ! [A5: set_Pr1811044260758604347_a_nat,B5: set_Pr1811044260758604347_a_nat,C3: produc4708774622424448987_a_nat] :
      ( ( ord_le1018141841545957851_a_nat @ A5 @ B5 )
     => ( ( member9062615507155100804_a_nat @ C3 @ A5 )
       => ( member9062615507155100804_a_nat @ C3 @ B5 ) ) ) ).

% subsetD
thf(fact_367_subsetD,axiom,
    ! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,C3: product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
     => ( ( member1426531477525435216od_a_a @ C3 @ A5 )
       => ( member1426531477525435216od_a_a @ C3 @ B5 ) ) ) ).

% subsetD
thf(fact_368_subsetD,axiom,
    ! [A5: set_term_a_b,B5: set_term_a_b,C3: term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ A5 @ B5 )
     => ( ( member_term_a_b @ C3 @ A5 )
       => ( member_term_a_b @ C3 @ B5 ) ) ) ).

% subsetD
thf(fact_369_subsetD,axiom,
    ! [A5: set_nat,B5: set_nat,C3: nat] :
      ( ( ord_less_eq_set_nat @ A5 @ B5 )
     => ( ( member_nat @ C3 @ A5 )
       => ( member_nat @ C3 @ B5 ) ) ) ).

% subsetD
thf(fact_370_subsetD,axiom,
    ! [A5: set_a,B5: set_a,C3: a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ( member_a @ C3 @ A5 )
       => ( member_a @ C3 @ B5 ) ) ) ).

% subsetD
thf(fact_371_subsetD,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A5 @ B5 )
     => ( ( member5869715511025134514rm_a_b @ C3 @ A5 )
       => ( member5869715511025134514rm_a_b @ C3 @ B5 ) ) ) ).

% subsetD
thf(fact_372_subsetD,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C3: product_prod_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( member5724188588386418708_a_nat @ C3 @ A5 )
       => ( member5724188588386418708_a_nat @ C3 @ B5 ) ) ) ).

% subsetD
thf(fact_373_in__mono,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,X: produc4523971788519308903rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ A5 @ B5 )
     => ( ( member8417600551952982416rm_a_b @ X @ A5 )
       => ( member8417600551952982416rm_a_b @ X @ B5 ) ) ) ).

% in_mono
thf(fact_374_in__mono,axiom,
    ! [A5: set_Pr1811044260758604347_a_nat,B5: set_Pr1811044260758604347_a_nat,X: produc4708774622424448987_a_nat] :
      ( ( ord_le1018141841545957851_a_nat @ A5 @ B5 )
     => ( ( member9062615507155100804_a_nat @ X @ A5 )
       => ( member9062615507155100804_a_nat @ X @ B5 ) ) ) ).

% in_mono
thf(fact_375_in__mono,axiom,
    ! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,X: product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
     => ( ( member1426531477525435216od_a_a @ X @ A5 )
       => ( member1426531477525435216od_a_a @ X @ B5 ) ) ) ).

% in_mono
thf(fact_376_in__mono,axiom,
    ! [A5: set_term_a_b,B5: set_term_a_b,X: term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ A5 @ B5 )
     => ( ( member_term_a_b @ X @ A5 )
       => ( member_term_a_b @ X @ B5 ) ) ) ).

% in_mono
thf(fact_377_in__mono,axiom,
    ! [A5: set_nat,B5: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A5 @ B5 )
     => ( ( member_nat @ X @ A5 )
       => ( member_nat @ X @ B5 ) ) ) ).

% in_mono
thf(fact_378_in__mono,axiom,
    ! [A5: set_a,B5: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ( member_a @ X @ A5 )
       => ( member_a @ X @ B5 ) ) ) ).

% in_mono
thf(fact_379_in__mono,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A5 @ B5 )
     => ( ( member5869715511025134514rm_a_b @ X @ A5 )
       => ( member5869715511025134514rm_a_b @ X @ B5 ) ) ) ).

% in_mono
thf(fact_380_in__mono,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( member5724188588386418708_a_nat @ X @ A5 )
       => ( member5724188588386418708_a_nat @ X @ B5 ) ) ) ).

% in_mono
thf(fact_381_srstep__monp,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F @ G )
     => ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ G @ ( rstep_a_b @ R2 ) ) ) ) ).

% srstep_monp
thf(fact_382_srsteps__with__root__step__sig__mono,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F @ G )
     => ( ord_le118470702582115849rm_a_b @ ( srstep7844470518422762656ep_a_b @ F @ R2 ) @ ( srstep7844470518422762656ep_a_b @ G @ R2 ) ) ) ).

% srsteps_with_root_step_sig_mono
thf(fact_383_restrict__SN__trancl__simp,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b] :
      ( ( transi9211502839322181930rm_a_b @ ( abstra5272529552662653398rm_a_b @ A5 @ A5 ) )
      = ( abstra5272529552662653398rm_a_b @ ( transi9211502839322181930rm_a_b @ A5 ) @ A5 ) ) ).

% restrict_SN_trancl_simp
thf(fact_384_restrict__SN__trancl__simp,axiom,
    ! [A5: set_Product_prod_a_a] :
      ( ( transitive_trancl_a @ ( abstra2152889483027340875t_SN_a @ A5 @ A5 ) )
      = ( abstra2152889483027340875t_SN_a @ ( transitive_trancl_a @ A5 ) @ A5 ) ) ).

% restrict_SN_trancl_simp
thf(fact_385_restrict__SN__trancl__simp,axiom,
    ! [A5: set_Pr1261947904930325089at_nat] :
      ( ( transi6264000038957366511cl_nat @ ( abstra2540617779794743107SN_nat @ A5 @ A5 ) )
      = ( abstra2540617779794743107SN_nat @ ( transi6264000038957366511cl_nat @ A5 ) @ A5 ) ) ).

% restrict_SN_trancl_simp
thf(fact_386_restrict__SN__trancl__simp,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7922773638565587891rm_a_b @ ( abstra4933154510623179783rm_a_b @ A5 @ A5 ) )
      = ( abstra4933154510623179783rm_a_b @ ( transi7922773638565587891rm_a_b @ A5 ) @ A5 ) ) ).

% restrict_SN_trancl_simp
thf(fact_387_srsteps__eq__monp,axiom,
    ! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F @ G )
     => ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R2 ) ) ) ) ) ).

% srsteps_eq_monp
thf(fact_388_same__fstI,axiom,
    ! [P: a > $o,X: a,Y6: nat,Y: nat,R3: a > set_Pr1261947904930325089at_nat] :
      ( ( P @ X )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y ) @ ( R3 @ X ) )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ X @ Y6 ) @ ( product_Pair_a_nat @ X @ Y ) ) @ ( same_fst_a_nat @ P @ R3 ) ) ) ) ).

% same_fstI
thf(fact_389_same__fstI,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,X: produc357393685978478089rm_a_b,Y6: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: produc357393685978478089rm_a_b > set_Pr2972776593051762503rm_a_b] :
      ( ( P @ X )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ Y ) @ ( R3 @ X ) )
       => ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y6 ) @ ( produc1763473618796451543rm_a_b @ X @ Y ) ) @ ( same_f7764617898092871340rm_a_b @ P @ R3 ) ) ) ) ).

% same_fstI
thf(fact_390_same__fstI,axiom,
    ! [P: product_prod_a_nat > $o,X: product_prod_a_nat,Y6: product_prod_a_nat,Y: product_prod_a_nat,R3: product_prod_a_nat > set_Pr1811044260758604347_a_nat] :
      ( ( P @ X )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y6 @ Y ) @ ( R3 @ X ) )
       => ( member3559618881423075344_a_nat @ ( produc1024945203614457431_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y6 ) @ ( produc2026711137822539155_a_nat @ X @ Y ) ) @ ( same_f6637753491745052904_a_nat @ P @ R3 ) ) ) ) ).

% same_fstI
thf(fact_391_same__fstI,axiom,
    ! [P: term_a_b > $o,X: term_a_b,Y6: term_a_b,Y: term_a_b,R3: term_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( P @ X )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Y ) @ ( R3 @ X ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y6 ) @ ( produc7020197800436672577rm_a_b @ X @ Y ) ) @ ( same_f843731131676418198rm_a_b @ P @ R3 ) ) ) ) ).

% same_fstI
thf(fact_392_relChain__def,axiom,
    ( bNF_Ca9222207453146494317_a_nat
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,As: term_a_b > set_Pr4934435412358123699_a_nat] :
        ! [I: term_a_b,J: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
         => ( ord_le8666007276011122963_a_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_393_relChain__def,axiom,
    ( bNF_Ca2413601583040309898_b_nat
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,As: term_a_b > nat] :
        ! [I: term_a_b,J: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_394_relChain__def,axiom,
    ( bNF_Ca2953780123467829363at_nat
    = ( ^ [R4: set_Pr1811044260758604347_a_nat,As: product_prod_a_nat > nat] :
        ! [I: product_prod_a_nat,J: product_prod_a_nat] :
          ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I @ J ) @ R4 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_395_relChain__def,axiom,
    ( bNF_Ca8621080293709114944et_nat
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,As: term_a_b > set_nat] :
        ! [I: term_a_b,J: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
         => ( ord_less_eq_set_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_396_relChain__def,axiom,
    ( bNF_Ca7371746721489711145et_nat
    = ( ^ [R4: set_Pr1811044260758604347_a_nat,As: product_prod_a_nat > set_nat] :
        ! [I: product_prod_a_nat,J: product_prod_a_nat] :
          ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I @ J ) @ R4 )
         => ( ord_less_eq_set_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_397_relChain__def,axiom,
    ( bNF_Ca1840730222860064804_set_a
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,As: term_a_b > set_a] :
        ! [I: term_a_b,J: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
         => ( ord_less_eq_set_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_398_relChain__def,axiom,
    ( bNF_Ca1323417287861452411_set_a
    = ( ^ [R4: set_Pr1811044260758604347_a_nat,As: product_prod_a_nat > set_a] :
        ! [I: product_prod_a_nat,J: product_prod_a_nat] :
          ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I @ J ) @ R4 )
         => ( ord_less_eq_set_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_399_relChain__def,axiom,
    ( bNF_Ca8408884726352799318_a_nat
    = ( ^ [R4: set_Pr1811044260758604347_a_nat,As: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
        ! [I: product_prod_a_nat,J: product_prod_a_nat] :
          ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I @ J ) @ R4 )
         => ( ord_le8666007276011122963_a_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_400_relChain__def,axiom,
    ( bNF_Ca2414252441828076237od_a_a
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,As: term_a_b > set_Product_prod_a_a] :
        ! [I: term_a_b,J: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
         => ( ord_le746702958409616551od_a_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_401_relChain__def,axiom,
    ( bNF_Ca2024383351977550372od_a_a
    = ( ^ [R4: set_Pr1811044260758604347_a_nat,As: product_prod_a_nat > set_Product_prod_a_a] :
        ! [I: product_prod_a_nat,J: product_prod_a_nat] :
          ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I @ J ) @ R4 )
         => ( ord_le746702958409616551od_a_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_402_Greatest__equality,axiom,
    ! [P: set_Product_prod_a_a > $o,X: set_Product_prod_a_a] :
      ( ( P @ X )
     => ( ! [Y3: set_Product_prod_a_a] :
            ( ( P @ Y3 )
           => ( ord_le746702958409616551od_a_a @ Y3 @ X ) )
       => ( ( order_6315983676290676192od_a_a @ P )
          = X ) ) ) ).

% Greatest_equality
thf(fact_403_Greatest__equality,axiom,
    ! [P: set_term_a_b > $o,X: set_term_a_b] :
      ( ( P @ X )
     => ( ! [Y3: set_term_a_b] :
            ( ( P @ Y3 )
           => ( ord_le2705286416250468010rm_a_b @ Y3 @ X ) )
       => ( ( order_8964606823743705265rm_a_b @ P )
          = X ) ) ) ).

% Greatest_equality
thf(fact_404_Greatest__equality,axiom,
    ! [P: set_nat > $o,X: set_nat] :
      ( ( P @ X )
     => ( ! [Y3: set_nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_set_nat @ Y3 @ X ) )
       => ( ( order_5724808138429204845et_nat @ P )
          = X ) ) ) ).

% Greatest_equality
thf(fact_405_Greatest__equality,axiom,
    ! [P: set_a > $o,X: set_a] :
      ( ( P @ X )
     => ( ! [Y3: set_a] :
            ( ( P @ Y3 )
           => ( ord_less_eq_set_a @ Y3 @ X ) )
       => ( ( order_Greatest_set_a @ P )
          = X ) ) ) ).

% Greatest_equality
thf(fact_406_Greatest__equality,axiom,
    ! [P: nat > $o,X: nat] :
      ( ( P @ X )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ X ) )
       => ( ( order_Greatest_nat @ P )
          = X ) ) ) ).

% Greatest_equality
thf(fact_407_Greatest__equality,axiom,
    ! [P: set_Pr4934435412358123699_a_nat > $o,X: set_Pr4934435412358123699_a_nat] :
      ( ( P @ X )
     => ( ! [Y3: set_Pr4934435412358123699_a_nat] :
            ( ( P @ Y3 )
           => ( ord_le8666007276011122963_a_nat @ Y3 @ X ) )
       => ( ( order_1399416596184996762_a_nat @ P )
          = X ) ) ) ).

% Greatest_equality
thf(fact_408_GreatestI2__order,axiom,
    ! [P: set_Product_prod_a_a > $o,X: set_Product_prod_a_a,Q: set_Product_prod_a_a > $o] :
      ( ( P @ X )
     => ( ! [Y3: set_Product_prod_a_a] :
            ( ( P @ Y3 )
           => ( ord_le746702958409616551od_a_a @ Y3 @ X ) )
       => ( ! [X3: set_Product_prod_a_a] :
              ( ( P @ X3 )
             => ( ! [Y7: set_Product_prod_a_a] :
                    ( ( P @ Y7 )
                   => ( ord_le746702958409616551od_a_a @ Y7 @ X3 ) )
               => ( Q @ X3 ) ) )
         => ( Q @ ( order_6315983676290676192od_a_a @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_409_GreatestI2__order,axiom,
    ! [P: set_term_a_b > $o,X: set_term_a_b,Q: set_term_a_b > $o] :
      ( ( P @ X )
     => ( ! [Y3: set_term_a_b] :
            ( ( P @ Y3 )
           => ( ord_le2705286416250468010rm_a_b @ Y3 @ X ) )
       => ( ! [X3: set_term_a_b] :
              ( ( P @ X3 )
             => ( ! [Y7: set_term_a_b] :
                    ( ( P @ Y7 )
                   => ( ord_le2705286416250468010rm_a_b @ Y7 @ X3 ) )
               => ( Q @ X3 ) ) )
         => ( Q @ ( order_8964606823743705265rm_a_b @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_410_GreatestI2__order,axiom,
    ! [P: set_nat > $o,X: set_nat,Q: set_nat > $o] :
      ( ( P @ X )
     => ( ! [Y3: set_nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_set_nat @ Y3 @ X ) )
       => ( ! [X3: set_nat] :
              ( ( P @ X3 )
             => ( ! [Y7: set_nat] :
                    ( ( P @ Y7 )
                   => ( ord_less_eq_set_nat @ Y7 @ X3 ) )
               => ( Q @ X3 ) ) )
         => ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_411_GreatestI2__order,axiom,
    ! [P: set_a > $o,X: set_a,Q: set_a > $o] :
      ( ( P @ X )
     => ( ! [Y3: set_a] :
            ( ( P @ Y3 )
           => ( ord_less_eq_set_a @ Y3 @ X ) )
       => ( ! [X3: set_a] :
              ( ( P @ X3 )
             => ( ! [Y7: set_a] :
                    ( ( P @ Y7 )
                   => ( ord_less_eq_set_a @ Y7 @ X3 ) )
               => ( Q @ X3 ) ) )
         => ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_412_GreatestI2__order,axiom,
    ! [P: nat > $o,X: nat,Q: nat > $o] :
      ( ( P @ X )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ X ) )
       => ( ! [X3: nat] :
              ( ( P @ X3 )
             => ( ! [Y7: nat] :
                    ( ( P @ Y7 )
                   => ( ord_less_eq_nat @ Y7 @ X3 ) )
               => ( Q @ X3 ) ) )
         => ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_413_GreatestI2__order,axiom,
    ! [P: set_Pr4934435412358123699_a_nat > $o,X: set_Pr4934435412358123699_a_nat,Q: set_Pr4934435412358123699_a_nat > $o] :
      ( ( P @ X )
     => ( ! [Y3: set_Pr4934435412358123699_a_nat] :
            ( ( P @ Y3 )
           => ( ord_le8666007276011122963_a_nat @ Y3 @ X ) )
       => ( ! [X3: set_Pr4934435412358123699_a_nat] :
              ( ( P @ X3 )
             => ( ! [Y7: set_Pr4934435412358123699_a_nat] :
                    ( ( P @ Y7 )
                   => ( ord_le8666007276011122963_a_nat @ Y7 @ X3 ) )
               => ( Q @ X3 ) ) )
         => ( Q @ ( order_1399416596184996762_a_nat @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_414_qc__imp__qc__trancl,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( abstra7274158693546625990rm_a_b @ R @ S )
     => ( abstra7274158693546625990rm_a_b @ R @ ( transi9211502839322181930rm_a_b @ S ) ) ) ).

% qc_imp_qc_trancl
thf(fact_415_qc__imp__qc__trancl,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( abstra929980004689193019mute_a @ R @ S )
     => ( abstra929980004689193019mute_a @ R @ ( transitive_trancl_a @ S ) ) ) ).

% qc_imp_qc_trancl
thf(fact_416_qc__imp__qc__trancl,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( abstra4838182543893801299te_nat @ R @ S )
     => ( abstra4838182543893801299te_nat @ R @ ( transi6264000038957366511cl_nat @ S ) ) ) ).

% qc_imp_qc_trancl
thf(fact_417_qc__imp__qc__trancl,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra4317384144805385751rm_a_b @ R @ S )
     => ( abstra4317384144805385751rm_a_b @ R @ ( transi7922773638565587891rm_a_b @ S ) ) ) ).

% qc_imp_qc_trancl
thf(fact_418_rstep_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rstep_a_b @ R2 ) )
     => ~ ! [S2: term_a_b,T3: term_a_b,C5: subterm_and_ctxt_a_b] :
            ( ( A1
              = ( subter2376574525758040790rm_a_b @ C5 @ S2 ) )
           => ( ( A22
                = ( subter2376574525758040790rm_a_b @ C5 @ T3 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S2 @ T3 ) @ ( rrstep_a_b @ R2 ) ) ) ) ) ).

% rstep.cases
thf(fact_419_rstep_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rstep_a_b @ R2 ) )
      = ( ? [S3: term_a_b,T2: term_a_b,C6: subterm_and_ctxt_a_b] :
            ( ( A1
              = ( subter2376574525758040790rm_a_b @ C6 @ S3 ) )
            & ( A22
              = ( subter2376574525758040790rm_a_b @ C6 @ T2 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T2 ) @ ( rrstep_a_b @ R2 ) ) ) ) ) ).

% rstep.simps
thf(fact_420_rstep_Ointros,axiom,
    ! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C4: subterm_and_ctxt_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R2 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( rstep_a_b @ R2 ) ) ) ).

% rstep.intros
thf(fact_421_r__into__rtrancl,axiom,
    ! [P2: produc4708774622424448987_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ P2 @ R )
     => ( member9062615507155100804_a_nat @ P2 @ ( transi2726145917338391738_a_nat @ R ) ) ) ).

% r_into_rtrancl
thf(fact_422_r__into__rtrancl,axiom,
    ! [P2: product_prod_a_a,R: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ P2 @ R )
     => ( member1426531477525435216od_a_a @ P2 @ ( transitive_rtrancl_a @ R ) ) ) ).

% r_into_rtrancl
thf(fact_423_r__into__rtrancl,axiom,
    ! [P2: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ R )
     => ( member8440522571783428010at_nat @ P2 @ ( transi2905341329935302413cl_nat @ R ) ) ) ).

% r_into_rtrancl
thf(fact_424_r__into__rtrancl,axiom,
    ! [P2: produc4523971788519308903rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ P2 @ R )
     => ( member8417600551952982416rm_a_b @ P2 @ ( transi2615809358984392588rm_a_b @ R ) ) ) ).

% r_into_rtrancl
thf(fact_425_r__into__rtrancl,axiom,
    ! [P2: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ R )
     => ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).

% r_into_rtrancl
thf(fact_426_rtrancl__idemp,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7742714808557438673rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) )
      = ( transi7742714808557438673rm_a_b @ R ) ) ).

% rtrancl_idemp
thf(fact_427_rtrancl__idemp,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ( transitive_rtrancl_a @ ( transitive_rtrancl_a @ R ) )
      = ( transitive_rtrancl_a @ R ) ) ).

% rtrancl_idemp
thf(fact_428_rtrancl__idemp,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( transi2905341329935302413cl_nat @ ( transi2905341329935302413cl_nat @ R ) )
      = ( transi2905341329935302413cl_nat @ R ) ) ).

% rtrancl_idemp
thf(fact_429_rtrancl__idemp,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b] :
      ( ( transi2615809358984392588rm_a_b @ ( transi2615809358984392588rm_a_b @ R ) )
      = ( transi2615809358984392588rm_a_b @ R ) ) ).

% rtrancl_idemp
thf(fact_430_trancl__rtrancl__absorb,axiom,
    ! [R3: set_Product_prod_a_a] :
      ( ( transitive_rtrancl_a @ ( transitive_trancl_a @ R3 ) )
      = ( transitive_rtrancl_a @ R3 ) ) ).

% trancl_rtrancl_absorb
thf(fact_431_trancl__rtrancl__absorb,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( transi2905341329935302413cl_nat @ ( transi6264000038957366511cl_nat @ R3 ) )
      = ( transi2905341329935302413cl_nat @ R3 ) ) ).

% trancl_rtrancl_absorb
thf(fact_432_trancl__rtrancl__absorb,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b] :
      ( ( transi2615809358984392588rm_a_b @ ( transi9211502839322181930rm_a_b @ R3 ) )
      = ( transi2615809358984392588rm_a_b @ R3 ) ) ).

% trancl_rtrancl_absorb
thf(fact_433_trancl__rtrancl__absorb,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7742714808557438673rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) )
      = ( transi7742714808557438673rm_a_b @ R3 ) ) ).

% trancl_rtrancl_absorb
thf(fact_434_rtrancl__trancl__absorb,axiom,
    ! [R3: set_Product_prod_a_a] :
      ( ( transitive_trancl_a @ ( transitive_rtrancl_a @ R3 ) )
      = ( transitive_rtrancl_a @ R3 ) ) ).

% rtrancl_trancl_absorb
thf(fact_435_rtrancl__trancl__absorb,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( transi6264000038957366511cl_nat @ ( transi2905341329935302413cl_nat @ R3 ) )
      = ( transi2905341329935302413cl_nat @ R3 ) ) ).

% rtrancl_trancl_absorb
thf(fact_436_rtrancl__trancl__absorb,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b] :
      ( ( transi9211502839322181930rm_a_b @ ( transi2615809358984392588rm_a_b @ R3 ) )
      = ( transi2615809358984392588rm_a_b @ R3 ) ) ).

% rtrancl_trancl_absorb
thf(fact_437_rtrancl__trancl__absorb,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7922773638565587891rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) )
      = ( transi7742714808557438673rm_a_b @ R3 ) ) ).

% rtrancl_trancl_absorb
thf(fact_438_rstep__ctxtI,axiom,
    ! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C4: subterm_and_ctxt_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( rstep_a_b @ R2 ) ) ) ).

% rstep_ctxtI
thf(fact_439_converse__rtrancl__into__rtrancl,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,C3: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ ( transi2726145917338391738_a_nat @ R ) )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C3 ) @ ( transi2726145917338391738_a_nat @ R ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_440_converse__rtrancl__into__rtrancl,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,C3: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C3 ) @ ( transitive_rtrancl_a @ R ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C3 ) @ ( transitive_rtrancl_a @ R ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_441_converse__rtrancl__into__rtrancl,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( transi2905341329935302413cl_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_442_converse__rtrancl__into__rtrancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_443_converse__rtrancl__into__rtrancl,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_444_converse__rtrancl__induct,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,P: product_prod_a_nat > $o] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi2726145917338391738_a_nat @ R ) )
     => ( ( P @ B )
       => ( ! [Y3: product_prod_a_nat,Z: product_prod_a_nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ Z ) @ R )
             => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Z @ B ) @ ( transi2726145917338391738_a_nat @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_445_converse__rtrancl__induct,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,P: a > $o] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R ) )
     => ( ( P @ B )
       => ( ! [Y3: a,Z: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z ) @ R )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ B ) @ ( transitive_rtrancl_a @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_446_converse__rtrancl__induct,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( ( P @ B )
       => ( ! [Y3: nat,Z: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_447_converse__rtrancl__induct,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( P @ B )
       => ( ! [Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ R )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_448_converse__rtrancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ( ( P @ B )
       => ( ! [Y3: term_a_b,Z: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_449_converse__rtranclE,axiom,
    ! [X: product_prod_a_nat,Z2: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z2 ) @ ( transi2726145917338391738_a_nat @ R ) )
     => ( ( X != Z2 )
       => ~ ! [Y3: product_prod_a_nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y3 ) @ R )
             => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ Z2 ) @ ( transi2726145917338391738_a_nat @ R ) ) ) ) ) ).

% converse_rtranclE
thf(fact_450_converse__rtranclE,axiom,
    ! [X: a,Z2: a,R: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ ( transitive_rtrancl_a @ R ) )
     => ( ( X != Z2 )
       => ~ ! [Y3: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
             => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z2 ) @ ( transitive_rtrancl_a @ R ) ) ) ) ) ).

% converse_rtranclE
thf(fact_451_converse__rtranclE,axiom,
    ! [X: nat,Z2: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( ( X != Z2 )
       => ~ ! [Y3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ) ).

% converse_rtranclE
thf(fact_452_converse__rtranclE,axiom,
    ! [X: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( X != Z2 )
       => ~ ! [Y3: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y3 ) @ R )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ).

% converse_rtranclE
thf(fact_453_converse__rtranclE,axiom,
    ! [X: term_a_b,Z2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ( ( X != Z2 )
       => ~ ! [Y3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ).

% converse_rtranclE
thf(fact_454_rtrancl__induct,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,P: product_prod_a_nat > $o] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi2726145917338391738_a_nat @ R ) )
     => ( ( P @ A )
       => ( ! [Y3: product_prod_a_nat,Z: product_prod_a_nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ Y3 ) @ ( transi2726145917338391738_a_nat @ R ) )
             => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ Z ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_455_rtrancl__induct,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,P: a > $o] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R ) )
     => ( ( P @ A )
       => ( ! [Y3: a,Z: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ Y3 ) @ ( transitive_rtrancl_a @ R ) )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_456_rtrancl__induct,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( ( P @ A )
       => ( ! [Y3: nat,Z: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi2905341329935302413cl_nat @ R ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_457_rtrancl__induct,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( P @ A )
       => ( ! [Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_458_rtrancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ( ( P @ A )
       => ( ! [Y3: term_a_b,Z: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_459_rtrancl__trans,axiom,
    ! [X: product_prod_a_nat,Y: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,Z2: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi2726145917338391738_a_nat @ R ) )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ Z2 ) @ ( transi2726145917338391738_a_nat @ R ) )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z2 ) @ ( transi2726145917338391738_a_nat @ R ) ) ) ) ).

% rtrancl_trans
thf(fact_460_rtrancl__trans,axiom,
    ! [X: a,Y: a,R: set_Product_prod_a_a,Z2: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_rtrancl_a @ R ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ ( transitive_rtrancl_a @ R ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ ( transitive_rtrancl_a @ R ) ) ) ) ).

% rtrancl_trans
thf(fact_461_rtrancl__trans,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Z2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ).

% rtrancl_trans
thf(fact_462_rtrancl__trans,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,Z2: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ).

% rtrancl_trans
thf(fact_463_rtrancl__trans,axiom,
    ! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).

% rtrancl_trans
thf(fact_464_rtranclE,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi2726145917338391738_a_nat @ R ) )
     => ( ( A != B )
       => ~ ! [Y3: product_prod_a_nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ Y3 ) @ ( transi2726145917338391738_a_nat @ R ) )
             => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ B ) @ R ) ) ) ) ).

% rtranclE
thf(fact_465_rtranclE,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R ) )
     => ( ( A != B )
       => ~ ! [Y3: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ Y3 ) @ ( transitive_rtrancl_a @ R ) )
             => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ B ) @ R ) ) ) ) ).

% rtranclE
thf(fact_466_rtranclE,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( ( A != B )
       => ~ ! [Y3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi2905341329935302413cl_nat @ R ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ B ) @ R ) ) ) ) ).

% rtranclE
thf(fact_467_rtranclE,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( A != B )
       => ~ ! [Y3: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ B ) @ R ) ) ) ) ).

% rtranclE
thf(fact_468_rtranclE,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ( ( A != B )
       => ~ ! [Y3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ B ) @ R ) ) ) ) ).

% rtranclE
thf(fact_469_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,C3: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi2726145917338391738_a_nat @ R ) )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ R )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C3 ) @ ( transi2726145917338391738_a_nat @ R ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_470_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,C3: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C3 ) @ R )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C3 ) @ ( transitive_rtrancl_a @ R ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_471_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_472_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ R )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_473_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ R )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_474_rtrancl_Ortrancl__refl,axiom,
    ! [A: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ A ) @ ( transi2726145917338391738_a_nat @ R ) ) ).

% rtrancl.rtrancl_refl
thf(fact_475_rtrancl_Ortrancl__refl,axiom,
    ! [A: a,R: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ ( transitive_rtrancl_a @ R ) ) ).

% rtrancl.rtrancl_refl
thf(fact_476_rtrancl_Ortrancl__refl,axiom,
    ! [A: nat,R: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( transi2905341329935302413cl_nat @ R ) ) ).

% rtrancl.rtrancl_refl
thf(fact_477_rtrancl_Ortrancl__refl,axiom,
    ! [A: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ ( transi2615809358984392588rm_a_b @ R ) ) ).

% rtrancl.rtrancl_refl
thf(fact_478_rtrancl_Ortrancl__refl,axiom,
    ! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( transi7742714808557438673rm_a_b @ R ) ) ).

% rtrancl.rtrancl_refl
thf(fact_479_rtrancl_Osimps,axiom,
    ! [A1: product_prod_a_nat,A22: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A1 @ A22 ) @ ( transi2726145917338391738_a_nat @ R ) )
      = ( ? [A3: product_prod_a_nat] :
            ( ( A1 = A3 )
            & ( A22 = A3 ) )
        | ? [A3: product_prod_a_nat,B4: product_prod_a_nat,C: product_prod_a_nat] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ B4 ) @ ( transi2726145917338391738_a_nat @ R ) )
            & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B4 @ C ) @ R ) ) ) ) ).

% rtrancl.simps
thf(fact_480_rtrancl_Osimps,axiom,
    ! [A1: a,A22: a,R: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( transitive_rtrancl_a @ R ) )
      = ( ? [A3: a] :
            ( ( A1 = A3 )
            & ( A22 = A3 ) )
        | ? [A3: a,B4: a,C: a] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B4 ) @ ( transitive_rtrancl_a @ R ) )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B4 @ C ) @ R ) ) ) ) ).

% rtrancl.simps
thf(fact_481_rtrancl_Osimps,axiom,
    ! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R ) )
      = ( ? [A3: nat] :
            ( ( A1 = A3 )
            & ( A22 = A3 ) )
        | ? [A3: nat,B4: nat,C: nat] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi2905341329935302413cl_nat @ R ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C ) @ R ) ) ) ) ).

% rtrancl.simps
thf(fact_482_rtrancl_Osimps,axiom,
    ! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( transi2615809358984392588rm_a_b @ R ) )
      = ( ? [A3: produc357393685978478089rm_a_b] :
            ( ( A1 = A3 )
            & ( A22 = A3 ) )
        | ? [A3: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B4 ) @ ( transi2615809358984392588rm_a_b @ R ) )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ C ) @ R ) ) ) ) ).

% rtrancl.simps
thf(fact_483_rtrancl_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R ) )
      = ( ? [A3: term_a_b] :
            ( ( A1 = A3 )
            & ( A22 = A3 ) )
        | ? [A3: term_a_b,B4: term_a_b,C: term_a_b] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C ) @ R ) ) ) ) ).

% rtrancl.simps
thf(fact_484_rtrancl_Ocases,axiom,
    ! [A1: product_prod_a_nat,A22: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A1 @ A22 ) @ ( transi2726145917338391738_a_nat @ R ) )
     => ( ( A22 != A1 )
       => ~ ! [B3: product_prod_a_nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A1 @ B3 ) @ ( transi2726145917338391738_a_nat @ R ) )
             => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B3 @ A22 ) @ R ) ) ) ) ).

% rtrancl.cases
thf(fact_485_rtrancl_Ocases,axiom,
    ! [A1: a,A22: a,R: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( transitive_rtrancl_a @ R ) )
     => ( ( A22 != A1 )
       => ~ ! [B3: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B3 ) @ ( transitive_rtrancl_a @ R ) )
             => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A22 ) @ R ) ) ) ) ).

% rtrancl.cases
thf(fact_486_rtrancl_Ocases,axiom,
    ! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( ( A22 != A1 )
       => ~ ! [B3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi2905341329935302413cl_nat @ R ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R ) ) ) ) ).

% rtrancl.cases
thf(fact_487_rtrancl_Ocases,axiom,
    ! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( A22 != A1 )
       => ~ ! [B3: produc357393685978478089rm_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ A22 ) @ R ) ) ) ) ).

% rtrancl.cases
thf(fact_488_rtrancl_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ( ( A22 != A1 )
       => ~ ! [B3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ R ) ) ) ) ).

% rtrancl.cases
thf(fact_489_rtrancl__mono,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R @ S )
     => ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).

% rtrancl_mono
thf(fact_490_rtrancl__mono,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ R @ S )
     => ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ R ) @ ( transi2905341329935302413cl_nat @ S ) ) ) ).

% rtrancl_mono
thf(fact_491_rtrancl__mono,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R @ S )
     => ( ord_le2593958535671617255rm_a_b @ ( transi2615809358984392588rm_a_b @ R ) @ ( transi2615809358984392588rm_a_b @ S ) ) ) ).

% rtrancl_mono
thf(fact_492_rtrancl__mono,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ R @ S )
     => ( ord_le746702958409616551od_a_a @ ( transitive_rtrancl_a @ R ) @ ( transitive_rtrancl_a @ S ) ) ) ).

% rtrancl_mono
thf(fact_493_rtrancl__subset,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R3 @ S4 )
     => ( ( ord_le118470702582115849rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ R3 ) )
       => ( ( transi7742714808557438673rm_a_b @ S4 )
          = ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).

% rtrancl_subset
thf(fact_494_rtrancl__subset,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ R3 @ S4 )
     => ( ( ord_le3146513528884898305at_nat @ S4 @ ( transi2905341329935302413cl_nat @ R3 ) )
       => ( ( transi2905341329935302413cl_nat @ S4 )
          = ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ).

% rtrancl_subset
thf(fact_495_rtrancl__subset,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,S4: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R3 @ S4 )
     => ( ( ord_le2593958535671617255rm_a_b @ S4 @ ( transi2615809358984392588rm_a_b @ R3 ) )
       => ( ( transi2615809358984392588rm_a_b @ S4 )
          = ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ).

% rtrancl_subset
thf(fact_496_rtrancl__subset,axiom,
    ! [R3: set_Product_prod_a_a,S4: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ R3 @ S4 )
     => ( ( ord_le746702958409616551od_a_a @ S4 @ ( transitive_rtrancl_a @ R3 ) )
       => ( ( transitive_rtrancl_a @ S4 )
          = ( transitive_rtrancl_a @ R3 ) ) ) ) ).

% rtrancl_subset
thf(fact_497_rtrancl__subset__rtrancl,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R @ ( transi7742714808557438673rm_a_b @ S ) )
     => ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).

% rtrancl_subset_rtrancl
thf(fact_498_rtrancl__subset__rtrancl,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ R @ ( transi2905341329935302413cl_nat @ S ) )
     => ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ R ) @ ( transi2905341329935302413cl_nat @ S ) ) ) ).

% rtrancl_subset_rtrancl
thf(fact_499_rtrancl__subset__rtrancl,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ R @ ( transi2615809358984392588rm_a_b @ S ) )
     => ( ord_le2593958535671617255rm_a_b @ ( transi2615809358984392588rm_a_b @ R ) @ ( transi2615809358984392588rm_a_b @ S ) ) ) ).

% rtrancl_subset_rtrancl
thf(fact_500_rtrancl__subset__rtrancl,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ R @ ( transitive_rtrancl_a @ S ) )
     => ( ord_le746702958409616551od_a_a @ ( transitive_rtrancl_a @ R ) @ ( transitive_rtrancl_a @ S ) ) ) ).

% rtrancl_subset_rtrancl
thf(fact_501_trancl__rtrancl__trancl,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,C3: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi6805586507763294364_a_nat @ R ) )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ ( transi2726145917338391738_a_nat @ R ) )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C3 ) @ ( transi6805586507763294364_a_nat @ R ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_502_trancl__rtrancl__trancl,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,C3: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C3 ) @ ( transitive_rtrancl_a @ R ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C3 ) @ ( transitive_trancl_a @ R ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_503_trancl__rtrancl__trancl,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( transi2905341329935302413cl_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_504_trancl__rtrancl__trancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( transi9211502839322181930rm_a_b @ R ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_505_trancl__rtrancl__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_506_rtrancl__trancl__trancl,axiom,
    ! [X: product_prod_a_nat,Y: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,Z2: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi2726145917338391738_a_nat @ R ) )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ Z2 ) @ ( transi6805586507763294364_a_nat @ R ) )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z2 ) @ ( transi6805586507763294364_a_nat @ R ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_507_rtrancl__trancl__trancl,axiom,
    ! [X: a,Y: a,R: set_Product_prod_a_a,Z2: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_rtrancl_a @ R ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ ( transitive_trancl_a @ R ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ ( transitive_trancl_a @ R ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_508_rtrancl__trancl__trancl,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Z2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( transi6264000038957366511cl_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_509_rtrancl__trancl__trancl,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,Z2: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_510_rtrancl__trancl__trancl,axiom,
    ! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_511_rtrancl__into__trancl2,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,C3: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ ( transi2726145917338391738_a_nat @ R ) )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C3 ) @ ( transi6805586507763294364_a_nat @ R ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_512_rtrancl__into__trancl2,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,C3: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C3 ) @ ( transitive_rtrancl_a @ R ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C3 ) @ ( transitive_trancl_a @ R ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_513_rtrancl__into__trancl2,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( transi2905341329935302413cl_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_514_rtrancl__into__trancl2,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( transi9211502839322181930rm_a_b @ R ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_515_rtrancl__into__trancl2,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_516_rtrancl__into__trancl1,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,C3: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi2726145917338391738_a_nat @ R ) )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ R )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C3 ) @ ( transi6805586507763294364_a_nat @ R ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_517_rtrancl__into__trancl1,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,C3: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C3 ) @ R )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C3 ) @ ( transitive_trancl_a @ R ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_518_rtrancl__into__trancl1,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_519_rtrancl__into__trancl1,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ R )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( transi9211502839322181930rm_a_b @ R ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_520_rtrancl__into__trancl1,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ R )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_521_rtrancl__eq__or__trancl,axiom,
    ! [X: product_prod_a_nat,Y: product_prod_a_nat,R3: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi2726145917338391738_a_nat @ R3 ) )
      = ( ( X = Y )
        | ( ( X != Y )
          & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi6805586507763294364_a_nat @ R3 ) ) ) ) ) ).

% rtrancl_eq_or_trancl
thf(fact_522_rtrancl__eq__or__trancl,axiom,
    ! [X: a,Y: a,R3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_rtrancl_a @ R3 ) )
      = ( ( X = Y )
        | ( ( X != Y )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R3 ) ) ) ) ) ).

% rtrancl_eq_or_trancl
thf(fact_523_rtrancl__eq__or__trancl,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) )
      = ( ( X = Y )
        | ( ( X != Y )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).

% rtrancl_eq_or_trancl
thf(fact_524_rtrancl__eq__or__trancl,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
      = ( ( X = Y )
        | ( ( X != Y )
          & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ) ).

% rtrancl_eq_or_trancl
thf(fact_525_rtrancl__eq__or__trancl,axiom,
    ! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
      = ( ( X = Y )
        | ( ( X != Y )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ) ).

% rtrancl_eq_or_trancl
thf(fact_526_trancl__into__rtrancl,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi6805586507763294364_a_nat @ R ) )
     => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi2726145917338391738_a_nat @ R ) ) ) ).

% trancl_into_rtrancl
thf(fact_527_trancl__into__rtrancl,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R ) )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R ) ) ) ).

% trancl_into_rtrancl
thf(fact_528_trancl__into__rtrancl,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ).

% trancl_into_rtrancl
thf(fact_529_trancl__into__rtrancl,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R ) )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ).

% trancl_into_rtrancl
thf(fact_530_trancl__into__rtrancl,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).

% trancl_into_rtrancl
thf(fact_531_tranclD2,axiom,
    ! [X: product_prod_a_nat,Y: product_prod_a_nat,R3: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi6805586507763294364_a_nat @ R3 ) )
     => ? [Z: product_prod_a_nat] :
          ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z ) @ ( transi2726145917338391738_a_nat @ R3 ) )
          & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Z @ Y ) @ R3 ) ) ) ).

% tranclD2
thf(fact_532_tranclD2,axiom,
    ! [X: a,Y: a,R3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R3 ) )
     => ? [Z: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z ) @ ( transitive_rtrancl_a @ R3 ) )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ R3 ) ) ) ).

% tranclD2
thf(fact_533_tranclD2,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
     => ? [Z: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi2905341329935302413cl_nat @ R3 ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ R3 ) ) ) ).

% tranclD2
thf(fact_534_tranclD2,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
     => ? [Z: produc357393685978478089rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
          & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ Y ) @ R3 ) ) ) ).

% tranclD2
thf(fact_535_tranclD2,axiom,
    ! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ? [Z: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ R3 ) ) ) ).

% tranclD2
thf(fact_536_rtranclD,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R3: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi2726145917338391738_a_nat @ R3 ) )
     => ( ( A = B )
        | ( ( A != B )
          & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi6805586507763294364_a_nat @ R3 ) ) ) ) ) ).

% rtranclD
thf(fact_537_rtranclD,axiom,
    ! [A: a,B: a,R3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R3 ) )
     => ( ( A = B )
        | ( ( A != B )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R3 ) ) ) ) ) ).

% rtranclD
thf(fact_538_rtranclD,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R3 ) )
     => ( ( A = B )
        | ( ( A != B )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).

% rtranclD
thf(fact_539_rtranclD,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
     => ( ( A = B )
        | ( ( A != B )
          & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ) ).

% rtranclD
thf(fact_540_rtranclD,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ( ( A = B )
        | ( ( A != B )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ) ).

% rtranclD
thf(fact_541_tranclD,axiom,
    ! [X: product_prod_a_nat,Y: product_prod_a_nat,R3: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi6805586507763294364_a_nat @ R3 ) )
     => ? [Z: product_prod_a_nat] :
          ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z ) @ R3 )
          & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Z @ Y ) @ ( transi2726145917338391738_a_nat @ R3 ) ) ) ) ).

% tranclD
thf(fact_542_tranclD,axiom,
    ! [X: a,Y: a,R3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R3 ) )
     => ? [Z: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R3 )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ ( transitive_rtrancl_a @ R3 ) ) ) ) ).

% tranclD
thf(fact_543_tranclD,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
     => ? [Z: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R3 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ).

% tranclD
thf(fact_544_tranclD,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
     => ? [Z: produc357393685978478089rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ R3 )
          & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ Y ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ).

% tranclD
thf(fact_545_tranclD,axiom,
    ! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ? [Z: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R3 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).

% tranclD
thf(fact_546_restrict__SN__subset,axiom,
    ! [R3: set_Product_prod_a_a,S4: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( abstra2152889483027340875t_SN_a @ R3 @ S4 ) @ R3 ) ).

% restrict_SN_subset
thf(fact_547_restrict__SN__subset,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra4933154510623179783rm_a_b @ R3 @ S4 ) @ R3 ) ).

% restrict_SN_subset
thf(fact_548_ctxt__eq,axiom,
    ! [C4: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
      ( ( ( subter2376574525758040790rm_a_b @ C4 @ S )
        = ( subter2376574525758040790rm_a_b @ C4 @ T ) )
      = ( S = T ) ) ).

% ctxt_eq
thf(fact_549_meetI,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,A5: set_Pr1811044260758604347_a_nat,C3: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi2726145917338391738_a_nat @ A5 ) )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C3 ) @ ( transi2726145917338391738_a_nat @ A5 ) )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ ( abstra5218407276271306624_a_nat @ A5 ) ) ) ) ).

% meetI
thf(fact_550_meetI,axiom,
    ! [A: a,B: a,A5: set_Product_prod_a_a,C3: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ A5 ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C3 ) @ ( transitive_rtrancl_a @ A5 ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C3 ) @ ( abstract_meet_a @ A5 ) ) ) ) ).

% meetI
thf(fact_551_meetI,axiom,
    ! [A: nat,B: nat,A5: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ A5 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi2905341329935302413cl_nat @ A5 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( abstract_meet_nat @ A5 ) ) ) ) ).

% meetI
thf(fact_552_meetI,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A5 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( transi2615809358984392588rm_a_b @ A5 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ ( abstra8296971427885378886rm_a_b @ A5 ) ) ) ) ).

% meetI
thf(fact_553_meetI,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( abstra1093061187967292567rm_a_b @ A5 ) ) ) ) ).

% meetI
thf(fact_554_restrict__rtrancl,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,S4: set_term_a_b] : ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( abstra6662565036008789004rm_a_b @ R @ S4 ) ) @ ( transi7742714808557438673rm_a_b @ R ) ) ).

% restrict_rtrancl
thf(fact_555_restrict__rtrancl,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S4: set_nat] : ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ ( abstra1219450721576609224ct_nat @ R @ S4 ) ) @ ( transi2905341329935302413cl_nat @ R ) ) ).

% restrict_rtrancl
thf(fact_556_restrict__rtrancl,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,S4: set_Pr4386577575007340137rm_a_b] : ( ord_le2593958535671617255rm_a_b @ ( transi2615809358984392588rm_a_b @ ( abstra2354209648719431185rm_a_b @ R @ S4 ) ) @ ( transi2615809358984392588rm_a_b @ R ) ) ).

% restrict_rtrancl
thf(fact_557_restrict__rtrancl,axiom,
    ! [R: set_Product_prod_a_a,S4: set_a] : ( ord_le746702958409616551od_a_a @ ( transitive_rtrancl_a @ ( abstract_restrict_a @ R @ S4 ) ) @ ( transitive_rtrancl_a @ R ) ) ).

% restrict_rtrancl
thf(fact_558_srsteps__with__root__step__def,axiom,
    ( srstep7844470518422762656ep_a_b
    = ( ^ [F3: set_Pr4934435412358123699_a_nat,R5: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R5 ) ) ) @ ( relcom370159955682700863rm_a_b @ ( sig_step_a_b @ F3 @ ( rrstep_a_b @ R5 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R5 ) ) ) ) ) ) ) ).

% srsteps_with_root_step_def
thf(fact_559_meetE,axiom,
    ! [B: product_prod_a_nat,C3: product_prod_a_nat,A5: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ ( abstra5218407276271306624_a_nat @ A5 ) )
     => ~ ! [A4: product_prod_a_nat] :
            ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ B ) @ ( transi2726145917338391738_a_nat @ A5 ) )
           => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ C3 ) @ ( transi2726145917338391738_a_nat @ A5 ) ) ) ) ).

% meetE
thf(fact_560_meetE,axiom,
    ! [B: a,C3: a,A5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C3 ) @ ( abstract_meet_a @ A5 ) )
     => ~ ! [A4: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B ) @ ( transitive_rtrancl_a @ A5 ) )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ C3 ) @ ( transitive_rtrancl_a @ A5 ) ) ) ) ).

% meetE
thf(fact_561_meetE,axiom,
    ! [B: nat,C3: nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( abstract_meet_nat @ A5 ) )
     => ~ ! [A4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B ) @ ( transi2905341329935302413cl_nat @ A5 ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ C3 ) @ ( transi2905341329935302413cl_nat @ A5 ) ) ) ) ).

% meetE
thf(fact_562_meetE,axiom,
    ! [B: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ ( abstra8296971427885378886rm_a_b @ A5 ) )
     => ~ ! [A4: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B ) @ ( transi2615809358984392588rm_a_b @ A5 ) )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C3 ) @ ( transi2615809358984392588rm_a_b @ A5 ) ) ) ) ).

% meetE
thf(fact_563_meetE,axiom,
    ! [B: term_a_b,C3: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( abstra1093061187967292567rm_a_b @ A5 ) )
     => ~ ! [A4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C3 ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).

% meetE
thf(fact_564_meetD,axiom,
    ! [B: product_prod_a_nat,C3: product_prod_a_nat,A5: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ ( abstra5218407276271306624_a_nat @ A5 ) )
     => ? [A4: product_prod_a_nat] :
          ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ B ) @ ( transi2726145917338391738_a_nat @ A5 ) )
          & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ C3 ) @ ( transi2726145917338391738_a_nat @ A5 ) ) ) ) ).

% meetD
thf(fact_565_meetD,axiom,
    ! [B: a,C3: a,A5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C3 ) @ ( abstract_meet_a @ A5 ) )
     => ? [A4: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B ) @ ( transitive_rtrancl_a @ A5 ) )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ C3 ) @ ( transitive_rtrancl_a @ A5 ) ) ) ) ).

% meetD
thf(fact_566_meetD,axiom,
    ! [B: nat,C3: nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( abstract_meet_nat @ A5 ) )
     => ? [A4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B ) @ ( transi2905341329935302413cl_nat @ A5 ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ C3 ) @ ( transi2905341329935302413cl_nat @ A5 ) ) ) ) ).

% meetD
thf(fact_567_meetD,axiom,
    ! [B: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ ( abstra8296971427885378886rm_a_b @ A5 ) )
     => ? [A4: produc357393685978478089rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B ) @ ( transi2615809358984392588rm_a_b @ A5 ) )
          & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C3 ) @ ( transi2615809358984392588rm_a_b @ A5 ) ) ) ) ).

% meetD
thf(fact_568_meetD,axiom,
    ! [B: term_a_b,C3: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( abstra1093061187967292567rm_a_b @ A5 ) )
     => ? [A4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C3 ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).

% meetD
thf(fact_569_normalizability__I_H,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,A5: set_Pr1811044260758604347_a_nat,C3: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi2726145917338391738_a_nat @ A5 ) )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ ( abstra3832961281062565333_a_nat @ A5 ) )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C3 ) @ ( abstra3832961281062565333_a_nat @ A5 ) ) ) ) ).

% normalizability_I'
thf(fact_570_normalizability__I_H,axiom,
    ! [A: a,B: a,A5: set_Product_prod_a_a,C3: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ A5 ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C3 ) @ ( abstra380364156431640998lity_a @ A5 ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C3 ) @ ( abstra380364156431640998lity_a @ A5 ) ) ) ) ).

% normalizability_I'
thf(fact_571_normalizability__I_H,axiom,
    ! [A: nat,B: nat,A5: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ A5 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( abstra7312431243108365096ty_nat @ A5 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( abstra7312431243108365096ty_nat @ A5 ) ) ) ) ).

% normalizability_I'
thf(fact_572_normalizability__I_H,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A5 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ ( abstra1496011390211842225rm_a_b @ A5 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( abstra1496011390211842225rm_a_b @ A5 ) ) ) ) ).

% normalizability_I'
thf(fact_573_normalizability__I_H,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( abstra2972651231392952684rm_a_b @ A5 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( abstra2972651231392952684rm_a_b @ A5 ) ) ) ) ).

% normalizability_I'
thf(fact_574_rstepI,axiom,
    ! [L: term_a_b,R: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,C4: subterm_and_ctxt_a_b,Sigma: b > term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R2 )
     => ( ( S
          = ( subter2376574525758040790rm_a_b @ C4 @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) ) )
       => ( ( T
            = ( subter2376574525758040790rm_a_b @ C4 @ ( subst_7999470309526761004_a_b_b @ R @ Sigma ) ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) ) ) ) ) ).

% rstepI
thf(fact_575_conversionI_H,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,A5: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi2726145917338391738_a_nat @ A5 ) )
     => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( abstra3054561309482028431_a_nat @ A5 ) ) ) ).

% conversionI'
thf(fact_576_conversionI_H,axiom,
    ! [A: a,B: a,A5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ A5 ) )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( abstra3918519255145857644sion_a @ A5 ) ) ) ).

% conversionI'
thf(fact_577_conversionI_H,axiom,
    ! [A: nat,B: nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ A5 ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( abstra2375765306220057826on_nat @ A5 ) ) ) ).

% conversionI'
thf(fact_578_conversionI_H,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A5 ) )
     => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra6125656530262095991rm_a_b @ A5 ) ) ) ).

% conversionI'
thf(fact_579_conversionI_H,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A5 ) ) ) ).

% conversionI'
thf(fact_580_conversion__conversion__idemp,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra1130568886744591910rm_a_b @ ( abstra1130568886744591910rm_a_b @ A5 ) )
      = ( abstra1130568886744591910rm_a_b @ A5 ) ) ).

% conversion_conversion_idemp
thf(fact_581_conversion__conversion__idemp,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b] :
      ( ( abstra6125656530262095991rm_a_b @ ( abstra6125656530262095991rm_a_b @ A5 ) )
      = ( abstra6125656530262095991rm_a_b @ A5 ) ) ).

% conversion_conversion_idemp
thf(fact_582_rtrancl__idemp__self__comp,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
      = ( transi7742714808557438673rm_a_b @ R3 ) ) ).

% rtrancl_idemp_self_comp
thf(fact_583_rtrancl__idemp__self__comp,axiom,
    ! [R3: set_Product_prod_a_a] :
      ( ( relcomp_a_a_a @ ( transitive_rtrancl_a @ R3 ) @ ( transitive_rtrancl_a @ R3 ) )
      = ( transitive_rtrancl_a @ R3 ) ) ).

% rtrancl_idemp_self_comp
thf(fact_584_rtrancl__idemp__self__comp,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( transi2905341329935302413cl_nat @ R3 ) )
      = ( transi2905341329935302413cl_nat @ R3 ) ) ).

% rtrancl_idemp_self_comp
thf(fact_585_rtrancl__idemp__self__comp,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b] :
      ( ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R3 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
      = ( transi2615809358984392588rm_a_b @ R3 ) ) ).

% rtrancl_idemp_self_comp
thf(fact_586_conversion__rtrancl,axiom,
    ! [A5: set_Product_prod_a_a] :
      ( ( transitive_rtrancl_a @ ( abstra3918519255145857644sion_a @ A5 ) )
      = ( abstra3918519255145857644sion_a @ A5 ) ) ).

% conversion_rtrancl
thf(fact_587_conversion__rtrancl,axiom,
    ! [A5: set_Pr1261947904930325089at_nat] :
      ( ( transi2905341329935302413cl_nat @ ( abstra2375765306220057826on_nat @ A5 ) )
      = ( abstra2375765306220057826on_nat @ A5 ) ) ).

% conversion_rtrancl
thf(fact_588_conversion__rtrancl,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7742714808557438673rm_a_b @ ( abstra1130568886744591910rm_a_b @ A5 ) )
      = ( abstra1130568886744591910rm_a_b @ A5 ) ) ).

% conversion_rtrancl
thf(fact_589_conversion__rtrancl,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b] :
      ( ( transi2615809358984392588rm_a_b @ ( abstra6125656530262095991rm_a_b @ A5 ) )
      = ( abstra6125656530262095991rm_a_b @ A5 ) ) ).

% conversion_rtrancl
thf(fact_590_conversion__O__conversion,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ ( abstra1130568886744591910rm_a_b @ A5 ) @ ( abstra1130568886744591910rm_a_b @ A5 ) )
      = ( abstra1130568886744591910rm_a_b @ A5 ) ) ).

% conversion_O_conversion
thf(fact_591_conversion__O__conversion,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b] :
      ( ( relcom6159724393391289352rm_a_b @ ( abstra6125656530262095991rm_a_b @ A5 ) @ ( abstra6125656530262095991rm_a_b @ A5 ) )
      = ( abstra6125656530262095991rm_a_b @ A5 ) ) ).

% conversion_O_conversion
thf(fact_592_rstep__substI,axiom,
    ! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( rstep_a_b @ R2 ) ) ) ).

% rstep_substI
thf(fact_593_O__assoc,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S4 ) @ T4 )
      = ( relcom370159955682700863rm_a_b @ R3 @ ( relcom370159955682700863rm_a_b @ S4 @ T4 ) ) ) ).

% O_assoc
thf(fact_594_O__assoc,axiom,
    ! [R3: set_Pr4934435412358123699_a_nat,S4: set_Pr1261947904930325089at_nat,T4: set_Pr1261947904930325089at_nat] :
      ( ( relcomp_a_nat_nat @ ( relcomp_a_nat_nat @ R3 @ S4 ) @ T4 )
      = ( relcomp_a_nat_nat @ R3 @ ( relcomp_nat_nat_nat @ S4 @ T4 ) ) ) ).

% O_assoc
thf(fact_595_O__assoc,axiom,
    ! [R3: set_Product_prod_a_a,S4: set_Pr4934435412358123699_a_nat,T4: set_Pr1261947904930325089at_nat] :
      ( ( relcomp_a_nat_nat @ ( relcomp_a_a_nat @ R3 @ S4 ) @ T4 )
      = ( relcomp_a_a_nat @ R3 @ ( relcomp_a_nat_nat @ S4 @ T4 ) ) ) ).

% O_assoc
thf(fact_596_O__assoc,axiom,
    ! [R3: set_Pr4934435412358123699_a_nat,S4: set_Pr4193341848836149977_nat_a,T4: set_Pr4934435412358123699_a_nat] :
      ( ( relcomp_a_a_nat @ ( relcomp_a_nat_a @ R3 @ S4 ) @ T4 )
      = ( relcomp_a_nat_nat @ R3 @ ( relcomp_nat_a_nat @ S4 @ T4 ) ) ) ).

% O_assoc
thf(fact_597_O__assoc,axiom,
    ! [R3: set_Product_prod_a_a,S4: set_Product_prod_a_a,T4: set_Pr4934435412358123699_a_nat] :
      ( ( relcomp_a_a_nat @ ( relcomp_a_a_a @ R3 @ S4 ) @ T4 )
      = ( relcomp_a_a_nat @ R3 @ ( relcomp_a_a_nat @ S4 @ T4 ) ) ) ).

% O_assoc
thf(fact_598_meet__imp__conversion,axiom,
    ! [R: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( abstract_meet_a @ R ) @ ( abstra3918519255145857644sion_a @ R ) ) ).

% meet_imp_conversion
thf(fact_599_meet__imp__conversion,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra1093061187967292567rm_a_b @ R ) @ ( abstra1130568886744591910rm_a_b @ R ) ) ).

% meet_imp_conversion
thf(fact_600_meet__imp__conversion,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b] : ( ord_le2593958535671617255rm_a_b @ ( abstra8296971427885378886rm_a_b @ R ) @ ( abstra6125656530262095991rm_a_b @ R ) ) ).

% meet_imp_conversion
thf(fact_601_relcomp_Ocases,axiom,
    ! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( relcom6159724393391289352rm_a_b @ R @ S ) )
     => ~ ! [B3: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B3 ) @ R )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_602_relcomp_Ocases,axiom,
    ! [A1: a,A22: nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_a_nat @ R @ S ) )
     => ~ ! [B3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B3 ) @ R )
           => ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_603_relcomp_Ocases,axiom,
    ! [A1: a,A22: nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_nat_nat @ R @ S ) )
     => ~ ! [B3: nat] :
            ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ B3 ) @ R )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_604_relcomp_Ocases,axiom,
    ! [A1: product_prod_a_nat,A22: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A1 @ A22 ) @ ( relcom3132138491631691642_a_nat @ R @ S ) )
     => ~ ! [B3: product_prod_a_nat] :
            ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A1 @ B3 ) @ R )
           => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_605_relcomp_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
     => ~ ! [B3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ R )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_606_relcomp_Osimps,axiom,
    ! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( relcom6159724393391289352rm_a_b @ R @ S ) )
      = ( ? [A3: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B4 ) @ R )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ C ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_607_relcomp_Osimps,axiom,
    ! [A1: a,A22: nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_a_nat @ R @ S ) )
      = ( ? [A3: a,B4: a,C: nat] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B4 ) @ R )
            & ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B4 @ C ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_608_relcomp_Osimps,axiom,
    ! [A1: a,A22: nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_nat_nat @ R @ S ) )
      = ( ? [A3: a,B4: nat,C: nat] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A3 @ B4 ) @ R )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_609_relcomp_Osimps,axiom,
    ! [A1: product_prod_a_nat,A22: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A1 @ A22 ) @ ( relcom3132138491631691642_a_nat @ R @ S ) )
      = ( ? [A3: product_prod_a_nat,B4: product_prod_a_nat,C: product_prod_a_nat] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ B4 ) @ R )
            & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B4 @ C ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_610_relcomp_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
      = ( ? [A3: term_a_b,B4: term_a_b,C: term_a_b] :
            ( ( A1 = A3 )
            & ( A22 = C )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ R )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_611_relcomp_OrelcompI,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,C3: nat,S: set_Pr4934435412358123699_a_nat] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
     => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B @ C3 ) @ S )
       => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C3 ) @ ( relcomp_a_a_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_612_relcomp_OrelcompI,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,C3: produc357393685978478089rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ S )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( relcom6159724393391289352rm_a_b @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_613_relcomp_OrelcompI,axiom,
    ! [A: a,B: nat,R: set_Pr4934435412358123699_a_nat,C3: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ S )
       => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C3 ) @ ( relcomp_a_nat_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_614_relcomp_OrelcompI,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,C3: product_prod_a_nat,S: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C3 ) @ S )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C3 ) @ ( relcom3132138491631691642_a_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_615_relcomp_OrelcompI,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C3: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ S )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_616_relcompE,axiom,
    ! [Xz: produc4523971788519308903rm_a_b,R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ Xz @ ( relcom6159724393391289352rm_a_b @ R @ S ) )
     => ~ ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
            ( ( Xz
              = ( produc1763473618796451543rm_a_b @ X3 @ Z ) )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_617_relcompE,axiom,
    ! [Xz: product_prod_a_nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( member5724188588386418708_a_nat @ Xz @ ( relcomp_a_a_nat @ R @ S ) )
     => ~ ! [X3: a,Y3: a,Z: nat] :
            ( ( Xz
              = ( product_Pair_a_nat @ X3 @ Z ) )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
             => ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y3 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_618_relcompE,axiom,
    ! [Xz: product_prod_a_nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member5724188588386418708_a_nat @ Xz @ ( relcomp_a_nat_nat @ R @ S ) )
     => ~ ! [X3: a,Y3: nat,Z: nat] :
            ( ( Xz
              = ( product_Pair_a_nat @ X3 @ Z ) )
           => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ R )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_619_relcompE,axiom,
    ! [Xz: produc4708774622424448987_a_nat,R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ Xz @ ( relcom3132138491631691642_a_nat @ R @ S ) )
     => ~ ! [X3: product_prod_a_nat,Y3: product_prod_a_nat,Z: product_prod_a_nat] :
            ( ( Xz
              = ( produc2026711137822539155_a_nat @ X3 @ Z ) )
           => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
             => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_620_relcompE,axiom,
    ! [Xz: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Xz @ ( relcom370159955682700863rm_a_b @ R @ S ) )
     => ~ ! [X3: term_a_b,Y3: term_a_b,Z: term_a_b] :
            ( ( Xz
              = ( produc7020197800436672577rm_a_b @ X3 @ Z ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_621_relcompEpair,axiom,
    ! [A: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( relcom6159724393391289352rm_a_b @ R @ S ) )
     => ~ ! [B3: produc357393685978478089rm_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ R )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C3 ) @ S ) ) ) ).

% relcompEpair
thf(fact_622_relcompEpair,axiom,
    ! [A: a,C3: nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C3 ) @ ( relcomp_a_a_nat @ R @ S ) )
     => ~ ! [B3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B3 ) @ R )
           => ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ C3 ) @ S ) ) ) ).

% relcompEpair
thf(fact_623_relcompEpair,axiom,
    ! [A: a,C3: nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C3 ) @ ( relcomp_a_nat_nat @ R @ S ) )
     => ~ ! [B3: nat] :
            ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B3 ) @ R )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C3 ) @ S ) ) ) ).

% relcompEpair
thf(fact_624_relcompEpair,axiom,
    ! [A: product_prod_a_nat,C3: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C3 ) @ ( relcom3132138491631691642_a_nat @ R @ S ) )
     => ~ ! [B3: product_prod_a_nat] :
            ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B3 ) @ R )
           => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B3 @ C3 ) @ S ) ) ) ).

% relcompEpair
thf(fact_625_relcompEpair,axiom,
    ! [A: term_a_b,C3: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
     => ~ ! [B3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C3 ) @ S ) ) ) ).

% relcompEpair
thf(fact_626_relcomp3__I,axiom,
    ! [T: nat,U: a,A5: set_Pr4193341848836149977_nat_a,S: a,B5: set_Pr4934435412358123699_a_nat,V: nat] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ T @ U ) @ A5 )
     => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ B5 )
       => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U @ V ) @ B5 )
         => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ V ) @ ( relcomp_a_nat_nat @ B5 @ ( relcomp_nat_a_nat @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_627_relcomp3__I,axiom,
    ! [T: produc357393685978478089rm_a_b,U: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b,S: produc357393685978478089rm_a_b,B5: set_Pr2972776593051762503rm_a_b,V: produc357393685978478089rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ A5 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ B5 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ V ) @ B5 )
         => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ V ) @ ( relcom6159724393391289352rm_a_b @ B5 @ ( relcom6159724393391289352rm_a_b @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_628_relcomp3__I,axiom,
    ! [T: a,U: nat,A5: set_Pr4934435412358123699_a_nat,S: nat,B5: set_Pr4193341848836149977_nat_a,V: a] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ T @ U ) @ A5 )
     => ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ T ) @ B5 )
       => ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ U @ V ) @ B5 )
         => ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ V ) @ ( relcomp_nat_a_a @ B5 @ ( relcomp_a_nat_a @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_629_relcomp3__I,axiom,
    ! [T: product_prod_a_nat,U: product_prod_a_nat,A5: set_Pr1811044260758604347_a_nat,S: product_prod_a_nat,B5: set_Pr1811044260758604347_a_nat,V: product_prod_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ T @ U ) @ A5 )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ S @ T ) @ B5 )
       => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ U @ V ) @ B5 )
         => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ S @ V ) @ ( relcom3132138491631691642_a_nat @ B5 @ ( relcom3132138491631691642_a_nat @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_630_relcomp3__I,axiom,
    ! [T: term_a_b,U: term_a_b,A5: set_Pr4386577575007340137rm_a_b,S: term_a_b,B5: set_Pr4386577575007340137rm_a_b,V: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ A5 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B5 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V ) @ B5 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_631_r__comp__rtrancl__eq,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ R @ ( transi7742714808557438673rm_a_b @ R ) )
      = ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ R ) ) ).

% r_comp_rtrancl_eq
thf(fact_632_r__comp__rtrancl__eq,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ( relcomp_a_a_a @ R @ ( transitive_rtrancl_a @ R ) )
      = ( relcomp_a_a_a @ ( transitive_rtrancl_a @ R ) @ R ) ) ).

% r_comp_rtrancl_eq
thf(fact_633_r__comp__rtrancl__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( relcomp_nat_nat_nat @ R @ ( transi2905341329935302413cl_nat @ R ) )
      = ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ R ) ) ).

% r_comp_rtrancl_eq
thf(fact_634_r__comp__rtrancl__eq,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b] :
      ( ( relcom6159724393391289352rm_a_b @ R @ ( transi2615809358984392588rm_a_b @ R ) )
      = ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R ) @ R ) ) ).

% r_comp_rtrancl_eq
thf(fact_635_rtrancl__O__shift,axiom,
    ! [S4: set_Product_prod_a_a,R3: set_Product_prod_a_a] :
      ( ( relcomp_a_a_a @ ( transitive_rtrancl_a @ ( relcomp_a_a_a @ S4 @ R3 ) ) @ S4 )
      = ( relcomp_a_a_a @ S4 @ ( transitive_rtrancl_a @ ( relcomp_a_a_a @ R3 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_636_rtrancl__O__shift,axiom,
    ! [S4: set_Pr4934435412358123699_a_nat,R3: set_Pr4193341848836149977_nat_a] :
      ( ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ ( relcomp_a_nat_a @ S4 @ R3 ) ) @ S4 )
      = ( relcomp_a_nat_nat @ S4 @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_a_nat @ R3 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_637_rtrancl__O__shift,axiom,
    ! [S4: set_Pr4193341848836149977_nat_a,R3: set_Pr4934435412358123699_a_nat] :
      ( ( relcomp_nat_nat_a @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_a_nat @ S4 @ R3 ) ) @ S4 )
      = ( relcomp_nat_a_a @ S4 @ ( transitive_rtrancl_a @ ( relcomp_a_nat_a @ R3 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_638_rtrancl__O__shift,axiom,
    ! [S4: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_nat_nat @ S4 @ R3 ) ) @ S4 )
      = ( relcomp_nat_nat_nat @ S4 @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_nat_nat @ R3 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_639_rtrancl__O__shift,axiom,
    ! [S4: set_Pr3298798960101569173_a_b_a,R3: set_Pr690693513201511543rm_a_b] :
      ( ( relcom5652721298402989587_a_b_a @ ( transi7742714808557438673rm_a_b @ ( relcom4147987028256916669rm_a_b @ S4 @ R3 ) ) @ S4 )
      = ( relcomp_term_a_b_a_a @ S4 @ ( transitive_rtrancl_a @ ( relcomp_a_term_a_b_a @ R3 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_640_rtrancl__O__shift,axiom,
    ! [S4: set_Pr6708357783813671845_b_nat,R3: set_Pr4549835640365387557rm_a_b] :
      ( ( relcom2286245697325151611_b_nat @ ( transi7742714808557438673rm_a_b @ ( relcom1324791570160179451rm_a_b @ S4 @ R3 ) ) @ S4 )
      = ( relcom2781103321250786871at_nat @ S4 @ ( transi2905341329935302413cl_nat @ ( relcom152770009037377975_b_nat @ R3 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_641_rtrancl__O__shift,axiom,
    ! [S4: set_Pr690693513201511543rm_a_b,R3: set_Pr3298798960101569173_a_b_a] :
      ( ( relcomp_a_a_term_a_b @ ( transitive_rtrancl_a @ ( relcomp_a_term_a_b_a @ S4 @ R3 ) ) @ S4 )
      = ( relcom7339816600239648359rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ ( relcom4147987028256916669rm_a_b @ R3 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_642_rtrancl__O__shift,axiom,
    ! [S4: set_Pr4549835640365387557rm_a_b,R3: set_Pr6708357783813671845_b_nat] :
      ( ( relcom8414687918727181623rm_a_b @ ( transi2905341329935302413cl_nat @ ( relcom152770009037377975_b_nat @ S4 @ R3 ) ) @ S4 )
      = ( relcom6565464767029197947rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ ( relcom1324791570160179451rm_a_b @ R3 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_643_rtrancl__O__shift,axiom,
    ! [S4: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R3 ) ) @ S4 )
      = ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_644_rtrancl__O__shift,axiom,
    ! [S4: set_Pr2303916211607585938rm_a_b,R3: set_Pr513079337203471164_a_b_a] :
      ( ( relcom6469880312746088648rm_a_b @ ( transitive_rtrancl_a @ ( relcom370660246271418610_a_b_a @ S4 @ R3 ) ) @ S4 )
      = ( relcom2073241673377248765rm_a_b @ S4 @ ( transi2615809358984392588rm_a_b @ ( relcom4164503515417244499rm_a_b @ R3 @ S4 ) ) ) ) ).

% rtrancl_O_shift
thf(fact_645_O__mono2,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R3 @ R6 )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ T4 ) @ ( relcom370159955682700863rm_a_b @ R6 @ T4 ) ) ) ).

% O_mono2
thf(fact_646_O__mono2,axiom,
    ! [R3: set_Pr4934435412358123699_a_nat,R6: set_Pr4934435412358123699_a_nat,T4: set_Pr4193341848836149977_nat_a] :
      ( ( ord_le8666007276011122963_a_nat @ R3 @ R6 )
     => ( ord_le746702958409616551od_a_a @ ( relcomp_a_nat_a @ R3 @ T4 ) @ ( relcomp_a_nat_a @ R6 @ T4 ) ) ) ).

% O_mono2
thf(fact_647_O__mono2,axiom,
    ! [R3: set_Product_prod_a_a,R6: set_Product_prod_a_a,T4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le746702958409616551od_a_a @ R3 @ R6 )
     => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R3 @ T4 ) @ ( relcomp_a_a_nat @ R6 @ T4 ) ) ) ).

% O_mono2
thf(fact_648_O__mono2,axiom,
    ! [R3: set_Product_prod_a_a,R6: set_Product_prod_a_a,T4: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ R3 @ R6 )
     => ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ R3 @ T4 ) @ ( relcomp_a_a_a @ R6 @ T4 ) ) ) ).

% O_mono2
thf(fact_649_O__mono2,axiom,
    ! [R3: set_Pr4934435412358123699_a_nat,R6: set_Pr4934435412358123699_a_nat,T4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8666007276011122963_a_nat @ R3 @ R6 )
     => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R3 @ T4 ) @ ( relcomp_a_nat_nat @ R6 @ T4 ) ) ) ).

% O_mono2
thf(fact_650_O__mono1,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R3 @ R6 )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R3 ) @ ( relcom370159955682700863rm_a_b @ S4 @ R6 ) ) ) ).

% O_mono1
thf(fact_651_O__mono1,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat,S4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le3146513528884898305at_nat @ R3 @ R6 )
     => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ S4 @ R3 ) @ ( relcomp_a_nat_nat @ S4 @ R6 ) ) ) ).

% O_mono1
thf(fact_652_O__mono1,axiom,
    ! [R3: set_Product_prod_a_a,R6: set_Product_prod_a_a,S4: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ R3 @ R6 )
     => ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ S4 @ R3 ) @ ( relcomp_a_a_a @ S4 @ R6 ) ) ) ).

% O_mono1
thf(fact_653_O__mono1,axiom,
    ! [R3: set_Pr4934435412358123699_a_nat,R6: set_Pr4934435412358123699_a_nat,S4: set_Product_prod_a_a] :
      ( ( ord_le8666007276011122963_a_nat @ R3 @ R6 )
     => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ S4 @ R3 ) @ ( relcomp_a_a_nat @ S4 @ R6 ) ) ) ).

% O_mono1
thf(fact_654_relcomp__mono,axiom,
    ! [R7: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R7 @ R )
     => ( ( ord_le118470702582115849rm_a_b @ S5 @ S )
       => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R7 @ S5 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) ) ) ) ).

% relcomp_mono
thf(fact_655_relcomp__mono,axiom,
    ! [R7: set_Pr4934435412358123699_a_nat,R: set_Pr4934435412358123699_a_nat,S5: set_Pr4193341848836149977_nat_a,S: set_Pr4193341848836149977_nat_a] :
      ( ( ord_le8666007276011122963_a_nat @ R7 @ R )
     => ( ( ord_le7924913712489149241_nat_a @ S5 @ S )
       => ( ord_le746702958409616551od_a_a @ ( relcomp_a_nat_a @ R7 @ S5 ) @ ( relcomp_a_nat_a @ R @ S ) ) ) ) ).

% relcomp_mono
thf(fact_656_relcomp__mono,axiom,
    ! [R7: set_Product_prod_a_a,R: set_Product_prod_a_a,S5: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ R7 @ R )
     => ( ( ord_le746702958409616551od_a_a @ S5 @ S )
       => ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ R7 @ S5 ) @ ( relcomp_a_a_a @ R @ S ) ) ) ) ).

% relcomp_mono
thf(fact_657_relcomp__mono,axiom,
    ! [R7: set_Product_prod_a_a,R: set_Product_prod_a_a,S5: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le746702958409616551od_a_a @ R7 @ R )
     => ( ( ord_le8666007276011122963_a_nat @ S5 @ S )
       => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R7 @ S5 ) @ ( relcomp_a_a_nat @ R @ S ) ) ) ) ).

% relcomp_mono
thf(fact_658_relcomp__mono,axiom,
    ! [R7: set_Pr4934435412358123699_a_nat,R: set_Pr4934435412358123699_a_nat,S5: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8666007276011122963_a_nat @ R7 @ R )
     => ( ( ord_le3146513528884898305at_nat @ S5 @ S )
       => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R7 @ S5 ) @ ( relcomp_a_nat_nat @ R @ S ) ) ) ) ).

% relcomp_mono
thf(fact_659_conversion__inv,axiom,
    ! [X: product_prod_a_nat,Y: product_prod_a_nat,R3: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( abstra3054561309482028431_a_nat @ R3 ) )
      = ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ X ) @ ( abstra3054561309482028431_a_nat @ R3 ) ) ) ).

% conversion_inv
thf(fact_660_conversion__inv,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra6125656530262095991rm_a_b @ R3 ) )
      = ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ ( abstra6125656530262095991rm_a_b @ R3 ) ) ) ).

% conversion_inv
thf(fact_661_conversion__inv,axiom,
    ! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R3 ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( abstra1130568886744591910rm_a_b @ R3 ) ) ) ).

% conversion_inv
thf(fact_662_conversion__refl,axiom,
    ! [A: product_prod_a_nat,A5: set_Pr1811044260758604347_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ A ) @ ( abstra3054561309482028431_a_nat @ A5 ) ) ).

% conversion_refl
thf(fact_663_conversion__refl,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ ( abstra6125656530262095991rm_a_b @ A5 ) ) ).

% conversion_refl
thf(fact_664_conversion__refl,axiom,
    ! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( abstra1130568886744591910rm_a_b @ A5 ) ) ).

% conversion_refl
thf(fact_665_conversion__mono,axiom,
    ! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
     => ( ord_le746702958409616551od_a_a @ ( abstra3918519255145857644sion_a @ A5 ) @ ( abstra3918519255145857644sion_a @ B5 ) ) ) ).

% conversion_mono
thf(fact_666_conversion__mono,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A5 @ B5 )
     => ( ord_le118470702582115849rm_a_b @ ( abstra1130568886744591910rm_a_b @ A5 ) @ ( abstra1130568886744591910rm_a_b @ B5 ) ) ) ).

% conversion_mono
thf(fact_667_conversion__mono,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ A5 @ B5 )
     => ( ord_le2593958535671617255rm_a_b @ ( abstra6125656530262095991rm_a_b @ A5 ) @ ( abstra6125656530262095991rm_a_b @ B5 ) ) ) ).

% conversion_mono
thf(fact_668_steps__map,axiom,
    ! [P: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F2: term_a_b > term_a_b,G2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
      ( ! [T3: term_a_b,U2: term_a_b,R8: set_Pr4386577575007340137rm_a_b] :
          ( ( P @ T3 )
         => ( ( Q @ R8 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U2 ) @ R8 )
             => ( ( P @ U2 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R8 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R3 )
         => ( ( Q @ S4 )
           => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G2 @ R3 ) ) ) )
              & ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G2 @ R3 ) ) @ ( relcom370159955682700863rm_a_b @ ( G2 @ S4 ) @ ( transi7742714808557438673rm_a_b @ ( G2 @ R3 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_669_steps__map,axiom,
    ! [P: a > $o,Q: set_Product_prod_a_a > $o,F2: a > a,G2: set_Product_prod_a_a > set_Product_prod_a_a,T: a,R3: set_Product_prod_a_a,S4: set_Product_prod_a_a,U: a] :
      ( ! [T3: a,U2: a,R8: set_Product_prod_a_a] :
          ( ( P @ T3 )
         => ( ( Q @ R8 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T3 @ U2 ) @ R8 )
             => ( ( P @ U2 )
                & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R8 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R3 )
         => ( ( Q @ S4 )
           => ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ ( transitive_rtrancl_a @ R3 ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transitive_rtrancl_a @ ( G2 @ R3 ) ) ) )
              & ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ ( relcomp_a_a_a @ ( transitive_rtrancl_a @ R3 ) @ ( relcomp_a_a_a @ S4 @ ( transitive_rtrancl_a @ R3 ) ) ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcomp_a_a_a @ ( transitive_rtrancl_a @ ( G2 @ R3 ) ) @ ( relcomp_a_a_a @ ( G2 @ S4 ) @ ( transitive_rtrancl_a @ ( G2 @ R3 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_670_steps__map,axiom,
    ! [P: a > $o,Q: set_Product_prod_a_a > $o,F2: a > nat,G2: set_Product_prod_a_a > set_Pr1261947904930325089at_nat,T: a,R3: set_Product_prod_a_a,S4: set_Product_prod_a_a,U: a] :
      ( ! [T3: a,U2: a,R8: set_Product_prod_a_a] :
          ( ( P @ T3 )
         => ( ( Q @ R8 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T3 @ U2 ) @ R8 )
             => ( ( P @ U2 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R8 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R3 )
         => ( ( Q @ S4 )
           => ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ ( transitive_rtrancl_a @ R3 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi2905341329935302413cl_nat @ ( G2 @ R3 ) ) ) )
              & ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ ( relcomp_a_a_a @ ( transitive_rtrancl_a @ R3 ) @ ( relcomp_a_a_a @ S4 @ ( transitive_rtrancl_a @ R3 ) ) ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( G2 @ R3 ) ) @ ( relcomp_nat_nat_nat @ ( G2 @ S4 ) @ ( transi2905341329935302413cl_nat @ ( G2 @ R3 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_671_steps__map,axiom,
    ! [P: nat > $o,Q: set_Pr1261947904930325089at_nat > $o,F2: nat > a,G2: set_Pr1261947904930325089at_nat > set_Product_prod_a_a,T: nat,R3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat,U: nat] :
      ( ! [T3: nat,U2: nat,R8: set_Pr1261947904930325089at_nat] :
          ( ( P @ T3 )
         => ( ( Q @ R8 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T3 @ U2 ) @ R8 )
             => ( ( P @ U2 )
                & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R8 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R3 )
         => ( ( Q @ S4 )
           => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ ( transi2905341329935302413cl_nat @ R3 ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transitive_rtrancl_a @ ( G2 @ R3 ) ) ) )
              & ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( relcomp_nat_nat_nat @ S4 @ ( transi2905341329935302413cl_nat @ R3 ) ) ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcomp_a_a_a @ ( transitive_rtrancl_a @ ( G2 @ R3 ) ) @ ( relcomp_a_a_a @ ( G2 @ S4 ) @ ( transitive_rtrancl_a @ ( G2 @ R3 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_672_steps__map,axiom,
    ! [P: nat > $o,Q: set_Pr1261947904930325089at_nat > $o,F2: nat > nat,G2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,T: nat,R3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat,U: nat] :
      ( ! [T3: nat,U2: nat,R8: set_Pr1261947904930325089at_nat] :
          ( ( P @ T3 )
         => ( ( Q @ R8 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T3 @ U2 ) @ R8 )
             => ( ( P @ U2 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R8 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R3 )
         => ( ( Q @ S4 )
           => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ ( transi2905341329935302413cl_nat @ R3 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi2905341329935302413cl_nat @ ( G2 @ R3 ) ) ) )
              & ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( relcomp_nat_nat_nat @ S4 @ ( transi2905341329935302413cl_nat @ R3 ) ) ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( G2 @ R3 ) ) @ ( relcomp_nat_nat_nat @ ( G2 @ S4 ) @ ( transi2905341329935302413cl_nat @ ( G2 @ R3 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_673_steps__map,axiom,
    ! [P: product_prod_a_nat > $o,Q: set_Pr1811044260758604347_a_nat > $o,F2: product_prod_a_nat > a,G2: set_Pr1811044260758604347_a_nat > set_Product_prod_a_a,T: product_prod_a_nat,R3: set_Pr1811044260758604347_a_nat,S4: set_Pr1811044260758604347_a_nat,U: product_prod_a_nat] :
      ( ! [T3: product_prod_a_nat,U2: product_prod_a_nat,R8: set_Pr1811044260758604347_a_nat] :
          ( ( P @ T3 )
         => ( ( Q @ R8 )
           => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ T3 @ U2 ) @ R8 )
             => ( ( P @ U2 )
                & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R8 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R3 )
         => ( ( Q @ S4 )
           => ( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ T @ U ) @ ( transi2726145917338391738_a_nat @ R3 ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transitive_rtrancl_a @ ( G2 @ R3 ) ) ) )
              & ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ T @ U ) @ ( relcom3132138491631691642_a_nat @ ( transi2726145917338391738_a_nat @ R3 ) @ ( relcom3132138491631691642_a_nat @ S4 @ ( transi2726145917338391738_a_nat @ R3 ) ) ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcomp_a_a_a @ ( transitive_rtrancl_a @ ( G2 @ R3 ) ) @ ( relcomp_a_a_a @ ( G2 @ S4 ) @ ( transitive_rtrancl_a @ ( G2 @ R3 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_674_steps__map,axiom,
    ! [P: product_prod_a_nat > $o,Q: set_Pr1811044260758604347_a_nat > $o,F2: product_prod_a_nat > nat,G2: set_Pr1811044260758604347_a_nat > set_Pr1261947904930325089at_nat,T: product_prod_a_nat,R3: set_Pr1811044260758604347_a_nat,S4: set_Pr1811044260758604347_a_nat,U: product_prod_a_nat] :
      ( ! [T3: product_prod_a_nat,U2: product_prod_a_nat,R8: set_Pr1811044260758604347_a_nat] :
          ( ( P @ T3 )
         => ( ( Q @ R8 )
           => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ T3 @ U2 ) @ R8 )
             => ( ( P @ U2 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R8 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R3 )
         => ( ( Q @ S4 )
           => ( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ T @ U ) @ ( transi2726145917338391738_a_nat @ R3 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi2905341329935302413cl_nat @ ( G2 @ R3 ) ) ) )
              & ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ T @ U ) @ ( relcom3132138491631691642_a_nat @ ( transi2726145917338391738_a_nat @ R3 ) @ ( relcom3132138491631691642_a_nat @ S4 @ ( transi2726145917338391738_a_nat @ R3 ) ) ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( G2 @ R3 ) ) @ ( relcomp_nat_nat_nat @ ( G2 @ S4 ) @ ( transi2905341329935302413cl_nat @ ( G2 @ R3 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_675_steps__map,axiom,
    ! [P: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F2: term_a_b > a,G2: set_Pr4386577575007340137rm_a_b > set_Product_prod_a_a,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
      ( ! [T3: term_a_b,U2: term_a_b,R8: set_Pr4386577575007340137rm_a_b] :
          ( ( P @ T3 )
         => ( ( Q @ R8 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U2 ) @ R8 )
             => ( ( P @ U2 )
                & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R8 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R3 )
         => ( ( Q @ S4 )
           => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transitive_rtrancl_a @ ( G2 @ R3 ) ) ) )
              & ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcomp_a_a_a @ ( transitive_rtrancl_a @ ( G2 @ R3 ) ) @ ( relcomp_a_a_a @ ( G2 @ S4 ) @ ( transitive_rtrancl_a @ ( G2 @ R3 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_676_steps__map,axiom,
    ! [P: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F2: term_a_b > nat,G2: set_Pr4386577575007340137rm_a_b > set_Pr1261947904930325089at_nat,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
      ( ! [T3: term_a_b,U2: term_a_b,R8: set_Pr4386577575007340137rm_a_b] :
          ( ( P @ T3 )
         => ( ( Q @ R8 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U2 ) @ R8 )
             => ( ( P @ U2 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R8 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R3 )
         => ( ( Q @ S4 )
           => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi2905341329935302413cl_nat @ ( G2 @ R3 ) ) ) )
              & ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( G2 @ R3 ) ) @ ( relcomp_nat_nat_nat @ ( G2 @ S4 ) @ ( transi2905341329935302413cl_nat @ ( G2 @ R3 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_677_steps__map,axiom,
    ! [P: a > $o,Q: set_Product_prod_a_a > $o,F2: a > product_prod_a_nat,G2: set_Product_prod_a_a > set_Pr1811044260758604347_a_nat,T: a,R3: set_Product_prod_a_a,S4: set_Product_prod_a_a,U: a] :
      ( ! [T3: a,U2: a,R8: set_Product_prod_a_a] :
          ( ( P @ T3 )
         => ( ( Q @ R8 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T3 @ U2 ) @ R8 )
             => ( ( P @ U2 )
                & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R8 ) ) ) ) ) )
     => ( ( P @ T )
       => ( ( Q @ R3 )
         => ( ( Q @ S4 )
           => ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ ( transitive_rtrancl_a @ R3 ) )
               => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi2726145917338391738_a_nat @ ( G2 @ R3 ) ) ) )
              & ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ ( relcomp_a_a_a @ ( transitive_rtrancl_a @ R3 ) @ ( relcomp_a_a_a @ S4 @ ( transitive_rtrancl_a @ R3 ) ) ) )
               => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom3132138491631691642_a_nat @ ( transi2726145917338391738_a_nat @ ( G2 @ R3 ) ) @ ( relcom3132138491631691642_a_nat @ ( G2 @ S4 ) @ ( transi2726145917338391738_a_nat @ ( G2 @ R3 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_678_rtrancl__O__push,axiom,
    ! [S4: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R3 ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( transi7742714808557438673rm_a_b @ S4 ) ) )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ R3 ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( transi7742714808557438673rm_a_b @ S4 ) ) ) ) ).

% rtrancl_O_push
thf(fact_679_rtrancl__O__push,axiom,
    ! [S4: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ S4 @ R3 ) @ ( relcomp_nat_nat_nat @ R3 @ ( transi2905341329935302413cl_nat @ S4 ) ) )
     => ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ S4 ) @ R3 ) @ ( relcomp_nat_nat_nat @ R3 @ ( transi2905341329935302413cl_nat @ S4 ) ) ) ) ).

% rtrancl_O_push
thf(fact_680_rtrancl__O__push,axiom,
    ! [S4: set_Pr2972776593051762503rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ S4 @ R3 ) @ ( relcom6159724393391289352rm_a_b @ R3 @ ( transi2615809358984392588rm_a_b @ S4 ) ) )
     => ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ S4 ) @ R3 ) @ ( relcom6159724393391289352rm_a_b @ R3 @ ( transi2615809358984392588rm_a_b @ S4 ) ) ) ) ).

% rtrancl_O_push
thf(fact_681_rtrancl__O__push,axiom,
    ! [S4: set_Product_prod_a_a,R3: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ S4 @ R3 ) @ ( relcomp_a_a_a @ R3 @ ( transitive_rtrancl_a @ S4 ) ) )
     => ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ ( transitive_rtrancl_a @ S4 ) @ R3 ) @ ( relcomp_a_a_a @ R3 @ ( transitive_rtrancl_a @ S4 ) ) ) ) ).

% rtrancl_O_push
thf(fact_682_compat__tr__compat,axiom,
    ! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S4 ) @ S4 ) ) ).

% compat_tr_compat
thf(fact_683_compat__tr__compat,axiom,
    ! [NS: set_Product_prod_a_a,S4: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ NS @ S4 ) @ S4 )
     => ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ ( transitive_rtrancl_a @ NS ) @ S4 ) @ S4 ) ) ).

% compat_tr_compat
thf(fact_684_compat__tr__compat,axiom,
    ! [NS: set_Product_prod_a_a,S4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S4 ) @ S4 )
     => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S4 ) @ S4 ) ) ).

% compat_tr_compat
thf(fact_685_kleene__induct,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,X5: set_Pr4934435412358123699_a_nat,B5: set_Product_prod_a_a,C4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ X5 )
     => ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B5 @ X5 ) @ X5 )
       => ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X5 @ C4 ) @ X5 )
         => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ B5 ) @ ( relcomp_a_nat_nat @ A5 @ ( transi2905341329935302413cl_nat @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_induct
thf(fact_686_kleene__induct,axiom,
    ! [A5: set_Pr4193341848836149977_nat_a,X5: set_Pr4193341848836149977_nat_a,B5: set_Pr1261947904930325089at_nat,C4: set_Product_prod_a_a] :
      ( ( ord_le7924913712489149241_nat_a @ A5 @ X5 )
     => ( ( ord_le7924913712489149241_nat_a @ ( relcomp_nat_nat_a @ B5 @ X5 ) @ X5 )
       => ( ( ord_le7924913712489149241_nat_a @ ( relcomp_nat_a_a @ X5 @ C4 ) @ X5 )
         => ( ord_le7924913712489149241_nat_a @ ( relcomp_nat_nat_a @ ( transi2905341329935302413cl_nat @ B5 ) @ ( relcomp_nat_a_a @ A5 @ ( transitive_rtrancl_a @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_induct
thf(fact_687_kleene__induct,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A5 @ X5 )
     => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ B5 @ X5 ) @ X5 )
       => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ X5 @ C4 ) @ X5 )
         => ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ B5 ) @ ( relcomp_nat_nat_nat @ A5 @ ( transi2905341329935302413cl_nat @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_induct
thf(fact_688_kleene__induct,axiom,
    ! [A5: set_Product_prod_a_a,X5: set_Product_prod_a_a,B5: set_Product_prod_a_a,C4: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A5 @ X5 )
     => ( ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ B5 @ X5 ) @ X5 )
       => ( ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ X5 @ C4 ) @ X5 )
         => ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ ( transitive_rtrancl_a @ B5 ) @ ( relcomp_a_a_a @ A5 @ ( transitive_rtrancl_a @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_induct
thf(fact_689_kleene__induct,axiom,
    ! [A5: set_Pr3298798960101569173_a_b_a,X5: set_Pr3298798960101569173_a_b_a,B5: set_Pr4386577575007340137rm_a_b,C4: set_Product_prod_a_a] :
      ( ( ord_le2921212089142715125_a_b_a @ A5 @ X5 )
     => ( ( ord_le2921212089142715125_a_b_a @ ( relcom5652721298402989587_a_b_a @ B5 @ X5 ) @ X5 )
       => ( ( ord_le2921212089142715125_a_b_a @ ( relcomp_term_a_b_a_a @ X5 @ C4 ) @ X5 )
         => ( ord_le2921212089142715125_a_b_a @ ( relcom5652721298402989587_a_b_a @ ( transi7742714808557438673rm_a_b @ B5 ) @ ( relcomp_term_a_b_a_a @ A5 @ ( transitive_rtrancl_a @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_induct
thf(fact_690_kleene__induct,axiom,
    ! [A5: set_Pr6708357783813671845_b_nat,X5: set_Pr6708357783813671845_b_nat,B5: set_Pr4386577575007340137rm_a_b,C4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le16013013860614981_b_nat @ A5 @ X5 )
     => ( ( ord_le16013013860614981_b_nat @ ( relcom2286245697325151611_b_nat @ B5 @ X5 ) @ X5 )
       => ( ( ord_le16013013860614981_b_nat @ ( relcom2781103321250786871at_nat @ X5 @ C4 ) @ X5 )
         => ( ord_le16013013860614981_b_nat @ ( relcom2286245697325151611_b_nat @ ( transi7742714808557438673rm_a_b @ B5 ) @ ( relcom2781103321250786871at_nat @ A5 @ ( transi2905341329935302413cl_nat @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_induct
thf(fact_691_kleene__induct,axiom,
    ! [A5: set_Pr690693513201511543rm_a_b,X5: set_Pr690693513201511543rm_a_b,B5: set_Product_prod_a_a,C4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le313106642242657495rm_a_b @ A5 @ X5 )
     => ( ( ord_le313106642242657495rm_a_b @ ( relcomp_a_a_term_a_b @ B5 @ X5 ) @ X5 )
       => ( ( ord_le313106642242657495rm_a_b @ ( relcom7339816600239648359rm_a_b @ X5 @ C4 ) @ X5 )
         => ( ord_le313106642242657495rm_a_b @ ( relcomp_a_a_term_a_b @ ( transitive_rtrancl_a @ B5 ) @ ( relcom7339816600239648359rm_a_b @ A5 @ ( transi7742714808557438673rm_a_b @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_induct
thf(fact_692_kleene__induct,axiom,
    ! [A5: set_Pr4549835640365387557rm_a_b,X5: set_Pr4549835640365387557rm_a_b,B5: set_Pr1261947904930325089at_nat,C4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le7080862907267106501rm_a_b @ A5 @ X5 )
     => ( ( ord_le7080862907267106501rm_a_b @ ( relcom8414687918727181623rm_a_b @ B5 @ X5 ) @ X5 )
       => ( ( ord_le7080862907267106501rm_a_b @ ( relcom6565464767029197947rm_a_b @ X5 @ C4 ) @ X5 )
         => ( ord_le7080862907267106501rm_a_b @ ( relcom8414687918727181623rm_a_b @ ( transi2905341329935302413cl_nat @ B5 ) @ ( relcom6565464767029197947rm_a_b @ A5 @ ( transi7742714808557438673rm_a_b @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_induct
thf(fact_693_kleene__induct,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,X5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A5 @ X5 )
     => ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B5 @ X5 ) @ X5 )
       => ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X5 @ C4 ) @ X5 )
         => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ B5 ) @ ( relcom370159955682700863rm_a_b @ A5 @ ( transi7742714808557438673rm_a_b @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_induct
thf(fact_694_kleene__induct,axiom,
    ! [A5: set_Pr2303916211607585938rm_a_b,X5: set_Pr2303916211607585938rm_a_b,B5: set_Product_prod_a_a,C4: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le1511013792394939442rm_a_b @ A5 @ X5 )
     => ( ( ord_le1511013792394939442rm_a_b @ ( relcom6469880312746088648rm_a_b @ B5 @ X5 ) @ X5 )
       => ( ( ord_le1511013792394939442rm_a_b @ ( relcom2073241673377248765rm_a_b @ X5 @ C4 ) @ X5 )
         => ( ord_le1511013792394939442rm_a_b @ ( relcom6469880312746088648rm_a_b @ ( transitive_rtrancl_a @ B5 ) @ ( relcom2073241673377248765rm_a_b @ A5 @ ( transi2615809358984392588rm_a_b @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_induct
thf(fact_695_rrstep_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R2 ) )
     => ~ ! [L3: term_a_b,R9: term_a_b,Sigma2: b > term_a_b] :
            ( ( A1
              = ( subst_7999470309526761004_a_b_b @ L3 @ Sigma2 ) )
           => ( ( A22
                = ( subst_7999470309526761004_a_b_b @ R9 @ Sigma2 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R9 ) @ R2 ) ) ) ) ).

% rrstep.cases
thf(fact_696_rrstep_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R2 ) )
      = ( ? [L4: term_a_b,R4: term_a_b,Sigma3: b > term_a_b] :
            ( ( A1
              = ( subst_7999470309526761004_a_b_b @ L4 @ Sigma3 ) )
            & ( A22
              = ( subst_7999470309526761004_a_b_b @ R4 @ Sigma3 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L4 @ R4 ) @ R2 ) ) ) ) ).

% rrstep.simps
thf(fact_697_rrstep_Ointros,axiom,
    ! [L: term_a_b,R: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R2 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ R @ Sigma ) ) @ ( rrstep_a_b @ R2 ) ) ) ).

% rrstep.intros
thf(fact_698_rrstep__subst,axiom,
    ! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R2 ) )
     => ~ ! [L3: term_a_b,R9: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R9 ) @ R2 )
           => ! [Sigma2: b > term_a_b] :
                ( ( S
                  = ( subst_7999470309526761004_a_b_b @ L3 @ Sigma2 ) )
               => ( T
                 != ( subst_7999470309526761004_a_b_b @ R9 @ Sigma2 ) ) ) ) ) ).

% rrstep_subst
thf(fact_699_O__rtrancl__O__O,axiom,
    ! [R3: set_Product_prod_a_a,S4: set_Product_prod_a_a] :
      ( ( relcomp_a_a_a @ R3 @ ( relcomp_a_a_a @ ( transitive_rtrancl_a @ ( relcomp_a_a_a @ S4 @ R3 ) ) @ S4 ) )
      = ( transitive_trancl_a @ ( relcomp_a_a_a @ R3 @ S4 ) ) ) ).

% O_rtrancl_O_O
thf(fact_700_O__rtrancl__O__O,axiom,
    ! [R3: set_Pr4193341848836149977_nat_a,S4: set_Pr4934435412358123699_a_nat] :
      ( ( relcomp_nat_a_nat @ R3 @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ ( relcomp_a_nat_a @ S4 @ R3 ) ) @ S4 ) )
      = ( transi6264000038957366511cl_nat @ ( relcomp_nat_a_nat @ R3 @ S4 ) ) ) ).

% O_rtrancl_O_O
thf(fact_701_O__rtrancl__O__O,axiom,
    ! [R3: set_Pr4934435412358123699_a_nat,S4: set_Pr4193341848836149977_nat_a] :
      ( ( relcomp_a_nat_a @ R3 @ ( relcomp_nat_nat_a @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_a_nat @ S4 @ R3 ) ) @ S4 ) )
      = ( transitive_trancl_a @ ( relcomp_a_nat_a @ R3 @ S4 ) ) ) ).

% O_rtrancl_O_O
thf(fact_702_O__rtrancl__O__O,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( relcomp_nat_nat_nat @ R3 @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_nat_nat @ S4 @ R3 ) ) @ S4 ) )
      = ( transi6264000038957366511cl_nat @ ( relcomp_nat_nat_nat @ R3 @ S4 ) ) ) ).

% O_rtrancl_O_O
thf(fact_703_O__rtrancl__O__O,axiom,
    ! [R3: set_Pr690693513201511543rm_a_b,S4: set_Pr3298798960101569173_a_b_a] :
      ( ( relcomp_a_term_a_b_a @ R3 @ ( relcom5652721298402989587_a_b_a @ ( transi7742714808557438673rm_a_b @ ( relcom4147987028256916669rm_a_b @ S4 @ R3 ) ) @ S4 ) )
      = ( transitive_trancl_a @ ( relcomp_a_term_a_b_a @ R3 @ S4 ) ) ) ).

% O_rtrancl_O_O
thf(fact_704_O__rtrancl__O__O,axiom,
    ! [R3: set_Pr4549835640365387557rm_a_b,S4: set_Pr6708357783813671845_b_nat] :
      ( ( relcom152770009037377975_b_nat @ R3 @ ( relcom2286245697325151611_b_nat @ ( transi7742714808557438673rm_a_b @ ( relcom1324791570160179451rm_a_b @ S4 @ R3 ) ) @ S4 ) )
      = ( transi6264000038957366511cl_nat @ ( relcom152770009037377975_b_nat @ R3 @ S4 ) ) ) ).

% O_rtrancl_O_O
thf(fact_705_O__rtrancl__O__O,axiom,
    ! [R3: set_Pr3298798960101569173_a_b_a,S4: set_Pr690693513201511543rm_a_b] :
      ( ( relcom4147987028256916669rm_a_b @ R3 @ ( relcomp_a_a_term_a_b @ ( transitive_rtrancl_a @ ( relcomp_a_term_a_b_a @ S4 @ R3 ) ) @ S4 ) )
      = ( transi7922773638565587891rm_a_b @ ( relcom4147987028256916669rm_a_b @ R3 @ S4 ) ) ) ).

% O_rtrancl_O_O
thf(fact_706_O__rtrancl__O__O,axiom,
    ! [R3: set_Pr6708357783813671845_b_nat,S4: set_Pr4549835640365387557rm_a_b] :
      ( ( relcom1324791570160179451rm_a_b @ R3 @ ( relcom8414687918727181623rm_a_b @ ( transi2905341329935302413cl_nat @ ( relcom152770009037377975_b_nat @ S4 @ R3 ) ) @ S4 ) )
      = ( transi7922773638565587891rm_a_b @ ( relcom1324791570160179451rm_a_b @ R3 @ S4 ) ) ) ).

% O_rtrancl_O_O
thf(fact_707_O__rtrancl__O__O,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ R3 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R3 ) ) @ S4 ) )
      = ( transi7922773638565587891rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S4 ) ) ) ).

% O_rtrancl_O_O
thf(fact_708_O__rtrancl__O__O,axiom,
    ! [R3: set_Pr513079337203471164_a_b_a,S4: set_Pr2303916211607585938rm_a_b] :
      ( ( relcom4164503515417244499rm_a_b @ R3 @ ( relcom6469880312746088648rm_a_b @ ( transitive_rtrancl_a @ ( relcom370660246271418610_a_b_a @ S4 @ R3 ) ) @ S4 ) )
      = ( transi9211502839322181930rm_a_b @ ( relcom4164503515417244499rm_a_b @ R3 @ S4 ) ) ) ).

% O_rtrancl_O_O
thf(fact_709_rtrancl__comp__trancl__conv,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ( relcomp_a_a_a @ ( transitive_rtrancl_a @ R ) @ R )
      = ( transitive_trancl_a @ R ) ) ).

% rtrancl_comp_trancl_conv
thf(fact_710_rtrancl__comp__trancl__conv,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ R )
      = ( transi6264000038957366511cl_nat @ R ) ) ).

% rtrancl_comp_trancl_conv
thf(fact_711_rtrancl__comp__trancl__conv,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b] :
      ( ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R ) @ R )
      = ( transi9211502839322181930rm_a_b @ R ) ) ).

% rtrancl_comp_trancl_conv
thf(fact_712_rtrancl__comp__trancl__conv,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ R )
      = ( transi7922773638565587891rm_a_b @ R ) ) ).

% rtrancl_comp_trancl_conv
thf(fact_713_trancl__unfold__left,axiom,
    ( transitive_trancl_a
    = ( ^ [R4: set_Product_prod_a_a] : ( relcomp_a_a_a @ R4 @ ( transitive_rtrancl_a @ R4 ) ) ) ) ).

% trancl_unfold_left
thf(fact_714_trancl__unfold__left,axiom,
    ( transi6264000038957366511cl_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat] : ( relcomp_nat_nat_nat @ R4 @ ( transi2905341329935302413cl_nat @ R4 ) ) ) ) ).

% trancl_unfold_left
thf(fact_715_trancl__unfold__left,axiom,
    ( transi9211502839322181930rm_a_b
    = ( ^ [R4: set_Pr2972776593051762503rm_a_b] : ( relcom6159724393391289352rm_a_b @ R4 @ ( transi2615809358984392588rm_a_b @ R4 ) ) ) ) ).

% trancl_unfold_left
thf(fact_716_trancl__unfold__left,axiom,
    ( transi7922773638565587891rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ).

% trancl_unfold_left
thf(fact_717_comp__trancl,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,S4: set_Pr2972776593051762503rm_a_b] :
      ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ R3 @ S4 ) @ S4 )
     => ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ R3 @ ( transi9211502839322181930rm_a_b @ S4 ) ) @ ( transi9211502839322181930rm_a_b @ S4 ) ) ) ).

% comp_trancl
thf(fact_718_comp__trancl,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ S4 ) @ S4 )
     => ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ ( transi6264000038957366511cl_nat @ S4 ) ) @ ( transi6264000038957366511cl_nat @ S4 ) ) ) ).

% comp_trancl
thf(fact_719_comp__trancl,axiom,
    ! [R3: set_Product_prod_a_a,S4: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ R3 @ S4 ) @ S4 )
     => ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ R3 @ ( transitive_trancl_a @ S4 ) ) @ ( transitive_trancl_a @ S4 ) ) ) ).

% comp_trancl
thf(fact_720_comp__trancl,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S4 ) @ S4 )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ ( transi7922773638565587891rm_a_b @ S4 ) ) @ ( transi7922773638565587891rm_a_b @ S4 ) ) ) ).

% comp_trancl
thf(fact_721_kleene__trancl__induct,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A5 @ X5 )
     => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ B5 @ X5 ) @ X5 )
       => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ X5 @ C4 ) @ X5 )
         => ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ ( transi6264000038957366511cl_nat @ B5 ) @ ( relcomp_nat_nat_nat @ A5 @ ( transi6264000038957366511cl_nat @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_trancl_induct
thf(fact_722_kleene__trancl__induct,axiom,
    ! [A5: set_Product_prod_a_a,X5: set_Product_prod_a_a,B5: set_Product_prod_a_a,C4: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A5 @ X5 )
     => ( ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ B5 @ X5 ) @ X5 )
       => ( ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ X5 @ C4 ) @ X5 )
         => ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ ( transitive_trancl_a @ B5 ) @ ( relcomp_a_a_a @ A5 @ ( transitive_trancl_a @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_trancl_induct
thf(fact_723_kleene__trancl__induct,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,X5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A5 @ X5 )
     => ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B5 @ X5 ) @ X5 )
       => ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X5 @ C4 ) @ X5 )
         => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ B5 ) @ ( relcom370159955682700863rm_a_b @ A5 @ ( transi7922773638565587891rm_a_b @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_trancl_induct
thf(fact_724_kleene__trancl__induct,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,X5: set_Pr4934435412358123699_a_nat,B5: set_Product_prod_a_a,C4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ X5 )
     => ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B5 @ X5 ) @ X5 )
       => ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X5 @ C4 ) @ X5 )
         => ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_trancl_a @ B5 ) @ ( relcomp_a_nat_nat @ A5 @ ( transi6264000038957366511cl_nat @ C4 ) ) ) @ X5 ) ) ) ) ).

% kleene_trancl_induct
thf(fact_725_rtrancl__induct2,axiom,
    ! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R: set_Pr2972776593051762503rm_a_b,P: term_a_b > term_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A4: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
               => ( ( P @ A4 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_726_converse__rtranclE2,axiom,
    ! [Xa: term_a_b,Xb: term_a_b,Za: term_a_b,Zb: term_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( ( produc7020197800436672577rm_a_b @ Xa @ Xb )
         != ( produc7020197800436672577rm_a_b @ Za @ Zb ) )
       => ~ ! [A4: term_a_b,B3: term_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ R )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_727_converse__rtrancl__induct2,axiom,
    ! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R: set_Pr2972776593051762503rm_a_b,P: term_a_b > term_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( P @ Bx @ By )
       => ( ! [A4: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A4 @ B3 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_728_rstep__imp__C__s__r,axiom,
    ! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
     => ? [C5: subterm_and_ctxt_a_b,Sigma2: b > term_a_b,L3: term_a_b,R9: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R9 ) @ R2 )
          & ( S
            = ( subter2376574525758040790rm_a_b @ C5 @ ( subst_7999470309526761004_a_b_b @ L3 @ Sigma2 ) ) )
          & ( T
            = ( subter2376574525758040790rm_a_b @ C5 @ ( subst_7999470309526761004_a_b_b @ R9 @ Sigma2 ) ) ) ) ) ).

% rstep_imp_C_s_r
thf(fact_729_rstep__imp__C__s__r_H,axiom,
    ! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
     => ~ ! [C5: subterm_and_ctxt_a_b,L3: term_a_b,R9: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R9 ) @ R2 )
           => ! [Sigma2: b > term_a_b] :
                ( ( S
                  = ( subter2376574525758040790rm_a_b @ C5 @ ( subst_7999470309526761004_a_b_b @ L3 @ Sigma2 ) ) )
               => ( T
                 != ( subter2376574525758040790rm_a_b @ C5 @ ( subst_7999470309526761004_a_b_b @ R9 @ Sigma2 ) ) ) ) ) ) ).

% rstep_imp_C_s_r'
thf(fact_730_non__strict__into__strict,axiom,
    ! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S4 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ S4 ) ) ) ).

% non_strict_into_strict
thf(fact_731_non__strict__into__strict,axiom,
    ! [NS: set_Product_prod_a_a,S4: set_Pr4934435412358123699_a_nat,S: a,T: nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S4 ) @ S4 )
     => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S4 ) )
       => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ S4 ) ) ) ).

% non_strict_into_strict
thf(fact_732_UNF__onI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A4 @ A5 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C2 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
             => ( B3 = C2 ) ) ) )
     => ( abstra1303798319023342519rm_a_b @ R @ A5 ) ) ).

% UNF_onI
thf(fact_733_UNF__onI,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [A4: term_a_b,B3: term_a_b,C2: term_a_b] :
          ( ( member_term_a_b @ A4 @ A5 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C2 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
             => ( B3 = C2 ) ) ) )
     => ( abstra3619855735679963878rm_a_b @ R @ A5 ) ) ).

% UNF_onI
thf(fact_734_WN__onI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ! [A4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A4 @ A5 )
         => ? [B7: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B7 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) )
     => ( abstra2721873542170697047rm_a_b @ R @ A5 ) ) ).

% WN_onI
thf(fact_735_WN__onI,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [A4: term_a_b] :
          ( ( member_term_a_b @ A4 @ A5 )
         => ? [B7: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B7 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) )
     => ( abstra4644735066934440262rm_a_b @ R @ A5 ) ) ).

% WN_onI
thf(fact_736_comp__rtrancl__trancl,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S4 ) @ S4 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ S4 ) ) @ S4 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ S4 ) ) ) ) ).

% comp_rtrancl_trancl
thf(fact_737_normalizability__I,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A5 ) )
     => ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A5 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ A5 ) ) ) ) ).

% normalizability_I
thf(fact_738_normalizability__I,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
     => ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A5 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ A5 ) ) ) ) ).

% normalizability_I
thf(fact_739_UNF__onD,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( abstra1303798319023342519rm_a_b @ R @ A5 )
     => ( ( member5869715511025134514rm_a_b @ A @ A5 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ R ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
           => ( B = C3 ) ) ) ) ) ).

% UNF_onD
thf(fact_740_UNF__onD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C3: term_a_b] :
      ( ( abstra3619855735679963878rm_a_b @ R @ A5 )
     => ( ( member_term_a_b @ A @ A5 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ R ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
           => ( B = C3 ) ) ) ) ) ).

% UNF_onD
thf(fact_741_UNF__onE,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( abstra1303798319023342519rm_a_b @ R @ A5 )
     => ( ( member5869715511025134514rm_a_b @ A @ A5 )
       => ( ( B != C3 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ R ) )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ) ) ).

% UNF_onE
thf(fact_742_UNF__onE,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C3: term_a_b] :
      ( ( abstra3619855735679963878rm_a_b @ R @ A5 )
     => ( ( member_term_a_b @ A @ A5 )
       => ( ( B != C3 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ R ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ) ) ).

% UNF_onE
thf(fact_743_UNF__on__def,axiom,
    ( abstra3619855735679963878rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
        ! [X4: term_a_b] :
          ( ( member_term_a_b @ X4 @ A6 )
         => ! [B4: term_a_b,C: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) )
             => ( B4 = C ) ) ) ) ) ).

% UNF_on_def
thf(fact_744_WN__onD,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( abstra2721873542170697047rm_a_b @ R @ A5 )
     => ( ( member5869715511025134514rm_a_b @ A @ A5 )
       => ? [B3: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ).

% WN_onD
thf(fact_745_WN__onD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b] :
      ( ( abstra4644735066934440262rm_a_b @ R @ A5 )
     => ( ( member_term_a_b @ A @ A5 )
       => ? [B3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ).

% WN_onD
thf(fact_746_Un__iff,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) )
      = ( ( member5869715511025134514rm_a_b @ C3 @ A5 )
        | ( member5869715511025134514rm_a_b @ C3 @ B5 ) ) ) ).

% Un_iff
thf(fact_747_UnCI,axiom,
    ! [C3: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( ~ ( member5869715511025134514rm_a_b @ C3 @ B5 )
       => ( member5869715511025134514rm_a_b @ C3 @ A5 ) )
     => ( member5869715511025134514rm_a_b @ C3 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) ) ) ).

% UnCI
thf(fact_748_Un__subset__iff,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A5 @ B5 ) @ C4 )
      = ( ( ord_le8666007276011122963_a_nat @ A5 @ C4 )
        & ( ord_le8666007276011122963_a_nat @ B5 @ C4 ) ) ) ).

% Un_subset_iff
thf(fact_749_NF__I,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
      ( ! [B3: produc357393685978478089rm_a_b] :
          ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ A5 )
     => ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) ) ) ).

% NF_I
thf(fact_750_NF__I,axiom,
    ! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ! [B3: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ A5 )
     => ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) ) ) ).

% NF_I
thf(fact_751_UnI2,axiom,
    ! [C3: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ B5 )
     => ( member5869715511025134514rm_a_b @ C3 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) ) ) ).

% UnI2
thf(fact_752_UnI1,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ A5 )
     => ( member5869715511025134514rm_a_b @ C3 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) ) ) ).

% UnI1
thf(fact_753_UnE,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) )
     => ( ~ ( member5869715511025134514rm_a_b @ C3 @ A5 )
       => ( member5869715511025134514rm_a_b @ C3 @ B5 ) ) ) ).

% UnE
thf(fact_754_Un__mono,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ C4 )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ D )
       => ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A5 @ B5 ) @ ( sup_su459911885395995103_a_nat @ C4 @ D ) ) ) ) ).

% Un_mono
thf(fact_755_Un__least,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ C4 )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ C4 )
       => ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A5 @ B5 ) @ C4 ) ) ) ).

% Un_least
thf(fact_756_Un__upper1,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A5 @ ( sup_su459911885395995103_a_nat @ A5 @ B5 ) ) ).

% Un_upper1
thf(fact_757_Un__upper2,axiom,
    ! [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B5 @ ( sup_su459911885395995103_a_nat @ A5 @ B5 ) ) ).

% Un_upper2
thf(fact_758_Un__absorb1,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( sup_su459911885395995103_a_nat @ A5 @ B5 )
        = B5 ) ) ).

% Un_absorb1
thf(fact_759_Un__absorb2,axiom,
    ! [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ B5 @ A5 )
     => ( ( sup_su459911885395995103_a_nat @ A5 @ B5 )
        = A5 ) ) ).

% Un_absorb2
thf(fact_760_subset__UnE,axiom,
    ! [C4: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ C4 @ ( sup_su459911885395995103_a_nat @ A5 @ B5 ) )
     => ~ ! [A7: set_Pr4934435412358123699_a_nat] :
            ( ( ord_le8666007276011122963_a_nat @ A7 @ A5 )
           => ! [B8: set_Pr4934435412358123699_a_nat] :
                ( ( ord_le8666007276011122963_a_nat @ B8 @ B5 )
               => ( C4
                 != ( sup_su459911885395995103_a_nat @ A7 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_761_subset__Un__eq,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
          ( ( sup_su459911885395995103_a_nat @ A6 @ B6 )
          = B6 ) ) ) ).

% subset_Un_eq
thf(fact_762_in__rtrancl__UnI,axiom,
    ! [X: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ R3 ) )
        | ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ S4 ) ) )
     => ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ S4 ) ) ) ) ).

% in_rtrancl_UnI
thf(fact_763_NF__iff__no__step,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
      = ( ! [B4: produc357393685978478089rm_a_b] :
            ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B4 ) @ A5 ) ) ) ).

% NF_iff_no_step
thf(fact_764_NF__iff__no__step,axiom,
    ! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
      = ( ! [B4: term_a_b] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ A5 ) ) ) ).

% NF_iff_no_step
thf(fact_765_NF__no__step,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
     => ! [B7: produc357393685978478089rm_a_b] :
          ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B7 ) @ A5 ) ) ).

% NF_no_step
thf(fact_766_NF__no__step,axiom,
    ! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
     => ! [B7: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B7 ) @ A5 ) ) ).

% NF_no_step
thf(fact_767_NF__E,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
     => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ A5 ) ) ).

% NF_E
thf(fact_768_NF__E,axiom,
    ! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
      ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ A5 ) ) ).

% NF_E
thf(fact_769_rtrancl__Un__separator__converseE,axiom,
    ! [A: term_a_b,B: term_a_b,P: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P @ Q ) ) )
     => ( ! [X3: term_a_b,Y3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B ) @ ( transi7742714808557438673rm_a_b @ P ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ Q )
             => ( Y3 = X3 ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_770_rtrancl__Un__separatorE,axiom,
    ! [A: term_a_b,B: term_a_b,P: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P @ Q ) ) )
     => ( ! [X3: term_a_b,Y3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X3 ) @ ( transi7742714808557438673rm_a_b @ P ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ Q )
             => ( X3 = Y3 ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_771_rtrancl__union__subset__rtrancl__union__trancl,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ ( transi7922773638565587891rm_a_b @ S ) ) )
      = ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ S ) ) ) ).

% rtrancl_union_subset_rtrancl_union_trancl
thf(fact_772_trancl__union__right,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ ( transi7922773638565587891rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S @ R ) ) ) ).

% trancl_union_right
thf(fact_773_trancl__unfold,axiom,
    ( transi7922773638565587891rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ R4 @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ R4 ) @ R4 ) ) ) ) ).

% trancl_unfold
thf(fact_774_NF__not__suc,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
     => ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R ) )
       => ( X = Y ) ) ) ).

% NF_not_suc
thf(fact_775_NF__not__suc,axiom,
    ! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R ) )
       => ( X = Y ) ) ) ).

% NF_not_suc
thf(fact_776_NF__anti__mono,axiom,
    ! [A5: set_Pr1811044260758604347_a_nat,B5: set_Pr1811044260758604347_a_nat] :
      ( ( ord_le1018141841545957851_a_nat @ A5 @ B5 )
     => ( ord_le8666007276011122963_a_nat @ ( abstra2685462378305987313_a_nat @ B5 ) @ ( abstra2685462378305987313_a_nat @ A5 ) ) ) ).

% NF_anti_mono
thf(fact_777_NF__no__trancl__step,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
     => ! [B7: produc357393685978478089rm_a_b] :
          ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B7 ) @ ( transi9211502839322181930rm_a_b @ A5 ) ) ) ).

% NF_no_trancl_step
thf(fact_778_NF__no__trancl__step,axiom,
    ! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
     => ! [B7: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B7 ) @ ( transi7922773638565587891rm_a_b @ A5 ) ) ) ).

% NF_no_trancl_step
thf(fact_779_firstStep,axiom,
    ! [L5: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( L5
        = ( sup_su6776935440552674877rm_a_b @ S4 @ R3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ L5 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
          | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ L5 ) ) ) ) ) ) ) ).

% firstStep
thf(fact_780_first__step,axiom,
    ! [C4: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( C4
        = ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ C4 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ B5 )
         => ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A5 ) @ B5 ) ) ) ) ) ).

% first_step
thf(fact_781_first__step__O,axiom,
    ! [C4: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( C4
        = ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ C4 ) @ B5 ) )
       => ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A5 ) @ B5 ) ) ) ) ).

% first_step_O
thf(fact_782_right__comp__S,axiom,
    ! [X: term_a_b,Y: term_a_b,S4: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).

% right_comp_S
thf(fact_783_normalizability__E,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ A5 ) )
     => ~ ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A5 ) )
         => ~ ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A5 ) ) ) ) ).

% normalizability_E
thf(fact_784_normalizability__E,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ A5 ) )
     => ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
         => ~ ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A5 ) ) ) ) ).

% normalizability_E
thf(fact_785_compatible__rtrancl__split,axiom,
    ! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).

% compatible_rtrancl_split
thf(fact_786_WN__on__def,axiom,
    ( abstra4644735066934440262rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
        ! [X4: term_a_b] :
          ( ( member_term_a_b @ X4 @ A6 )
         => ? [B4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) ) ) ) ).

% WN_on_def
thf(fact_787_WN__onE,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( abstra2721873542170697047rm_a_b @ R @ A5 )
     => ( ( member5869715511025134514rm_a_b @ A @ A5 )
       => ~ ! [B3: produc357393685978478089rm_a_b] :
              ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ).

% WN_onE
thf(fact_788_WN__onE,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b] :
      ( ( abstra4644735066934440262rm_a_b @ R @ A5 )
     => ( ( member_term_a_b @ A @ A5 )
       => ~ ! [B3: term_a_b] :
              ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ).

% WN_onE
thf(fact_789_sup_Obounded__iff,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B @ C3 ) @ A )
      = ( ( ord_le8666007276011122963_a_nat @ B @ A )
        & ( ord_le8666007276011122963_a_nat @ C3 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_790_le__sup__iff,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X @ Y ) @ Z2 )
      = ( ( ord_le8666007276011122963_a_nat @ X @ Z2 )
        & ( ord_le8666007276011122963_a_nat @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_791_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_792_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_793_le__supE,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ X )
     => ~ ( ( ord_le8666007276011122963_a_nat @ A @ X )
         => ~ ( ord_le8666007276011122963_a_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_794_le__supI,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ X )
     => ( ( ord_le8666007276011122963_a_nat @ B @ X )
       => ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_795_sup__ge1,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_796_sup__ge2,axiom,
    ! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_797_le__supI1,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ A )
     => ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_798_le__supI2,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ B )
     => ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_799_sup_Omono,axiom,
    ! [C3: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ C3 @ A )
     => ( ( ord_le8666007276011122963_a_nat @ D2 @ B )
       => ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ C3 @ D2 ) @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_800_sup__mono,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ C3 )
     => ( ( ord_le8666007276011122963_a_nat @ B @ D2 )
       => ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ ( sup_su459911885395995103_a_nat @ C3 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_801_sup__least,axiom,
    ! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ Y @ X )
     => ( ( ord_le8666007276011122963_a_nat @ Z2 @ X )
       => ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ Y @ Z2 ) @ X ) ) ) ).

% sup_least
thf(fact_802_le__iff__sup,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
          ( ( sup_su459911885395995103_a_nat @ X4 @ Y5 )
          = Y5 ) ) ) ).

% le_iff_sup
thf(fact_803_sup_OorderE,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ B @ A )
     => ( A
        = ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_804_sup_OorderI,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( A
        = ( sup_su459911885395995103_a_nat @ A @ B ) )
     => ( ord_le8666007276011122963_a_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_805_sup__unique,axiom,
    ! [F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
      ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ ( F2 @ X3 @ Y3 ) )
     => ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y3 @ ( F2 @ X3 @ Y3 ) )
       => ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] :
              ( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
             => ( ( ord_le8666007276011122963_a_nat @ Z @ X3 )
               => ( ord_le8666007276011122963_a_nat @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
         => ( ( sup_su459911885395995103_a_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_806_sup_Oabsorb1,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ B @ A )
     => ( ( sup_su459911885395995103_a_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_807_sup_Oabsorb2,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ B )
     => ( ( sup_su459911885395995103_a_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_808_sup__absorb1,axiom,
    ! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ Y @ X )
     => ( ( sup_su459911885395995103_a_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_809_sup__absorb2,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ Y )
     => ( ( sup_su459911885395995103_a_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_810_sup_OboundedE,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B @ C3 ) @ A )
     => ~ ( ( ord_le8666007276011122963_a_nat @ B @ A )
         => ~ ( ord_le8666007276011122963_a_nat @ C3 @ A ) ) ) ).

% sup.boundedE
thf(fact_811_sup_OboundedI,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ B @ A )
     => ( ( ord_le8666007276011122963_a_nat @ C3 @ A )
       => ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B @ C3 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_812_sup_Oorder__iff,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [B4: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
          ( A3
          = ( sup_su459911885395995103_a_nat @ A3 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_813_sup_Ocobounded1,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_814_sup_Ocobounded2,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_815_sup_Oabsorb__iff1,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [B4: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
          ( ( sup_su459911885395995103_a_nat @ A3 @ B4 )
          = A3 ) ) ) ).

% sup.absorb_iff1
thf(fact_816_sup_Oabsorb__iff2,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
          ( ( sup_su459911885395995103_a_nat @ A3 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_817_sup_OcoboundedI1,axiom,
    ! [C3: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ C3 @ A )
     => ( ord_le8666007276011122963_a_nat @ C3 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_818_sup_OcoboundedI2,axiom,
    ! [C3: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ C3 @ B )
     => ( ord_le8666007276011122963_a_nat @ C3 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_819_UNC__def,axiom,
    ( abstra8480497335821202873rm_a_b
    = ( ^ [A6: set_Pr2972776593051762503rm_a_b] :
        ! [A3: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
          ( ( ( member5869715511025134514rm_a_b @ A3 @ ( abstra1902783451776770197rm_a_b @ A6 ) )
            & ( member5869715511025134514rm_a_b @ B4 @ ( abstra1902783451776770197rm_a_b @ A6 ) )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B4 ) @ ( abstra6125656530262095991rm_a_b @ A6 ) ) )
         => ( A3 = B4 ) ) ) ) ).

% UNC_def
thf(fact_820_UNC__def,axiom,
    ( abstra4649771476585718372rm_a_b
    = ( ^ [A6: set_Pr4386577575007340137rm_a_b] :
        ! [A3: term_a_b,B4: term_a_b] :
          ( ( ( member_term_a_b @ A3 @ ( abstract_NF_term_a_b @ A6 ) )
            & ( member_term_a_b @ B4 @ ( abstract_NF_term_a_b @ A6 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ ( abstra1130568886744591910rm_a_b @ A6 ) ) )
         => ( A3 = B4 ) ) ) ) ).

% UNC_def
thf(fact_821_UNC__I,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b] :
      ( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A4 @ ( abstra1902783451776770197rm_a_b @ A5 ) )
         => ( ( member5869715511025134514rm_a_b @ B3 @ ( abstra1902783451776770197rm_a_b @ A5 ) )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) @ ( abstra6125656530262095991rm_a_b @ A5 ) )
             => ( A4 = B3 ) ) ) )
     => ( abstra8480497335821202873rm_a_b @ A5 ) ) ).

% UNC_I
thf(fact_822_UNC__I,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ! [A4: term_a_b,B3: term_a_b] :
          ( ( member_term_a_b @ A4 @ ( abstract_NF_term_a_b @ A5 ) )
         => ( ( member_term_a_b @ B3 @ ( abstract_NF_term_a_b @ A5 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( abstra1130568886744591910rm_a_b @ A5 ) )
             => ( A4 = B3 ) ) ) )
     => ( abstra4649771476585718372rm_a_b @ A5 ) ) ).

% UNC_I
thf(fact_823_UNC__E,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( abstra8480497335821202873rm_a_b @ A5 )
     => ( ( A != B )
       => ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
         => ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A5 ) )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra6125656530262095991rm_a_b @ A5 ) ) ) ) ) ) ).

% UNC_E
thf(fact_824_UNC__E,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( abstra4649771476585718372rm_a_b @ A5 )
     => ( ( A != B )
       => ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
         => ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A5 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A5 ) ) ) ) ) ) ).

% UNC_E
thf(fact_825_conversionI,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A5 @ ( conver7608559976937845093rm_a_b @ A5 ) ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A5 ) ) ) ).

% conversionI
thf(fact_826_trancl__reflcl,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7922773638565587891rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) )
      = ( transi7742714808557438673rm_a_b @ R ) ) ).

% trancl_reflcl
thf(fact_827_conversionE,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A5 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A5 @ ( conver7608559976937845093rm_a_b @ A5 ) ) ) ) ) ).

% conversionE
thf(fact_828_converse__iff,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( conver7608559976937845093rm_a_b @ R ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R ) ) ).

% converse_iff
thf(fact_829_pair__in__Id__conv,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ id_term_a_b )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_830_IdI,axiom,
    ! [A: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ id_term_a_b ) ).

% IdI
thf(fact_831_converse__mono,axiom,
    ! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le7924913712489149241_nat_a @ ( converse_a_nat @ R ) @ ( converse_a_nat @ S ) )
      = ( ord_le8666007276011122963_a_nat @ R @ S ) ) ).

% converse_mono
thf(fact_832_converse__mono,axiom,
    ! [R: set_Pr4193341848836149977_nat_a,S: set_Pr4193341848836149977_nat_a] :
      ( ( ord_le8666007276011122963_a_nat @ ( converse_nat_a @ R ) @ ( converse_nat_a @ S ) )
      = ( ord_le7924913712489149241_nat_a @ R @ S ) ) ).

% converse_mono
thf(fact_833_converseI,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).

% converseI
thf(fact_834_converseE,axiom,
    ! [Yx: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Yx @ ( conver7608559976937845093rm_a_b @ R ) )
     => ~ ! [X3: term_a_b,Y3: term_a_b] :
            ( ( Yx
              = ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R ) ) ) ).

% converseE
thf(fact_835_converseD,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( conver7608559976937845093rm_a_b @ R ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R ) ) ).

% converseD
thf(fact_836_converse_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R ) )
      = ( ? [A3: term_a_b,B4: term_a_b] :
            ( ( A1 = B4 )
            & ( A22 = A3 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ R ) ) ) ) ).

% converse.simps
thf(fact_837_converse_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A22 @ A1 ) @ R ) ) ).

% converse.cases
thf(fact_838_converse__subset__swap,axiom,
    ! [R: set_Pr4193341848836149977_nat_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le7924913712489149241_nat_a @ R @ ( converse_a_nat @ S ) )
      = ( ord_le8666007276011122963_a_nat @ ( converse_nat_a @ R ) @ S ) ) ).

% converse_subset_swap
thf(fact_839_converse__subset__swap,axiom,
    ! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4193341848836149977_nat_a] :
      ( ( ord_le8666007276011122963_a_nat @ R @ ( converse_nat_a @ S ) )
      = ( ord_le7924913712489149241_nat_a @ ( converse_a_nat @ R ) @ S ) ) ).

% converse_subset_swap
thf(fact_840_trancl__converse,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) )
      = ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).

% trancl_converse
thf(fact_841_IdE,axiom,
    ! [P2: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ id_term_a_b )
     => ~ ! [X3: term_a_b] :
            ( P2
           != ( produc7020197800436672577rm_a_b @ X3 @ X3 ) ) ) ).

% IdE
thf(fact_842_rtrancl__converseI,axiom,
    ! [Y: term_a_b,X: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).

% rtrancl_converseI
thf(fact_843_rtrancl__converseD,axiom,
    ! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).

% rtrancl_converseD
thf(fact_844_peak__iff,axiom,
    ! [X: term_a_b,Y: term_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A5 ) @ B5 ) )
      = ( ? [U3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ X ) @ A5 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ Y ) @ B5 ) ) ) ) ).

% peak_iff
thf(fact_845_trancl__converseI,axiom,
    ! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).

% trancl_converseI
thf(fact_846_trancl__converseD,axiom,
    ! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).

% trancl_converseD
thf(fact_847_funas__rel__converse,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F )
     => ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ F ) ) ).

% funas_rel_converse
thf(fact_848_rtrancl__trancl__reflcl,axiom,
    ( transi7742714808557438673rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ ( transi7922773638565587891rm_a_b @ R4 ) @ id_term_a_b ) ) ) ).

% rtrancl_trancl_reflcl
thf(fact_849_trancl__o__refl__is__trancl,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) )
      = ( transi7922773638565587891rm_a_b @ R ) ) ).

% trancl_o_refl_is_trancl
thf(fact_850_strongly__confluent__on__E11,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
      ( ( abstra837640964651051114rm_a_b @ R @ A5 )
     => ( ( member5869715511025134514rm_a_b @ X @ A5 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R )
           => ? [U2: produc357393685978478089rm_a_b] :
                ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ U2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
                & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z2 @ U2 ) @ ( sup_su1845815533690797339rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_851_strongly__confluent__on__E11,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( abstra5664096641628173427rm_a_b @ R @ A5 )
     => ( ( member_term_a_b @ X @ A5 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R )
           => ? [U2: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_852_strongly__confluent__on__def,axiom,
    ( abstra5664096641628173427rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
        ! [X4: term_a_b] :
          ( ( member_term_a_b @ X4 @ A6 )
         => ! [Y5: term_a_b,Z4: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R4 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z4 ) @ R4 ) )
             => ? [U3: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ U3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
                  & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ U3 ) @ ( sup_su6776935440552674877rm_a_b @ R4 @ id_term_a_b ) ) ) ) ) ) ) ).

% strongly_confluent_on_def
thf(fact_853_strongly__confluentI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y3: term_a_b,Z: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z ) @ R )
           => ? [U4: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ U4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) )
     => ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b ) ) ).

% strongly_confluentI
thf(fact_854_IdD,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ id_term_a_b )
     => ( A = B ) ) ).

% IdD
thf(fact_855_NF__join__imp__reach,axiom,
    ! [Y: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ A5 ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ A5 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ A5 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_856_NF__join__imp__reach,axiom,
    ! [Y: term_a_b,A5: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
      ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ A5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ A5 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).

% NF_join_imp_reach
thf(fact_857_UNIV__I,axiom,
    ! [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ top_to1314267278846557113rm_a_b ) ).

% UNIV_I
thf(fact_858_Int__iff,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ ( inf_in1693821330460682071rm_a_b @ A5 @ B5 ) )
      = ( ( member5869715511025134514rm_a_b @ C3 @ A5 )
        & ( member5869715511025134514rm_a_b @ C3 @ B5 ) ) ) ).

% Int_iff
thf(fact_859_IntI,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ A5 )
     => ( ( member5869715511025134514rm_a_b @ C3 @ B5 )
       => ( member5869715511025134514rm_a_b @ C3 @ ( inf_in1693821330460682071rm_a_b @ A5 @ B5 ) ) ) ) ).

% IntI
thf(fact_860_inf_Obounded__iff,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ ( inf_in3813033988066259269_a_nat @ B @ C3 ) )
      = ( ( ord_le8666007276011122963_a_nat @ A @ B )
        & ( ord_le8666007276011122963_a_nat @ A @ C3 ) ) ) ).

% inf.bounded_iff
thf(fact_861_le__inf__iff,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ Y @ Z2 ) )
      = ( ( ord_le8666007276011122963_a_nat @ X @ Y )
        & ( ord_le8666007276011122963_a_nat @ X @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_862_Int__subset__iff,axiom,
    ! [C4: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ C4 @ ( inf_in3813033988066259269_a_nat @ A5 @ B5 ) )
      = ( ( ord_le8666007276011122963_a_nat @ C4 @ A5 )
        & ( ord_le8666007276011122963_a_nat @ C4 @ B5 ) ) ) ).

% Int_subset_iff
thf(fact_863_joinI__right,axiom,
    ! [B: term_a_b,A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A5 ) ) ) ).

% joinI_right
thf(fact_864_joinI__left,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A5 ) ) ) ).

% joinI_left
thf(fact_865_joinI,axiom,
    ! [A: term_a_b,C3: term_a_b,A5: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A5 ) ) ) ) ).

% joinI
thf(fact_866_UNIV__witness,axiom,
    ? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ top_to1314267278846557113rm_a_b ) ).

% UNIV_witness
thf(fact_867_UNIV__eq__I,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ A5 )
     => ( top_to1314267278846557113rm_a_b = A5 ) ) ).

% UNIV_eq_I
thf(fact_868_IntD2,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ ( inf_in1693821330460682071rm_a_b @ A5 @ B5 ) )
     => ( member5869715511025134514rm_a_b @ C3 @ B5 ) ) ).

% IntD2
thf(fact_869_IntD1,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ ( inf_in1693821330460682071rm_a_b @ A5 @ B5 ) )
     => ( member5869715511025134514rm_a_b @ C3 @ A5 ) ) ).

% IntD1
thf(fact_870_IntE,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ ( inf_in1693821330460682071rm_a_b @ A5 @ B5 ) )
     => ~ ( ( member5869715511025134514rm_a_b @ C3 @ A5 )
         => ~ ( member5869715511025134514rm_a_b @ C3 @ B5 ) ) ) ).

% IntE
thf(fact_871_top__greatest,axiom,
    ! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ top_to3353692345378799459_a_nat ) ).

% top_greatest
thf(fact_872_top_Oextremum__unique,axiom,
    ! [A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A )
      = ( A = top_to3353692345378799459_a_nat ) ) ).

% top.extremum_unique
thf(fact_873_top_Oextremum__uniqueI,axiom,
    ! [A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A )
     => ( A = top_to3353692345378799459_a_nat ) ) ).

% top.extremum_uniqueI
thf(fact_874_inf_OcoboundedI2,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
     => ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ C3 ) ) ).

% inf.coboundedI2
thf(fact_875_inf_OcoboundedI1,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ C3 )
     => ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ C3 ) ) ).

% inf.coboundedI1
thf(fact_876_inf_Oabsorb__iff2,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [B4: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
          ( ( inf_in3813033988066259269_a_nat @ A3 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_877_inf_Oabsorb__iff1,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
          ( ( inf_in3813033988066259269_a_nat @ A3 @ B4 )
          = A3 ) ) ) ).

% inf.absorb_iff1
thf(fact_878_inf_Ocobounded2,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_879_inf_Ocobounded1,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_880_inf_Oorder__iff,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
          ( A3
          = ( inf_in3813033988066259269_a_nat @ A3 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_881_inf__greatest,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ Y )
     => ( ( ord_le8666007276011122963_a_nat @ X @ Z2 )
       => ( ord_le8666007276011122963_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_882_inf_OboundedI,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ B )
     => ( ( ord_le8666007276011122963_a_nat @ A @ C3 )
       => ( ord_le8666007276011122963_a_nat @ A @ ( inf_in3813033988066259269_a_nat @ B @ C3 ) ) ) ) ).

% inf.boundedI
thf(fact_883_inf_OboundedE,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ ( inf_in3813033988066259269_a_nat @ B @ C3 ) )
     => ~ ( ( ord_le8666007276011122963_a_nat @ A @ B )
         => ~ ( ord_le8666007276011122963_a_nat @ A @ C3 ) ) ) ).

% inf.boundedE
thf(fact_884_inf__absorb2,axiom,
    ! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ Y @ X )
     => ( ( inf_in3813033988066259269_a_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_885_inf__absorb1,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ Y )
     => ( ( inf_in3813033988066259269_a_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_886_inf_Oabsorb2,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ B @ A )
     => ( ( inf_in3813033988066259269_a_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_887_inf_Oabsorb1,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ B )
     => ( ( inf_in3813033988066259269_a_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_888_le__iff__inf,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
          ( ( inf_in3813033988066259269_a_nat @ X4 @ Y5 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_889_inf__unique,axiom,
    ! [F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
      ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] :
              ( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
             => ( ( ord_le8666007276011122963_a_nat @ X3 @ Z )
               => ( ord_le8666007276011122963_a_nat @ X3 @ ( F2 @ Y3 @ Z ) ) ) )
         => ( ( inf_in3813033988066259269_a_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_890_inf_OorderI,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( A
        = ( inf_in3813033988066259269_a_nat @ A @ B ) )
     => ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_891_inf_OorderE,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ B )
     => ( A
        = ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_892_le__infI2,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ B @ X )
     => ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_893_le__infI1,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ X )
     => ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_894_inf__mono,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ C3 )
     => ( ( ord_le8666007276011122963_a_nat @ B @ D2 )
       => ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ ( inf_in3813033988066259269_a_nat @ C3 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_895_le__infI,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ A )
     => ( ( ord_le8666007276011122963_a_nat @ X @ B )
       => ( ord_le8666007276011122963_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_896_le__infE,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ A @ B ) )
     => ~ ( ( ord_le8666007276011122963_a_nat @ X @ A )
         => ~ ( ord_le8666007276011122963_a_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_897_inf__le2,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_898_inf__le1,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_899_inf__sup__ord_I1_J,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_900_inf__sup__ord_I2_J,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_901_Int__mono,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ C4 )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ D )
       => ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A5 @ B5 ) @ ( inf_in3813033988066259269_a_nat @ C4 @ D ) ) ) ) ).

% Int_mono
thf(fact_902_Int__lower1,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A5 @ B5 ) @ A5 ) ).

% Int_lower1
thf(fact_903_Int__lower2,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A5 @ B5 ) @ B5 ) ).

% Int_lower2
thf(fact_904_Int__absorb1,axiom,
    ! [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ B5 @ A5 )
     => ( ( inf_in3813033988066259269_a_nat @ A5 @ B5 )
        = B5 ) ) ).

% Int_absorb1
thf(fact_905_Int__absorb2,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( inf_in3813033988066259269_a_nat @ A5 @ B5 )
        = A5 ) ) ).

% Int_absorb2
thf(fact_906_Int__greatest,axiom,
    ! [C4: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ C4 @ A5 )
     => ( ( ord_le8666007276011122963_a_nat @ C4 @ B5 )
       => ( ord_le8666007276011122963_a_nat @ C4 @ ( inf_in3813033988066259269_a_nat @ A5 @ B5 ) ) ) ) ).

% Int_greatest
thf(fact_907_Int__Collect__mono,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
      ( ( ord_le118470702582115849rm_a_b @ A5 @ B5 )
     => ( ! [X3: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A5 @ ( collec99177395925924084rm_a_b @ P ) ) @ ( inf_in1693821330460682071rm_a_b @ B5 @ ( collec99177395925924084rm_a_b @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_908_Int__Collect__mono,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ! [X3: product_prod_a_nat] :
            ( ( member5724188588386418708_a_nat @ X3 @ A5 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A5 @ ( collec4464134535221767506_a_nat @ P ) ) @ ( inf_in3813033988066259269_a_nat @ B5 @ ( collec4464134535221767506_a_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_909_subset__UNIV,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A5 @ top_to3353692345378799459_a_nat ) ).

% subset_UNIV
thf(fact_910_join__sym,axiom,
    ! [S: term_a_b,T: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ A5 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ ( abstra4096080454567261402rm_a_b @ A5 ) ) ) ).

% join_sym
thf(fact_911_distrib__sup__le,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ Y @ Z2 ) ) @ ( inf_in3813033988066259269_a_nat @ ( sup_su459911885395995103_a_nat @ X @ Y ) @ ( sup_su459911885395995103_a_nat @ X @ Z2 ) ) ) ).

% distrib_sup_le
thf(fact_912_distrib__inf__le,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ ( inf_in3813033988066259269_a_nat @ X @ Z2 ) ) @ ( inf_in3813033988066259269_a_nat @ X @ ( sup_su459911885395995103_a_nat @ Y @ Z2 ) ) ) ).

% distrib_inf_le
thf(fact_913_Un__Int__assoc__eq,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
      ( ( ( sup_su459911885395995103_a_nat @ ( inf_in3813033988066259269_a_nat @ A5 @ B5 ) @ C4 )
        = ( inf_in3813033988066259269_a_nat @ A5 @ ( sup_su459911885395995103_a_nat @ B5 @ C4 ) ) )
      = ( ord_le8666007276011122963_a_nat @ C4 @ A5 ) ) ).

% Un_Int_assoc_eq
thf(fact_914_rtrancl__join__join,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( abstra4096080454567261402rm_a_b @ A5 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( abstra4096080454567261402rm_a_b @ A5 ) ) ) ) ).

% rtrancl_join_join
thf(fact_915_join__rtrancl__join,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C3 @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( abstra4096080454567261402rm_a_b @ A5 ) ) ) ) ).

% join_rtrancl_join
thf(fact_916_joinE,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A5 ) )
     => ~ ! [C2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).

% joinE
thf(fact_917_joinD,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A5 ) )
     => ? [C2: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).

% joinD
thf(fact_918_join__NF__imp__eq,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ R ) )
     => ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R ) )
         => ( X = Y ) ) ) ) ).

% join_NF_imp_eq
thf(fact_919_join__NF__imp__eq,axiom,
    ! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R ) )
     => ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R ) )
       => ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R ) )
         => ( X = Y ) ) ) ) ).

% join_NF_imp_eq
thf(fact_920_trancl__Int__subset,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R @ S )
     => ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( inf_in1693821330460682071rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ S ) @ R ) @ S )
       => ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ S ) ) ) ).

% trancl_Int_subset
thf(fact_921_some__NF__UNF,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( abstra1303798319023342519rm_a_b @ R @ top_to1314267278846557113rm_a_b )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R ) )
         => ( Y
            = ( abstra8203652949708803681rm_a_b @ R @ X ) ) ) ) ) ).

% some_NF_UNF
thf(fact_922_some__NF__UNF,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( abstra3619855735679963878rm_a_b @ R @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
       => ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R ) )
         => ( Y
            = ( abstra2589370493620291516rm_a_b @ R @ X ) ) ) ) ) ).

% some_NF_UNF
thf(fact_923_semi__complete__imp__conversionIff__same__NF,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( abstra7162307272211697846rm_a_b @ R @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R ) )
        = ( ! [U3: term_a_b,V2: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ U3 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ V2 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) )
             => ( U3 = V2 ) ) ) ) ) ).

% semi_complete_imp_conversionIff_same_NF
thf(fact_924_WCR__onI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A4 @ A5 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) @ R )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C2 ) @ R )
             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) )
     => ( abstra875157153669124638rm_a_b @ R @ A5 ) ) ).

% WCR_onI
thf(fact_925_WCR__onI,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [A4: term_a_b,B3: term_a_b,C2: term_a_b] :
          ( ( member_term_a_b @ A4 @ A5 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ R )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C2 ) @ R )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) )
     => ( abstra5026849107050860991rm_a_b @ R @ A5 ) ) ).

% WCR_onI
thf(fact_926_CR__onI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A4 @ A5 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) )
     => ( abstra7359694574113289279rm_a_b @ R @ A5 ) ) ).

% CR_onI
thf(fact_927_CR__onI,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [A4: term_a_b,B3: term_a_b,C2: term_a_b] :
          ( ( member_term_a_b @ A4 @ A5 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) )
     => ( abstra8448919418672941150rm_a_b @ R @ A5 ) ) ).

% CR_onI
thf(fact_928_top__empty__eq,axiom,
    ( top_to1262317115727556708_a_b_o
    = ( ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ top_to1314267278846557113rm_a_b ) ) ) ).

% top_empty_eq
thf(fact_929_CR__on__def,axiom,
    ( abstra8448919418672941150rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
        ! [X4: term_a_b] :
          ( ( member_term_a_b @ X4 @ A6 )
         => ! [B4: term_a_b,C: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C ) @ ( abstra4096080454567261402rm_a_b @ R4 ) ) ) ) ) ) ).

% CR_on_def
thf(fact_930_CR__onE,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( abstra7359694574113289279rm_a_b @ R @ A5 )
     => ( ( member5869715511025134514rm_a_b @ A @ A5 )
       => ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ ( abstra4889682940153293507rm_a_b @ R ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ) ).

% CR_onE
thf(fact_931_CR__onE,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C3: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R @ A5 )
     => ( ( member_term_a_b @ A @ A5 )
       => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ) ).

% CR_onE
thf(fact_932_CR__onD,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( abstra7359694574113289279rm_a_b @ R @ A5 )
     => ( ( member5869715511025134514rm_a_b @ A @ A5 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
           => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ) ).

% CR_onD
thf(fact_933_CR__onD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C3: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R @ A5 )
     => ( ( member_term_a_b @ A @ A5 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ) ).

% CR_onD
thf(fact_934_WCR__on__def,axiom,
    ( abstra5026849107050860991rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
        ! [X4: term_a_b] :
          ( ( member_term_a_b @ X4 @ A6 )
         => ! [B4: term_a_b,C: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ R4 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C ) @ R4 ) )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C ) @ ( abstra4096080454567261402rm_a_b @ R4 ) ) ) ) ) ) ).

% WCR_on_def
thf(fact_935_WCR__onE,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( abstra875157153669124638rm_a_b @ R @ A5 )
     => ( ( member5869715511025134514rm_a_b @ A @ A5 )
       => ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ ( abstra4889682940153293507rm_a_b @ R ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ R ) ) ) ) ) ).

% WCR_onE
thf(fact_936_WCR__onE,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C3: term_a_b] :
      ( ( abstra5026849107050860991rm_a_b @ R @ A5 )
     => ( ( member_term_a_b @ A @ A5 )
       => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ R ) ) ) ) ) ).

% WCR_onE
thf(fact_937_WCR__onD,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( abstra875157153669124638rm_a_b @ R @ A5 )
     => ( ( member5869715511025134514rm_a_b @ A @ A5 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ R )
           => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C3 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ) ).

% WCR_onD
thf(fact_938_WCR__onD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C3: term_a_b] :
      ( ( abstra5026849107050860991rm_a_b @ R @ A5 )
     => ( ( member_term_a_b @ A @ A5 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ R )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ) ).

% WCR_onD
thf(fact_939_CR__join__left__I,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).

% CR_join_left_I
thf(fact_940_CR__join__right__I,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).

% CR_join_right_I
thf(fact_941_partially__localize__CR,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
      = ( ! [X4: term_a_b,Y5: term_a_b,Z4: term_a_b] :
            ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R )
              & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z4 ) @ ( transi7742714808557438673rm_a_b @ R ) ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Z4 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).

% partially_localize_CR
thf(fact_942_CR__divergence__imp__join,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).

% CR_divergence_imp_join
thf(fact_943_CR__NF__conv,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,T: produc357393685978478089rm_a_b,U: produc357393685978478089rm_a_b] :
      ( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
     => ( ( member5869715511025134514rm_a_b @ T @ ( abstra1902783451776770197rm_a_b @ R ) )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra6125656530262095991rm_a_b @ R ) )
         => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ) ).

% CR_NF_conv
thf(fact_944_CR__NF__conv,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,T: term_a_b,U: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
     => ( ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra1130568886744591910rm_a_b @ R ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ) ).

% CR_NF_conv
thf(fact_945_strongly__confluent__E1n,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b,N: nat] :
      ( ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R ) )
         => ? [U2: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
              & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ).

% strongly_confluent_E1n
thf(fact_946_irrefl__tranclI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
      ( ( ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ R ) )
        = bot_bo197521221353338581rm_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).

% irrefl_tranclI
thf(fact_947_all__not__in__conv,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ( ! [X4: produc357393685978478089rm_a_b] :
            ~ ( member5869715511025134514rm_a_b @ X4 @ A5 ) )
      = ( A5 = bot_bo197521221353338581rm_a_b ) ) ).

% all_not_in_conv
thf(fact_948_empty__iff,axiom,
    ! [C3: produc357393685978478089rm_a_b] :
      ~ ( member5869715511025134514rm_a_b @ C3 @ bot_bo197521221353338581rm_a_b ) ).

% empty_iff
thf(fact_949_subset__empty,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ bot_bo9049108969261143879_a_nat )
      = ( A5 = bot_bo9049108969261143879_a_nat ) ) ).

% subset_empty
thf(fact_950_empty__subsetI,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A5 ) ).

% empty_subsetI
thf(fact_951_trancl__empty,axiom,
    ( ( transi7922773638565587891rm_a_b @ bot_bo197521221353338581rm_a_b )
    = bot_bo197521221353338581rm_a_b ) ).

% trancl_empty
thf(fact_952_relpow__refl__mono,axiom,
    ! [Rel: set_Pr4386577575007340137rm_a_b,M: nat,N: nat,A: term_a_b,B: term_a_b] :
      ( ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ Rel )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ M @ Rel ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ N @ Rel ) ) ) ) ) ).

% relpow_refl_mono
thf(fact_953_relpow__image,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,R7: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,N: nat] :
      ( ! [S2: term_a_b,T3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S2 @ T3 ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ S2 ) @ ( F2 @ T3 ) ) @ R7 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( compow4057154403645558940rm_a_b @ N @ R ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ S ) @ ( F2 @ T ) ) @ ( compow4057154403645558940rm_a_b @ N @ R7 ) ) ) ) ).

% relpow_image
thf(fact_954_relpow__Suc__D2_H,axiom,
    ! [N: nat,R3: set_Pr4386577575007340137rm_a_b,X6: term_a_b,Y7: term_a_b,Z5: term_a_b] :
      ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Y7 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ Z5 ) @ R3 ) )
     => ? [W: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ W ) @ R3 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ W @ Z5 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) ) ) ) ).

% relpow_Suc_D2'
thf(fact_955_SN__on__trancl,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R @ A5 )
     => ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ A5 ) ) ).

% SN_on_trancl
thf(fact_956_SN__on__trancl__imp__SN__on,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,T4: set_term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ T4 )
     => ( abstra4720023341729745482rm_a_b @ R3 @ T4 ) ) ).

% SN_on_trancl_imp_SN_on
thf(fact_957_SN__on__trancl__SN__on__conv,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,T4: set_term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ T4 )
      = ( abstra4720023341729745482rm_a_b @ R3 @ T4 ) ) ).

% SN_on_trancl_SN_on_conv
thf(fact_958_rtrancl__power,axiom,
    ! [P2: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R3 ) )
      = ( ? [N2: nat] : ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N2 @ R3 ) ) ) ) ).

% rtrancl_power
thf(fact_959_relpow__imp__rtrancl,axiom,
    ! [P2: produc357393685978478089rm_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
     => ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ).

% relpow_imp_rtrancl
thf(fact_960_rtrancl__imp__relpow,axiom,
    ! [P2: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ? [N3: nat] : ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N3 @ R3 ) ) ) ).

% rtrancl_imp_relpow
thf(fact_961_Int__emptyI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
         => ~ ( member5869715511025134514rm_a_b @ X3 @ B5 ) )
     => ( ( inf_in1693821330460682071rm_a_b @ A5 @ B5 )
        = bot_bo197521221353338581rm_a_b ) ) ).

% Int_emptyI
thf(fact_962_disjoint__iff,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ( inf_in1693821330460682071rm_a_b @ A5 @ B5 )
        = bot_bo197521221353338581rm_a_b )
      = ( ! [X4: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X4 @ A5 )
           => ~ ( member5869715511025134514rm_a_b @ X4 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_963_SN__on__induct,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,S: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( abstra2398554102055911763rm_a_b @ R3 @ A5 )
     => ( ( member5869715511025134514rm_a_b @ S @ A5 )
       => ( ! [T3: produc357393685978478089rm_a_b] :
              ( ! [U4: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T3 @ U4 ) @ R3 )
                 => ( P @ U4 ) )
             => ( P @ T3 ) )
         => ( P @ S ) ) ) ) ).

% SN_on_induct
thf(fact_964_SN__on__induct,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,S: term_a_b,P: term_a_b > $o] :
      ( ( abstra4720023341729745482rm_a_b @ R3 @ A5 )
     => ( ( member_term_a_b @ S @ A5 )
       => ( ! [T3: term_a_b] :
              ( ! [U4: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U4 ) @ R3 )
                 => ( P @ U4 ) )
             => ( P @ T3 ) )
         => ( P @ S ) ) ) ) ).

% SN_on_induct
thf(fact_965_SN__on__irrefl,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R @ A5 )
     => ! [X6: term_a_b] :
          ( ( member_term_a_b @ X6 @ A5 )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ X6 ) @ R ) ) ) ).

% SN_on_irrefl
thf(fact_966_ex__in__conv,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A5 ) )
      = ( A5 != bot_bo197521221353338581rm_a_b ) ) ).

% ex_in_conv
thf(fact_967_equals0I,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ! [Y3: produc357393685978478089rm_a_b] :
          ~ ( member5869715511025134514rm_a_b @ Y3 @ A5 )
     => ( A5 = bot_bo197521221353338581rm_a_b ) ) ).

% equals0I
thf(fact_968_equals0D,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( A5 = bot_bo197521221353338581rm_a_b )
     => ~ ( member5869715511025134514rm_a_b @ A @ A5 ) ) ).

% equals0D
thf(fact_969_emptyE,axiom,
    ! [A: produc357393685978478089rm_a_b] :
      ~ ( member5869715511025134514rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ).

% emptyE
thf(fact_970_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat )
     => ( A = bot_bo9049108969261143879_a_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_971_bot_Oextremum__unique,axiom,
    ! [A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat )
      = ( A = bot_bo9049108969261143879_a_nat ) ) ).

% bot.extremum_unique
thf(fact_972_bot_Oextremum,axiom,
    ! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A ) ).

% bot.extremum
thf(fact_973_SN__on__subset2,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( abstra8653715922312955827_a_nat @ R @ B5 )
       => ( abstra8653715922312955827_a_nat @ R @ A5 ) ) ) ).

% SN_on_subset2
thf(fact_974_refl__not__SN,axiom,
    ! [X: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R3 )
     => ~ ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b ) ) ).

% refl_not_SN
thf(fact_975_SN__imp__minimal,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ A5 @ top_to1314267278846557113rm_a_b )
     => ! [Q2: set_Pr4386577575007340137rm_a_b] :
          ( ? [X6: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X6 @ Q2 )
         => ? [X3: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X3 @ Q2 )
              & ! [Y7: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y7 ) @ A5 )
                 => ~ ( member5869715511025134514rm_a_b @ Y7 @ Q2 ) ) ) ) ) ).

% SN_imp_minimal
thf(fact_976_SN__imp__minimal,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ A5 @ top_top_set_term_a_b )
     => ! [Q2: set_term_a_b] :
          ( ? [X6: term_a_b] : ( member_term_a_b @ X6 @ Q2 )
         => ? [X3: term_a_b] :
              ( ( member_term_a_b @ X3 @ Q2 )
              & ! [Y7: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y7 ) @ A5 )
                 => ~ ( member_term_a_b @ Y7 @ Q2 ) ) ) ) ) ).

% SN_imp_minimal
thf(fact_977_SN__induct__rule,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R @ top_top_set_term_a_b )
     => ( ! [A4: term_a_b] :
            ( ! [B7: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B7 ) @ R )
               => ( P @ B7 ) )
           => ( P @ A4 ) )
       => ( P @ A ) ) ) ).

% SN_induct_rule
thf(fact_978_SN__imp__SN__trancl,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b )
     => ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ top_top_set_term_a_b ) ) ).

% SN_imp_SN_trancl
thf(fact_979_SN__trancl__imp__SN,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ top_top_set_term_a_b )
     => ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b ) ) ).

% SN_trancl_imp_SN
thf(fact_980_SN__trancl__SN__conv,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ top_top_set_term_a_b )
      = ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b ) ) ).

% SN_trancl_SN_conv
thf(fact_981_rtrancl__len__E,axiom,
    ! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ~ ! [N3: nat] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N3 @ R ) ) ) ).

% rtrancl_len_E
thf(fact_982_trancl__steps__relpow,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,N: nat] :
      ( ( ord_le118470702582115849rm_a_b @ A @ ( transi7922773638565587891rm_a_b @ B ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ A ) )
       => ? [M2: nat] :
            ( ( ord_less_eq_nat @ N @ M2 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ M2 @ B ) ) ) ) ) ).

% trancl_steps_relpow
thf(fact_983_trancl__Un2__separatorE,axiom,
    ! [B5: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( ( relcom370159955682700863rm_a_b @ B5 @ A5 )
        = bot_bo197521221353338581rm_a_b )
     => ( ( transi7922773638565587891rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) )
        = ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( transi7922773638565587891rm_a_b @ A5 ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ A5 ) @ ( transi7922773638565587891rm_a_b @ B5 ) ) ) @ ( transi7922773638565587891rm_a_b @ B5 ) ) ) ) ).

% trancl_Un2_separatorE
thf(fact_984_the__NF,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ A5 @ top_top_set_term_a_b )
     => ( ( abstra8448919418672941150rm_a_b @ A5 @ top_top_set_term_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( abstra2242782261322123075rm_a_b @ A5 @ A ) ) @ ( abstra2972651231392952684rm_a_b @ A5 ) ) ) ) ).

% the_NF
thf(fact_985_bot__empty__eq,axiom,
    ( bot_bo23760184229683272_a_b_o
    = ( ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ bot_bo197521221353338581rm_a_b ) ) ) ).

% bot_empty_eq
thf(fact_986_the__NF__step,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ A5 @ top_top_set_term_a_b )
     => ( ( abstra8448919418672941150rm_a_b @ A5 @ top_top_set_term_a_b )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ A5 )
         => ( ( abstra2242782261322123075rm_a_b @ A5 @ A )
            = ( abstra2242782261322123075rm_a_b @ A5 @ B ) ) ) ) ) ).

% the_NF_step
thf(fact_987_the__NF__NF,axiom,
    ! [A5: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ A5 @ top_to1314267278846557113rm_a_b )
     => ( ( abstra7359694574113289279rm_a_b @ A5 @ top_to1314267278846557113rm_a_b )
       => ( member5869715511025134514rm_a_b @ ( abstra8972549633426078106rm_a_b @ A5 @ A ) @ ( abstra1902783451776770197rm_a_b @ A5 ) ) ) ) ).

% the_NF_NF
thf(fact_988_the__NF__steps,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ A5 @ top_top_set_term_a_b )
     => ( ( abstra8448919418672941150rm_a_b @ A5 @ top_top_set_term_a_b )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
         => ( ( abstra2242782261322123075rm_a_b @ A5 @ A )
            = ( abstra2242782261322123075rm_a_b @ A5 @ B ) ) ) ) ) ).

% the_NF_steps
thf(fact_989_the__NF__conv,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ A5 @ top_top_set_term_a_b )
     => ( ( abstra8448919418672941150rm_a_b @ A5 @ top_top_set_term_a_b )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A5 ) )
         => ( ( abstra2242782261322123075rm_a_b @ A5 @ A )
            = ( abstra2242782261322123075rm_a_b @ A5 @ B ) ) ) ) ) ).

% the_NF_conv
thf(fact_990_some__NF,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ ( abstra8203652949708803681rm_a_b @ R @ X ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
        & ( member5869715511025134514rm_a_b @ ( abstra8203652949708803681rm_a_b @ R @ X ) @ ( abstra1902783451776770197rm_a_b @ R ) ) ) ) ).

% some_NF
thf(fact_991_some__NF,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ ( abstra2589370493620291516rm_a_b @ R @ X ) ) @ ( transi7742714808557438673rm_a_b @ R ) )
        & ( member_term_a_b @ ( abstra2589370493620291516rm_a_b @ R @ X ) @ ( abstract_NF_term_a_b @ R ) ) ) ) ).

% some_NF
thf(fact_992_subset__emptyI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b] :
          ~ ( member5869715511025134514rm_a_b @ X3 @ A5 )
     => ( ord_le118470702582115849rm_a_b @ A5 @ bot_bo197521221353338581rm_a_b ) ) ).

% subset_emptyI
thf(fact_993_subset__emptyI,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat] :
      ( ! [X3: product_prod_a_nat] :
          ~ ( member5724188588386418708_a_nat @ X3 @ A5 )
     => ( ord_le8666007276011122963_a_nat @ A5 @ bot_bo9049108969261143879_a_nat ) ) ).

% subset_emptyI
thf(fact_994_insert__iff,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ A5 ) )
      = ( ( A = B )
        | ( member5869715511025134514rm_a_b @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_995_insertCI,axiom,
    ! [A: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( ~ ( member5869715511025134514rm_a_b @ A @ B5 )
       => ( A = B ) )
     => ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ B5 ) ) ) ).

% insertCI
thf(fact_996_singletonI,axiom,
    ! [A: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ).

% singletonI
thf(fact_997_insert__subset,axiom,
    ! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A5 ) @ B5 )
      = ( ( member5869715511025134514rm_a_b @ X @ B5 )
        & ( ord_le118470702582115849rm_a_b @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_998_insert__subset,axiom,
    ! [X: product_prod_a_nat,A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ X @ A5 ) @ B5 )
      = ( ( member5724188588386418708_a_nat @ X @ B5 )
        & ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_999_Int__insert__left__if0,axiom,
    ! [A: produc357393685978478089rm_a_b,C4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ A @ C4 )
     => ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B5 ) @ C4 )
        = ( inf_in1693821330460682071rm_a_b @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_1000_Int__insert__left__if1,axiom,
    ! [A: produc357393685978478089rm_a_b,C4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ C4 )
     => ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B5 ) @ C4 )
        = ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1001_Int__insert__right__if0,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ A @ A5 )
     => ( ( inf_in1693821330460682071rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ A @ B5 ) )
        = ( inf_in1693821330460682071rm_a_b @ A5 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_1002_Int__insert__right__if1,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ A5 )
     => ( ( inf_in1693821330460682071rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ A @ B5 ) )
        = ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ A5 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1003_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_a_nat,A: product_prod_a_nat,A5: set_Pr4934435412358123699_a_nat] :
      ( ( ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat )
        = ( insert8054603423593749677_a_nat @ A @ A5 ) )
      = ( ( A = B )
        & ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1004_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_a_nat,A5: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat] :
      ( ( ( insert8054603423593749677_a_nat @ A @ A5 )
        = ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) )
      = ( ( A = B )
        & ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1005_insert__disjoint_I1_J,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ A5 ) @ B5 )
        = bot_bo197521221353338581rm_a_b )
      = ( ~ ( member5869715511025134514rm_a_b @ A @ B5 )
        & ( ( inf_in1693821330460682071rm_a_b @ A5 @ B5 )
          = bot_bo197521221353338581rm_a_b ) ) ) ).

% insert_disjoint(1)
thf(fact_1006_insert__disjoint_I2_J,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( bot_bo197521221353338581rm_a_b
        = ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ A5 ) @ B5 ) )
      = ( ~ ( member5869715511025134514rm_a_b @ A @ B5 )
        & ( bot_bo197521221353338581rm_a_b
          = ( inf_in1693821330460682071rm_a_b @ A5 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1007_disjoint__insert_I1_J,axiom,
    ! [B5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( ( inf_in1693821330460682071rm_a_b @ B5 @ ( insert7009541432154983385rm_a_b @ A @ A5 ) )
        = bot_bo197521221353338581rm_a_b )
      = ( ~ ( member5869715511025134514rm_a_b @ A @ B5 )
        & ( ( inf_in1693821330460682071rm_a_b @ B5 @ A5 )
          = bot_bo197521221353338581rm_a_b ) ) ) ).

% disjoint_insert(1)
thf(fact_1008_disjoint__insert_I2_J,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( bot_bo197521221353338581rm_a_b
        = ( inf_in1693821330460682071rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ B @ B5 ) ) )
      = ( ~ ( member5869715511025134514rm_a_b @ B @ A5 )
        & ( bot_bo197521221353338581rm_a_b
          = ( inf_in1693821330460682071rm_a_b @ A5 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1009_CR__on__singletonI,axiom,
    ! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [B3: term_a_b,C2: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) )
     => ( abstra8448919418672941150rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).

% CR_on_singletonI
thf(fact_1010_singleton__iff,axiom,
    ! [B: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_1011_singletonD,axiom,
    ! [B: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_1012_insert__subsetI,axiom,
    ! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,X5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X @ A5 )
     => ( ( ord_le118470702582115849rm_a_b @ X5 @ A5 )
       => ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ X5 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_1013_insert__subsetI,axiom,
    ! [X: product_prod_a_nat,A5: set_Pr4934435412358123699_a_nat,X5: set_Pr4934435412358123699_a_nat] :
      ( ( member5724188588386418708_a_nat @ X @ A5 )
     => ( ( ord_le8666007276011122963_a_nat @ X5 @ A5 )
       => ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ X @ X5 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_1014_insert__mono,axiom,
    ! [C4: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ C4 @ D )
     => ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ A @ C4 ) @ ( insert8054603423593749677_a_nat @ A @ D ) ) ) ).

% insert_mono
thf(fact_1015_subset__insert,axiom,
    ! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ X @ A5 )
     => ( ( ord_le118470702582115849rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ X @ B5 ) )
        = ( ord_le118470702582115849rm_a_b @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_1016_subset__insert,axiom,
    ! [X: product_prod_a_nat,A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ~ ( member5724188588386418708_a_nat @ X @ A5 )
     => ( ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ B5 ) )
        = ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_1017_subset__insertI,axiom,
    ! [B5: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ B5 @ ( insert8054603423593749677_a_nat @ A @ B5 ) ) ).

% subset_insertI
thf(fact_1018_subset__insertI2,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_1019_Int__insert__left,axiom,
    ! [A: produc357393685978478089rm_a_b,C4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ( member5869715511025134514rm_a_b @ A @ C4 )
       => ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B5 ) @ C4 )
          = ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ B5 @ C4 ) ) ) )
      & ( ~ ( member5869715511025134514rm_a_b @ A @ C4 )
       => ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B5 ) @ C4 )
          = ( inf_in1693821330460682071rm_a_b @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_1020_Int__insert__right,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ( member5869715511025134514rm_a_b @ A @ A5 )
       => ( ( inf_in1693821330460682071rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ A @ B5 ) )
          = ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ A5 @ B5 ) ) ) )
      & ( ~ ( member5869715511025134514rm_a_b @ A @ A5 )
       => ( ( inf_in1693821330460682071rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ A @ B5 ) )
          = ( inf_in1693821330460682071rm_a_b @ A5 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_1021_mk__disjoint__insert,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ A5 )
     => ? [B9: set_Pr4386577575007340137rm_a_b] :
          ( ( A5
            = ( insert7009541432154983385rm_a_b @ A @ B9 ) )
          & ~ ( member5869715511025134514rm_a_b @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_1022_insert__eq__iff,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ A @ A5 )
     => ( ~ ( member5869715511025134514rm_a_b @ B @ B5 )
       => ( ( ( insert7009541432154983385rm_a_b @ A @ A5 )
            = ( insert7009541432154983385rm_a_b @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A5 = B5 ) )
            & ( ( A != B )
             => ? [C6: set_Pr4386577575007340137rm_a_b] :
                  ( ( A5
                    = ( insert7009541432154983385rm_a_b @ B @ C6 ) )
                  & ~ ( member5869715511025134514rm_a_b @ B @ C6 )
                  & ( B5
                    = ( insert7009541432154983385rm_a_b @ A @ C6 ) )
                  & ~ ( member5869715511025134514rm_a_b @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1023_insert__absorb,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ A5 )
     => ( ( insert7009541432154983385rm_a_b @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_1024_insert__ident,axiom,
    ! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ X @ A5 )
     => ( ~ ( member5869715511025134514rm_a_b @ X @ B5 )
       => ( ( ( insert7009541432154983385rm_a_b @ X @ A5 )
            = ( insert7009541432154983385rm_a_b @ X @ B5 ) )
          = ( A5 = B5 ) ) ) ) ).

% insert_ident
thf(fact_1025_Set_Oset__insert,axiom,
    ! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X @ A5 )
     => ~ ! [B9: set_Pr4386577575007340137rm_a_b] :
            ( ( A5
              = ( insert7009541432154983385rm_a_b @ X @ B9 ) )
           => ( member5869715511025134514rm_a_b @ X @ B9 ) ) ) ).

% Set.set_insert
thf(fact_1026_insertI2,axiom,
    ! [A: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ B5 )
     => ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ B5 ) ) ) ).

% insertI2
thf(fact_1027_insertI1,axiom,
    ! [A: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A @ B5 ) ) ).

% insertI1
thf(fact_1028_insertE,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ A5 ) )
     => ( ( A != B )
       => ( member5869715511025134514rm_a_b @ A @ A5 ) ) ) ).

% insertE
thf(fact_1029_subset__singletonD,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
     => ( ( A5 = bot_bo9049108969261143879_a_nat )
        | ( A5
          = ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) ) ) ).

% subset_singletonD
thf(fact_1030_subset__singleton__iff,axiom,
    ! [X5: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X5 @ ( insert8054603423593749677_a_nat @ A @ bot_bo9049108969261143879_a_nat ) )
      = ( ( X5 = bot_bo9049108969261143879_a_nat )
        | ( X5
          = ( insert8054603423593749677_a_nat @ A @ bot_bo9049108969261143879_a_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_1031_SN__on__induct_H,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,P: term_a_b > $o] :
      ( ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) )
     => ( ! [T3: term_a_b] :
            ( ! [U4: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U4 ) @ R3 )
               => ( P @ U4 ) )
           => ( P @ T3 ) )
       => ( P @ S ) ) ) ).

% SN_on_induct'
thf(fact_1032_step__reflects__SN__on,axiom,
    ! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [B3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R )
         => ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B3 @ bot_bot_set_term_a_b ) ) )
     => ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).

% step_reflects_SN_on
thf(fact_1033_SN__on__imp__on__minimal,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
     => ! [Q2: set_Pr4386577575007340137rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ Q2 )
         => ? [X3: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X3 @ Q2 )
              & ! [Y7: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y7 ) @ R )
                 => ~ ( member5869715511025134514rm_a_b @ Y7 @ Q2 ) ) ) ) ) ).

% SN_on_imp_on_minimal
thf(fact_1034_SN__on__imp__on__minimal,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
     => ! [Q2: set_term_a_b] :
          ( ( member_term_a_b @ X @ Q2 )
         => ? [X3: term_a_b] :
              ( ( member_term_a_b @ X3 @ Q2 )
              & ! [Y7: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y7 ) @ R )
                 => ~ ( member_term_a_b @ Y7 @ Q2 ) ) ) ) ) ).

% SN_on_imp_on_minimal
thf(fact_1035_step__preserves__SN__on,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
       => ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ).

% step_preserves_SN_on
thf(fact_1036_SN__on__induct__acc__style,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,P: term_a_b > $o] :
      ( ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
     => ( ! [X3: term_a_b] :
            ( ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) )
           => ( ! [Y7: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y7 ) @ R3 )
                 => ( P @ Y7 ) )
             => ( P @ X3 ) ) )
       => ( P @ A ) ) ) ).

% SN_on_induct_acc_style
thf(fact_1037_lower__set__imp__not__SN__on,axiom,
    ! [S: produc357393685978478089rm_a_b,X5: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ S @ X5 )
     => ( ! [X3: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X3 @ X5 )
           => ? [Xa2: produc357393685978478089rm_a_b] :
                ( ( member5869715511025134514rm_a_b @ Xa2 @ X5 )
                & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Xa2 ) @ R3 ) ) )
       => ~ ( abstra2398554102055911763rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ S @ bot_bo197521221353338581rm_a_b ) ) ) ) ).

% lower_set_imp_not_SN_on
thf(fact_1038_lower__set__imp__not__SN__on,axiom,
    ! [S: term_a_b,X5: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ S @ X5 )
     => ( ! [X3: term_a_b] :
            ( ( member_term_a_b @ X3 @ X5 )
           => ? [Xa2: term_a_b] :
                ( ( member_term_a_b @ Xa2 @ X5 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Xa2 ) @ R3 ) ) )
       => ~ ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) ) ) ) ).

% lower_set_imp_not_SN_on
thf(fact_1039_SN__on__all__reducts__SN__on__conv,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
      = ( ! [B4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ R )
           => ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B4 @ bot_bot_set_term_a_b ) ) ) ) ) ).

% SN_on_all_reducts_SN_on_conv
thf(fact_1040_ssubst__Pair__rhs,axiom,
    ! [R: term_a_b,S: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S5: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S ) @ R3 )
     => ( ( S5 = S )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1041_NF__imp__SN__on,axiom,
    ! [A: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ R3 ) )
     => ( abstra2398554102055911763rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ) ).

% NF_imp_SN_on
thf(fact_1042_steps__reflect__SN__on,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,B: term_a_b,A: term_a_b] :
      ( ~ ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
       => ~ ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ) ).

% steps_reflect_SN_on
thf(fact_1043_steps__preserve__SN__on,axiom,
    ! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
     => ( ( abstra4720023341729745482rm_a_b @ A5 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
       => ( abstra4720023341729745482rm_a_b @ A5 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ).

% steps_preserve_SN_on
thf(fact_1044_SN__on__imp__normalizability,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
     => ? [B3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ).

% SN_on_imp_normalizability
thf(fact_1045_SN__reaches__NF,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
     => ? [Y3: produc357393685978478089rm_a_b] :
          ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
          & ( member5869715511025134514rm_a_b @ Y3 @ ( abstra1902783451776770197rm_a_b @ R ) ) ) ) ).

% SN_reaches_NF
thf(fact_1046_SN__reaches__NF,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
     => ? [Y3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
          & ( member_term_a_b @ Y3 @ ( abstract_NF_term_a_b @ R ) ) ) ) ).

% SN_reaches_NF
thf(fact_1047_is__singletonI_H,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ( A5 != bot_bo197521221353338581rm_a_b )
     => ( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
           => ( ( member5869715511025134514rm_a_b @ Y3 @ A5 )
             => ( X3 = Y3 ) ) )
       => ( is_sin332545968047684349rm_a_b @ A5 ) ) ) ).

% is_singletonI'
thf(fact_1048_Field__insert,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( field_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R ) )
      = ( sup_sup_set_term_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) ) ) ).

% Field_insert
thf(fact_1049_Diff__iff,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ ( minus_5192120951422937424rm_a_b @ A5 @ B5 ) )
      = ( ( member5869715511025134514rm_a_b @ C3 @ A5 )
        & ~ ( member5869715511025134514rm_a_b @ C3 @ B5 ) ) ) ).

% Diff_iff
thf(fact_1050_DiffI,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ A5 )
     => ( ~ ( member5869715511025134514rm_a_b @ C3 @ B5 )
       => ( member5869715511025134514rm_a_b @ C3 @ ( minus_5192120951422937424rm_a_b @ A5 @ B5 ) ) ) ) ).

% DiffI
thf(fact_1051_insert__Diff1,axiom,
    ! [X: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X @ B5 )
     => ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A5 ) @ B5 )
        = ( minus_5192120951422937424rm_a_b @ A5 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_1052_Diff__insert0,axiom,
    ! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ X @ A5 )
     => ( ( minus_5192120951422937424rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ X @ B5 ) )
        = ( minus_5192120951422937424rm_a_b @ A5 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_1053_Diff__eq__empty__iff,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ( minus_8630901905721465484_a_nat @ A5 @ B5 )
        = bot_bo9049108969261143879_a_nat )
      = ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_1054_insert__Diff__if,axiom,
    ! [X: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( ( member5869715511025134514rm_a_b @ X @ B5 )
       => ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A5 ) @ B5 )
          = ( minus_5192120951422937424rm_a_b @ A5 @ B5 ) ) )
      & ( ~ ( member5869715511025134514rm_a_b @ X @ B5 )
       => ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A5 ) @ B5 )
          = ( insert7009541432154983385rm_a_b @ X @ ( minus_5192120951422937424rm_a_b @ A5 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1055_FieldI2,axiom,
    ! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R3 )
     => ( member5869715511025134514rm_a_b @ J2 @ ( field_6884932134483023318rm_a_b @ R3 ) ) ) ).

% FieldI2
thf(fact_1056_FieldI2,axiom,
    ! [I2: term_a_b,J2: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R3 )
     => ( member_term_a_b @ J2 @ ( field_term_a_b @ R3 ) ) ) ).

% FieldI2
thf(fact_1057_FieldI1,axiom,
    ! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R3 )
     => ( member5869715511025134514rm_a_b @ I2 @ ( field_6884932134483023318rm_a_b @ R3 ) ) ) ).

% FieldI1
thf(fact_1058_FieldI1,axiom,
    ! [I2: term_a_b,J2: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R3 )
     => ( member_term_a_b @ I2 @ ( field_term_a_b @ R3 ) ) ) ).

% FieldI1
thf(fact_1059_double__diff,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ C4 )
       => ( ( minus_8630901905721465484_a_nat @ B5 @ ( minus_8630901905721465484_a_nat @ C4 @ A5 ) )
          = A5 ) ) ) ).

% double_diff
thf(fact_1060_Diff__subset,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A5 @ B5 ) @ A5 ) ).

% Diff_subset
thf(fact_1061_Diff__mono,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ C4 )
     => ( ( ord_le8666007276011122963_a_nat @ D @ B5 )
       => ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A5 @ B5 ) @ ( minus_8630901905721465484_a_nat @ C4 @ D ) ) ) ) ).

% Diff_mono
thf(fact_1062_DiffD2,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ ( minus_5192120951422937424rm_a_b @ A5 @ B5 ) )
     => ~ ( member5869715511025134514rm_a_b @ C3 @ B5 ) ) ).

% DiffD2
thf(fact_1063_DiffD1,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ ( minus_5192120951422937424rm_a_b @ A5 @ B5 ) )
     => ( member5869715511025134514rm_a_b @ C3 @ A5 ) ) ).

% DiffD1
thf(fact_1064_DiffE,axiom,
    ! [C3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C3 @ ( minus_5192120951422937424rm_a_b @ A5 @ B5 ) )
     => ~ ( ( member5869715511025134514rm_a_b @ C3 @ A5 )
         => ( member5869715511025134514rm_a_b @ C3 @ B5 ) ) ) ).

% DiffE
thf(fact_1065_diff__shunt__var,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
      ( ( ( minus_8630901905721465484_a_nat @ X @ Y )
        = bot_bo9049108969261143879_a_nat )
      = ( ord_le8666007276011122963_a_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_1066_Diff__insert__absorb,axiom,
    ! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ X @ A5 )
     => ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A5 ) @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
        = A5 ) ) ).

% Diff_insert_absorb
thf(fact_1067_insert__Diff,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ A5 )
     => ( ( insert7009541432154983385rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) )
        = A5 ) ) ).

% insert_Diff
thf(fact_1068_subset__Diff__insert,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,C4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A5 @ ( minus_5192120951422937424rm_a_b @ B5 @ ( insert7009541432154983385rm_a_b @ X @ C4 ) ) )
      = ( ( ord_le118470702582115849rm_a_b @ A5 @ ( minus_5192120951422937424rm_a_b @ B5 @ C4 ) )
        & ~ ( member5869715511025134514rm_a_b @ X @ A5 ) ) ) ).

% subset_Diff_insert
thf(fact_1069_subset__Diff__insert,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat,C4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ ( minus_8630901905721465484_a_nat @ B5 @ ( insert8054603423593749677_a_nat @ X @ C4 ) ) )
      = ( ( ord_le8666007276011122963_a_nat @ A5 @ ( minus_8630901905721465484_a_nat @ B5 @ C4 ) )
        & ~ ( member5724188588386418708_a_nat @ X @ A5 ) ) ) ).

% subset_Diff_insert
thf(fact_1070_Diff__partition,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( sup_su459911885395995103_a_nat @ A5 @ ( minus_8630901905721465484_a_nat @ B5 @ A5 ) )
        = B5 ) ) ).

% Diff_partition
thf(fact_1071_Diff__subset__conv,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A5 @ B5 ) @ C4 )
      = ( ord_le8666007276011122963_a_nat @ A5 @ ( sup_su459911885395995103_a_nat @ B5 @ C4 ) ) ) ).

% Diff_subset_conv
thf(fact_1072_mono__Field,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
      ( ( ord_le1018141841545957851_a_nat @ R @ S )
     => ( ord_le8666007276011122963_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ ( field_8954927560578634480_a_nat @ S ) ) ) ).

% mono_Field
thf(fact_1073_Diff__single__insert,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) @ B5 )
     => ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_1074_subset__insert__iff,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ X @ B5 ) )
      = ( ( ( member5869715511025134514rm_a_b @ X @ A5 )
         => ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) @ B5 ) )
        & ( ~ ( member5869715511025134514rm_a_b @ X @ A5 )
         => ( ord_le118470702582115849rm_a_b @ A5 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_1075_subset__insert__iff,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ B5 ) )
      = ( ( ( member5724188588386418708_a_nat @ X @ A5 )
         => ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) @ B5 ) )
        & ( ~ ( member5724188588386418708_a_nat @ X @ A5 )
         => ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_1076_rtrancl__diff__decomp,axiom,
    ! [X: term_a_b,Y: term_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( minus_5192120951422937424rm_a_b @ ( transi7742714808557438673rm_a_b @ A5 ) @ ( transi7742714808557438673rm_a_b @ B5 ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A5 ) @ ( relcom370159955682700863rm_a_b @ ( minus_5192120951422937424rm_a_b @ A5 @ B5 ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ) ).

% rtrancl_diff_decomp
thf(fact_1077_cofinal__def,axiom,
    ( bNF_Ca4789820777293777539rm_a_b
    = ( ^ [A6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
        ! [X4: term_a_b] :
          ( ( member_term_a_b @ X4 @ ( field_term_a_b @ R4 ) )
         => ? [Y5: term_a_b] :
              ( ( member_term_a_b @ Y5 @ A6 )
              & ( X4 != Y5 )
              & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R4 ) ) ) ) ) ).

% cofinal_def
thf(fact_1078_acyclic__insert,axiom,
    ! [Y: term_a_b,X: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( transi5314701259734769157rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) )
      = ( ( transi5314701259734769157rm_a_b @ R )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).

% acyclic_insert
thf(fact_1079_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,Phi: term_a_b > term_a_b > $o] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) )
       => ( ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) )
              | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_1080_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat,B: product_prod_a_nat,Phi: product_prod_a_nat > product_prod_a_nat > $o] :
      ( ( bNF_We4298404315170990993_a_nat @ R )
     => ( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ A @ ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) ) @ ( field_8954927560578634480_a_nat @ R ) )
       => ( ( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( minus_1341117088723627554_a_nat @ R @ id_Pro5207055338379199009_a_nat ) )
              | ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ A ) @ ( minus_1341117088723627554_a_nat @ R @ id_Pro5207055338379199009_a_nat ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_1081_member__remove,axiom,
    ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X @ ( remove8523906646656265582rm_a_b @ Y @ A5 ) )
      = ( ( member5869715511025134514rm_a_b @ X @ A5 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_1082_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,J2: produc357393685978478089rm_a_b,I2: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R )
     => ( ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ J2 @ I2 ) @ R )
          | ( J2 = I2 ) )
       => ( ( member5869715511025134514rm_a_b @ I2 @ ( field_6884932134483023318rm_a_b @ R ) )
         => ( ( member5869715511025134514rm_a_b @ J2 @ ( field_6884932134483023318rm_a_b @ R ) )
           => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1083_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,J2: term_a_b,I2: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ J2 @ I2 ) @ R )
          | ( J2 = I2 ) )
       => ( ( member_term_a_b @ I2 @ ( field_term_a_b @ R ) )
         => ( ( member_term_a_b @ J2 @ ( field_term_a_b @ R ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1084_wo__rel_OTOTALS,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ! [X6: term_a_b] :
          ( ( member_term_a_b @ X6 @ ( field_term_a_b @ R ) )
         => ! [Xa2: term_a_b] :
              ( ( member_term_a_b @ Xa2 @ ( field_term_a_b @ R ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Xa2 ) @ R )
                | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa2 @ X6 ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_1085_wo__rel_Owell__order__induct,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ! [X3: term_a_b] :
            ( ! [Y7: term_a_b] :
                ( ( ( Y7 != X3 )
                  & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ X3 ) @ R ) )
               => ( P @ Y7 ) )
           => ( P @ X3 ) )
       => ( P @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_1086_well__order__induct__imp,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o,A: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R )
     => ( ! [X3: produc357393685978478089rm_a_b] :
            ( ! [Y7: produc357393685978478089rm_a_b] :
                ( ( ( Y7 != X3 )
                  & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y7 @ X3 ) @ R ) )
               => ( ( member5869715511025134514rm_a_b @ Y7 @ ( field_6884932134483023318rm_a_b @ R ) )
                 => ( P @ Y7 ) ) )
           => ( ( member5869715511025134514rm_a_b @ X3 @ ( field_6884932134483023318rm_a_b @ R ) )
             => ( P @ X3 ) ) )
       => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1087_well__order__induct__imp,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ! [X3: term_a_b] :
            ( ! [Y7: term_a_b] :
                ( ( ( Y7 != X3 )
                  & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ X3 ) @ R ) )
               => ( ( member_term_a_b @ Y7 @ ( field_term_a_b @ R ) )
                 => ( P @ Y7 ) ) )
           => ( ( member_term_a_b @ X3 @ ( field_term_a_b @ R ) )
             => ( P @ X3 ) ) )
       => ( ( member_term_a_b @ A @ ( field_term_a_b @ R ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1088_acyclicI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
     => ( transi5314701259734769157rm_a_b @ R ) ) ).

% acyclicI
thf(fact_1089_acyclic__def,axiom,
    ( transi5314701259734769157rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
        ! [X4: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ X4 ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ).

% acyclic_def
thf(fact_1090_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,Phi: term_a_b > term_a_b > $o] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) )
       => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
           => ( Phi @ A @ B ) )
         => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_1091_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat,B: product_prod_a_nat,Phi: product_prod_a_nat > product_prod_a_nat > $o] :
      ( ( bNF_We4298404315170990993_a_nat @ R )
     => ( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ A @ ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) ) @ ( field_8954927560578634480_a_nat @ R ) )
       => ( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R )
           => ( Phi @ A @ B ) )
         => ( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ A ) @ R )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_1092_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ ( field_8954927560578634480_a_nat @ R ) )
       => ( ( B5 != bot_bo9049108969261143879_a_nat )
         => ? [X_1: product_prod_a_nat] : ( bNF_We4498669257762480967_a_nat @ R @ B5 @ X_1 ) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
thf(fact_1093_wo__rel_Ominim__inField,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R )
     => ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R ) )
       => ( ( B5 != bot_bo197521221353338581rm_a_b )
         => ( member5869715511025134514rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R @ B5 ) @ ( field_6884932134483023318rm_a_b @ R ) ) ) ) ) ).

% wo_rel.minim_inField
thf(fact_1094_wo__rel_Ominim__inField,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ ( field_8954927560578634480_a_nat @ R ) )
       => ( ( B5 != bot_bo9049108969261143879_a_nat )
         => ( member5724188588386418708_a_nat @ ( bNF_We3895015349946260913_a_nat @ R @ B5 ) @ ( field_8954927560578634480_a_nat @ R ) ) ) ) ) ).

% wo_rel.minim_inField
thf(fact_1095_wo__rel_Ominim__in,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R )
     => ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R ) )
       => ( ( B5 != bot_bo197521221353338581rm_a_b )
         => ( member5869715511025134514rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R @ B5 ) @ B5 ) ) ) ) ).

% wo_rel.minim_in
thf(fact_1096_wo__rel_Ominim__in,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ ( field_8954927560578634480_a_nat @ R ) )
       => ( ( B5 != bot_bo9049108969261143879_a_nat )
         => ( member5724188588386418708_a_nat @ ( bNF_We3895015349946260913_a_nat @ R @ B5 ) @ B5 ) ) ) ) ).

% wo_rel.minim_in
thf(fact_1097_wo__rel_Ominim__isMinim,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ ( field_8954927560578634480_a_nat @ R ) )
       => ( ( B5 != bot_bo9049108969261143879_a_nat )
         => ( bNF_We4498669257762480967_a_nat @ R @ B5 @ ( bNF_We3895015349946260913_a_nat @ R @ B5 ) ) ) ) ) ).

% wo_rel.minim_isMinim
thf(fact_1098_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R )
     => ( ( bNF_We8430659725849320639rm_a_b @ R @ A5 @ B )
        = ( ( member5869715511025134514rm_a_b @ B @ A5 )
          & ! [X4: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X4 @ A5 )
             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ X4 ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_1099_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ( bNF_We695392074589568990rm_a_b @ R @ A5 @ B )
        = ( ( member_term_a_b @ B @ A5 )
          & ! [X4: term_a_b] :
              ( ( member_term_a_b @ X4 @ A5 )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ X4 ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_1100_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R )
     => ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ B @ B5 )
         => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_1101_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,B5: set_term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ( ord_le2705286416250468010rm_a_b @ B5 @ ( field_term_a_b @ R ) )
       => ( ( member_term_a_b @ B @ B5 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We6258903063523145544rm_a_b @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_1102_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,B5: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ ( field_8954927560578634480_a_nat @ R ) )
       => ( ( member5724188588386418708_a_nat @ B @ B5 )
         => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( bNF_We3895015349946260913_a_nat @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_1103_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R )
     => ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ A @ B5 )
         => ( ! [B3: produc357393685978478089rm_a_b] :
                ( ( member5869715511025134514rm_a_b @ B3 @ B5 )
               => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ R ) )
           => ( A
              = ( bNF_We1338740440619677141rm_a_b @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_1104_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,B5: set_term_a_b,A: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ( ord_le2705286416250468010rm_a_b @ B5 @ ( field_term_a_b @ R ) )
       => ( ( member_term_a_b @ A @ B5 )
         => ( ! [B3: term_a_b] :
                ( ( member_term_a_b @ B3 @ B5 )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R ) )
           => ( A
              = ( bNF_We6258903063523145544rm_a_b @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_1105_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,B5: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ ( field_8954927560578634480_a_nat @ R ) )
       => ( ( member5724188588386418708_a_nat @ A @ B5 )
         => ( ! [B3: product_prod_a_nat] :
                ( ( member5724188588386418708_a_nat @ B3 @ B5 )
               => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B3 ) @ R ) )
           => ( A
              = ( bNF_We3895015349946260913_a_nat @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_1106_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ ( bNF_We1369222903935127127rm_a_b @ R @ A @ B ) ) @ R )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ ( bNF_We1369222903935127127rm_a_b @ R @ A @ B ) ) @ R )
            & ( member5869715511025134514rm_a_b @ ( bNF_We1369222903935127127rm_a_b @ R @ A @ B ) @ ( insert7009541432154983385rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_1107_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( bNF_We4618759378340593734rm_a_b @ R @ A @ B ) ) @ R )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ ( bNF_We4618759378340593734rm_a_b @ R @ A @ B ) ) @ R )
            & ( member_term_a_b @ ( bNF_We4618759378340593734rm_a_b @ R @ A @ B ) @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_1108_refl__on__singleton,axiom,
    ! [X: term_a_b] : ( refl_on_term_a_b @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ bot_bo197521221353338581rm_a_b ) ) ).

% refl_on_singleton
thf(fact_1109_refl__onD,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A5 @ R )
     => ( ( member5869715511025134514rm_a_b @ A @ A5 )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_1110_refl__onD,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( refl_on_term_a_b @ A5 @ R )
     => ( ( member_term_a_b @ A @ A5 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_1111_refl__onD1,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A5 @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
       => ( member5869715511025134514rm_a_b @ X @ A5 ) ) ) ).

% refl_onD1
thf(fact_1112_refl__onD1,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( refl_on_term_a_b @ A5 @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
       => ( member_term_a_b @ X @ A5 ) ) ) ).

% refl_onD1
thf(fact_1113_refl__onD2,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A5 @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
       => ( member5869715511025134514rm_a_b @ Y @ A5 ) ) ) ).

% refl_onD2
thf(fact_1114_refl__onD2,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
      ( ( refl_on_term_a_b @ A5 @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
       => ( member_term_a_b @ Y @ A5 ) ) ) ).

% refl_onD2
thf(fact_1115_reflD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( refl_on_term_a_b @ top_top_set_term_a_b @ R )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R ) ) ).

% reflD
thf(fact_1116_reflI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R )
     => ( refl_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).

% reflI
thf(fact_1117_wo__rel_Omax2__def,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
         => ( ( bNF_We4618759378340593734rm_a_b @ R @ A @ B )
            = B ) )
        & ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
         => ( ( bNF_We4618759378340593734rm_a_b @ R @ A @ B )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_1118_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ ( bNF_We1369222903935127127rm_a_b @ R @ A @ B ) ) @ R )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ ( bNF_We1369222903935127127rm_a_b @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1119_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( bNF_We4618759378340593734rm_a_b @ R @ A @ B ) ) @ R )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ ( bNF_We4618759378340593734rm_a_b @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1120_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
         => ( ( ( bNF_We1369222903935127127rm_a_b @ R @ A @ B )
              = B )
            = ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1121_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R ) )
         => ( ( ( bNF_We4618759378340593734rm_a_b @ R @ A @ B )
              = B )
            = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1122_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
         => ( ( ( bNF_We1369222903935127127rm_a_b @ R @ A @ B )
              = A )
            = ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_1123_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R ) )
         => ( ( ( bNF_We4618759378340593734rm_a_b @ R @ A @ B )
              = A )
            = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_1124_refl__on__domain,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A5 @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
       => ( ( member5869715511025134514rm_a_b @ A @ A5 )
          & ( member5869715511025134514rm_a_b @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_1125_refl__on__domain,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( refl_on_term_a_b @ A5 @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
       => ( ( member_term_a_b @ A @ A5 )
          & ( member_term_a_b @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_1126_Total__subset__Id,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( total_on_term_a_b @ ( field_term_a_b @ R ) @ R )
     => ( ( ord_le118470702582115849rm_a_b @ R @ id_term_a_b )
       => ( ( R = bot_bo197521221353338581rm_a_b )
          | ? [A4: term_a_b] :
              ( R
              = ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).

% Total_subset_Id
thf(fact_1127_linear__order__on__singleton,axiom,
    ! [X: term_a_b] : ( order_5388802246213473311rm_a_b @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ bot_bo197521221353338581rm_a_b ) ) ).

% linear_order_on_singleton
thf(fact_1128_total__on__trancl,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( total_on_term_a_b @ A5 @ R )
     => ( total_on_term_a_b @ A5 @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).

% total_on_trancl
thf(fact_1129_total__on__subset,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( total_9110697612923228665_a_nat @ A5 @ R )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ A5 )
       => ( total_9110697612923228665_a_nat @ B5 @ R ) ) ) ).

% total_on_subset
thf(fact_1130_total__onI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
         => ( ( member5869715511025134514rm_a_b @ Y3 @ A5 )
           => ( ( X3 != Y3 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
                | ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ X3 ) @ R ) ) ) ) )
     => ( total_4010809703584253837rm_a_b @ A5 @ R ) ) ).

% total_onI
thf(fact_1131_total__onI,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y3: term_a_b] :
          ( ( member_term_a_b @ X3 @ A5 )
         => ( ( member_term_a_b @ Y3 @ A5 )
           => ( ( X3 != Y3 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
                | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R ) ) ) ) )
     => ( total_on_term_a_b @ A5 @ R ) ) ).

% total_onI
thf(fact_1132_total__on__def,axiom,
    ( total_on_term_a_b
    = ( ^ [A6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
        ! [X4: term_a_b] :
          ( ( member_term_a_b @ X4 @ A6 )
         => ! [Y5: term_a_b] :
              ( ( member_term_a_b @ Y5 @ A6 )
             => ( ( X4 != Y5 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R4 )
                  | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_1133_totalI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Y3: term_a_b] :
          ( ( X3 != Y3 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
            | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R ) ) )
     => ( total_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).

% totalI
thf(fact_1134_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
            = ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ ( minus_45395518312058030rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_1135_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R ) @ R )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
            = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_1136_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R ) @ R )
     => ( ( wf_term_a_b @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) )
        = ( ! [A6: set_term_a_b] :
              ( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R ) )
             => ( ( A6 != bot_bot_set_term_a_b )
               => ? [X4: term_a_b] :
                    ( ( member_term_a_b @ X4 @ A6 )
                    & ! [Y5: term_a_b] :
                        ( ( member_term_a_b @ Y5 @ A6 )
                       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_1137_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr1811044260758604347_a_nat] :
      ( ( order_6388386362885710600_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
     => ( ( wf_Pro8109884346892000340_a_nat @ ( minus_1341117088723627554_a_nat @ R @ id_Pro5207055338379199009_a_nat ) )
        = ( ! [A6: set_Pr4934435412358123699_a_nat] :
              ( ( ord_le8666007276011122963_a_nat @ A6 @ ( field_8954927560578634480_a_nat @ R ) )
             => ( ( A6 != bot_bo9049108969261143879_a_nat )
               => ? [X4: product_prod_a_nat] :
                    ( ( member5724188588386418708_a_nat @ X4 @ A6 )
                    & ! [Y5: product_prod_a_nat] :
                        ( ( member5724188588386418708_a_nat @ Y5 @ A6 )
                       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_1138_in__lex__prod,axiom,
    ! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) ) @ ( lex_pr6163557265797435481rm_a_b @ R @ S ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1139_wf__insert,axiom,
    ! [Y: term_a_b,X: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( wf_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) )
      = ( ( wf_term_a_b @ R )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).

% wf_insert
thf(fact_1140_wf__converse__trancl,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ R ) )
     => ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).

% wf_converse_trancl
thf(fact_1141_wf__trancl,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( wf_term_a_b @ R )
     => ( wf_term_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).

% wf_trancl
thf(fact_1142_wf__def,axiom,
    ( wf_term_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
        ! [P3: term_a_b > $o] :
          ( ! [X4: term_a_b] :
              ( ! [Y5: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 )
                 => ( P3 @ Y5 ) )
             => ( P3 @ X4 ) )
         => ! [X7: term_a_b] : ( P3 @ X7 ) ) ) ) ).

% wf_def
thf(fact_1143_wfE__min,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
      ( ( wf_Pro2335863617654816626rm_a_b @ R3 )
     => ( ( member5869715511025134514rm_a_b @ X @ Q )
       => ~ ! [Z: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ Z @ Q )
             => ~ ! [Y7: produc357393685978478089rm_a_b] :
                    ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y7 @ Z ) @ R3 )
                   => ~ ( member5869715511025134514rm_a_b @ Y7 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_1144_wfE__min,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Q: set_term_a_b] :
      ( ( wf_term_a_b @ R3 )
     => ( ( member_term_a_b @ X @ Q )
       => ~ ! [Z: term_a_b] :
              ( ( member_term_a_b @ Z @ Q )
             => ~ ! [Y7: term_a_b] :
                    ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ Z ) @ R3 )
                   => ~ ( member_term_a_b @ Y7 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_1145_wfI__min,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b] :
      ( ! [X3: produc357393685978478089rm_a_b,Q3: set_Pr4386577575007340137rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X3 @ Q3 )
         => ? [Xa2: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ Xa2 @ Q3 )
              & ! [Y3: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Xa2 ) @ R3 )
                 => ~ ( member5869715511025134514rm_a_b @ Y3 @ Q3 ) ) ) )
     => ( wf_Pro2335863617654816626rm_a_b @ R3 ) ) ).

% wfI_min
thf(fact_1146_wfI__min,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [X3: term_a_b,Q3: set_term_a_b] :
          ( ( member_term_a_b @ X3 @ Q3 )
         => ? [Xa2: term_a_b] :
              ( ( member_term_a_b @ Xa2 @ Q3 )
              & ! [Y3: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Xa2 ) @ R3 )
                 => ~ ( member_term_a_b @ Y3 @ Q3 ) ) ) )
     => ( wf_term_a_b @ R3 ) ) ).

% wfI_min
thf(fact_1147_wfUNIVI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [P4: term_a_b > $o,X3: term_a_b] :
          ( ! [Xa2: term_a_b] :
              ( ! [Y3: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Xa2 ) @ R )
                 => ( P4 @ Y3 ) )
             => ( P4 @ Xa2 ) )
         => ( P4 @ X3 ) )
     => ( wf_term_a_b @ R ) ) ).

% wfUNIVI
thf(fact_1148_wf__asym,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,X: term_a_b] :
      ( ( wf_term_a_b @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X ) @ R )
       => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A ) @ R ) ) ) ).

% wf_asym
thf(fact_1149_wf__induct,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
      ( ( wf_term_a_b @ R )
     => ( ! [X3: term_a_b] :
            ( ! [Y7: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ X3 ) @ R )
               => ( P @ Y7 ) )
           => ( P @ X3 ) )
       => ( P @ A ) ) ) ).

% wf_induct
thf(fact_1150_wf__irrefl,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( wf_term_a_b @ R )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R ) ) ).

% wf_irrefl
thf(fact_1151_wf__not__sym,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,X: term_a_b] :
      ( ( wf_term_a_b @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X ) @ R )
       => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A ) @ R ) ) ) ).

% wf_not_sym
thf(fact_1152_wf__not__refl,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( wf_term_a_b @ R )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R ) ) ).

% wf_not_refl
thf(fact_1153_wf__eq__minimal,axiom,
    ( wf_Pro2335863617654816626rm_a_b
    = ( ^ [R4: set_Pr2972776593051762503rm_a_b] :
        ! [Q4: set_Pr4386577575007340137rm_a_b] :
          ( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ Q4 )
         => ? [X4: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X4 @ Q4 )
              & ! [Y5: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ X4 ) @ R4 )
                 => ~ ( member5869715511025134514rm_a_b @ Y5 @ Q4 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1154_wf__eq__minimal,axiom,
    ( wf_term_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
        ! [Q4: set_term_a_b] :
          ( ? [X4: term_a_b] : ( member_term_a_b @ X4 @ Q4 )
         => ? [X4: term_a_b] :
              ( ( member_term_a_b @ X4 @ Q4 )
              & ! [Y5: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 )
                 => ~ ( member_term_a_b @ Y5 @ Q4 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1155_wf__induct__rule,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
      ( ( wf_term_a_b @ R )
     => ( ! [X3: term_a_b] :
            ( ! [Y7: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ X3 ) @ R )
               => ( P @ Y7 ) )
           => ( P @ X3 ) )
       => ( P @ A ) ) ) ).

% wf_induct_rule
thf(fact_1156_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b] :
      ( ( wf_term_a_b @ S )
     => ( ! [X3: term_a_b,Y3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
       => ( wf_term_a_b @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_1157_wfE__min_H,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
      ( ( wf_Pro2335863617654816626rm_a_b @ R3 )
     => ( ( Q != bot_bo197521221353338581rm_a_b )
       => ~ ! [Z: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ Z @ Q )
             => ~ ! [Y7: produc357393685978478089rm_a_b] :
                    ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y7 @ Z ) @ R3 )
                   => ~ ( member5869715511025134514rm_a_b @ Y7 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_1158_wfE__min_H,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,Q: set_term_a_b] :
      ( ( wf_term_a_b @ R3 )
     => ( ( Q != bot_bot_set_term_a_b )
       => ~ ! [Z: term_a_b] :
              ( ( member_term_a_b @ Z @ Q )
             => ~ ! [Y7: term_a_b] :
                    ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ Z ) @ R3 )
                   => ~ ( member_term_a_b @ Y7 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_1159_minimal__imp__wf,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b] :
      ( ! [Q3: set_Pr4386577575007340137rm_a_b] :
          ( ? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ Q3 )
         => ? [X6: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X6 @ Q3 )
              & ! [Y3: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X6 @ Y3 ) @ R )
                 => ~ ( member5869715511025134514rm_a_b @ Y3 @ Q3 ) ) ) )
     => ( wf_Pro2335863617654816626rm_a_b @ ( conver1575690562132235771rm_a_b @ R ) ) ) ).

% minimal_imp_wf
thf(fact_1160_minimal__imp__wf,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [Q3: set_term_a_b] :
          ( ? [X3: term_a_b] : ( member_term_a_b @ X3 @ Q3 )
         => ? [X6: term_a_b] :
              ( ( member_term_a_b @ X6 @ Q3 )
              & ! [Y3: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Y3 ) @ R )
                 => ~ ( member_term_a_b @ Y3 @ Q3 ) ) ) )
     => ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).

% minimal_imp_wf
thf(fact_1161_wf__eq__minimal2,axiom,
    ( wf_term_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
        ! [A6: set_term_a_b] :
          ( ( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R4 ) )
            & ( A6 != bot_bot_set_term_a_b ) )
         => ? [X4: term_a_b] :
              ( ( member_term_a_b @ X4 @ A6 )
              & ! [Y5: term_a_b] :
                  ( ( member_term_a_b @ Y5 @ A6 )
                 => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_1162_wf__eq__minimal2,axiom,
    ( wf_Pro8109884346892000340_a_nat
    = ( ^ [R4: set_Pr1811044260758604347_a_nat] :
        ! [A6: set_Pr4934435412358123699_a_nat] :
          ( ( ( ord_le8666007276011122963_a_nat @ A6 @ ( field_8954927560578634480_a_nat @ R4 ) )
            & ( A6 != bot_bo9049108969261143879_a_nat ) )
         => ? [X4: product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ X4 @ A6 )
              & ! [Y5: product_prod_a_nat] :
                  ( ( member5724188588386418708_a_nat @ Y5 @ A6 )
                 => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y5 @ X4 ) @ R4 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_1163_non__strict__ending,axiom,
    ! [T: nat > term_a_b,R3: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ S4 ) )
     => ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S4 ) @ S4 )
       => ( ( abstra4720023341729745482rm_a_b @ S4 @ ( insert_term_a_b @ ( T @ zero_zero_nat ) @ bot_bot_set_term_a_b ) )
         => ? [J3: nat] :
            ! [I4: nat] :
              ( ( ord_less_eq_nat @ J3 @ I4 )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I4 ) @ ( T @ ( suc @ I4 ) ) ) @ ( minus_5192120951422937424rm_a_b @ R3 @ S4 ) ) ) ) ) ) ).

% non_strict_ending
thf(fact_1164_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R ) @ R )
     => ( ( order_5719896216891381494rm_a_b @ ( field_term_a_b @ R ) @ R )
        = ( ! [A6: set_term_a_b] :
              ( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R ) )
             => ( ( A6 != bot_bot_set_term_a_b )
               => ? [X4: term_a_b] :
                    ( ( member_term_a_b @ X4 @ A6 )
                    & ! [Y5: term_a_b] :
                        ( ( member_term_a_b @ Y5 @ A6 )
                       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1165_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr1811044260758604347_a_nat] :
      ( ( order_6388386362885710600_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
     => ( ( order_7753855512070164063_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
        = ( ! [A6: set_Pr4934435412358123699_a_nat] :
              ( ( ord_le8666007276011122963_a_nat @ A6 @ ( field_8954927560578634480_a_nat @ R ) )
             => ( ( A6 != bot_bo9049108969261143879_a_nat )
               => ? [X4: product_prod_a_nat] :
                    ( ( member5724188588386418708_a_nat @ X4 @ A6 )
                    & ! [Y5: product_prod_a_nat] :
                        ( ( member5724188588386418708_a_nat @ Y5 @ A6 )
                       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1166_SN__onI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ! [F4: nat > produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( F4 @ zero_zero_nat ) @ A5 )
         => ~ ! [I4: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) )
     => ( abstra2398554102055911763rm_a_b @ R @ A5 ) ) ).

% SN_onI
thf(fact_1167_SN__onI,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [F4: nat > term_a_b] :
          ( ( member_term_a_b @ ( F4 @ zero_zero_nat ) @ A5 )
         => ~ ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) )
     => ( abstra4720023341729745482rm_a_b @ R @ A5 ) ) ).

% SN_onI
thf(fact_1168_not__SN__onI,axiom,
    ! [F2: nat > produc357393685978478089rm_a_b,X5: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( F2 @ zero_zero_nat ) @ X5 )
     => ( ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) ) @ R3 )
       => ~ ( abstra2398554102055911763rm_a_b @ R3 @ X5 ) ) ) ).

% not_SN_onI
thf(fact_1169_not__SN__onI,axiom,
    ! [F2: nat > term_a_b,X5: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ ( F2 @ zero_zero_nat ) @ X5 )
     => ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) ) @ R3 )
       => ~ ( abstra4720023341729745482rm_a_b @ R3 @ X5 ) ) ) ).

% not_SN_onI
thf(fact_1170_relpow__0__I,axiom,
    ! [X: term_a_b,R3: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R3 ) ) ).

% relpow_0_I
thf(fact_1171_relpow__0__E,axiom,
    ! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R3 ) )
     => ( X = Y ) ) ).

% relpow_0_E
thf(fact_1172_Abstract__Rewriting_Ochain__mono,axiom,
    ! [R6: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b,Seq: nat > term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R6 @ R3 )
     => ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R6 )
       => ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I4 ) @ ( Seq @ ( suc @ I4 ) ) ) @ R3 ) ) ) ).

% Abstract_Rewriting.chain_mono
thf(fact_1173_SN__onE,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ R @ A5 )
     => ~ ? [F5: nat > produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( F5 @ zero_zero_nat ) @ A5 )
            & ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R ) ) ) ).

% SN_onE
thf(fact_1174_SN__onE,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R @ A5 )
     => ~ ? [F5: nat > term_a_b] :
            ( ( member_term_a_b @ ( F5 @ zero_zero_nat ) @ A5 )
            & ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R ) ) ) ).

% SN_onE
thf(fact_1175_relpow__E,axiom,
    ! [X: term_a_b,Z2: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
     => ( ( ( N = zero_zero_nat )
         => ( X != Z2 ) )
       => ~ ! [Y3: term_a_b,M2: nat] :
              ( ( N
                = ( suc @ M2 ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( compow4057154403645558940rm_a_b @ M2 @ R3 ) )
               => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ R3 ) ) ) ) ) ).

% relpow_E
thf(fact_1176_SN__on__def,axiom,
    ( abstra2398554102055911763rm_a_b
    = ( ^ [R4: set_Pr2972776593051762503rm_a_b,A6: set_Pr4386577575007340137rm_a_b] :
          ~ ? [F6: nat > produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( F6 @ zero_zero_nat ) @ A6 )
              & ! [I: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F6 @ I ) @ ( F6 @ ( suc @ I ) ) ) @ R4 ) ) ) ) ).

% SN_on_def
thf(fact_1177_SN__on__def,axiom,
    ( abstra4720023341729745482rm_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
          ~ ? [F6: nat > term_a_b] :
              ( ( member_term_a_b @ ( F6 @ zero_zero_nat ) @ A6 )
              & ! [I: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F6 @ I ) @ ( F6 @ ( suc @ I ) ) ) @ R4 ) ) ) ) ).

% SN_on_def
thf(fact_1178_relpow__E2,axiom,
    ! [X: term_a_b,Z2: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
     => ( ( ( N = zero_zero_nat )
         => ( X != Z2 ) )
       => ~ ! [Y3: term_a_b,M2: nat] :
              ( ( N
                = ( suc @ M2 ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R3 )
               => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ M2 @ R3 ) ) ) ) ) ) ).

% relpow_E2
thf(fact_1179_not__SN__onE,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ~ ( abstra2398554102055911763rm_a_b @ R @ A5 )
     => ~ ! [F4: nat > produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( F4 @ zero_zero_nat ) @ A5 )
           => ~ ! [I4: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) ) ) ).

% not_SN_onE
thf(fact_1180_not__SN__onE,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b] :
      ( ~ ( abstra4720023341729745482rm_a_b @ R @ A5 )
     => ~ ! [F4: nat > term_a_b] :
            ( ( member_term_a_b @ ( F4 @ zero_zero_nat ) @ A5 )
           => ~ ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) ) ) ).

% not_SN_onE
thf(fact_1181_relpow__Suc__E,axiom,
    ! [X: term_a_b,Z2: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 ) )
     => ~ ! [Y3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ R3 ) ) ) ).

% relpow_Suc_E
thf(fact_1182_relpow__Suc__I,axiom,
    ! [X: term_a_b,Y: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R3 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 ) ) ) ) ).

% relpow_Suc_I
thf(fact_1183_relpow__Suc__D2,axiom,
    ! [X: term_a_b,Z2: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 ) )
     => ? [Y3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R3 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) ) ) ) ).

% relpow_Suc_D2
thf(fact_1184_relpow__Suc__E2,axiom,
    ! [X: term_a_b,Z2: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 ) )
     => ~ ! [Y3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R3 )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) ) ) ) ).

% relpow_Suc_E2
thf(fact_1185_relpow__Suc__I2,axiom,
    ! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Z2: term_a_b,N: nat] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 ) ) ) ) ).

% relpow_Suc_I2
thf(fact_1186_well__order__on__domain,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( order_8637870607158948775rm_a_b @ A5 @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
       => ( ( member5869715511025134514rm_a_b @ A @ A5 )
          & ( member5869715511025134514rm_a_b @ B @ A5 ) ) ) ) ).

% well_order_on_domain
thf(fact_1187_well__order__on__domain,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( order_5719896216891381494rm_a_b @ A5 @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
       => ( ( member_term_a_b @ A @ A5 )
          & ( member_term_a_b @ B @ A5 ) ) ) ) ).

% well_order_on_domain
thf(fact_1188_relpow__Suc__E2_H,axiom,
    ! [X: term_a_b,Z2: term_a_b,N: nat,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ A5 ) )
     => ~ ! [Y3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ A5 )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).

% relpow_Suc_E2'
thf(fact_1189_pow__Suc__subset__trancl,axiom,
    ! [N: nat,R3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ).

% pow_Suc_subset_trancl
thf(fact_1190_chain__imp__not__SN__on,axiom,
    ! [F2: nat > term_a_b,R: set_Pr4386577575007340137rm_a_b,I2: nat] :
      ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) ) @ R )
     => ~ ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ ( F2 @ I2 ) @ bot_bot_set_term_a_b ) ) ) ).

% chain_imp_not_SN_on
thf(fact_1191_chain__imp__rtrancl,axiom,
    ! [S4: nat > term_a_b,R: set_Pr4386577575007340137rm_a_b,I2: nat,J2: nat] :
      ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I3 ) @ ( S4 @ ( suc @ I3 ) ) ) @ R )
     => ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I2 ) @ ( S4 @ J2 ) ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).

% chain_imp_rtrancl
thf(fact_1192_wf__iff__no__infinite__down__chain,axiom,
    ( wf_term_a_b
    = ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
          ~ ? [F6: nat > term_a_b] :
            ! [I: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F6 @ ( suc @ I ) ) @ ( F6 @ I ) ) @ R4 ) ) ) ).

% wf_iff_no_infinite_down_chain
thf(fact_1193_wf__no__infinite__down__chainE,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F2: nat > term_a_b] :
      ( ( wf_term_a_b @ R )
     => ~ ! [K: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ K ) ) @ ( F2 @ K ) ) @ R ) ) ).

% wf_no_infinite_down_chainE
thf(fact_1194_lift__Suc__mono__le,axiom,
    ! [F2: nat > set_Pr4934435412358123699_a_nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_le8666007276011122963_a_nat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_le8666007276011122963_a_nat @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1195_lift__Suc__antimono__le,axiom,
    ! [F2: nat > set_Pr4934435412358123699_a_nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_le8666007276011122963_a_nat @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_le8666007276011122963_a_nat @ ( F2 @ N4 ) @ ( F2 @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1196_chain__Un__SN__on__imp__first__step,axiom,
    ! [T: nat > term_a_b,R3: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ S4 ) )
     => ( ( abstra4720023341729745482rm_a_b @ S4 @ ( insert_term_a_b @ ( T @ zero_zero_nat ) @ bot_bot_set_term_a_b ) )
       => ? [I3: nat] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ R3 )
            & ! [J4: nat] :
                ( ( ord_less_nat @ J4 @ I3 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ J4 ) @ ( T @ ( suc @ J4 ) ) ) @ S4 )
                  & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ J4 ) @ ( T @ ( suc @ J4 ) ) ) @ R3 ) ) ) ) ) ) ).

% chain_Un_SN_on_imp_first_step
thf(fact_1197_leD,axiom,
    ! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ Y @ X )
     => ~ ( ord_le2482536469199093511_a_nat @ X @ Y ) ) ).

% leD
thf(fact_1198_nless__le,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ~ ( ord_le2482536469199093511_a_nat @ A @ B ) )
      = ( ~ ( ord_le8666007276011122963_a_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1199_antisym__conv1,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
      ( ~ ( ord_le2482536469199093511_a_nat @ X @ Y )
     => ( ( ord_le8666007276011122963_a_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_1200_antisym__conv2,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ Y )
     => ( ( ~ ( ord_le2482536469199093511_a_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_1201_less__le__not__le,axiom,
    ( ord_le2482536469199093511_a_nat
    = ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ X4 @ Y5 )
          & ~ ( ord_le8666007276011122963_a_nat @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_1202_order_Oorder__iff__strict,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le2482536469199093511_a_nat @ A3 @ B4 )
          | ( A3 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1203_order_Ostrict__iff__order,axiom,
    ( ord_le2482536469199093511_a_nat
    = ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ A3 @ B4 )
          & ( A3 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1204_order_Ostrict__trans1,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ B )
     => ( ( ord_le2482536469199093511_a_nat @ B @ C3 )
       => ( ord_le2482536469199093511_a_nat @ A @ C3 ) ) ) ).

% order.strict_trans1
thf(fact_1205_order_Ostrict__trans2,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le2482536469199093511_a_nat @ A @ B )
     => ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
       => ( ord_le2482536469199093511_a_nat @ A @ C3 ) ) ) ).

% order.strict_trans2
thf(fact_1206_order_Ostrict__iff__not,axiom,
    ( ord_le2482536469199093511_a_nat
    = ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ A3 @ B4 )
          & ~ ( ord_le8666007276011122963_a_nat @ B4 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1207_dual__order_Oorder__iff__strict,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [B4: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le2482536469199093511_a_nat @ B4 @ A3 )
          | ( A3 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1208_dual__order_Ostrict__iff__order,axiom,
    ( ord_le2482536469199093511_a_nat
    = ( ^ [B4: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ B4 @ A3 )
          & ( A3 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1209_dual__order_Ostrict__trans1,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ B @ A )
     => ( ( ord_le2482536469199093511_a_nat @ C3 @ B )
       => ( ord_le2482536469199093511_a_nat @ C3 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1210_dual__order_Ostrict__trans2,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le2482536469199093511_a_nat @ B @ A )
     => ( ( ord_le8666007276011122963_a_nat @ C3 @ B )
       => ( ord_le2482536469199093511_a_nat @ C3 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1211_dual__order_Ostrict__iff__not,axiom,
    ( ord_le2482536469199093511_a_nat
    = ( ^ [B4: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ B4 @ A3 )
          & ~ ( ord_le8666007276011122963_a_nat @ A3 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1212_order_Ostrict__implies__order,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le2482536469199093511_a_nat @ A @ B )
     => ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1213_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le2482536469199093511_a_nat @ B @ A )
     => ( ord_le8666007276011122963_a_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1214_order__le__less,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le2482536469199093511_a_nat @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1215_order__less__le,axiom,
    ( ord_le2482536469199093511_a_nat
    = ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1216_order__less__imp__le,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le2482536469199093511_a_nat @ X @ Y )
     => ( ord_le8666007276011122963_a_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1217_order__le__neq__trans,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ B )
     => ( ( A != B )
       => ( ord_le2482536469199093511_a_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1218_order__neq__le__trans,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
      ( ( A != B )
     => ( ( ord_le8666007276011122963_a_nat @ A @ B )
       => ( ord_le2482536469199093511_a_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1219_order__le__less__trans,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ Y )
     => ( ( ord_le2482536469199093511_a_nat @ Y @ Z2 )
       => ( ord_le2482536469199093511_a_nat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1220_order__less__le__trans,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le2482536469199093511_a_nat @ X @ Y )
     => ( ( ord_le8666007276011122963_a_nat @ Y @ Z2 )
       => ( ord_le2482536469199093511_a_nat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1221_order__le__less__subst2,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A @ B )
     => ( ( ord_le2482536469199093511_a_nat @ ( F2 @ B ) @ C3 )
       => ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
              ( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
             => ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2482536469199093511_a_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1222_order__less__le__subst1,axiom,
    ! [A: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le2482536469199093511_a_nat @ A @ ( F2 @ B ) )
     => ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
       => ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
              ( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
             => ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le2482536469199093511_a_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1223_order__le__imp__less__or__eq,axiom,
    ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ X @ Y )
     => ( ( ord_le2482536469199093511_a_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1224_chain__imp__trancl,axiom,
    ! [S4: nat > term_a_b,R: set_Pr4386577575007340137rm_a_b,I2: nat,J2: nat] :
      ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I3 ) @ ( S4 @ ( suc @ I3 ) ) ) @ R )
     => ( ( ord_less_nat @ I2 @ J2 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I2 ) @ ( S4 @ J2 ) ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).

% chain_imp_trancl
thf(fact_1225_wf__bounded__measure,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,Ub: term_a_b > nat,F2: term_a_b > nat] :
      ( ! [A4: term_a_b,B3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A4 ) @ R )
         => ( ( ord_less_eq_nat @ ( Ub @ B3 ) @ ( Ub @ A4 ) )
            & ( ord_less_eq_nat @ ( F2 @ B3 ) @ ( Ub @ A4 ) )
            & ( ord_less_nat @ ( F2 @ A4 ) @ ( F2 @ B3 ) ) ) )
     => ( wf_term_a_b @ R ) ) ).

% wf_bounded_measure
thf(fact_1226_trancl__power,axiom,
    ! [P2: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P2 @ ( transi7922773638565587891rm_a_b @ R3 ) )
      = ( ? [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
            & ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N2 @ R3 ) ) ) ) ) ).

% trancl_power
thf(fact_1227_rtrancl__fun__conv,axiom,
    ! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
      = ( ? [F6: nat > term_a_b,N2: nat] :
            ( ( ( F6 @ zero_zero_nat )
              = S )
            & ( ( F6 @ N2 )
              = T )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ N2 )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F6 @ I ) @ ( F6 @ ( suc @ I ) ) ) @ R3 ) ) ) ) ) ).

% rtrancl_fun_conv
thf(fact_1228_rtrancl__imp__seq,axiom,
    ! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
     => ? [F4: nat > term_a_b,N3: nat] :
          ( ( ( F4 @ zero_zero_nat )
            = X )
          & ( ( F4 @ N3 )
            = Y )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ N3 )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) ) ) ) ).

% rtrancl_imp_seq
thf(fact_1229_relpow__seq,axiom,
    ! [X: term_a_b,Y: term_a_b,N: nat,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ R ) )
     => ? [F4: nat > term_a_b] :
          ( ( ( F4 @ zero_zero_nat )
            = X )
          & ( ( F4 @ N )
            = Y )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ N )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) ) ) ) ).

% relpow_seq
thf(fact_1230_relpow__fun__conv,axiom,
    ! [A: term_a_b,B: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
      = ( ? [F6: nat > term_a_b] :
            ( ( ( F6 @ zero_zero_nat )
              = A )
            & ( ( F6 @ N )
              = B )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ N )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F6 @ I ) @ ( F6 @ ( suc @ I ) ) ) @ R3 ) ) ) ) ) ).

% relpow_fun_conv
thf(fact_1231_in__measure,axiom,
    ! [X: term_a_b,Y: term_a_b,F2: term_a_b > nat] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( measure_term_a_b @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_1232_psubsetI,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( A5 != B5 )
       => ( ord_le2482536469199093511_a_nat @ A5 @ B5 ) ) ) ).

% psubsetI
thf(fact_1233_psubsetE,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le2482536469199093511_a_nat @ A5 @ B5 )
     => ~ ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
         => ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ) ).

% psubsetE
thf(fact_1234_psubset__eq,axiom,
    ( ord_le2482536469199093511_a_nat
    = ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ A6 @ B6 )
          & ( A6 != B6 ) ) ) ) ).

% psubset_eq
thf(fact_1235_psubset__imp__subset,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le2482536469199093511_a_nat @ A5 @ B5 )
     => ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ).

% psubset_imp_subset
thf(fact_1236_psubset__subset__trans,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le2482536469199093511_a_nat @ A5 @ B5 )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ C4 )
       => ( ord_le2482536469199093511_a_nat @ A5 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_1237_subset__not__subset__eq,axiom,
    ( ord_le2482536469199093511_a_nat
    = ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ A6 @ B6 )
          & ~ ( ord_le8666007276011122963_a_nat @ B6 @ A6 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_1238_subset__psubset__trans,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( ord_le2482536469199093511_a_nat @ B5 @ C4 )
       => ( ord_le2482536469199093511_a_nat @ A5 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_1239_subset__iff__psubset__eq,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le2482536469199093511_a_nat @ A6 @ B6 )
          | ( A6 = B6 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_1240_psubset__imp__ex__mem,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le4883060154188259093rm_a_b @ A5 @ B5 )
     => ? [B3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ B3 @ ( minus_5192120951422937424rm_a_b @ B5 @ A5 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1241_psubset__insert__iff,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le4883060154188259093rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ X @ B5 ) )
      = ( ( ( member5869715511025134514rm_a_b @ X @ B5 )
         => ( ord_le4883060154188259093rm_a_b @ A5 @ B5 ) )
        & ( ~ ( member5869715511025134514rm_a_b @ X @ B5 )
         => ( ( ( member5869715511025134514rm_a_b @ X @ A5 )
             => ( ord_le4883060154188259093rm_a_b @ ( minus_5192120951422937424rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) @ B5 ) )
            & ( ~ ( member5869715511025134514rm_a_b @ X @ A5 )
             => ( ord_le118470702582115849rm_a_b @ A5 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1242_psubset__insert__iff,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le2482536469199093511_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ B5 ) )
      = ( ( ( member5724188588386418708_a_nat @ X @ B5 )
         => ( ord_le2482536469199093511_a_nat @ A5 @ B5 ) )
        & ( ~ ( member5724188588386418708_a_nat @ X @ B5 )
         => ( ( ( member5724188588386418708_a_nat @ X @ A5 )
             => ( ord_le2482536469199093511_a_nat @ ( minus_8630901905721465484_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) @ B5 ) )
            & ( ~ ( member5724188588386418708_a_nat @ X @ A5 )
             => ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1243_psubsetD,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C3: produc357393685978478089rm_a_b] :
      ( ( ord_le4883060154188259093rm_a_b @ A5 @ B5 )
     => ( ( member5869715511025134514rm_a_b @ C3 @ A5 )
       => ( member5869715511025134514rm_a_b @ C3 @ B5 ) ) ) ).

% psubsetD
thf(fact_1244_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1245_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila3720225504097595269_a_nat @ inf_in3813033988066259269_a_nat @ top_to3353692345378799459_a_nat @ ord_le8666007276011122963_a_nat @ ord_le2482536469199093511_a_nat ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_1246_bsqr__max2,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A1: term_a_b,A22: term_a_b,B1: term_a_b,B22: term_a_b] :
      ( ( order_5719896216891381494rm_a_b @ ( field_term_a_b @ R ) @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( produc7020197800436672577rm_a_b @ B1 @ B22 ) ) @ ( bNF_We8522098035112346167rm_a_b @ R ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We4618759378340593734rm_a_b @ R @ A1 @ A22 ) @ ( bNF_We4618759378340593734rm_a_b @ R @ B1 @ B22 ) ) @ R ) ) ) ).

% bsqr_max2
thf(fact_1247_top_Oordering__top__axioms,axiom,
    orderi2411975122467621314_a_nat @ ord_le8666007276011122963_a_nat @ ord_le2482536469199093511_a_nat @ top_to3353692345378799459_a_nat ).

% top.ordering_top_axioms
thf(fact_1248_underS__incl__iff,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
         => ( ( ord_le118470702582115849rm_a_b @ ( order_8163520290499871439rm_a_b @ R @ A ) @ ( order_8163520290499871439rm_a_b @ R @ B ) )
            = ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1249_underS__incl__iff,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R ) @ R )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R ) )
         => ( ( ord_le2705286416250468010rm_a_b @ ( order_1619770106104096206rm_a_b @ R @ A ) @ ( order_1619770106104096206rm_a_b @ R @ B ) )
            = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1250_underS__incl__iff,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat,B: product_prod_a_nat] :
      ( ( order_6388386362885710600_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
     => ( ( member5724188588386418708_a_nat @ A @ ( field_8954927560578634480_a_nat @ R ) )
       => ( ( member5724188588386418708_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
         => ( ( ord_le8666007276011122963_a_nat @ ( order_2219920454636572983_a_nat @ R @ A ) @ ( order_2219920454636572983_a_nat @ R @ B ) )
            = ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1251_underS__I,axiom,
    ! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( I2 != J2 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R3 )
       => ( member5869715511025134514rm_a_b @ I2 @ ( order_8163520290499871439rm_a_b @ R3 @ J2 ) ) ) ) ).

% underS_I
thf(fact_1252_underS__I,axiom,
    ! [I2: term_a_b,J2: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( I2 != J2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R3 )
       => ( member_term_a_b @ I2 @ ( order_1619770106104096206rm_a_b @ R3 @ J2 ) ) ) ) ).

% underS_I
thf(fact_1253_underS__E,axiom,
    ! [I2: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,J2: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ I2 @ ( order_8163520290499871439rm_a_b @ R3 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1254_underS__E,axiom,
    ! [I2: term_a_b,R3: set_Pr4386577575007340137rm_a_b,J2: term_a_b] :
      ( ( member_term_a_b @ I2 @ ( order_1619770106104096206rm_a_b @ R3 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1255_Order__Relation_OunderS__Field,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ ( order_2219920454636572983_a_nat @ R @ A ) @ ( field_8954927560578634480_a_nat @ R ) ) ).

% Order_Relation.underS_Field
thf(fact_1256_Refl__under__underS,axiom,
    ! [R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R ) )
       => ( ( order_346605463002971118rm_a_b @ R @ A )
          = ( sup_su6776935440552674877rm_a_b @ ( order_8163520290499871439rm_a_b @ R @ A ) @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).

% Refl_under_underS
thf(fact_1257_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic4070998268164056665_a_nat @ inf_in3813033988066259269_a_nat @ ord_le8666007276011122963_a_nat @ ord_le2482536469199093511_a_nat ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_1258_underS__subset__under,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ ( order_2219920454636572983_a_nat @ R @ A ) @ ( order_6535219843107416792_a_nat @ R @ A ) ) ).

% underS_subset_under
thf(fact_1259_under__Field,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ ( order_6535219843107416792_a_nat @ R @ A ) @ ( field_8954927560578634480_a_nat @ R ) ) ).

% under_Field
thf(fact_1260_wo__rel_Oofilter__def,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,A5: set_Pr4934435412358123699_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R )
     => ( ( order_7295016824049152615_a_nat @ R @ A5 )
        = ( ( ord_le8666007276011122963_a_nat @ A5 @ ( field_8954927560578634480_a_nat @ R ) )
          & ! [X4: product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ X4 @ A5 )
             => ( ord_le8666007276011122963_a_nat @ ( order_6535219843107416792_a_nat @ R @ X4 ) @ A5 ) ) ) ) ) ).

% wo_rel.ofilter_def
thf(fact_1261_Range__insert,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( range_8688746797885257759rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R ) )
      = ( insert_term_a_b @ B @ ( range_8688746797885257759rm_a_b @ R ) ) ) ).

% Range_insert
thf(fact_1262_trancl__range,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( range_8688746797885257759rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) )
      = ( range_8688746797885257759rm_a_b @ R ) ) ).

% trancl_range
thf(fact_1263_Range_Ocases,axiom,
    ! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( range_8688746797885257759rm_a_b @ R ) )
     => ~ ! [A4: term_a_b] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A ) @ R ) ) ).

% Range.cases
thf(fact_1264_Range_Osimps,axiom,
    ! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( range_8688746797885257759rm_a_b @ R ) )
      = ( ? [A3: term_a_b,B4: term_a_b] :
            ( ( A = B4 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_1265_Range_Ointros,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( member_term_a_b @ B @ ( range_8688746797885257759rm_a_b @ R ) ) ) ).

% Range.intros
thf(fact_1266_RangeE,axiom,
    ! [B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ B @ ( range_8688746797885257759rm_a_b @ R ) )
     => ~ ! [A4: term_a_b] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ R ) ) ).

% RangeE
thf(fact_1267_Range__iff,axiom,
    ! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( range_8688746797885257759rm_a_b @ R ) )
      = ( ? [Y5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_1268_wo__rel_Oofilter__linord,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R )
     => ( ( order_7295016824049152615_a_nat @ R @ A5 )
       => ( ( order_7295016824049152615_a_nat @ R @ B5 )
         => ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
            | ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ) ) ) ).

% wo_rel.ofilter_linord
thf(fact_1269_Range__mono,axiom,
    ! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ R @ S )
     => ( ord_less_eq_set_nat @ ( range_a_nat @ R ) @ ( range_a_nat @ S ) ) ) ).

% Range_mono
thf(fact_1270_ofilter__def,axiom,
    ( order_7295016824049152615_a_nat
    = ( ^ [R4: set_Pr1811044260758604347_a_nat,A6: set_Pr4934435412358123699_a_nat] :
          ( ( ord_le8666007276011122963_a_nat @ A6 @ ( field_8954927560578634480_a_nat @ R4 ) )
          & ! [X4: product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ X4 @ A6 )
             => ( ord_le8666007276011122963_a_nat @ ( order_6535219843107416792_a_nat @ R4 @ X4 ) @ A6 ) ) ) ) ) ).

% ofilter_def
thf(fact_1271_trancl__domain,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( domain6518119340328324150rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) )
      = ( domain6518119340328324150rm_a_b @ R ) ) ).

% trancl_domain
thf(fact_1272_Domain__insert,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( domain6518119340328324150rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R ) )
      = ( insert_term_a_b @ A @ ( domain6518119340328324150rm_a_b @ R ) ) ) ).

% Domain_insert
thf(fact_1273_Domain__mono,axiom,
    ! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ R @ S )
     => ( ord_less_eq_set_a @ ( domain_a_nat @ R ) @ ( domain_a_nat @ S ) ) ) ).

% Domain_mono
thf(fact_1274_Domain__iff,axiom,
    ! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ A @ ( domain6518119340328324150rm_a_b @ R ) )
      = ( ? [Y5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y5 ) @ R ) ) ) ).

% Domain_iff

% Conjectures (1)
thf(conj_0,conjecture,
    member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ f @ v @ ta ) @ t ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) ) ).

%------------------------------------------------------------------------------