%------------------------------------------------------------------------------
% 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    : FO_Theory_Rewriting/0082_FOR_Check/prob_00328_016719__18919742_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1795 ( 460 unt; 513 typ;   0 def)
%            Number of atoms       : 3993 (1008 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 11284 ( 408   ~;  68   |; 301   &;8660   @)
%                                         (   0 <=>;1847  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   7 avg)
%            Number of types       :  123 ( 122 usr)
%            Number of type conns  : 1144 (1144   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  394 ( 391 usr;  25 con; 0-4 aty)
%            Number of variables   : 3831 ( 600   ^;3138   !;  93   ?;3831   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:03:11.861
%------------------------------------------------------------------------------
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__f,type,
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% Explicit typings (391)
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thf(sy_c_FOR__Certificate_OR1Fin_001t__FOR____Certificate__Oftrs,type,
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thf(sy_c_FOR__Certificate_OR2NonRootStep_001t__FOR____Certificate__Oftrs,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J_Mt__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
    member2505966486127315062rm_f_v: produc3175510285795157069rm_f_v > set_Pr2075880423747436333rm_f_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J_Mt__Nat__Onat_J,type,
    member3662771866509663549_f_nat: produc2194508710173438886_f_nat > set_Pr5080083045707565404_f_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J_Mt__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J_J,type,
    member3777300322277616400term_f: produc8979644124618241255term_f > set_Pr1766941801908210887term_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
    member2678038196313615385_f_nat: product_prod_f_nat > set_Pr5245412377734449720_f_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Ground____Terms__Ogterm_Itf__f_J_J,type,
    member9219338147399679421term_f: set_Ground_gterm_f > set_se421094335941524212term_f > $o ).

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

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J_J,type,
    member6919674941555055664term_f: set_Pr989862937836626183term_f > set_se5570125628664547815term_f > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
    member4041562125048333488rm_f_v: set_Pr8827868859434726151rm_f_v > set_se4583834864486174823rm_f_v > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
    member8345558847462883279_f_nat: set_Pr5245412377734449720_f_nat > set_se1451291994713068270_f_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Ground____Terms__Ogterm_Itf__f_J_J_J,type,
    member7620211966956398237term_f: set_se421094335941524212term_f > set_se94132152784077780term_f > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    member_set_set_nat: set_set_nat > set_set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J_J_J,type,
    member4407653303224918288term_f: set_se5570125628664547815term_f > set_se4781744943471505095term_f > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J_J,type,
    member5502175894719687813_f_nat: set_se1451291994713068270_f_nat > set_se8249591372659029156_f_nat > $o ).

thf(sy_c_member_001tf__f,type,
    member_f: f > set_f > $o ).

thf(sy_v_Rs,type,
    rs: list_f1824981274722084755rm_f_v ).

thf(sy_v__092_060F_062,type,
    f2: fset_P6228066233360383026_f_nat ).

thf(sy_v_r____,type,
    r: fOR_rr2_rel_FOR_ftrs ).

thf(sy_v_t____,type,
    t: ground_gterm_f ).

% Relevant facts (1273)
thf(fact_0_not__finite__existsD,axiom,
    ! [P: product_prod_f_nat > $o] :
      ( ~ ( finite3598747971073327385_f_nat @ ( collec1417984143148964183_f_nat @ P ) )
     => ? [X_1: product_prod_f_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1_not__finite__existsD,axiom,
    ! [P: set_Ground_gterm_f > $o] :
      ( ~ ( finite6824738741894949053term_f @ ( collec1979342819066322687term_f @ P ) )
     => ? [X_1: set_Ground_gterm_f] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_2_not__finite__existsD,axiom,
    ! [P: set_Pr989862937836626183term_f > $o] :
      ( ~ ( finite5058142709724841776term_f @ ( collec4490547877585280114term_f @ P ) )
     => ? [X_1: set_Pr989862937836626183term_f] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_3_not__finite__existsD,axiom,
    ! [P: set_Pr5245412377734449720_f_nat > $o] :
      ( ~ ( finite879454794548843727_f_nat @ ( collec6964094325902095629_f_nat @ P ) )
     => ? [X_1: set_Pr5245412377734449720_f_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_4_not__finite__existsD,axiom,
    ! [P: set_nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
     => ? [X_1: set_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_5_not__finite__existsD,axiom,
    ! [P: ground_gterm_f > $o] :
      ( ~ ( finite1923210418194103261term_f @ ( collec3522850625759001119term_f @ P ) )
     => ? [X_1: ground_gterm_f] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7__092_060open_062eval__rr2__rel_A_Ifset_A_092_060F_062_J_A_Imap_Afset_ARs_J_Ar_A_092_060subseteq_062_A_092_060T_062_092_060_094sub_062G_A_Ifset_A_092_060F_062_J_A_092_060times_062_A_092_060T_062_092_060_094sub_062G_A_Ifset_A_092_060F_062_J_092_060close_062,axiom,
    ( ord_le263819222746101927term_f @ ( fOR_eval_rr2_rel_f_v @ ( fset_P3576968334923099475_f_nat @ f2 ) @ ( map_fs8602507653405230974rm_f_v @ fset_P4617584883882644886rm_f_v @ rs ) @ r )
    @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ ( fset_P3576968334923099475_f_nat @ f2 ) )
      @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ ( fset_P3576968334923099475_f_nat @ f2 ) ) ) ) ).

% \<open>eval_rr2_rel (fset \<F>) (map fset Rs) r \<subseteq> \<T>\<^sub>G (fset \<F>) \<times> \<T>\<^sub>G (fset \<F>)\<close>
thf(fact_8__092_060open_062_092_060And_062t_O_A_123u_O_A_It_M_Au_J_A_092_060in_062_Aeval__rr2__rel_A_Ifset_A_092_060F_062_J_A_Imap_Afset_ARs_J_Ar_A_092_060and_062_Afunas__gterm_Au_A_092_060subseteq_062_Afset_A_092_060F_062_125_A_061_A_123u_O_A_It_M_Au_J_A_092_060in_062_Aeval__rr2__rel_A_Ifset_A_092_060F_062_J_A_Imap_Afset_ARs_J_Ar_125_092_060close_062,axiom,
    ! [T: ground_gterm_f] :
      ( ( collec3522850625759001119term_f
        @ ^ [U: ground_gterm_f] :
            ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T @ U ) @ ( fOR_eval_rr2_rel_f_v @ ( fset_P3576968334923099475_f_nat @ f2 ) @ ( map_fs8602507653405230974rm_f_v @ fset_P4617584883882644886rm_f_v @ rs ) @ r ) )
            & ( ord_le8976984241387448984_f_nat @ ( ground_funas_gterm_f @ U ) @ ( fset_P3576968334923099475_f_nat @ f2 ) ) ) )
      = ( collec3522850625759001119term_f
        @ ^ [U: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T @ U ) @ ( fOR_eval_rr2_rel_f_v @ ( fset_P3576968334923099475_f_nat @ f2 ) @ ( map_fs8602507653405230974rm_f_v @ fset_P4617584883882644886rm_f_v @ rs ) @ r ) ) ) ) ).

% \<open>\<And>t. {u. (t, u) \<in> eval_rr2_rel (fset \<F>) (map fset Rs) r \<and> funas_gterm u \<subseteq> fset \<F>} = {u. (t, u) \<in> eval_rr2_rel (fset \<F>) (map fset Rs) r}\<close>
thf(fact_9_finite__Collect__subsets,axiom,
    ! [A: set_se421094335941524212term_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( finite6023745176923099549term_f
        @ ( collec8921018771386908639term_f
          @ ^ [B: set_se421094335941524212term_f] : ( ord_le2325249104654787220term_f @ B @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_10_finite__Collect__subsets,axiom,
    ! [A: set_se5570125628664547815term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( finite9150278737398483984term_f
        @ ( collec6999422248803473746term_f
          @ ^ [B: set_se5570125628664547815term_f] : ( ord_le5727726935103693191term_f @ B @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_11_finite__Collect__subsets,axiom,
    ! [A: set_se1451291994713068270_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( finite1774089887677170053_f_nat
        @ ( collec2226960723385805251_f_nat
          @ ^ [B: set_se1451291994713068270_f_nat] : ( ord_le5386115319958680398_f_nat @ B @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_12_finite__Collect__subsets,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( finite6739761609112101331et_nat
        @ ( collect_set_set_nat
          @ ^ [B: set_set_nat] : ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_13_finite__Collect__subsets,axiom,
    ! [A: set_Ground_gterm_f] :
      ( ( finite1923210418194103261term_f @ A )
     => ( finite6824738741894949053term_f
        @ ( collec1979342819066322687term_f
          @ ^ [B: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ B @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_14_finite__Collect__subsets,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B: set_nat] : ( ord_less_eq_set_nat @ B @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_15_finite__Collect__subsets,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( finite879454794548843727_f_nat
        @ ( collec6964094325902095629_f_nat
          @ ^ [B: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ B @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_16_finite__Collect__subsets,axiom,
    ! [A: set_Pr989862937836626183term_f] :
      ( ( finite4071141763865312848term_f @ A )
     => ( finite5058142709724841776term_f
        @ ( collec4490547877585280114term_f
          @ ^ [B: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ B @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_17_finite__snd,axiom,
    ! [S: set_Pr5245412377734449720_f_nat,Y: f] :
      ( ( finite3598747971073327385_f_nat @ S )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ Y @ X ) @ S ) ) ) ) ).

% finite_snd
thf(fact_18_finite__snd,axiom,
    ! [S: set_Pr989862937836626183term_f,Y: ground_gterm_f] :
      ( ( finite4071141763865312848term_f @ S )
     => ( finite1923210418194103261term_f
        @ ( collec3522850625759001119term_f
          @ ^ [X: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y @ X ) @ S ) ) ) ) ).

% finite_snd
thf(fact_19_finite__Collect__conjI,axiom,
    ! [P: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
      ( ( ( finite3598747971073327385_f_nat @ ( collec1417984143148964183_f_nat @ P ) )
        | ( finite3598747971073327385_f_nat @ ( collec1417984143148964183_f_nat @ Q ) ) )
     => ( finite3598747971073327385_f_nat
        @ ( collec1417984143148964183_f_nat
          @ ^ [X: product_prod_f_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_20_finite__Collect__conjI,axiom,
    ! [P: set_Ground_gterm_f > $o,Q: set_Ground_gterm_f > $o] :
      ( ( ( finite6824738741894949053term_f @ ( collec1979342819066322687term_f @ P ) )
        | ( finite6824738741894949053term_f @ ( collec1979342819066322687term_f @ Q ) ) )
     => ( finite6824738741894949053term_f
        @ ( collec1979342819066322687term_f
          @ ^ [X: set_Ground_gterm_f] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_21_finite__Collect__conjI,axiom,
    ! [P: set_Pr989862937836626183term_f > $o,Q: set_Pr989862937836626183term_f > $o] :
      ( ( ( finite5058142709724841776term_f @ ( collec4490547877585280114term_f @ P ) )
        | ( finite5058142709724841776term_f @ ( collec4490547877585280114term_f @ Q ) ) )
     => ( finite5058142709724841776term_f
        @ ( collec4490547877585280114term_f
          @ ^ [X: set_Pr989862937836626183term_f] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_22_finite__Collect__conjI,axiom,
    ! [P: set_Pr5245412377734449720_f_nat > $o,Q: set_Pr5245412377734449720_f_nat > $o] :
      ( ( ( finite879454794548843727_f_nat @ ( collec6964094325902095629_f_nat @ P ) )
        | ( finite879454794548843727_f_nat @ ( collec6964094325902095629_f_nat @ Q ) ) )
     => ( finite879454794548843727_f_nat
        @ ( collec6964094325902095629_f_nat
          @ ^ [X: set_Pr5245412377734449720_f_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_23_finite__Collect__conjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_24_finite__Collect__conjI,axiom,
    ! [P: ground_gterm_f > $o,Q: ground_gterm_f > $o] :
      ( ( ( finite1923210418194103261term_f @ ( collec3522850625759001119term_f @ P ) )
        | ( finite1923210418194103261term_f @ ( collec3522850625759001119term_f @ Q ) ) )
     => ( finite1923210418194103261term_f
        @ ( collec3522850625759001119term_f
          @ ^ [X: ground_gterm_f] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_25_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_26_finite__Collect__disjI,axiom,
    ! [P: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
      ( ( finite3598747971073327385_f_nat
        @ ( collec1417984143148964183_f_nat
          @ ^ [X: product_prod_f_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite3598747971073327385_f_nat @ ( collec1417984143148964183_f_nat @ P ) )
        & ( finite3598747971073327385_f_nat @ ( collec1417984143148964183_f_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_27_finite__Collect__disjI,axiom,
    ! [P: set_Ground_gterm_f > $o,Q: set_Ground_gterm_f > $o] :
      ( ( finite6824738741894949053term_f
        @ ( collec1979342819066322687term_f
          @ ^ [X: set_Ground_gterm_f] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite6824738741894949053term_f @ ( collec1979342819066322687term_f @ P ) )
        & ( finite6824738741894949053term_f @ ( collec1979342819066322687term_f @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_28_finite__Collect__disjI,axiom,
    ! [P: set_Pr989862937836626183term_f > $o,Q: set_Pr989862937836626183term_f > $o] :
      ( ( finite5058142709724841776term_f
        @ ( collec4490547877585280114term_f
          @ ^ [X: set_Pr989862937836626183term_f] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite5058142709724841776term_f @ ( collec4490547877585280114term_f @ P ) )
        & ( finite5058142709724841776term_f @ ( collec4490547877585280114term_f @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_29_finite__Collect__disjI,axiom,
    ! [P: set_Pr5245412377734449720_f_nat > $o,Q: set_Pr5245412377734449720_f_nat > $o] :
      ( ( finite879454794548843727_f_nat
        @ ( collec6964094325902095629_f_nat
          @ ^ [X: set_Pr5245412377734449720_f_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite879454794548843727_f_nat @ ( collec6964094325902095629_f_nat @ P ) )
        & ( finite879454794548843727_f_nat @ ( collec6964094325902095629_f_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_30_finite__Collect__disjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_31_finite__Collect__disjI,axiom,
    ! [P: ground_gterm_f > $o,Q: ground_gterm_f > $o] :
      ( ( finite1923210418194103261term_f
        @ ( collec3522850625759001119term_f
          @ ^ [X: ground_gterm_f] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite1923210418194103261term_f @ ( collec3522850625759001119term_f @ P ) )
        & ( finite1923210418194103261term_f @ ( collec3522850625759001119term_f @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_32_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_33_a,axiom,
    ? [A2: tree_r733329426570293750tion_f] :
      ( ( fOR_rr7226795042121552277el_f_v @ f2 @ rs @ r )
      = ( some_T4055341017772447441tion_f @ A2 ) ) ).

% a
thf(fact_34_Inf__branching__terms__def,axiom,
    ( rR2_In312037779070622676erms_f
    = ( ^ [R: set_Pr989862937836626183term_f,F: fset_P6228066233360383026_f_nat] :
          ( collec3522850625759001119term_f
          @ ^ [T2: ground_gterm_f] :
              ( ~ ( finite1923210418194103261term_f
                  @ ( collec3522850625759001119term_f
                    @ ^ [U: ground_gterm_f] :
                        ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T2 @ U ) @ R )
                        & ( ord_le8976984241387448984_f_nat @ ( ground_funas_gterm_f @ U ) @ ( fset_P3576968334923099475_f_nat @ F ) ) ) ) )
              & ( ord_le8976984241387448984_f_nat @ ( ground_funas_gterm_f @ T2 ) @ ( fset_P3576968334923099475_f_nat @ F ) ) ) ) ) ) ).

% Inf_branching_terms_def
thf(fact_35_subsetI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ! [X2: product_prod_f_nat] :
          ( ( member2678038196313615385_f_nat @ X2 @ A )
         => ( member2678038196313615385_f_nat @ X2 @ B2 ) )
     => ( ord_le8976984241387448984_f_nat @ A @ B2 ) ) ).

% subsetI
thf(fact_36_subsetI,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ! [X2: produc7245736746747425831term_f] :
          ( ( member848276444142703440term_f @ X2 @ A )
         => ( member848276444142703440term_f @ X2 @ B2 ) )
     => ( ord_le263819222746101927term_f @ A @ B2 ) ) ).

% subsetI
thf(fact_37_subsetI,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ! [X2: nat] :
          ( ( member_nat @ X2 @ A )
         => ( member_nat @ X2 @ B2 ) )
     => ( ord_less_eq_set_nat @ A @ B2 ) ) ).

% subsetI
thf(fact_38_subsetI,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ! [X2: ground_gterm_f] :
          ( ( member5261315044688711901term_f @ X2 @ A )
         => ( member5261315044688711901term_f @ X2 @ B2 ) )
     => ( ord_le2735537439747282356term_f @ A @ B2 ) ) ).

% subsetI
thf(fact_39_subset__antisym,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
     => ( ( ord_le8976984241387448984_f_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% subset_antisym
thf(fact_40_subset__antisym,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A @ B2 )
     => ( ( ord_le263819222746101927term_f @ B2 @ A )
       => ( A = B2 ) ) ) ).

% subset_antisym
thf(fact_41_subset__antisym,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% subset_antisym
thf(fact_42_subset__antisym,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A @ B2 )
     => ( ( ord_le2735537439747282356term_f @ B2 @ A )
       => ( A = B2 ) ) ) ).

% subset_antisym
thf(fact_43_fset,axiom,
    ! [X3: fset_P5324236452418035597term_f] : ( member6919674941555055664term_f @ ( fset_P5939342050162774678term_f @ X3 ) @ ( collec4490547877585280114term_f @ finite4071141763865312848term_f ) ) ).

% fset
thf(fact_44_fset,axiom,
    ! [X3: fset_P8018961893305114765rm_f_v] : ( member4041562125048333488rm_f_v @ ( fset_P4617584883882644886rm_f_v @ X3 ) @ ( collec107294023721454066rm_f_v @ finite6110293842680836688rm_f_v ) ) ).

% fset
thf(fact_45_fset,axiom,
    ! [X3: fset_Ground_gterm_f] : ( member9219338147399679421term_f @ ( fset_Ground_gterm_f2 @ X3 ) @ ( collec1979342819066322687term_f @ finite1923210418194103261term_f ) ) ).

% fset
thf(fact_46_fset,axiom,
    ! [X3: fset_nat] : ( member_set_nat @ ( fset_nat2 @ X3 ) @ ( collect_set_nat @ finite_finite_nat ) ) ).

% fset
thf(fact_47_fset,axiom,
    ! [X3: fset_P6228066233360383026_f_nat] : ( member8345558847462883279_f_nat @ ( fset_P3576968334923099475_f_nat @ X3 ) @ ( collec6964094325902095629_f_nat @ finite3598747971073327385_f_nat ) ) ).

% fset
thf(fact_48_fset,axiom,
    ! [X3: fset_s250637050704127098term_f] : ( member7620211966956398237term_f @ ( fset_s3373063111670843395term_f @ X3 ) @ ( collec8921018771386908639term_f @ finite6824738741894949053term_f ) ) ).

% fset
thf(fact_49_fset,axiom,
    ! [X3: fset_s9063456872818339949term_f] : ( member4407653303224918288term_f @ ( fset_s6989464911991047030term_f @ X3 ) @ ( collec6999422248803473746term_f @ finite5058142709724841776term_f ) ) ).

% fset
thf(fact_50_fset,axiom,
    ! [X3: fset_s3012505784449309928_f_nat] : ( member5502175894719687813_f_nat @ ( fset_s10630165170805001_f_nat @ X3 ) @ ( collec2226960723385805251_f_nat @ finite879454794548843727_f_nat ) ) ).

% fset
thf(fact_51_fset,axiom,
    ! [X3: fset_set_nat] : ( member_set_set_nat @ ( fset_set_nat2 @ X3 ) @ ( collect_set_set_nat @ finite1152437895449049373et_nat ) ) ).

% fset
thf(fact_52_fset__cases,axiom,
    ! [Y: set_Pr989862937836626183term_f] :
      ( ( member6919674941555055664term_f @ Y @ ( collec4490547877585280114term_f @ finite4071141763865312848term_f ) )
     => ~ ! [X2: fset_P5324236452418035597term_f] :
            ( Y
           != ( fset_P5939342050162774678term_f @ X2 ) ) ) ).

% fset_cases
thf(fact_53_fset__cases,axiom,
    ! [Y: set_Pr8827868859434726151rm_f_v] :
      ( ( member4041562125048333488rm_f_v @ Y @ ( collec107294023721454066rm_f_v @ finite6110293842680836688rm_f_v ) )
     => ~ ! [X2: fset_P8018961893305114765rm_f_v] :
            ( Y
           != ( fset_P4617584883882644886rm_f_v @ X2 ) ) ) ).

% fset_cases
thf(fact_54_fset__cases,axiom,
    ! [Y: set_Ground_gterm_f] :
      ( ( member9219338147399679421term_f @ Y @ ( collec1979342819066322687term_f @ finite1923210418194103261term_f ) )
     => ~ ! [X2: fset_Ground_gterm_f] :
            ( Y
           != ( fset_Ground_gterm_f2 @ X2 ) ) ) ).

% fset_cases
thf(fact_55_fset__cases,axiom,
    ! [Y: set_nat] :
      ( ( member_set_nat @ Y @ ( collect_set_nat @ finite_finite_nat ) )
     => ~ ! [X2: fset_nat] :
            ( Y
           != ( fset_nat2 @ X2 ) ) ) ).

% fset_cases
thf(fact_56_fset__cases,axiom,
    ! [Y: set_Pr5245412377734449720_f_nat] :
      ( ( member8345558847462883279_f_nat @ Y @ ( collec6964094325902095629_f_nat @ finite3598747971073327385_f_nat ) )
     => ~ ! [X2: fset_P6228066233360383026_f_nat] :
            ( Y
           != ( fset_P3576968334923099475_f_nat @ X2 ) ) ) ).

% fset_cases
thf(fact_57_fset__cases,axiom,
    ! [Y: set_se421094335941524212term_f] :
      ( ( member7620211966956398237term_f @ Y @ ( collec8921018771386908639term_f @ finite6824738741894949053term_f ) )
     => ~ ! [X2: fset_s250637050704127098term_f] :
            ( Y
           != ( fset_s3373063111670843395term_f @ X2 ) ) ) ).

% fset_cases
thf(fact_58_fset__cases,axiom,
    ! [Y: set_se5570125628664547815term_f] :
      ( ( member4407653303224918288term_f @ Y @ ( collec6999422248803473746term_f @ finite5058142709724841776term_f ) )
     => ~ ! [X2: fset_s9063456872818339949term_f] :
            ( Y
           != ( fset_s6989464911991047030term_f @ X2 ) ) ) ).

% fset_cases
thf(fact_59_fset__cases,axiom,
    ! [Y: set_se1451291994713068270_f_nat] :
      ( ( member5502175894719687813_f_nat @ Y @ ( collec2226960723385805251_f_nat @ finite879454794548843727_f_nat ) )
     => ~ ! [X2: fset_s3012505784449309928_f_nat] :
            ( Y
           != ( fset_s10630165170805001_f_nat @ X2 ) ) ) ).

% fset_cases
thf(fact_60_fset__cases,axiom,
    ! [Y: set_set_nat] :
      ( ( member_set_set_nat @ Y @ ( collect_set_set_nat @ finite1152437895449049373et_nat ) )
     => ~ ! [X2: fset_set_nat] :
            ( Y
           != ( fset_set_nat2 @ X2 ) ) ) ).

% fset_cases
thf(fact_61_finite__SigmaI,axiom,
    ! [A: set_f,B2: f > set_nat] :
      ( ( finite_finite_f @ A )
     => ( ! [A3: f] :
            ( ( member_f @ A3 @ A )
           => ( finite_finite_nat @ ( B2 @ A3 ) ) )
       => ( finite3598747971073327385_f_nat @ ( product_Sigma_f_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI
thf(fact_62_finite__SigmaI,axiom,
    ! [A: set_nat,B2: nat > set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B2 @ A3 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI
thf(fact_63_finite__SigmaI,axiom,
    ! [A: set_Ground_gterm_f,B2: ground_gterm_f > set_nat] :
      ( ( finite1923210418194103261term_f @ A )
     => ( ! [A3: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ A3 @ A )
           => ( finite_finite_nat @ ( B2 @ A3 ) ) )
       => ( finite311254710766229066_f_nat @ ( produc4746993288989285692_f_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI
thf(fact_64_finite__SigmaI,axiom,
    ! [A: set_nat,B2: nat > set_Ground_gterm_f] :
      ( ( finite_finite_nat @ A )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite1923210418194103261term_f @ ( B2 @ A3 ) ) )
       => ( finite4078170753649457648term_f @ ( produc3100878011685873178term_f @ A @ B2 ) ) ) ) ).

% finite_SigmaI
thf(fact_65_finite__SigmaI,axiom,
    ! [A: set_nat,B2: nat > set_set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite1152437895449049373et_nat @ ( B2 @ A3 ) ) )
       => ( finite962763233036973920et_nat @ ( produc8883945523214541856et_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI
thf(fact_66_finite__SigmaI,axiom,
    ! [A: set_set_nat,B2: set_nat > set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ! [A3: set_nat] :
            ( ( member_set_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B2 @ A3 ) ) )
       => ( finite6054027020396858976at_nat @ ( produc5318310618559172640at_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI
thf(fact_67_finite__SigmaI,axiom,
    ! [A: set_Ground_gterm_f,B2: ground_gterm_f > set_set_nat] :
      ( ( finite1923210418194103261term_f @ A )
     => ( ! [A3: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ A3 @ A )
           => ( finite1152437895449049373et_nat @ ( B2 @ A3 ) ) )
       => ( finite3836040682740435712et_nat @ ( produc2061566356152752370et_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI
thf(fact_68_finite__SigmaI,axiom,
    ! [A: set_nat,B2: nat > set_Pr5245412377734449720_f_nat] :
      ( ( finite_finite_nat @ A )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite3598747971073327385_f_nat @ ( B2 @ A3 ) ) )
       => ( finite5167703378860198236_f_nat @ ( produc1258315153622678812_f_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI
thf(fact_69_finite__SigmaI,axiom,
    ! [A: set_nat,B2: nat > set_se421094335941524212term_f] :
      ( ( finite_finite_nat @ A )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite6824738741894949053term_f @ ( B2 @ A3 ) ) )
       => ( finite2583538073261184720term_f @ ( produc5271949834123747066term_f @ A @ B2 ) ) ) ) ).

% finite_SigmaI
thf(fact_70_finite__SigmaI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: product_prod_f_nat > set_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ! [A3: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B2 @ A3 ) ) )
       => ( finite1611537298927112028at_nat @ ( produc4600693851930136092at_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI
thf(fact_71_R1Inf,axiom,
    ! [Ta2: tree_r733329426570293750tion_f] :
      ( ( ( fOR_rr7226795042121552277el_f_v @ f2 @ rs @ r )
        = ( some_T4055341017772447441tion_f @ Ta2 ) )
     => ( rRn_RR2_spec_nat_f_f @ Ta2 @ ( fOR_eval_rr2_rel_f_v @ ( fset_P3576968334923099475_f_nat @ f2 ) @ ( map_fs8602507653405230974rm_f_v @ fset_P4617584883882644886rm_f_v @ rs ) @ r ) ) ) ).

% R1Inf
thf(fact_72_finite__cartesian__product,axiom,
    ! [A: set_f,B2: set_nat] :
      ( ( finite_finite_f @ A )
     => ( ( finite_finite_nat @ B2 )
       => ( finite3598747971073327385_f_nat
          @ ( product_Sigma_f_nat @ A
            @ ^ [Uu: f] : B2 ) ) ) ) ).

% finite_cartesian_product
thf(fact_73_finite__cartesian__product,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B2 )
       => ( finite6177210948735845034at_nat
          @ ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : B2 ) ) ) ) ).

% finite_cartesian_product
thf(fact_74_finite__cartesian__product,axiom,
    ! [A: set_Ground_gterm_f,B2: set_nat] :
      ( ( finite1923210418194103261term_f @ A )
     => ( ( finite_finite_nat @ B2 )
       => ( finite311254710766229066_f_nat
          @ ( produc4746993288989285692_f_nat @ A
            @ ^ [Uu: ground_gterm_f] : B2 ) ) ) ) ).

% finite_cartesian_product
thf(fact_75_finite__cartesian__product,axiom,
    ! [A: set_nat,B2: set_Ground_gterm_f] :
      ( ( finite_finite_nat @ A )
     => ( ( finite1923210418194103261term_f @ B2 )
       => ( finite4078170753649457648term_f
          @ ( produc3100878011685873178term_f @ A
            @ ^ [Uu: nat] : B2 ) ) ) ) ).

% finite_cartesian_product
thf(fact_76_finite__cartesian__product,axiom,
    ! [A: set_nat,B2: set_set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( finite962763233036973920et_nat
          @ ( produc8883945523214541856et_nat @ A
            @ ^ [Uu: nat] : B2 ) ) ) ) ).

% finite_cartesian_product
thf(fact_77_finite__cartesian__product,axiom,
    ! [A: set_set_nat,B2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( finite_finite_nat @ B2 )
       => ( finite6054027020396858976at_nat
          @ ( produc5318310618559172640at_nat @ A
            @ ^ [Uu: set_nat] : B2 ) ) ) ) ).

% finite_cartesian_product
thf(fact_78_finite__cartesian__product,axiom,
    ! [A: set_Ground_gterm_f,B2: set_set_nat] :
      ( ( finite1923210418194103261term_f @ A )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( finite3836040682740435712et_nat
          @ ( produc2061566356152752370et_nat @ A
            @ ^ [Uu: ground_gterm_f] : B2 ) ) ) ) ).

% finite_cartesian_product
thf(fact_79_finite__cartesian__product,axiom,
    ! [A: set_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite3598747971073327385_f_nat @ B2 )
       => ( finite5167703378860198236_f_nat
          @ ( produc1258315153622678812_f_nat @ A
            @ ^ [Uu: nat] : B2 ) ) ) ) ).

% finite_cartesian_product
thf(fact_80_finite__cartesian__product,axiom,
    ! [A: set_nat,B2: set_se421094335941524212term_f] :
      ( ( finite_finite_nat @ A )
     => ( ( finite6824738741894949053term_f @ B2 )
       => ( finite2583538073261184720term_f
          @ ( produc5271949834123747066term_f @ A
            @ ^ [Uu: nat] : B2 ) ) ) ) ).

% finite_cartesian_product
thf(fact_81_finite__cartesian__product,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( finite_finite_nat @ B2 )
       => ( finite1611537298927112028at_nat
          @ ( produc4600693851930136092at_nat @ A
            @ ^ [Uu: product_prod_f_nat] : B2 ) ) ) ) ).

% finite_cartesian_product
thf(fact_82_Collect__mono__iff,axiom,
    ! [P: set_Pr989862937836626183term_f > $o,Q: set_Pr989862937836626183term_f > $o] :
      ( ( ord_le5727726935103693191term_f @ ( collec4490547877585280114term_f @ P ) @ ( collec4490547877585280114term_f @ Q ) )
      = ( ! [X: set_Pr989862937836626183term_f] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_83_Collect__mono__iff,axiom,
    ! [P: set_Pr5245412377734449720_f_nat > $o,Q: set_Pr5245412377734449720_f_nat > $o] :
      ( ( ord_le5386115319958680398_f_nat @ ( collec6964094325902095629_f_nat @ P ) @ ( collec6964094325902095629_f_nat @ Q ) )
      = ( ! [X: set_Pr5245412377734449720_f_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_84_Collect__mono__iff,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
      = ( ! [X: set_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_85_Collect__mono__iff,axiom,
    ! [P: set_Ground_gterm_f > $o,Q: set_Ground_gterm_f > $o] :
      ( ( ord_le2325249104654787220term_f @ ( collec1979342819066322687term_f @ P ) @ ( collec1979342819066322687term_f @ Q ) )
      = ( ! [X: set_Ground_gterm_f] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_86_Collect__mono__iff,axiom,
    ! [P: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
      ( ( ord_le8976984241387448984_f_nat @ ( collec1417984143148964183_f_nat @ P ) @ ( collec1417984143148964183_f_nat @ Q ) )
      = ( ! [X: product_prod_f_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_87_Collect__mono__iff,axiom,
    ! [P: produc7245736746747425831term_f > $o,Q: produc7245736746747425831term_f > $o] :
      ( ( ord_le263819222746101927term_f @ ( collec3209569126566747026term_f @ P ) @ ( collec3209569126566747026term_f @ Q ) )
      = ( ! [X: produc7245736746747425831term_f] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_88_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_89_Collect__mono__iff,axiom,
    ! [P: ground_gterm_f > $o,Q: ground_gterm_f > $o] :
      ( ( ord_le2735537439747282356term_f @ ( collec3522850625759001119term_f @ P ) @ ( collec3522850625759001119term_f @ Q ) )
      = ( ! [X: ground_gterm_f] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_90_set__eq__subset,axiom,
    ( ( ^ [Y2: set_Pr5245412377734449720_f_nat,Z: set_Pr5245412377734449720_f_nat] : ( Y2 = Z ) )
    = ( ^ [A4: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ A4 @ B )
          & ( ord_le8976984241387448984_f_nat @ B @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_91_set__eq__subset,axiom,
    ( ( ^ [Y2: set_Pr989862937836626183term_f,Z: set_Pr989862937836626183term_f] : ( Y2 = Z ) )
    = ( ^ [A4: set_Pr989862937836626183term_f,B: set_Pr989862937836626183term_f] :
          ( ( ord_le263819222746101927term_f @ A4 @ B )
          & ( ord_le263819222746101927term_f @ B @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_92_set__eq__subset,axiom,
    ( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
    = ( ^ [A4: set_nat,B: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B )
          & ( ord_less_eq_set_nat @ B @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_93_set__eq__subset,axiom,
    ( ( ^ [Y2: set_Ground_gterm_f,Z: set_Ground_gterm_f] : ( Y2 = Z ) )
    = ( ^ [A4: set_Ground_gterm_f,B: set_Ground_gterm_f] :
          ( ( ord_le2735537439747282356term_f @ A4 @ B )
          & ( ord_le2735537439747282356term_f @ B @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_94_subset__trans,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
     => ( ( ord_le8976984241387448984_f_nat @ B2 @ C )
       => ( ord_le8976984241387448984_f_nat @ A @ C ) ) ) ).

% subset_trans
thf(fact_95_subset__trans,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f,C: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A @ B2 )
     => ( ( ord_le263819222746101927term_f @ B2 @ C )
       => ( ord_le263819222746101927term_f @ A @ C ) ) ) ).

% subset_trans
thf(fact_96_subset__trans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% subset_trans
thf(fact_97_subset__trans,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f,C: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A @ B2 )
     => ( ( ord_le2735537439747282356term_f @ B2 @ C )
       => ( ord_le2735537439747282356term_f @ A @ C ) ) ) ).

% subset_trans
thf(fact_98_Collect__mono,axiom,
    ! [P: set_Pr989862937836626183term_f > $o,Q: set_Pr989862937836626183term_f > $o] :
      ( ! [X2: set_Pr989862937836626183term_f] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le5727726935103693191term_f @ ( collec4490547877585280114term_f @ P ) @ ( collec4490547877585280114term_f @ Q ) ) ) ).

% Collect_mono
thf(fact_99_Collect__mono,axiom,
    ! [P: set_Pr5245412377734449720_f_nat > $o,Q: set_Pr5245412377734449720_f_nat > $o] :
      ( ! [X2: set_Pr5245412377734449720_f_nat] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le5386115319958680398_f_nat @ ( collec6964094325902095629_f_nat @ P ) @ ( collec6964094325902095629_f_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_100_Collect__mono,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X2: set_nat] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_101_Collect__mono,axiom,
    ! [P: set_Ground_gterm_f > $o,Q: set_Ground_gterm_f > $o] :
      ( ! [X2: set_Ground_gterm_f] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le2325249104654787220term_f @ ( collec1979342819066322687term_f @ P ) @ ( collec1979342819066322687term_f @ Q ) ) ) ).

% Collect_mono
thf(fact_102_Collect__mono,axiom,
    ! [P: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
      ( ! [X2: product_prod_f_nat] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le8976984241387448984_f_nat @ ( collec1417984143148964183_f_nat @ P ) @ ( collec1417984143148964183_f_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_103_Collect__mono,axiom,
    ! [P: produc7245736746747425831term_f > $o,Q: produc7245736746747425831term_f > $o] :
      ( ! [X2: produc7245736746747425831term_f] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le263819222746101927term_f @ ( collec3209569126566747026term_f @ P ) @ ( collec3209569126566747026term_f @ Q ) ) ) ).

% Collect_mono
thf(fact_104_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X2: nat] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_105_Collect__mono,axiom,
    ! [P: ground_gterm_f > $o,Q: ground_gterm_f > $o] :
      ( ! [X2: ground_gterm_f] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le2735537439747282356term_f @ ( collec3522850625759001119term_f @ P ) @ ( collec3522850625759001119term_f @ Q ) ) ) ).

% Collect_mono
thf(fact_106_subset__refl,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A @ A ) ).

% subset_refl
thf(fact_107_subset__refl,axiom,
    ! [A: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ A @ A ) ).

% subset_refl
thf(fact_108_subset__refl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% subset_refl
thf(fact_109_subset__refl,axiom,
    ! [A: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ A @ A ) ).

% subset_refl
thf(fact_110_subset__iff,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A4: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
        ! [T2: product_prod_f_nat] :
          ( ( member2678038196313615385_f_nat @ T2 @ A4 )
         => ( member2678038196313615385_f_nat @ T2 @ B ) ) ) ) ).

% subset_iff
thf(fact_111_subset__iff,axiom,
    ( ord_le263819222746101927term_f
    = ( ^ [A4: set_Pr989862937836626183term_f,B: set_Pr989862937836626183term_f] :
        ! [T2: produc7245736746747425831term_f] :
          ( ( member848276444142703440term_f @ T2 @ A4 )
         => ( member848276444142703440term_f @ T2 @ B ) ) ) ) ).

% subset_iff
thf(fact_112_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A4 )
         => ( member_nat @ T2 @ B ) ) ) ) ).

% subset_iff
thf(fact_113_subset__iff,axiom,
    ( ord_le2735537439747282356term_f
    = ( ^ [A4: set_Ground_gterm_f,B: set_Ground_gterm_f] :
        ! [T2: ground_gterm_f] :
          ( ( member5261315044688711901term_f @ T2 @ A4 )
         => ( member5261315044688711901term_f @ T2 @ B ) ) ) ) ).

% subset_iff
thf(fact_114_Set_OequalityD2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( A = B2 )
     => ( ord_le8976984241387448984_f_nat @ B2 @ A ) ) ).

% Set.equalityD2
thf(fact_115_Set_OequalityD2,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( A = B2 )
     => ( ord_le263819222746101927term_f @ B2 @ A ) ) ).

% Set.equalityD2
thf(fact_116_Set_OequalityD2,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A = B2 )
     => ( ord_less_eq_set_nat @ B2 @ A ) ) ).

% Set.equalityD2
thf(fact_117_Set_OequalityD2,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( A = B2 )
     => ( ord_le2735537439747282356term_f @ B2 @ A ) ) ).

% Set.equalityD2
thf(fact_118_equalityD1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( A = B2 )
     => ( ord_le8976984241387448984_f_nat @ A @ B2 ) ) ).

% equalityD1
thf(fact_119_equalityD1,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( A = B2 )
     => ( ord_le263819222746101927term_f @ A @ B2 ) ) ).

% equalityD1
thf(fact_120_equalityD1,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A = B2 )
     => ( ord_less_eq_set_nat @ A @ B2 ) ) ).

% equalityD1
thf(fact_121_equalityD1,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( A = B2 )
     => ( ord_le2735537439747282356term_f @ A @ B2 ) ) ).

% equalityD1
thf(fact_122_subset__eq,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A4: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
        ! [X: product_prod_f_nat] :
          ( ( member2678038196313615385_f_nat @ X @ A4 )
         => ( member2678038196313615385_f_nat @ X @ B ) ) ) ) ).

% subset_eq
thf(fact_123_subset__eq,axiom,
    ( ord_le263819222746101927term_f
    = ( ^ [A4: set_Pr989862937836626183term_f,B: set_Pr989862937836626183term_f] :
        ! [X: produc7245736746747425831term_f] :
          ( ( member848276444142703440term_f @ X @ A4 )
         => ( member848276444142703440term_f @ X @ B ) ) ) ) ).

% subset_eq
thf(fact_124_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A4 )
         => ( member_nat @ X @ B ) ) ) ) ).

% subset_eq
thf(fact_125_subset__eq,axiom,
    ( ord_le2735537439747282356term_f
    = ( ^ [A4: set_Ground_gterm_f,B: set_Ground_gterm_f] :
        ! [X: ground_gterm_f] :
          ( ( member5261315044688711901term_f @ X @ A4 )
         => ( member5261315044688711901term_f @ X @ B ) ) ) ) ).

% subset_eq
thf(fact_126_equalityE,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( A = B2 )
     => ~ ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
         => ~ ( ord_le8976984241387448984_f_nat @ B2 @ A ) ) ) ).

% equalityE
thf(fact_127_equalityE,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( A = B2 )
     => ~ ( ( ord_le263819222746101927term_f @ A @ B2 )
         => ~ ( ord_le263819222746101927term_f @ B2 @ A ) ) ) ).

% equalityE
thf(fact_128_equalityE,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A = B2 )
     => ~ ( ( ord_less_eq_set_nat @ A @ B2 )
         => ~ ( ord_less_eq_set_nat @ B2 @ A ) ) ) ).

% equalityE
thf(fact_129_equalityE,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( A = B2 )
     => ~ ( ( ord_le2735537439747282356term_f @ A @ B2 )
         => ~ ( ord_le2735537439747282356term_f @ B2 @ A ) ) ) ).

% equalityE
thf(fact_130_subsetD,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat,C2: product_prod_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
     => ( ( member2678038196313615385_f_nat @ C2 @ A )
       => ( member2678038196313615385_f_nat @ C2 @ B2 ) ) ) ).

% subsetD
thf(fact_131_subsetD,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f,C2: produc7245736746747425831term_f] :
      ( ( ord_le263819222746101927term_f @ A @ B2 )
     => ( ( member848276444142703440term_f @ C2 @ A )
       => ( member848276444142703440term_f @ C2 @ B2 ) ) ) ).

% subsetD
thf(fact_132_subsetD,axiom,
    ! [A: set_nat,B2: set_nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( member_nat @ C2 @ A )
       => ( member_nat @ C2 @ B2 ) ) ) ).

% subsetD
thf(fact_133_subsetD,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f,C2: ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A @ B2 )
     => ( ( member5261315044688711901term_f @ C2 @ A )
       => ( member5261315044688711901term_f @ C2 @ B2 ) ) ) ).

% subsetD
thf(fact_134_in__mono,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
     => ( ( member2678038196313615385_f_nat @ X3 @ A )
       => ( member2678038196313615385_f_nat @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_135_in__mono,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f,X3: produc7245736746747425831term_f] :
      ( ( ord_le263819222746101927term_f @ A @ B2 )
     => ( ( member848276444142703440term_f @ X3 @ A )
       => ( member848276444142703440term_f @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_136_in__mono,axiom,
    ! [A: set_nat,B2: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( member_nat @ X3 @ A )
       => ( member_nat @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_137_in__mono,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f,X3: ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A @ B2 )
     => ( ( member5261315044688711901term_f @ X3 @ A )
       => ( member5261315044688711901term_f @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_138_fset__cong,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
      ( ( ( fset_P3576968334923099475_f_nat @ X3 )
        = ( fset_P3576968334923099475_f_nat @ Y ) )
      = ( X3 = Y ) ) ).

% fset_cong
thf(fact_139_fset__cong,axiom,
    ! [X3: fset_P8018961893305114765rm_f_v,Y: fset_P8018961893305114765rm_f_v] :
      ( ( ( fset_P4617584883882644886rm_f_v @ X3 )
        = ( fset_P4617584883882644886rm_f_v @ Y ) )
      = ( X3 = Y ) ) ).

% fset_cong
thf(fact_140_Collect__subset,axiom,
    ! [A: set_se5570125628664547815term_f,P: set_Pr989862937836626183term_f > $o] :
      ( ord_le5727726935103693191term_f
      @ ( collec4490547877585280114term_f
        @ ^ [X: set_Pr989862937836626183term_f] :
            ( ( member6919674941555055664term_f @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_141_Collect__subset,axiom,
    ! [A: set_se1451291994713068270_f_nat,P: set_Pr5245412377734449720_f_nat > $o] :
      ( ord_le5386115319958680398_f_nat
      @ ( collec6964094325902095629_f_nat
        @ ^ [X: set_Pr5245412377734449720_f_nat] :
            ( ( member8345558847462883279_f_nat @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_142_Collect__subset,axiom,
    ! [A: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( member_set_nat @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_143_Collect__subset,axiom,
    ! [A: set_se421094335941524212term_f,P: set_Ground_gterm_f > $o] :
      ( ord_le2325249104654787220term_f
      @ ( collec1979342819066322687term_f
        @ ^ [X: set_Ground_gterm_f] :
            ( ( member9219338147399679421term_f @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_144_Collect__subset,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,P: product_prod_f_nat > $o] :
      ( ord_le8976984241387448984_f_nat
      @ ( collec1417984143148964183_f_nat
        @ ^ [X: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_145_Collect__subset,axiom,
    ! [A: set_Pr989862937836626183term_f,P: produc7245736746747425831term_f > $o] :
      ( ord_le263819222746101927term_f
      @ ( collec3209569126566747026term_f
        @ ^ [X: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_146_Collect__subset,axiom,
    ! [A: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_147_Collect__subset,axiom,
    ! [A: set_Ground_gterm_f,P: ground_gterm_f > $o] :
      ( ord_le2735537439747282356term_f
      @ ( collec3522850625759001119term_f
        @ ^ [X: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_148_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B2: set_nat,R2: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R2 @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A )
                        & ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_149_pigeonhole__infinite__rel,axiom,
    ! [A: set_Ground_gterm_f,B2: set_nat,R2: ground_gterm_f > nat > $o] :
      ( ~ ( finite1923210418194103261term_f @ A )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X2: ground_gterm_f] :
              ( ( member5261315044688711901term_f @ X2 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R2 @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B2 )
              & ~ ( finite1923210418194103261term_f
                  @ ( collec3522850625759001119term_f
                    @ ^ [A5: ground_gterm_f] :
                        ( ( member5261315044688711901term_f @ A5 @ A )
                        & ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_150_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B2: set_Ground_gterm_f,R2: nat > ground_gterm_f > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite1923210418194103261term_f @ B2 )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A )
             => ? [Xa: ground_gterm_f] :
                  ( ( member5261315044688711901term_f @ Xa @ B2 )
                  & ( R2 @ X2 @ Xa ) ) )
         => ? [X2: ground_gterm_f] :
              ( ( member5261315044688711901term_f @ X2 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A )
                        & ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_151_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B2: set_set_nat,R2: nat > set_nat > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A )
             => ? [Xa: set_nat] :
                  ( ( member_set_nat @ Xa @ B2 )
                  & ( R2 @ X2 @ Xa ) ) )
         => ? [X2: set_nat] :
              ( ( member_set_nat @ X2 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A )
                        & ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_152_pigeonhole__infinite__rel,axiom,
    ! [A: set_set_nat,B2: set_nat,R2: set_nat > nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ A )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X2: set_nat] :
              ( ( member_set_nat @ X2 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R2 @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B2 )
              & ~ ( finite1152437895449049373et_nat
                  @ ( collect_set_nat
                    @ ^ [A5: set_nat] :
                        ( ( member_set_nat @ A5 @ A )
                        & ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_153_pigeonhole__infinite__rel,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f,R2: ground_gterm_f > ground_gterm_f > $o] :
      ( ~ ( finite1923210418194103261term_f @ A )
     => ( ( finite1923210418194103261term_f @ B2 )
       => ( ! [X2: ground_gterm_f] :
              ( ( member5261315044688711901term_f @ X2 @ A )
             => ? [Xa: ground_gterm_f] :
                  ( ( member5261315044688711901term_f @ Xa @ B2 )
                  & ( R2 @ X2 @ Xa ) ) )
         => ? [X2: ground_gterm_f] :
              ( ( member5261315044688711901term_f @ X2 @ B2 )
              & ~ ( finite1923210418194103261term_f
                  @ ( collec3522850625759001119term_f
                    @ ^ [A5: ground_gterm_f] :
                        ( ( member5261315044688711901term_f @ A5 @ A )
                        & ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_154_pigeonhole__infinite__rel,axiom,
    ! [A: set_Ground_gterm_f,B2: set_set_nat,R2: ground_gterm_f > set_nat > $o] :
      ( ~ ( finite1923210418194103261term_f @ A )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( ! [X2: ground_gterm_f] :
              ( ( member5261315044688711901term_f @ X2 @ A )
             => ? [Xa: set_nat] :
                  ( ( member_set_nat @ Xa @ B2 )
                  & ( R2 @ X2 @ Xa ) ) )
         => ? [X2: set_nat] :
              ( ( member_set_nat @ X2 @ B2 )
              & ~ ( finite1923210418194103261term_f
                  @ ( collec3522850625759001119term_f
                    @ ^ [A5: ground_gterm_f] :
                        ( ( member5261315044688711901term_f @ A5 @ A )
                        & ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_155_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B2: set_Pr5245412377734449720_f_nat,R2: nat > product_prod_f_nat > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite3598747971073327385_f_nat @ B2 )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A )
             => ? [Xa: product_prod_f_nat] :
                  ( ( member2678038196313615385_f_nat @ Xa @ B2 )
                  & ( R2 @ X2 @ Xa ) ) )
         => ? [X2: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ X2 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A )
                        & ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_156_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B2: set_se421094335941524212term_f,R2: nat > set_Ground_gterm_f > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite6824738741894949053term_f @ B2 )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A )
             => ? [Xa: set_Ground_gterm_f] :
                  ( ( member9219338147399679421term_f @ Xa @ B2 )
                  & ( R2 @ X2 @ Xa ) ) )
         => ? [X2: set_Ground_gterm_f] :
              ( ( member9219338147399679421term_f @ X2 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A )
                        & ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_157_pigeonhole__infinite__rel,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_nat,R2: product_prod_f_nat > nat > $o] :
      ( ~ ( finite3598747971073327385_f_nat @ A )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X2: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ X2 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R2 @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B2 )
              & ~ ( finite3598747971073327385_f_nat
                  @ ( collec1417984143148964183_f_nat
                    @ ^ [A5: product_prod_f_nat] :
                        ( ( member2678038196313615385_f_nat @ A5 @ A )
                        & ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_158_finite__has__minimal2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,A6: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( member2678038196313615385_f_nat @ A6 @ A )
       => ? [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A )
            & ( ord_le2234929214960958178_f_nat @ X2 @ A6 )
            & ! [Xa: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ Xa @ A )
               => ( ( ord_le2234929214960958178_f_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_159_finite__has__minimal2,axiom,
    ! [A: set_se1451291994713068270_f_nat,A6: set_Pr5245412377734449720_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( member8345558847462883279_f_nat @ A6 @ A )
       => ? [X2: set_Pr5245412377734449720_f_nat] :
            ( ( member8345558847462883279_f_nat @ X2 @ A )
            & ( ord_le8976984241387448984_f_nat @ X2 @ A6 )
            & ! [Xa: set_Pr5245412377734449720_f_nat] :
                ( ( member8345558847462883279_f_nat @ Xa @ A )
               => ( ( ord_le8976984241387448984_f_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_160_finite__has__minimal2,axiom,
    ! [A: set_se5570125628664547815term_f,A6: set_Pr989862937836626183term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( member6919674941555055664term_f @ A6 @ A )
       => ? [X2: set_Pr989862937836626183term_f] :
            ( ( member6919674941555055664term_f @ X2 @ A )
            & ( ord_le263819222746101927term_f @ X2 @ A6 )
            & ! [Xa: set_Pr989862937836626183term_f] :
                ( ( member6919674941555055664term_f @ Xa @ A )
               => ( ( ord_le263819222746101927term_f @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_161_finite__has__minimal2,axiom,
    ! [A: set_fs7237271039932420712_f_nat,A6: fset_P6228066233360383026_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( member6677119207206147657_f_nat @ A6 @ A )
       => ? [X2: fset_P6228066233360383026_f_nat] :
            ( ( member6677119207206147657_f_nat @ X2 @ A )
            & ( ord_le1552505484586773650_f_nat @ X2 @ A6 )
            & ! [Xa: fset_P6228066233360383026_f_nat] :
                ( ( member6677119207206147657_f_nat @ Xa @ A )
               => ( ( ord_le1552505484586773650_f_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_162_finite__has__minimal2,axiom,
    ! [A: set_nat,A6: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A6 @ A )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( ord_less_eq_nat @ X2 @ A6 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_163_finite__has__minimal2,axiom,
    ! [A: set_set_nat,A6: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ A6 @ A )
       => ? [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A )
            & ( ord_less_eq_set_nat @ X2 @ A6 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A )
               => ( ( ord_less_eq_set_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_164_finite__has__minimal2,axiom,
    ! [A: set_se421094335941524212term_f,A6: set_Ground_gterm_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( member9219338147399679421term_f @ A6 @ A )
       => ? [X2: set_Ground_gterm_f] :
            ( ( member9219338147399679421term_f @ X2 @ A )
            & ( ord_le2735537439747282356term_f @ X2 @ A6 )
            & ! [Xa: set_Ground_gterm_f] :
                ( ( member9219338147399679421term_f @ Xa @ A )
               => ( ( ord_le2735537439747282356term_f @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_165_finite__has__maximal2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,A6: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( member2678038196313615385_f_nat @ A6 @ A )
       => ? [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A )
            & ( ord_le2234929214960958178_f_nat @ A6 @ X2 )
            & ! [Xa: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ Xa @ A )
               => ( ( ord_le2234929214960958178_f_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_166_finite__has__maximal2,axiom,
    ! [A: set_se1451291994713068270_f_nat,A6: set_Pr5245412377734449720_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( member8345558847462883279_f_nat @ A6 @ A )
       => ? [X2: set_Pr5245412377734449720_f_nat] :
            ( ( member8345558847462883279_f_nat @ X2 @ A )
            & ( ord_le8976984241387448984_f_nat @ A6 @ X2 )
            & ! [Xa: set_Pr5245412377734449720_f_nat] :
                ( ( member8345558847462883279_f_nat @ Xa @ A )
               => ( ( ord_le8976984241387448984_f_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_167_finite__has__maximal2,axiom,
    ! [A: set_se5570125628664547815term_f,A6: set_Pr989862937836626183term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( member6919674941555055664term_f @ A6 @ A )
       => ? [X2: set_Pr989862937836626183term_f] :
            ( ( member6919674941555055664term_f @ X2 @ A )
            & ( ord_le263819222746101927term_f @ A6 @ X2 )
            & ! [Xa: set_Pr989862937836626183term_f] :
                ( ( member6919674941555055664term_f @ Xa @ A )
               => ( ( ord_le263819222746101927term_f @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_168_finite__has__maximal2,axiom,
    ! [A: set_fs7237271039932420712_f_nat,A6: fset_P6228066233360383026_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( member6677119207206147657_f_nat @ A6 @ A )
       => ? [X2: fset_P6228066233360383026_f_nat] :
            ( ( member6677119207206147657_f_nat @ X2 @ A )
            & ( ord_le1552505484586773650_f_nat @ A6 @ X2 )
            & ! [Xa: fset_P6228066233360383026_f_nat] :
                ( ( member6677119207206147657_f_nat @ Xa @ A )
               => ( ( ord_le1552505484586773650_f_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_169_finite__has__maximal2,axiom,
    ! [A: set_nat,A6: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A6 @ A )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( ord_less_eq_nat @ A6 @ X2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_170_finite__has__maximal2,axiom,
    ! [A: set_set_nat,A6: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ A6 @ A )
       => ? [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A )
            & ( ord_less_eq_set_nat @ A6 @ X2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A )
               => ( ( ord_less_eq_set_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_171_finite__has__maximal2,axiom,
    ! [A: set_se421094335941524212term_f,A6: set_Ground_gterm_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( member9219338147399679421term_f @ A6 @ A )
       => ? [X2: set_Ground_gterm_f] :
            ( ( member9219338147399679421term_f @ X2 @ A )
            & ( ord_le2735537439747282356term_f @ A6 @ X2 )
            & ! [Xa: set_Ground_gterm_f] :
                ( ( member9219338147399679421term_f @ Xa @ A )
               => ( ( ord_le2735537439747282356term_f @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_172_rev__finite__subset,axiom,
    ! [B2: set_se421094335941524212term_f,A: set_se421094335941524212term_f] :
      ( ( finite6824738741894949053term_f @ B2 )
     => ( ( ord_le2325249104654787220term_f @ A @ B2 )
       => ( finite6824738741894949053term_f @ A ) ) ) ).

% rev_finite_subset
thf(fact_173_rev__finite__subset,axiom,
    ! [B2: set_se5570125628664547815term_f,A: set_se5570125628664547815term_f] :
      ( ( finite5058142709724841776term_f @ B2 )
     => ( ( ord_le5727726935103693191term_f @ A @ B2 )
       => ( finite5058142709724841776term_f @ A ) ) ) ).

% rev_finite_subset
thf(fact_174_rev__finite__subset,axiom,
    ! [B2: set_se1451291994713068270_f_nat,A: set_se1451291994713068270_f_nat] :
      ( ( finite879454794548843727_f_nat @ B2 )
     => ( ( ord_le5386115319958680398_f_nat @ A @ B2 )
       => ( finite879454794548843727_f_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_175_rev__finite__subset,axiom,
    ! [B2: set_set_nat,A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ A @ B2 )
       => ( finite1152437895449049373et_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_176_rev__finite__subset,axiom,
    ! [B2: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( finite3598747971073327385_f_nat @ B2 )
     => ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
       => ( finite3598747971073327385_f_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_177_rev__finite__subset,axiom,
    ! [B2: set_Pr989862937836626183term_f,A: set_Pr989862937836626183term_f] :
      ( ( finite4071141763865312848term_f @ B2 )
     => ( ( ord_le263819222746101927term_f @ A @ B2 )
       => ( finite4071141763865312848term_f @ A ) ) ) ).

% rev_finite_subset
thf(fact_178_rev__finite__subset,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A @ B2 )
       => ( finite_finite_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_179_rev__finite__subset,axiom,
    ! [B2: set_Ground_gterm_f,A: set_Ground_gterm_f] :
      ( ( finite1923210418194103261term_f @ B2 )
     => ( ( ord_le2735537439747282356term_f @ A @ B2 )
       => ( finite1923210418194103261term_f @ A ) ) ) ).

% rev_finite_subset
thf(fact_180_infinite__super,axiom,
    ! [S: set_se421094335941524212term_f,T3: set_se421094335941524212term_f] :
      ( ( ord_le2325249104654787220term_f @ S @ T3 )
     => ( ~ ( finite6824738741894949053term_f @ S )
       => ~ ( finite6824738741894949053term_f @ T3 ) ) ) ).

% infinite_super
thf(fact_181_infinite__super,axiom,
    ! [S: set_se5570125628664547815term_f,T3: set_se5570125628664547815term_f] :
      ( ( ord_le5727726935103693191term_f @ S @ T3 )
     => ( ~ ( finite5058142709724841776term_f @ S )
       => ~ ( finite5058142709724841776term_f @ T3 ) ) ) ).

% infinite_super
thf(fact_182_infinite__super,axiom,
    ! [S: set_se1451291994713068270_f_nat,T3: set_se1451291994713068270_f_nat] :
      ( ( ord_le5386115319958680398_f_nat @ S @ T3 )
     => ( ~ ( finite879454794548843727_f_nat @ S )
       => ~ ( finite879454794548843727_f_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_183_infinite__super,axiom,
    ! [S: set_set_nat,T3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ S @ T3 )
     => ( ~ ( finite1152437895449049373et_nat @ S )
       => ~ ( finite1152437895449049373et_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_184_infinite__super,axiom,
    ! [S: set_Pr5245412377734449720_f_nat,T3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ S @ T3 )
     => ( ~ ( finite3598747971073327385_f_nat @ S )
       => ~ ( finite3598747971073327385_f_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_185_infinite__super,axiom,
    ! [S: set_Pr989862937836626183term_f,T3: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ S @ T3 )
     => ( ~ ( finite4071141763865312848term_f @ S )
       => ~ ( finite4071141763865312848term_f @ T3 ) ) ) ).

% infinite_super
thf(fact_186_infinite__super,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T3 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_187_infinite__super,axiom,
    ! [S: set_Ground_gterm_f,T3: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ S @ T3 )
     => ( ~ ( finite1923210418194103261term_f @ S )
       => ~ ( finite1923210418194103261term_f @ T3 ) ) ) ).

% infinite_super
thf(fact_188_finite__subset,axiom,
    ! [A: set_se421094335941524212term_f,B2: set_se421094335941524212term_f] :
      ( ( ord_le2325249104654787220term_f @ A @ B2 )
     => ( ( finite6824738741894949053term_f @ B2 )
       => ( finite6824738741894949053term_f @ A ) ) ) ).

% finite_subset
thf(fact_189_finite__subset,axiom,
    ! [A: set_se5570125628664547815term_f,B2: set_se5570125628664547815term_f] :
      ( ( ord_le5727726935103693191term_f @ A @ B2 )
     => ( ( finite5058142709724841776term_f @ B2 )
       => ( finite5058142709724841776term_f @ A ) ) ) ).

% finite_subset
thf(fact_190_finite__subset,axiom,
    ! [A: set_se1451291994713068270_f_nat,B2: set_se1451291994713068270_f_nat] :
      ( ( ord_le5386115319958680398_f_nat @ A @ B2 )
     => ( ( finite879454794548843727_f_nat @ B2 )
       => ( finite879454794548843727_f_nat @ A ) ) ) ).

% finite_subset
thf(fact_191_finite__subset,axiom,
    ! [A: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B2 )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( finite1152437895449049373et_nat @ A ) ) ) ).

% finite_subset
thf(fact_192_finite__subset,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
     => ( ( finite3598747971073327385_f_nat @ B2 )
       => ( finite3598747971073327385_f_nat @ A ) ) ) ).

% finite_subset
thf(fact_193_finite__subset,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A @ B2 )
     => ( ( finite4071141763865312848term_f @ B2 )
       => ( finite4071141763865312848term_f @ A ) ) ) ).

% finite_subset
thf(fact_194_finite__subset,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( finite_finite_nat @ B2 )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_subset
thf(fact_195_finite__subset,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A @ B2 )
     => ( ( finite1923210418194103261term_f @ B2 )
       => ( finite1923210418194103261term_f @ A ) ) ) ).

% finite_subset
thf(fact_196_finite__fset,axiom,
    ! [S: fset_P8018961893305114765rm_f_v] : ( finite6110293842680836688rm_f_v @ ( fset_P4617584883882644886rm_f_v @ S ) ) ).

% finite_fset
thf(fact_197_finite__fset,axiom,
    ! [S: fset_Ground_gterm_f] : ( finite1923210418194103261term_f @ ( fset_Ground_gterm_f2 @ S ) ) ).

% finite_fset
thf(fact_198_finite__fset,axiom,
    ! [S: fset_nat] : ( finite_finite_nat @ ( fset_nat2 @ S ) ) ).

% finite_fset
thf(fact_199_finite__fset,axiom,
    ! [S: fset_P6228066233360383026_f_nat] : ( finite3598747971073327385_f_nat @ ( fset_P3576968334923099475_f_nat @ S ) ) ).

% finite_fset
thf(fact_200_finite__fset,axiom,
    ! [S: fset_s250637050704127098term_f] : ( finite6824738741894949053term_f @ ( fset_s3373063111670843395term_f @ S ) ) ).

% finite_fset
thf(fact_201_finite__fset,axiom,
    ! [S: fset_s9063456872818339949term_f] : ( finite5058142709724841776term_f @ ( fset_s6989464911991047030term_f @ S ) ) ).

% finite_fset
thf(fact_202_finite__fset,axiom,
    ! [S: fset_s3012505784449309928_f_nat] : ( finite879454794548843727_f_nat @ ( fset_s10630165170805001_f_nat @ S ) ) ).

% finite_fset
thf(fact_203_finite__fset,axiom,
    ! [S: fset_set_nat] : ( finite1152437895449049373et_nat @ ( fset_set_nat2 @ S ) ) ).

% finite_fset
thf(fact_204_fset_Ofset__induct,axiom,
    ! [Y: set_Pr989862937836626183term_f,P: set_Pr989862937836626183term_f > $o] :
      ( ( member6919674941555055664term_f @ Y @ ( collec4490547877585280114term_f @ finite4071141763865312848term_f ) )
     => ( ! [X2: fset_P5324236452418035597term_f] : ( P @ ( fset_P5939342050162774678term_f @ X2 ) )
       => ( P @ Y ) ) ) ).

% fset.fset_induct
thf(fact_205_fset_Ofset__induct,axiom,
    ! [Y: set_Pr8827868859434726151rm_f_v,P: set_Pr8827868859434726151rm_f_v > $o] :
      ( ( member4041562125048333488rm_f_v @ Y @ ( collec107294023721454066rm_f_v @ finite6110293842680836688rm_f_v ) )
     => ( ! [X2: fset_P8018961893305114765rm_f_v] : ( P @ ( fset_P4617584883882644886rm_f_v @ X2 ) )
       => ( P @ Y ) ) ) ).

% fset.fset_induct
thf(fact_206_fset_Ofset__induct,axiom,
    ! [Y: set_Ground_gterm_f,P: set_Ground_gterm_f > $o] :
      ( ( member9219338147399679421term_f @ Y @ ( collec1979342819066322687term_f @ finite1923210418194103261term_f ) )
     => ( ! [X2: fset_Ground_gterm_f] : ( P @ ( fset_Ground_gterm_f2 @ X2 ) )
       => ( P @ Y ) ) ) ).

% fset.fset_induct
thf(fact_207_fset_Ofset__induct,axiom,
    ! [Y: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ Y @ ( collect_set_nat @ finite_finite_nat ) )
     => ( ! [X2: fset_nat] : ( P @ ( fset_nat2 @ X2 ) )
       => ( P @ Y ) ) ) ).

% fset.fset_induct
thf(fact_208_fset_Ofset__induct,axiom,
    ! [Y: set_Pr5245412377734449720_f_nat,P: set_Pr5245412377734449720_f_nat > $o] :
      ( ( member8345558847462883279_f_nat @ Y @ ( collec6964094325902095629_f_nat @ finite3598747971073327385_f_nat ) )
     => ( ! [X2: fset_P6228066233360383026_f_nat] : ( P @ ( fset_P3576968334923099475_f_nat @ X2 ) )
       => ( P @ Y ) ) ) ).

% fset.fset_induct
thf(fact_209_fset_Ofset__induct,axiom,
    ! [Y: set_se421094335941524212term_f,P: set_se421094335941524212term_f > $o] :
      ( ( member7620211966956398237term_f @ Y @ ( collec8921018771386908639term_f @ finite6824738741894949053term_f ) )
     => ( ! [X2: fset_s250637050704127098term_f] : ( P @ ( fset_s3373063111670843395term_f @ X2 ) )
       => ( P @ Y ) ) ) ).

% fset.fset_induct
thf(fact_210_fset_Ofset__induct,axiom,
    ! [Y: set_se5570125628664547815term_f,P: set_se5570125628664547815term_f > $o] :
      ( ( member4407653303224918288term_f @ Y @ ( collec6999422248803473746term_f @ finite5058142709724841776term_f ) )
     => ( ! [X2: fset_s9063456872818339949term_f] : ( P @ ( fset_s6989464911991047030term_f @ X2 ) )
       => ( P @ Y ) ) ) ).

% fset.fset_induct
thf(fact_211_fset_Ofset__induct,axiom,
    ! [Y: set_se1451291994713068270_f_nat,P: set_se1451291994713068270_f_nat > $o] :
      ( ( member5502175894719687813_f_nat @ Y @ ( collec2226960723385805251_f_nat @ finite879454794548843727_f_nat ) )
     => ( ! [X2: fset_s3012505784449309928_f_nat] : ( P @ ( fset_s10630165170805001_f_nat @ X2 ) )
       => ( P @ Y ) ) ) ).

% fset.fset_induct
thf(fact_212_fset_Ofset__induct,axiom,
    ! [Y: set_set_nat,P: set_set_nat > $o] :
      ( ( member_set_set_nat @ Y @ ( collect_set_set_nat @ finite1152437895449049373et_nat ) )
     => ( ! [X2: fset_set_nat] : ( P @ ( fset_set_nat2 @ X2 ) )
       => ( P @ Y ) ) ) ).

% fset.fset_induct
thf(fact_213__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062_092_060exists_062A_O_Arr2__of__rr2__rel_A_092_060F_062_ARs_Ar_A_061_ASome_AA_A_092_060Longrightarrow_062_Athesis_059_Arr2__of__rr2__rel_A_092_060F_062_ARs_Ar_A_061_ANone_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ( ~ ? [A2: tree_r733329426570293750tion_f] :
          ( ( fOR_rr7226795042121552277el_f_v @ f2 @ rs @ r )
          = ( some_T4055341017772447441tion_f @ A2 ) )
   => ( ( fOR_rr7226795042121552277el_f_v @ f2 @ rs @ r )
      = none_T5277256714714431317tion_f ) ) ).

% \<open>\<And>thesis. \<lbrakk>\<exists>A. rr2_of_rr2_rel \<F> Rs r = Some A \<Longrightarrow> thesis; rr2_of_rr2_rel \<F> Rs r = None \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_214_mem__Sigma__iff,axiom,
    ! [A6: produc7245736746747425831term_f,B3: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f,B2: produc7245736746747425831term_f > set_Pr989862937836626183term_f] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ A6 @ B3 ) @ ( produc4509621232857779464term_f @ A @ B2 ) )
      = ( ( member848276444142703440term_f @ A6 @ A )
        & ( member848276444142703440term_f @ B3 @ ( B2 @ A6 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_215_mem__Sigma__iff,axiom,
    ! [A6: produc7245736746747425831term_f,B3: nat,A: set_Pr989862937836626183term_f,B2: produc7245736746747425831term_f > set_nat] :
      ( ( member3662771866509663549_f_nat @ ( produc5217865480721399968_f_nat @ A6 @ B3 ) @ ( produc8380933303712717999_f_nat @ A @ B2 ) )
      = ( ( member848276444142703440term_f @ A6 @ A )
        & ( member_nat @ B3 @ ( B2 @ A6 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_216_mem__Sigma__iff,axiom,
    ! [A6: nat,B3: produc7245736746747425831term_f,A: set_nat,B2: nat > set_Pr989862937836626183term_f] :
      ( ( member4003629742723934691term_f @ ( produc6263668818416315902term_f @ A6 @ B3 ) @ ( produc203364604552858125term_f @ A @ B2 ) )
      = ( ( member_nat @ A6 @ A )
        & ( member848276444142703440term_f @ B3 @ ( B2 @ A6 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_217_mem__Sigma__iff,axiom,
    ! [A6: nat,B3: nat,A: set_nat,B2: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B3 ) @ ( produc457027306803732586at_nat @ A @ B2 ) )
      = ( ( member_nat @ A6 @ A )
        & ( member_nat @ B3 @ ( B2 @ A6 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_218_mem__Sigma__iff,axiom,
    ! [A6: ground_gterm_f,B3: ground_gterm_f,A: set_Ground_gterm_f,B2: ground_gterm_f > set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A6 @ B3 ) @ ( produc6043514909933191368term_f @ A @ B2 ) )
      = ( ( member5261315044688711901term_f @ A6 @ A )
        & ( member5261315044688711901term_f @ B3 @ ( B2 @ A6 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_219_mem__Collect__eq,axiom,
    ! [A6: produc7245736746747425831term_f,P: produc7245736746747425831term_f > $o] :
      ( ( member848276444142703440term_f @ A6 @ ( collec3209569126566747026term_f @ P ) )
      = ( P @ A6 ) ) ).

% mem_Collect_eq
thf(fact_220_mem__Collect__eq,axiom,
    ! [A6: ground_gterm_f,P: ground_gterm_f > $o] :
      ( ( member5261315044688711901term_f @ A6 @ ( collec3522850625759001119term_f @ P ) )
      = ( P @ A6 ) ) ).

% mem_Collect_eq
thf(fact_221_mem__Collect__eq,axiom,
    ! [A6: nat,P: nat > $o] :
      ( ( member_nat @ A6 @ ( collect_nat @ P ) )
      = ( P @ A6 ) ) ).

% mem_Collect_eq
thf(fact_222_mem__Collect__eq,axiom,
    ! [A6: set_Pr989862937836626183term_f,P: set_Pr989862937836626183term_f > $o] :
      ( ( member6919674941555055664term_f @ A6 @ ( collec4490547877585280114term_f @ P ) )
      = ( P @ A6 ) ) ).

% mem_Collect_eq
thf(fact_223_mem__Collect__eq,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,P: set_Pr5245412377734449720_f_nat > $o] :
      ( ( member8345558847462883279_f_nat @ A6 @ ( collec6964094325902095629_f_nat @ P ) )
      = ( P @ A6 ) ) ).

% mem_Collect_eq
thf(fact_224_mem__Collect__eq,axiom,
    ! [A6: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A6 @ ( collect_set_nat @ P ) )
      = ( P @ A6 ) ) ).

% mem_Collect_eq
thf(fact_225_mem__Collect__eq,axiom,
    ! [A6: set_Ground_gterm_f,P: set_Ground_gterm_f > $o] :
      ( ( member9219338147399679421term_f @ A6 @ ( collec1979342819066322687term_f @ P ) )
      = ( P @ A6 ) ) ).

% mem_Collect_eq
thf(fact_226_Collect__mem__eq,axiom,
    ! [A: set_Pr989862937836626183term_f] :
      ( ( collec3209569126566747026term_f
        @ ^ [X: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_227_Collect__mem__eq,axiom,
    ! [A: set_Ground_gterm_f] :
      ( ( collec3522850625759001119term_f
        @ ^ [X: ground_gterm_f] : ( member5261315044688711901term_f @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_228_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_229_Collect__mem__eq,axiom,
    ! [A: set_se5570125628664547815term_f] :
      ( ( collec4490547877585280114term_f
        @ ^ [X: set_Pr989862937836626183term_f] : ( member6919674941555055664term_f @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_230_Collect__mem__eq,axiom,
    ! [A: set_se1451291994713068270_f_nat] :
      ( ( collec6964094325902095629_f_nat
        @ ^ [X: set_Pr5245412377734449720_f_nat] : ( member8345558847462883279_f_nat @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_231_Collect__mem__eq,axiom,
    ! [A: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] : ( member_set_nat @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_232_Collect__mem__eq,axiom,
    ! [A: set_se421094335941524212term_f] :
      ( ( collec1979342819066322687term_f
        @ ^ [X: set_Ground_gterm_f] : ( member9219338147399679421term_f @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_233_Collect__cong,axiom,
    ! [P: ground_gterm_f > $o,Q: ground_gterm_f > $o] :
      ( ! [X2: ground_gterm_f] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec3522850625759001119term_f @ P )
        = ( collec3522850625759001119term_f @ Q ) ) ) ).

% Collect_cong
thf(fact_234_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X2: nat] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_235_Collect__cong,axiom,
    ! [P: set_Pr989862937836626183term_f > $o,Q: set_Pr989862937836626183term_f > $o] :
      ( ! [X2: set_Pr989862937836626183term_f] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec4490547877585280114term_f @ P )
        = ( collec4490547877585280114term_f @ Q ) ) ) ).

% Collect_cong
thf(fact_236_Collect__cong,axiom,
    ! [P: set_Pr5245412377734449720_f_nat > $o,Q: set_Pr5245412377734449720_f_nat > $o] :
      ( ! [X2: set_Pr5245412377734449720_f_nat] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec6964094325902095629_f_nat @ P )
        = ( collec6964094325902095629_f_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_237_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X2: set_nat] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_238_Collect__cong,axiom,
    ! [P: set_Ground_gterm_f > $o,Q: set_Ground_gterm_f > $o] :
      ( ! [X2: set_Ground_gterm_f] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec1979342819066322687term_f @ P )
        = ( collec1979342819066322687term_f @ Q ) ) ) ).

% Collect_cong
thf(fact_239_SigmaI,axiom,
    ! [A6: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f,B3: produc7245736746747425831term_f,B2: produc7245736746747425831term_f > set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ A6 @ A )
     => ( ( member848276444142703440term_f @ B3 @ ( B2 @ A6 ) )
       => ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ A6 @ B3 ) @ ( produc4509621232857779464term_f @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_240_SigmaI,axiom,
    ! [A6: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f,B3: nat,B2: produc7245736746747425831term_f > set_nat] :
      ( ( member848276444142703440term_f @ A6 @ A )
     => ( ( member_nat @ B3 @ ( B2 @ A6 ) )
       => ( member3662771866509663549_f_nat @ ( produc5217865480721399968_f_nat @ A6 @ B3 ) @ ( produc8380933303712717999_f_nat @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_241_SigmaI,axiom,
    ! [A6: nat,A: set_nat,B3: produc7245736746747425831term_f,B2: nat > set_Pr989862937836626183term_f] :
      ( ( member_nat @ A6 @ A )
     => ( ( member848276444142703440term_f @ B3 @ ( B2 @ A6 ) )
       => ( member4003629742723934691term_f @ ( produc6263668818416315902term_f @ A6 @ B3 ) @ ( produc203364604552858125term_f @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_242_SigmaI,axiom,
    ! [A6: nat,A: set_nat,B3: nat,B2: nat > set_nat] :
      ( ( member_nat @ A6 @ A )
     => ( ( member_nat @ B3 @ ( B2 @ A6 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B3 ) @ ( produc457027306803732586at_nat @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_243_SigmaI,axiom,
    ! [A6: ground_gterm_f,A: set_Ground_gterm_f,B3: ground_gterm_f,B2: ground_gterm_f > set_Ground_gterm_f] :
      ( ( member5261315044688711901term_f @ A6 @ A )
     => ( ( member5261315044688711901term_f @ B3 @ ( B2 @ A6 ) )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A6 @ B3 ) @ ( produc6043514909933191368term_f @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_244_less__eq__option__Some,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le6049919707012941544_f_nat @ ( some_s2191777106535767827_f_nat @ X3 ) @ ( some_s2191777106535767827_f_nat @ Y ) )
      = ( ord_le8976984241387448984_f_nat @ X3 @ Y ) ) ).

% less_eq_option_Some
thf(fact_245_less__eq__option__Some,axiom,
    ! [X3: set_Pr989862937836626183term_f,Y: set_Pr989862937836626183term_f] :
      ( ( ord_le1859086841259116141term_f @ ( some_s6688184489335603692term_f @ X3 ) @ ( some_s6688184489335603692term_f @ Y ) )
      = ( ord_le263819222746101927term_f @ X3 @ Y ) ) ).

% less_eq_option_Some
thf(fact_246_less__eq__option__Some,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1231092944882968930_f_nat @ ( some_f8740016242142974861_f_nat @ X3 ) @ ( some_f8740016242142974861_f_nat @ Y ) )
      = ( ord_le1552505484586773650_f_nat @ X3 @ Y ) ) ).

% less_eq_option_Some
thf(fact_247_less__eq__option__Some,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y ) )
      = ( ord_less_eq_nat @ X3 @ Y ) ) ).

% less_eq_option_Some
thf(fact_248_less__eq__option__Some,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_le2843612097646854710et_nat @ ( some_set_nat @ X3 ) @ ( some_set_nat @ Y ) )
      = ( ord_less_eq_set_nat @ X3 @ Y ) ) ).

% less_eq_option_Some
thf(fact_249_less__eq__option__Some,axiom,
    ! [X3: set_Ground_gterm_f,Y: set_Ground_gterm_f] :
      ( ( ord_le9076208610553477754term_f @ ( some_s5628733402641973625term_f @ X3 ) @ ( some_s5628733402641973625term_f @ Y ) )
      = ( ord_le2735537439747282356term_f @ X3 @ Y ) ) ).

% less_eq_option_Some
thf(fact_250_eval__rr12__rel__sig_I2_J,axiom,
    ! [F2: set_Pr5245412377734449720_f_nat,Rs: list_s8746099396510718605rm_f_v,R22: fOR_rr2_rel_FOR_ftrs] :
      ( ord_le263819222746101927term_f @ ( fOR_eval_rr2_rel_f_v @ F2 @ Rs @ R22 )
      @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
        @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) ) ).

% eval_rr12_rel_sig(2)
thf(fact_251__092_060T_062_092_060_094sub_062G__equivalent__def,axiom,
    ( ground_T_G_f
    = ( ^ [F: set_Pr5245412377734449720_f_nat] :
          ( collec3522850625759001119term_f
          @ ^ [S2: ground_gterm_f] : ( ord_le8976984241387448984_f_nat @ ( ground_funas_gterm_f @ S2 ) @ F ) ) ) ) ).

% \<T>\<^sub>G_equivalent_def
thf(fact_252_infinite__cartesian__product,axiom,
    ! [A: set_f,B2: set_nat] :
      ( ~ ( finite_finite_f @ A )
     => ( ~ ( finite_finite_nat @ B2 )
       => ~ ( finite3598747971073327385_f_nat
            @ ( product_Sigma_f_nat @ A
              @ ^ [Uu: f] : B2 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_253_infinite__cartesian__product,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ~ ( finite_finite_nat @ B2 )
       => ~ ( finite6177210948735845034at_nat
            @ ( produc457027306803732586at_nat @ A
              @ ^ [Uu: nat] : B2 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_254_infinite__cartesian__product,axiom,
    ! [A: set_Ground_gterm_f,B2: set_nat] :
      ( ~ ( finite1923210418194103261term_f @ A )
     => ( ~ ( finite_finite_nat @ B2 )
       => ~ ( finite311254710766229066_f_nat
            @ ( produc4746993288989285692_f_nat @ A
              @ ^ [Uu: ground_gterm_f] : B2 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_255_infinite__cartesian__product,axiom,
    ! [A: set_nat,B2: set_Ground_gterm_f] :
      ( ~ ( finite_finite_nat @ A )
     => ( ~ ( finite1923210418194103261term_f @ B2 )
       => ~ ( finite4078170753649457648term_f
            @ ( produc3100878011685873178term_f @ A
              @ ^ [Uu: nat] : B2 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_256_infinite__cartesian__product,axiom,
    ! [A: set_nat,B2: set_set_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ~ ( finite1152437895449049373et_nat @ B2 )
       => ~ ( finite962763233036973920et_nat
            @ ( produc8883945523214541856et_nat @ A
              @ ^ [Uu: nat] : B2 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_257_infinite__cartesian__product,axiom,
    ! [A: set_set_nat,B2: set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ A )
     => ( ~ ( finite_finite_nat @ B2 )
       => ~ ( finite6054027020396858976at_nat
            @ ( produc5318310618559172640at_nat @ A
              @ ^ [Uu: set_nat] : B2 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_258_infinite__cartesian__product,axiom,
    ! [A: set_Ground_gterm_f,B2: set_set_nat] :
      ( ~ ( finite1923210418194103261term_f @ A )
     => ( ~ ( finite1152437895449049373et_nat @ B2 )
       => ~ ( finite3836040682740435712et_nat
            @ ( produc2061566356152752370et_nat @ A
              @ ^ [Uu: ground_gterm_f] : B2 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_259_infinite__cartesian__product,axiom,
    ! [A: set_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ~ ( finite3598747971073327385_f_nat @ B2 )
       => ~ ( finite5167703378860198236_f_nat
            @ ( produc1258315153622678812_f_nat @ A
              @ ^ [Uu: nat] : B2 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_260_infinite__cartesian__product,axiom,
    ! [A: set_nat,B2: set_se421094335941524212term_f] :
      ( ~ ( finite_finite_nat @ A )
     => ( ~ ( finite6824738741894949053term_f @ B2 )
       => ~ ( finite2583538073261184720term_f
            @ ( produc5271949834123747066term_f @ A
              @ ^ [Uu: nat] : B2 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_261_infinite__cartesian__product,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_nat] :
      ( ~ ( finite3598747971073327385_f_nat @ A )
     => ( ~ ( finite_finite_nat @ B2 )
       => ~ ( finite1611537298927112028at_nat
            @ ( produc4600693851930136092at_nat @ A
              @ ^ [Uu: product_prod_f_nat] : B2 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_262_Times__subset__cancel2,axiom,
    ! [X3: produc7245736746747425831term_f,C: set_Pr989862937836626183term_f,A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( member848276444142703440term_f @ X3 @ C )
     => ( ( ord_le8624717471634568240term_f
          @ ( produc4719704871242175707term_f @ A
            @ ^ [Uu: product_prod_f_nat] : C )
          @ ( produc4719704871242175707term_f @ B2
            @ ^ [Uu: product_prod_f_nat] : C ) )
        = ( ord_le8976984241387448984_f_nat @ A @ B2 ) ) ) ).

% Times_subset_cancel2
thf(fact_263_Times__subset__cancel2,axiom,
    ! [X3: nat,C: set_nat,A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( member_nat @ X3 @ C )
     => ( ( ord_le703013924939395507at_nat
          @ ( produc4600693851930136092at_nat @ A
            @ ^ [Uu: product_prod_f_nat] : C )
          @ ( produc4600693851930136092at_nat @ B2
            @ ^ [Uu: product_prod_f_nat] : C ) )
        = ( ord_le8976984241387448984_f_nat @ A @ B2 ) ) ) ).

% Times_subset_cancel2
thf(fact_264_Times__subset__cancel2,axiom,
    ! [X3: produc7245736746747425831term_f,C: set_Pr989862937836626183term_f,A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ X3 @ C )
     => ( ( ord_le2215455632037200999term_f
          @ ( produc4509621232857779464term_f @ A
            @ ^ [Uu: produc7245736746747425831term_f] : C )
          @ ( produc4509621232857779464term_f @ B2
            @ ^ [Uu: produc7245736746747425831term_f] : C ) )
        = ( ord_le263819222746101927term_f @ A @ B2 ) ) ) ).

% Times_subset_cancel2
thf(fact_265_Times__subset__cancel2,axiom,
    ! [X3: nat,C: set_nat,A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( member_nat @ X3 @ C )
     => ( ( ord_le3531222934869404092_f_nat
          @ ( produc8380933303712717999_f_nat @ A
            @ ^ [Uu: produc7245736746747425831term_f] : C )
          @ ( produc8380933303712717999_f_nat @ B2
            @ ^ [Uu: produc7245736746747425831term_f] : C ) )
        = ( ord_le263819222746101927term_f @ A @ B2 ) ) ) ).

% Times_subset_cancel2
thf(fact_266_Times__subset__cancel2,axiom,
    ! [X3: produc7245736746747425831term_f,C: set_Pr989862937836626183term_f,A: set_nat,B2: set_nat] :
      ( ( member848276444142703440term_f @ X3 @ C )
     => ( ( ord_le354178036722549602term_f
          @ ( produc203364604552858125term_f @ A
            @ ^ [Uu: nat] : C )
          @ ( produc203364604552858125term_f @ B2
            @ ^ [Uu: nat] : C ) )
        = ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).

% Times_subset_cancel2
thf(fact_267_Times__subset__cancel2,axiom,
    ! [X3: nat,C: set_nat,A: set_nat,B2: set_nat] :
      ( ( member_nat @ X3 @ C )
     => ( ( ord_le3146513528884898305at_nat
          @ ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : C )
          @ ( produc457027306803732586at_nat @ B2
            @ ^ [Uu: nat] : C ) )
        = ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).

% Times_subset_cancel2
thf(fact_268_Times__subset__cancel2,axiom,
    ! [X3: produc7245736746747425831term_f,C: set_Pr989862937836626183term_f,A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ X3 @ C )
     => ( ( ord_le3661770224690064922term_f
          @ ( produc5635716683150298555term_f @ A
            @ ^ [Uu: ground_gterm_f] : C )
          @ ( produc5635716683150298555term_f @ B2
            @ ^ [Uu: ground_gterm_f] : C ) )
        = ( ord_le2735537439747282356term_f @ A @ B2 ) ) ) ).

% Times_subset_cancel2
thf(fact_269_Times__subset__cancel2,axiom,
    ! [X3: nat,C: set_nat,A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( member_nat @ X3 @ C )
     => ( ( ord_le6666376546550546761_f_nat
          @ ( produc4746993288989285692_f_nat @ A
            @ ^ [Uu: ground_gterm_f] : C )
          @ ( produc4746993288989285692_f_nat @ B2
            @ ^ [Uu: ground_gterm_f] : C ) )
        = ( ord_le2735537439747282356term_f @ A @ B2 ) ) ) ).

% Times_subset_cancel2
thf(fact_270_Times__subset__cancel2,axiom,
    ! [X3: nat,C: set_nat,A: set_f,B2: set_f] :
      ( ( member_nat @ X3 @ C )
     => ( ( ord_le8976984241387448984_f_nat
          @ ( product_Sigma_f_nat @ A
            @ ^ [Uu: f] : C )
          @ ( product_Sigma_f_nat @ B2
            @ ^ [Uu: f] : C ) )
        = ( ord_less_eq_set_f @ A @ B2 ) ) ) ).

% Times_subset_cancel2
thf(fact_271_Times__subset__cancel2,axiom,
    ! [X3: ground_gterm_f,C: set_Ground_gterm_f,A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( member5261315044688711901term_f @ X3 @ C )
     => ( ( ord_le263819222746101927term_f
          @ ( produc6043514909933191368term_f @ A
            @ ^ [Uu: ground_gterm_f] : C )
          @ ( produc6043514909933191368term_f @ B2
            @ ^ [Uu: ground_gterm_f] : C ) )
        = ( ord_le2735537439747282356term_f @ A @ B2 ) ) ) ).

% Times_subset_cancel2
thf(fact_272__092_060T_062_092_060_094sub_062G__funas__gterm__conv,axiom,
    ! [S3: ground_gterm_f,F2: set_Pr5245412377734449720_f_nat] :
      ( ( member5261315044688711901term_f @ S3 @ ( ground_T_G_f @ F2 ) )
      = ( ord_le8976984241387448984_f_nat @ ( ground_funas_gterm_f @ S3 ) @ F2 ) ) ).

% \<T>\<^sub>G_funas_gterm_conv
thf(fact_273__092_060T_062_092_060_094sub_062G__complete,axiom,
    ! [S3: ground_gterm_f,F2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( ground_funas_gterm_f @ S3 ) @ F2 )
     => ( member5261315044688711901term_f @ S3 @ ( ground_T_G_f @ F2 ) ) ) ).

% \<T>\<^sub>G_complete
thf(fact_274__092_060T_062_092_060_094sub_062G__sound,axiom,
    ! [S3: ground_gterm_f,F2: set_Pr5245412377734449720_f_nat] :
      ( ( member5261315044688711901term_f @ S3 @ ( ground_T_G_f @ F2 ) )
     => ( ord_le8976984241387448984_f_nat @ ( ground_funas_gterm_f @ S3 ) @ F2 ) ) ).

% \<T>\<^sub>G_sound
thf(fact_275_Sigma__mono,axiom,
    ! [A: set_f,C: set_f,B2: f > set_nat,D: f > set_nat] :
      ( ( ord_less_eq_set_f @ A @ C )
     => ( ! [X2: f] :
            ( ( member_f @ X2 @ A )
           => ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le8976984241387448984_f_nat @ ( product_Sigma_f_nat @ A @ B2 ) @ ( product_Sigma_f_nat @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_276_Sigma__mono,axiom,
    ! [A: set_nat,C: set_nat,B2: nat > set_nat,D: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le3146513528884898305at_nat @ ( produc457027306803732586at_nat @ A @ B2 ) @ ( produc457027306803732586at_nat @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_277_Sigma__mono,axiom,
    ! [A: set_nat,C: set_nat,B2: nat > set_Ground_gterm_f,D: nat > set_Ground_gterm_f] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ( ord_le2735537439747282356term_f @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le4584778628501795311term_f @ ( produc3100878011685873178term_f @ A @ B2 ) @ ( produc3100878011685873178term_f @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_278_Sigma__mono,axiom,
    ! [A: set_Ground_gterm_f,C: set_Ground_gterm_f,B2: ground_gterm_f > set_nat,D: ground_gterm_f > set_nat] :
      ( ( ord_le2735537439747282356term_f @ A @ C )
     => ( ! [X2: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X2 @ A )
           => ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le6666376546550546761_f_nat @ ( produc4746993288989285692_f_nat @ A @ B2 ) @ ( produc4746993288989285692_f_nat @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_279_Sigma__mono,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat,B2: product_prod_f_nat > set_nat,D: product_prod_f_nat > set_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ C )
     => ( ! [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A )
           => ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le703013924939395507at_nat @ ( produc4600693851930136092at_nat @ A @ B2 ) @ ( produc4600693851930136092at_nat @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_280_Sigma__mono,axiom,
    ! [A: set_nat,C: set_nat,B2: nat > set_Pr5245412377734449720_f_nat,D: nat > set_Pr5245412377734449720_f_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ( ord_le8976984241387448984_f_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le697507202650467251_f_nat @ ( produc1258315153622678812_f_nat @ A @ B2 ) @ ( produc1258315153622678812_f_nat @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_281_Sigma__mono,axiom,
    ! [A: set_Ground_gterm_f,C: set_Ground_gterm_f,B2: ground_gterm_f > set_Ground_gterm_f,D: ground_gterm_f > set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A @ C )
     => ( ! [X2: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X2 @ A )
           => ( ord_le2735537439747282356term_f @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le263819222746101927term_f @ ( produc6043514909933191368term_f @ A @ B2 ) @ ( produc6043514909933191368term_f @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_282_Sigma__mono,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat,B2: product_prod_f_nat > set_Ground_gterm_f,D: product_prod_f_nat > set_Ground_gterm_f] :
      ( ( ord_le8976984241387448984_f_nat @ A @ C )
     => ( ! [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A )
           => ( ord_le2735537439747282356term_f @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le6888376801088063933term_f @ ( produc2338249810576627688term_f @ A @ B2 ) @ ( produc2338249810576627688term_f @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_283_Sigma__mono,axiom,
    ! [A: set_Ground_gterm_f,C: set_Ground_gterm_f,B2: ground_gterm_f > set_Pr5245412377734449720_f_nat,D: ground_gterm_f > set_Pr5245412377734449720_f_nat] :
      ( ( ord_le2735537439747282356term_f @ A @ C )
     => ( ! [X2: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X2 @ A )
           => ( ord_le8976984241387448984_f_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le3590369893943491323_f_nat @ ( produc5546477881898244078_f_nat @ A @ B2 ) @ ( produc5546477881898244078_f_nat @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_284_Sigma__mono,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat,B2: product_prod_f_nat > set_Pr5245412377734449720_f_nat,D: product_prod_f_nat > set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ C )
     => ( ! [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A )
           => ( ord_le8976984241387448984_f_nat @ ( B2 @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le8946695678602162533_f_nat @ ( produc7667213020494768334_f_nat @ A @ B2 ) @ ( produc7667213020494768334_f_nat @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_285_prod_Oinject,axiom,
    ! [X1: ground_gterm_f,X22: ground_gterm_f,Y1: ground_gterm_f,Y22: ground_gterm_f] :
      ( ( ( produc3560254623552331287term_f @ X1 @ X22 )
        = ( produc3560254623552331287term_f @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_286_old_Oprod_Oinject,axiom,
    ! [A6: ground_gterm_f,B3: ground_gterm_f,A7: ground_gterm_f,B4: ground_gterm_f] :
      ( ( ( produc3560254623552331287term_f @ A6 @ B3 )
        = ( produc3560254623552331287term_f @ A7 @ B4 ) )
      = ( ( A6 = A7 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_287_finite__Collect__less__eq,axiom,
    ! [Q: ground_gterm_f > $o,P: ground_gterm_f > $o] :
      ( ( ord_le4300966928821188649rm_f_o @ Q @ P )
     => ( ( finite1923210418194103261term_f @ ( collec3522850625759001119term_f @ P ) )
       => ( finite1923210418194103261term_f @ ( collec3522850625759001119term_f @ Q ) ) ) ) ).

% finite_Collect_less_eq
thf(fact_288_finite__Collect__less__eq,axiom,
    ! [Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_nat_o @ Q @ P )
     => ( ( finite_finite_nat @ ( collect_nat @ P ) )
       => ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_less_eq
thf(fact_289_finite__Collect__less__eq,axiom,
    ! [Q: product_prod_f_nat > $o,P: product_prod_f_nat > $o] :
      ( ( ord_le3748900571519617005_nat_o @ Q @ P )
     => ( ( finite3598747971073327385_f_nat @ ( collec1417984143148964183_f_nat @ P ) )
       => ( finite3598747971073327385_f_nat @ ( collec1417984143148964183_f_nat @ Q ) ) ) ) ).

% finite_Collect_less_eq
thf(fact_290_finite__Collect__less__eq,axiom,
    ! [Q: set_Ground_gterm_f > $o,P: set_Ground_gterm_f > $o] :
      ( ( ord_le3364362122161230665rm_f_o @ Q @ P )
     => ( ( finite6824738741894949053term_f @ ( collec1979342819066322687term_f @ P ) )
       => ( finite6824738741894949053term_f @ ( collec1979342819066322687term_f @ Q ) ) ) ) ).

% finite_Collect_less_eq
thf(fact_291_finite__Collect__less__eq,axiom,
    ! [Q: set_Pr989862937836626183term_f > $o,P: set_Pr989862937836626183term_f > $o] :
      ( ( ord_le8472472344239181334rm_f_o @ Q @ P )
     => ( ( finite5058142709724841776term_f @ ( collec4490547877585280114term_f @ P ) )
       => ( finite5058142709724841776term_f @ ( collec4490547877585280114term_f @ Q ) ) ) ) ).

% finite_Collect_less_eq
thf(fact_292_finite__Collect__less__eq,axiom,
    ! [Q: set_Pr5245412377734449720_f_nat > $o,P: set_Pr5245412377734449720_f_nat > $o] :
      ( ( ord_le7946898928936407735_nat_o @ Q @ P )
     => ( ( finite879454794548843727_f_nat @ ( collec6964094325902095629_f_nat @ P ) )
       => ( finite879454794548843727_f_nat @ ( collec6964094325902095629_f_nat @ Q ) ) ) ) ).

% finite_Collect_less_eq
thf(fact_293_finite__Collect__less__eq,axiom,
    ! [Q: set_nat > $o,P: set_nat > $o] :
      ( ( ord_le3964352015994296041_nat_o @ Q @ P )
     => ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
       => ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).

% finite_Collect_less_eq
thf(fact_294_less__eq__set__def,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A4: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
          ( ord_le3748900571519617005_nat_o
          @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ A4 )
          @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ B ) ) ) ) ).

% less_eq_set_def
thf(fact_295_less__eq__set__def,axiom,
    ( ord_le263819222746101927term_f
    = ( ^ [A4: set_Pr989862937836626183term_f,B: set_Pr989862937836626183term_f] :
          ( ord_le1842562795780935414rm_f_o
          @ ^ [X: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X @ A4 )
          @ ^ [X: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X @ B ) ) ) ) ).

% less_eq_set_def
thf(fact_296_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A4 )
          @ ^ [X: nat] : ( member_nat @ X @ B ) ) ) ) ).

% less_eq_set_def
thf(fact_297_less__eq__set__def,axiom,
    ( ord_le2735537439747282356term_f
    = ( ^ [A4: set_Ground_gterm_f,B: set_Ground_gterm_f] :
          ( ord_le4300966928821188649rm_f_o
          @ ^ [X: ground_gterm_f] : ( member5261315044688711901term_f @ X @ A4 )
          @ ^ [X: ground_gterm_f] : ( member5261315044688711901term_f @ X @ B ) ) ) ) ).

% less_eq_set_def
thf(fact_298_less__eq__fset_Orep__eq,axiom,
    ( ord_le4587745213494032429rm_f_v
    = ( ^ [X: fset_P8018961893305114765rm_f_v,Xa2: fset_P8018961893305114765rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( fset_P4617584883882644886rm_f_v @ X ) @ ( fset_P4617584883882644886rm_f_v @ Xa2 ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_299_less__eq__fset_Orep__eq,axiom,
    ( ord_le8532657618882012973term_f
    = ( ^ [X: fset_P5324236452418035597term_f,Xa2: fset_P5324236452418035597term_f] : ( ord_le263819222746101927term_f @ ( fset_P5939342050162774678term_f @ X ) @ ( fset_P5939342050162774678term_f @ Xa2 ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_300_less__eq__fset_Orep__eq,axiom,
    ( ord_less_eq_fset_nat
    = ( ^ [X: fset_nat,Xa2: fset_nat] : ( ord_less_eq_set_nat @ ( fset_nat2 @ X ) @ ( fset_nat2 @ Xa2 ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_301_less__eq__fset_Orep__eq,axiom,
    ( ord_le8198718236663383866term_f
    = ( ^ [X: fset_Ground_gterm_f,Xa2: fset_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ ( fset_Ground_gterm_f2 @ X ) @ ( fset_Ground_gterm_f2 @ Xa2 ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_302_less__eq__fset_Orep__eq,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [X: fset_P6228066233360383026_f_nat,Xa2: fset_P6228066233360383026_f_nat] : ( ord_le8976984241387448984_f_nat @ ( fset_P3576968334923099475_f_nat @ X ) @ ( fset_P3576968334923099475_f_nat @ Xa2 ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_303_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7245736746747425831term_f] :
      ~ ! [A3: ground_gterm_f,B5: ground_gterm_f] :
          ( Y
         != ( produc3560254623552331287term_f @ A3 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_304_surj__pair,axiom,
    ! [P2: produc7245736746747425831term_f] :
    ? [X2: ground_gterm_f,Y3: ground_gterm_f] :
      ( P2
      = ( produc3560254623552331287term_f @ X2 @ Y3 ) ) ).

% surj_pair
thf(fact_305_prod__cases,axiom,
    ! [P: produc7245736746747425831term_f > $o,P2: produc7245736746747425831term_f] :
      ( ! [A3: ground_gterm_f,B5: ground_gterm_f] : ( P @ ( produc3560254623552331287term_f @ A3 @ B5 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_306_Pair__inject,axiom,
    ! [A6: ground_gterm_f,B3: ground_gterm_f,A7: ground_gterm_f,B4: ground_gterm_f] :
      ( ( ( produc3560254623552331287term_f @ A6 @ B3 )
        = ( produc3560254623552331287term_f @ A7 @ B4 ) )
     => ~ ( ( A6 = A7 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_307_finite__funas__gterm,axiom,
    ! [T: ground_gterm_f] : ( finite3598747971073327385_f_nat @ ( ground_funas_gterm_f @ T ) ) ).

% finite_funas_gterm
thf(fact_308_Sigma__cong,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f,C: ground_gterm_f > set_Ground_gterm_f,D: ground_gterm_f > set_Ground_gterm_f] :
      ( ( A = B2 )
     => ( ! [X2: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X2 @ B2 )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( produc6043514909933191368term_f @ A @ C )
          = ( produc6043514909933191368term_f @ B2 @ D ) ) ) ) ).

% Sigma_cong
thf(fact_309_Times__eq__cancel2,axiom,
    ! [X3: ground_gterm_f,C: set_Ground_gterm_f,A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( member5261315044688711901term_f @ X3 @ C )
     => ( ( ( produc6043514909933191368term_f @ A
            @ ^ [Uu: ground_gterm_f] : C )
          = ( produc6043514909933191368term_f @ B2
            @ ^ [Uu: ground_gterm_f] : C ) )
        = ( A = B2 ) ) ) ).

% Times_eq_cancel2
thf(fact_310_SigmaE,axiom,
    ! [C2: produc8979644124618241255term_f,A: set_Pr989862937836626183term_f,B2: produc7245736746747425831term_f > set_Pr989862937836626183term_f] :
      ( ( member3777300322277616400term_f @ C2 @ ( produc4509621232857779464term_f @ A @ B2 ) )
     => ~ ! [X2: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X2 @ A )
           => ! [Y3: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ Y3 @ ( B2 @ X2 ) )
               => ( C2
                 != ( produc7348823797596265303term_f @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_311_SigmaE,axiom,
    ! [C2: produc2194508710173438886_f_nat,A: set_Pr989862937836626183term_f,B2: produc7245736746747425831term_f > set_nat] :
      ( ( member3662771866509663549_f_nat @ C2 @ ( produc8380933303712717999_f_nat @ A @ B2 ) )
     => ~ ! [X2: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X2 @ A )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B2 @ X2 ) )
               => ( C2
                 != ( produc5217865480721399968_f_nat @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_312_SigmaE,axiom,
    ! [C2: produc2535366586387710028term_f,A: set_nat,B2: nat > set_Pr989862937836626183term_f] :
      ( ( member4003629742723934691term_f @ C2 @ ( produc203364604552858125term_f @ A @ B2 ) )
     => ~ ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ! [Y3: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ Y3 @ ( B2 @ X2 ) )
               => ( C2
                 != ( produc6263668818416315902term_f @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_313_SigmaE,axiom,
    ! [C2: product_prod_nat_nat,A: set_nat,B2: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( produc457027306803732586at_nat @ A @ B2 ) )
     => ~ ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B2 @ X2 ) )
               => ( C2
                 != ( product_Pair_nat_nat @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_314_SigmaE,axiom,
    ! [C2: produc7245736746747425831term_f,A: set_Ground_gterm_f,B2: ground_gterm_f > set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ C2 @ ( produc6043514909933191368term_f @ A @ B2 ) )
     => ~ ! [X2: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X2 @ A )
           => ! [Y3: ground_gterm_f] :
                ( ( member5261315044688711901term_f @ Y3 @ ( B2 @ X2 ) )
               => ( C2
                 != ( produc3560254623552331287term_f @ X2 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_315_SigmaD1,axiom,
    ! [A6: ground_gterm_f,B3: ground_gterm_f,A: set_Ground_gterm_f,B2: ground_gterm_f > set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A6 @ B3 ) @ ( produc6043514909933191368term_f @ A @ B2 ) )
     => ( member5261315044688711901term_f @ A6 @ A ) ) ).

% SigmaD1
thf(fact_316_SigmaD2,axiom,
    ! [A6: ground_gterm_f,B3: ground_gterm_f,A: set_Ground_gterm_f,B2: ground_gterm_f > set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A6 @ B3 ) @ ( produc6043514909933191368term_f @ A @ B2 ) )
     => ( member5261315044688711901term_f @ B3 @ ( B2 @ A6 ) ) ) ).

% SigmaD2
thf(fact_317_SigmaE2,axiom,
    ! [A6: produc7245736746747425831term_f,B3: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f,B2: produc7245736746747425831term_f > set_Pr989862937836626183term_f] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ A6 @ B3 ) @ ( produc4509621232857779464term_f @ A @ B2 ) )
     => ~ ( ( member848276444142703440term_f @ A6 @ A )
         => ~ ( member848276444142703440term_f @ B3 @ ( B2 @ A6 ) ) ) ) ).

% SigmaE2
thf(fact_318_SigmaE2,axiom,
    ! [A6: produc7245736746747425831term_f,B3: nat,A: set_Pr989862937836626183term_f,B2: produc7245736746747425831term_f > set_nat] :
      ( ( member3662771866509663549_f_nat @ ( produc5217865480721399968_f_nat @ A6 @ B3 ) @ ( produc8380933303712717999_f_nat @ A @ B2 ) )
     => ~ ( ( member848276444142703440term_f @ A6 @ A )
         => ~ ( member_nat @ B3 @ ( B2 @ A6 ) ) ) ) ).

% SigmaE2
thf(fact_319_SigmaE2,axiom,
    ! [A6: nat,B3: produc7245736746747425831term_f,A: set_nat,B2: nat > set_Pr989862937836626183term_f] :
      ( ( member4003629742723934691term_f @ ( produc6263668818416315902term_f @ A6 @ B3 ) @ ( produc203364604552858125term_f @ A @ B2 ) )
     => ~ ( ( member_nat @ A6 @ A )
         => ~ ( member848276444142703440term_f @ B3 @ ( B2 @ A6 ) ) ) ) ).

% SigmaE2
thf(fact_320_SigmaE2,axiom,
    ! [A6: nat,B3: nat,A: set_nat,B2: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B3 ) @ ( produc457027306803732586at_nat @ A @ B2 ) )
     => ~ ( ( member_nat @ A6 @ A )
         => ~ ( member_nat @ B3 @ ( B2 @ A6 ) ) ) ) ).

% SigmaE2
thf(fact_321_SigmaE2,axiom,
    ! [A6: ground_gterm_f,B3: ground_gterm_f,A: set_Ground_gterm_f,B2: ground_gterm_f > set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A6 @ B3 ) @ ( produc6043514909933191368term_f @ A @ B2 ) )
     => ~ ( ( member5261315044688711901term_f @ A6 @ A )
         => ~ ( member5261315044688711901term_f @ B3 @ ( B2 @ A6 ) ) ) ) ).

% SigmaE2
thf(fact_322__092_060T_062_092_060_094sub_062G__mono,axiom,
    ! [G: set_Pr5245412377734449720_f_nat,F2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ G @ F2 )
     => ( ord_le2735537439747282356term_f @ ( ground_T_G_f @ G ) @ ( ground_T_G_f @ F2 ) ) ) ).

% \<T>\<^sub>G_mono
thf(fact_323_gterms__mono,axiom,
    ! [G: set_Pr5245412377734449720_f_nat,F2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ G @ F2 )
     => ( ord_le2735537439747282356term_f
        @ ( collec3522850625759001119term_f
          @ ^ [S2: ground_gterm_f] : ( ord_le8976984241387448984_f_nat @ ( ground_funas_gterm_f @ S2 ) @ G ) )
        @ ( collec3522850625759001119term_f
          @ ^ [S2: ground_gterm_f] : ( ord_le8976984241387448984_f_nat @ ( ground_funas_gterm_f @ S2 ) @ F2 ) ) ) ) ).

% gterms_mono
thf(fact_324_not__Some__eq,axiom,
    ! [X3: option3789488934265196358tion_f] :
      ( ( ! [Y4: tree_r733329426570293750tion_f] :
            ( X3
           != ( some_T4055341017772447441tion_f @ Y4 ) ) )
      = ( X3 = none_T5277256714714431317tion_f ) ) ).

% not_Some_eq
thf(fact_325_not__None__eq,axiom,
    ! [X3: option3789488934265196358tion_f] :
      ( ( X3 != none_T5277256714714431317tion_f )
      = ( ? [Y4: tree_r733329426570293750tion_f] :
            ( X3
            = ( some_T4055341017772447441tion_f @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_326_RR2__spec__simplify__reg,axiom,
    ! [A8: tree_r733329426570293750tion_f,R2: set_Pr989862937836626183term_f] :
      ( ( rRn_RR2_spec_nat_f_f @ ( fOR_si5451137711280541013tion_f @ A8 ) @ R2 )
      = ( rRn_RR2_spec_nat_f_f @ A8 @ R2 ) ) ).

% RR2_spec_simplify_reg
thf(fact_327_relChain__def,axiom,
    ( bNF_Ca6898542283492539976_f_nat
    = ( ^ [R3: set_Pr989862937836626183term_f,As: ground_gterm_f > set_Pr5245412377734449720_f_nat] :
        ! [I: ground_gterm_f,J: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ I @ J ) @ R3 )
         => ( ord_le8976984241387448984_f_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_328_relChain__def,axiom,
    ( bNF_Ca4795745921695105271term_f
    = ( ^ [R3: set_Pr989862937836626183term_f,As: ground_gterm_f > set_Pr989862937836626183term_f] :
        ! [I: ground_gterm_f,J: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ I @ J ) @ R3 )
         => ( ord_le263819222746101927term_f @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_329_relChain__def,axiom,
    ( bNF_Ca5107865095533045826_f_nat
    = ( ^ [R3: set_Pr989862937836626183term_f,As: ground_gterm_f > fset_P6228066233360383026_f_nat] :
        ! [I: ground_gterm_f,J: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ I @ J ) @ R3 )
         => ( ord_le1552505484586773650_f_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_330_relChain__def,axiom,
    ( bNF_Ca3335553290143045600_f_nat
    = ( ^ [R3: set_Pr989862937836626183term_f,As: ground_gterm_f > nat] :
        ! [I: ground_gterm_f,J: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ I @ J ) @ R3 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_331_relChain__def,axiom,
    ( bNF_Ca6437535090678629270et_nat
    = ( ^ [R3: set_Pr989862937836626183term_f,As: ground_gterm_f > set_nat] :
        ! [I: ground_gterm_f,J: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ I @ J ) @ R3 )
         => ( ord_less_eq_set_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_332_relChain__def,axiom,
    ( bNF_Ca4600563243626923524term_f
    = ( ^ [R3: set_Pr989862937836626183term_f,As: ground_gterm_f > set_Ground_gterm_f] :
        ! [I: ground_gterm_f,J: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ I @ J ) @ R3 )
         => ( ord_le2735537439747282356term_f @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_333_option_Oinject,axiom,
    ! [X22: tree_r733329426570293750tion_f,Y22: tree_r733329426570293750tion_f] :
      ( ( ( some_T4055341017772447441tion_f @ X22 )
        = ( some_T4055341017772447441tion_f @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_334_order__refl,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_335_order__refl,axiom,
    ! [X3: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ X3 @ X3 ) ).

% order_refl
thf(fact_336_order__refl,axiom,
    ! [X3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_337_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_338_order__refl,axiom,
    ! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_339_order__refl,axiom,
    ! [X3: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ X3 @ X3 ) ).

% order_refl
thf(fact_340_dual__order_Orefl,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A6 @ A6 ) ).

% dual_order.refl
thf(fact_341_dual__order_Orefl,axiom,
    ! [A6: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ A6 @ A6 ) ).

% dual_order.refl
thf(fact_342_dual__order_Orefl,axiom,
    ! [A6: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ A6 @ A6 ) ).

% dual_order.refl
thf(fact_343_dual__order_Orefl,axiom,
    ! [A6: nat] : ( ord_less_eq_nat @ A6 @ A6 ) ).

% dual_order.refl
thf(fact_344_dual__order_Orefl,axiom,
    ! [A6: set_nat] : ( ord_less_eq_set_nat @ A6 @ A6 ) ).

% dual_order.refl
thf(fact_345_dual__order_Orefl,axiom,
    ! [A6: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ A6 @ A6 ) ).

% dual_order.refl
thf(fact_346_pred__subset__eq2,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,S: set_Pr5245412377734449720_f_nat] :
      ( ( ord_less_eq_f_nat_o
        @ ^ [X: f,Y4: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y4 ) @ R2 )
        @ ^ [X: f,Y4: nat] : ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y4 ) @ S ) )
      = ( ord_le8976984241387448984_f_nat @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_347_pred__subset__eq2,axiom,
    ! [R2: set_Pr989862937836626183term_f,S: set_Pr989862937836626183term_f] :
      ( ( ord_le8892309489402766734rm_f_o
        @ ^ [X: ground_gterm_f,Y4: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y4 ) @ R2 )
        @ ^ [X: ground_gterm_f,Y4: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y4 ) @ S ) )
      = ( ord_le263819222746101927term_f @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_348_fsubset__antisym,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ B2 )
     => ( ( ord_le1552505484586773650_f_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% fsubset_antisym
thf(fact_349_fequalityE,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B2: fset_P6228066233360383026_f_nat] :
      ( ( A = B2 )
     => ~ ( ( ord_le1552505484586773650_f_nat @ A @ B2 )
         => ~ ( ord_le1552505484586773650_f_nat @ B2 @ A ) ) ) ).

% fequalityE
thf(fact_350_fequalityD1,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B2: fset_P6228066233360383026_f_nat] :
      ( ( A = B2 )
     => ( ord_le1552505484586773650_f_nat @ A @ B2 ) ) ).

% fequalityD1
thf(fact_351_fequalityD2,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B2: fset_P6228066233360383026_f_nat] :
      ( ( A = B2 )
     => ( ord_le1552505484586773650_f_nat @ B2 @ A ) ) ).

% fequalityD2
thf(fact_352_fsubset__refl,axiom,
    ! [A: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ A @ A ) ).

% fsubset_refl
thf(fact_353_fsubset__trans,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B2: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ B2 )
     => ( ( ord_le1552505484586773650_f_nat @ B2 @ C )
       => ( ord_le1552505484586773650_f_nat @ A @ C ) ) ) ).

% fsubset_trans
thf(fact_354_fset__eq__fsubset,axiom,
    ( ( ^ [Y2: fset_P6228066233360383026_f_nat,Z: fset_P6228066233360383026_f_nat] : ( Y2 = Z ) )
    = ( ^ [A4: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
          ( ( ord_le1552505484586773650_f_nat @ A4 @ B )
          & ( ord_le1552505484586773650_f_nat @ B @ A4 ) ) ) ) ).

% fset_eq_fsubset
thf(fact_355_order__antisym__conv,axiom,
    ! [Y: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ Y @ X3 )
     => ( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_356_order__antisym__conv,axiom,
    ! [Y: set_Pr989862937836626183term_f,X3: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ Y @ X3 )
     => ( ( ord_le263819222746101927term_f @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_357_order__antisym__conv,axiom,
    ! [Y: fset_P6228066233360383026_f_nat,X3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ Y @ X3 )
     => ( ( ord_le1552505484586773650_f_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_358_order__antisym__conv,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_359_order__antisym__conv,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X3 )
     => ( ( ord_less_eq_set_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_360_order__antisym__conv,axiom,
    ! [Y: set_Ground_gterm_f,X3: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ Y @ X3 )
     => ( ( ord_le2735537439747282356term_f @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_361_linorder__le__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_le_cases
thf(fact_362_ord__le__eq__subst,axiom,
    ! [A6: nat,B3: nat,F3: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ( F3 @ B3 )
          = C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_363_ord__le__eq__subst,axiom,
    ! [A6: nat,B3: nat,F3: nat > set_nat,C2: set_nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ( F3 @ B3 )
          = C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_364_ord__le__eq__subst,axiom,
    ! [A6: set_nat,B3: set_nat,F3: set_nat > nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A6 @ B3 )
     => ( ( ( F3 @ B3 )
          = C2 )
       => ( ! [X2: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_365_ord__le__eq__subst,axiom,
    ! [A6: nat,B3: nat,F3: nat > set_Ground_gterm_f,C2: set_Ground_gterm_f] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ( F3 @ B3 )
          = C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_le2735537439747282356term_f @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_le2735537439747282356term_f @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_366_ord__le__eq__subst,axiom,
    ! [A6: set_nat,B3: set_nat,F3: set_nat > set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ B3 )
     => ( ( ( F3 @ B3 )
          = C2 )
       => ( ! [X2: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X2 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_367_ord__le__eq__subst,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f,F3: set_Ground_gterm_f > nat,C2: nat] :
      ( ( ord_le2735537439747282356term_f @ A6 @ B3 )
     => ( ( ( F3 @ B3 )
          = C2 )
       => ( ! [X2: set_Ground_gterm_f,Y3: set_Ground_gterm_f] :
              ( ( ord_le2735537439747282356term_f @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_368_ord__le__eq__subst,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat,F3: set_Pr5245412377734449720_f_nat > nat,C2: nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ B3 )
     => ( ( ( F3 @ B3 )
          = C2 )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_369_ord__le__eq__subst,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,F3: fset_P6228066233360383026_f_nat > nat,C2: nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ B3 )
     => ( ( ( F3 @ B3 )
          = C2 )
       => ( ! [X2: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] :
              ( ( ord_le1552505484586773650_f_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_370_ord__le__eq__subst,axiom,
    ! [A6: nat,B3: nat,F3: nat > set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ( F3 @ B3 )
          = C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_le8976984241387448984_f_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_371_ord__le__eq__subst,axiom,
    ! [A6: nat,B3: nat,F3: nat > fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ( F3 @ B3 )
          = C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_le1552505484586773650_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_le1552505484586773650_f_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_372_ord__eq__le__subst,axiom,
    ! [A6: nat,F3: nat > nat,B3: nat,C2: nat] :
      ( ( A6
        = ( F3 @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_373_ord__eq__le__subst,axiom,
    ! [A6: set_nat,F3: nat > set_nat,B3: nat,C2: nat] :
      ( ( A6
        = ( F3 @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_374_ord__eq__le__subst,axiom,
    ! [A6: nat,F3: set_nat > nat,B3: set_nat,C2: set_nat] :
      ( ( A6
        = ( F3 @ B3 ) )
     => ( ( ord_less_eq_set_nat @ B3 @ C2 )
       => ( ! [X2: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_375_ord__eq__le__subst,axiom,
    ! [A6: set_Ground_gterm_f,F3: nat > set_Ground_gterm_f,B3: nat,C2: nat] :
      ( ( A6
        = ( F3 @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_le2735537439747282356term_f @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_le2735537439747282356term_f @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_376_ord__eq__le__subst,axiom,
    ! [A6: set_nat,F3: set_nat > set_nat,B3: set_nat,C2: set_nat] :
      ( ( A6
        = ( F3 @ B3 ) )
     => ( ( ord_less_eq_set_nat @ B3 @ C2 )
       => ( ! [X2: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X2 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_377_ord__eq__le__subst,axiom,
    ! [A6: nat,F3: set_Ground_gterm_f > nat,B3: set_Ground_gterm_f,C2: set_Ground_gterm_f] :
      ( ( A6
        = ( F3 @ B3 ) )
     => ( ( ord_le2735537439747282356term_f @ B3 @ C2 )
       => ( ! [X2: set_Ground_gterm_f,Y3: set_Ground_gterm_f] :
              ( ( ord_le2735537439747282356term_f @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_378_ord__eq__le__subst,axiom,
    ! [A6: nat,F3: set_Pr5245412377734449720_f_nat > nat,B3: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
      ( ( A6
        = ( F3 @ B3 ) )
     => ( ( ord_le8976984241387448984_f_nat @ B3 @ C2 )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_379_ord__eq__le__subst,axiom,
    ! [A6: nat,F3: fset_P6228066233360383026_f_nat > nat,B3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( A6
        = ( F3 @ B3 ) )
     => ( ( ord_le1552505484586773650_f_nat @ B3 @ C2 )
       => ( ! [X2: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] :
              ( ( ord_le1552505484586773650_f_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_380_ord__eq__le__subst,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,F3: nat > set_Pr5245412377734449720_f_nat,B3: nat,C2: nat] :
      ( ( A6
        = ( F3 @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_le8976984241387448984_f_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_381_ord__eq__le__subst,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,F3: nat > fset_P6228066233360383026_f_nat,B3: nat,C2: nat] :
      ( ( A6
        = ( F3 @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_le1552505484586773650_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_le1552505484586773650_f_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_382_linorder__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
      | ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_linear
thf(fact_383_order__eq__refl,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
      ( ( X3 = Y )
     => ( ord_le8976984241387448984_f_nat @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_384_order__eq__refl,axiom,
    ! [X3: set_Pr989862937836626183term_f,Y: set_Pr989862937836626183term_f] :
      ( ( X3 = Y )
     => ( ord_le263819222746101927term_f @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_385_order__eq__refl,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
      ( ( X3 = Y )
     => ( ord_le1552505484586773650_f_nat @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_386_order__eq__refl,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 = Y )
     => ( ord_less_eq_nat @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_387_order__eq__refl,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( X3 = Y )
     => ( ord_less_eq_set_nat @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_388_order__eq__refl,axiom,
    ! [X3: set_Ground_gterm_f,Y: set_Ground_gterm_f] :
      ( ( X3 = Y )
     => ( ord_le2735537439747282356term_f @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_389_order__subst2,axiom,
    ! [A6: nat,B3: nat,F3: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ord_less_eq_nat @ ( F3 @ B3 ) @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_390_order__subst2,axiom,
    ! [A6: nat,B3: nat,F3: nat > set_nat,C2: set_nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ord_less_eq_set_nat @ ( F3 @ B3 ) @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_391_order__subst2,axiom,
    ! [A6: set_nat,B3: set_nat,F3: set_nat > nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A6 @ B3 )
     => ( ( ord_less_eq_nat @ ( F3 @ B3 ) @ C2 )
       => ( ! [X2: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_392_order__subst2,axiom,
    ! [A6: nat,B3: nat,F3: nat > set_Ground_gterm_f,C2: set_Ground_gterm_f] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ord_le2735537439747282356term_f @ ( F3 @ B3 ) @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_le2735537439747282356term_f @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_le2735537439747282356term_f @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_393_order__subst2,axiom,
    ! [A6: set_nat,B3: set_nat,F3: set_nat > set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ B3 )
     => ( ( ord_less_eq_set_nat @ ( F3 @ B3 ) @ C2 )
       => ( ! [X2: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X2 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_394_order__subst2,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f,F3: set_Ground_gterm_f > nat,C2: nat] :
      ( ( ord_le2735537439747282356term_f @ A6 @ B3 )
     => ( ( ord_less_eq_nat @ ( F3 @ B3 ) @ C2 )
       => ( ! [X2: set_Ground_gterm_f,Y3: set_Ground_gterm_f] :
              ( ( ord_le2735537439747282356term_f @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_395_order__subst2,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat,F3: set_Pr5245412377734449720_f_nat > nat,C2: nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ B3 )
     => ( ( ord_less_eq_nat @ ( F3 @ B3 ) @ C2 )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_396_order__subst2,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,F3: fset_P6228066233360383026_f_nat > nat,C2: nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ B3 )
     => ( ( ord_less_eq_nat @ ( F3 @ B3 ) @ C2 )
       => ( ! [X2: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] :
              ( ( ord_le1552505484586773650_f_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_397_order__subst2,axiom,
    ! [A6: nat,B3: nat,F3: nat > set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ord_le8976984241387448984_f_nat @ ( F3 @ B3 ) @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_le8976984241387448984_f_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_398_order__subst2,axiom,
    ! [A6: nat,B3: nat,F3: nat > fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ord_le1552505484586773650_f_nat @ ( F3 @ B3 ) @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_le1552505484586773650_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_le1552505484586773650_f_nat @ ( F3 @ A6 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_399_order__subst1,axiom,
    ! [A6: nat,F3: nat > nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A6 @ ( F3 @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_400_order__subst1,axiom,
    ! [A6: nat,F3: set_nat > nat,B3: set_nat,C2: set_nat] :
      ( ( ord_less_eq_nat @ A6 @ ( F3 @ B3 ) )
     => ( ( ord_less_eq_set_nat @ B3 @ C2 )
       => ( ! [X2: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_401_order__subst1,axiom,
    ! [A6: set_nat,F3: nat > set_nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A6 @ ( F3 @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_402_order__subst1,axiom,
    ! [A6: nat,F3: set_Ground_gterm_f > nat,B3: set_Ground_gterm_f,C2: set_Ground_gterm_f] :
      ( ( ord_less_eq_nat @ A6 @ ( F3 @ B3 ) )
     => ( ( ord_le2735537439747282356term_f @ B3 @ C2 )
       => ( ! [X2: set_Ground_gterm_f,Y3: set_Ground_gterm_f] :
              ( ( ord_le2735537439747282356term_f @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_403_order__subst1,axiom,
    ! [A6: set_nat,F3: set_nat > set_nat,B3: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ ( F3 @ B3 ) )
     => ( ( ord_less_eq_set_nat @ B3 @ C2 )
       => ( ! [X2: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X2 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_404_order__subst1,axiom,
    ! [A6: set_Ground_gterm_f,F3: nat > set_Ground_gterm_f,B3: nat,C2: nat] :
      ( ( ord_le2735537439747282356term_f @ A6 @ ( F3 @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_le2735537439747282356term_f @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_le2735537439747282356term_f @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_405_order__subst1,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,F3: nat > set_Pr5245412377734449720_f_nat,B3: nat,C2: nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ ( F3 @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_le8976984241387448984_f_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_406_order__subst1,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,F3: nat > fset_P6228066233360383026_f_nat,B3: nat,C2: nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ ( F3 @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_le1552505484586773650_f_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_le1552505484586773650_f_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_407_order__subst1,axiom,
    ! [A6: nat,F3: set_Pr5245412377734449720_f_nat > nat,B3: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_less_eq_nat @ A6 @ ( F3 @ B3 ) )
     => ( ( ord_le8976984241387448984_f_nat @ B3 @ C2 )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_408_order__subst1,axiom,
    ! [A6: nat,F3: fset_P6228066233360383026_f_nat > nat,B3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( ord_less_eq_nat @ A6 @ ( F3 @ B3 ) )
     => ( ( ord_le1552505484586773650_f_nat @ B3 @ C2 )
       => ( ! [X2: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] :
              ( ( ord_le1552505484586773650_f_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A6 @ ( F3 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_409_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_Pr5245412377734449720_f_nat,Z: set_Pr5245412377734449720_f_nat] : ( Y2 = Z ) )
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B6: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ A5 @ B6 )
          & ( ord_le8976984241387448984_f_nat @ B6 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_410_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_Pr989862937836626183term_f,Z: set_Pr989862937836626183term_f] : ( Y2 = Z ) )
    = ( ^ [A5: set_Pr989862937836626183term_f,B6: set_Pr989862937836626183term_f] :
          ( ( ord_le263819222746101927term_f @ A5 @ B6 )
          & ( ord_le263819222746101927term_f @ B6 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_411_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: fset_P6228066233360383026_f_nat,Z: fset_P6228066233360383026_f_nat] : ( Y2 = Z ) )
    = ( ^ [A5: fset_P6228066233360383026_f_nat,B6: fset_P6228066233360383026_f_nat] :
          ( ( ord_le1552505484586773650_f_nat @ A5 @ B6 )
          & ( ord_le1552505484586773650_f_nat @ B6 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_412_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [A5: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A5 @ B6 )
          & ( ord_less_eq_nat @ B6 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_413_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
    = ( ^ [A5: set_nat,B6: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B6 )
          & ( ord_less_eq_set_nat @ B6 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_414_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_Ground_gterm_f,Z: set_Ground_gterm_f] : ( Y2 = Z ) )
    = ( ^ [A5: set_Ground_gterm_f,B6: set_Ground_gterm_f] :
          ( ( ord_le2735537439747282356term_f @ A5 @ B6 )
          & ( ord_le2735537439747282356term_f @ B6 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_415_antisym,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ B3 )
     => ( ( ord_le8976984241387448984_f_nat @ B3 @ A6 )
       => ( A6 = B3 ) ) ) ).

% antisym
thf(fact_416_antisym,axiom,
    ! [A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ B3 )
     => ( ( ord_le263819222746101927term_f @ B3 @ A6 )
       => ( A6 = B3 ) ) ) ).

% antisym
thf(fact_417_antisym,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ B3 )
     => ( ( ord_le1552505484586773650_f_nat @ B3 @ A6 )
       => ( A6 = B3 ) ) ) ).

% antisym
thf(fact_418_antisym,axiom,
    ! [A6: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ A6 )
       => ( A6 = B3 ) ) ) ).

% antisym
thf(fact_419_antisym,axiom,
    ! [A6: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ A6 )
       => ( A6 = B3 ) ) ) ).

% antisym
thf(fact_420_antisym,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ B3 )
     => ( ( ord_le2735537439747282356term_f @ B3 @ A6 )
       => ( A6 = B3 ) ) ) ).

% antisym
thf(fact_421_dual__order_Otrans,axiom,
    ! [B3: set_Pr5245412377734449720_f_nat,A6: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B3 @ A6 )
     => ( ( ord_le8976984241387448984_f_nat @ C2 @ B3 )
       => ( ord_le8976984241387448984_f_nat @ C2 @ A6 ) ) ) ).

% dual_order.trans
thf(fact_422_dual__order_Otrans,axiom,
    ! [B3: set_Pr989862937836626183term_f,A6: set_Pr989862937836626183term_f,C2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ B3 @ A6 )
     => ( ( ord_le263819222746101927term_f @ C2 @ B3 )
       => ( ord_le263819222746101927term_f @ C2 @ A6 ) ) ) ).

% dual_order.trans
thf(fact_423_dual__order_Otrans,axiom,
    ! [B3: fset_P6228066233360383026_f_nat,A6: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B3 @ A6 )
     => ( ( ord_le1552505484586773650_f_nat @ C2 @ B3 )
       => ( ord_le1552505484586773650_f_nat @ C2 @ A6 ) ) ) ).

% dual_order.trans
thf(fact_424_dual__order_Otrans,axiom,
    ! [B3: nat,A6: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A6 )
     => ( ( ord_less_eq_nat @ C2 @ B3 )
       => ( ord_less_eq_nat @ C2 @ A6 ) ) ) ).

% dual_order.trans
thf(fact_425_dual__order_Otrans,axiom,
    ! [B3: set_nat,A6: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A6 )
     => ( ( ord_less_eq_set_nat @ C2 @ B3 )
       => ( ord_less_eq_set_nat @ C2 @ A6 ) ) ) ).

% dual_order.trans
thf(fact_426_dual__order_Otrans,axiom,
    ! [B3: set_Ground_gterm_f,A6: set_Ground_gterm_f,C2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ B3 @ A6 )
     => ( ( ord_le2735537439747282356term_f @ C2 @ B3 )
       => ( ord_le2735537439747282356term_f @ C2 @ A6 ) ) ) ).

% dual_order.trans
thf(fact_427_dual__order_Oantisym,axiom,
    ! [B3: set_Pr5245412377734449720_f_nat,A6: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B3 @ A6 )
     => ( ( ord_le8976984241387448984_f_nat @ A6 @ B3 )
       => ( A6 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_428_dual__order_Oantisym,axiom,
    ! [B3: set_Pr989862937836626183term_f,A6: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ B3 @ A6 )
     => ( ( ord_le263819222746101927term_f @ A6 @ B3 )
       => ( A6 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_429_dual__order_Oantisym,axiom,
    ! [B3: fset_P6228066233360383026_f_nat,A6: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B3 @ A6 )
     => ( ( ord_le1552505484586773650_f_nat @ A6 @ B3 )
       => ( A6 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_430_dual__order_Oantisym,axiom,
    ! [B3: nat,A6: nat] :
      ( ( ord_less_eq_nat @ B3 @ A6 )
     => ( ( ord_less_eq_nat @ A6 @ B3 )
       => ( A6 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_431_dual__order_Oantisym,axiom,
    ! [B3: set_nat,A6: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A6 )
     => ( ( ord_less_eq_set_nat @ A6 @ B3 )
       => ( A6 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_432_dual__order_Oantisym,axiom,
    ! [B3: set_Ground_gterm_f,A6: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ B3 @ A6 )
     => ( ( ord_le2735537439747282356term_f @ A6 @ B3 )
       => ( A6 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_433_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_Pr5245412377734449720_f_nat,Z: set_Pr5245412377734449720_f_nat] : ( Y2 = Z ) )
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B6: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ B6 @ A5 )
          & ( ord_le8976984241387448984_f_nat @ A5 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_434_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_Pr989862937836626183term_f,Z: set_Pr989862937836626183term_f] : ( Y2 = Z ) )
    = ( ^ [A5: set_Pr989862937836626183term_f,B6: set_Pr989862937836626183term_f] :
          ( ( ord_le263819222746101927term_f @ B6 @ A5 )
          & ( ord_le263819222746101927term_f @ A5 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_435_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: fset_P6228066233360383026_f_nat,Z: fset_P6228066233360383026_f_nat] : ( Y2 = Z ) )
    = ( ^ [A5: fset_P6228066233360383026_f_nat,B6: fset_P6228066233360383026_f_nat] :
          ( ( ord_le1552505484586773650_f_nat @ B6 @ A5 )
          & ( ord_le1552505484586773650_f_nat @ A5 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_436_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [A5: nat,B6: nat] :
          ( ( ord_less_eq_nat @ B6 @ A5 )
          & ( ord_less_eq_nat @ A5 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_437_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
    = ( ^ [A5: set_nat,B6: set_nat] :
          ( ( ord_less_eq_set_nat @ B6 @ A5 )
          & ( ord_less_eq_set_nat @ A5 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_438_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_Ground_gterm_f,Z: set_Ground_gterm_f] : ( Y2 = Z ) )
    = ( ^ [A5: set_Ground_gterm_f,B6: set_Ground_gterm_f] :
          ( ( ord_le2735537439747282356term_f @ B6 @ A5 )
          & ( ord_le2735537439747282356term_f @ A5 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_439_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A6: nat,B3: nat] :
      ( ! [A3: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A3 @ B5 )
         => ( P @ A3 @ B5 ) )
     => ( ! [A3: nat,B5: nat] :
            ( ( P @ B5 @ A3 )
           => ( P @ A3 @ B5 ) )
       => ( P @ A6 @ B3 ) ) ) ).

% linorder_wlog
thf(fact_440_order__trans,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
     => ( ( ord_le8976984241387448984_f_nat @ Y @ Z2 )
       => ( ord_le8976984241387448984_f_nat @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_441_order__trans,axiom,
    ! [X3: set_Pr989862937836626183term_f,Y: set_Pr989862937836626183term_f,Z2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ X3 @ Y )
     => ( ( ord_le263819222746101927term_f @ Y @ Z2 )
       => ( ord_le263819222746101927term_f @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_442_order__trans,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat,Z2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X3 @ Y )
     => ( ( ord_le1552505484586773650_f_nat @ Y @ Z2 )
       => ( ord_le1552505484586773650_f_nat @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_443_order__trans,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_444_order__trans,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z2 )
       => ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_445_order__trans,axiom,
    ! [X3: set_Ground_gterm_f,Y: set_Ground_gterm_f,Z2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ X3 @ Y )
     => ( ( ord_le2735537439747282356term_f @ Y @ Z2 )
       => ( ord_le2735537439747282356term_f @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_446_order_Otrans,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ B3 )
     => ( ( ord_le8976984241387448984_f_nat @ B3 @ C2 )
       => ( ord_le8976984241387448984_f_nat @ A6 @ C2 ) ) ) ).

% order.trans
thf(fact_447_order_Otrans,axiom,
    ! [A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f,C2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ B3 )
     => ( ( ord_le263819222746101927term_f @ B3 @ C2 )
       => ( ord_le263819222746101927term_f @ A6 @ C2 ) ) ) ).

% order.trans
thf(fact_448_order_Otrans,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ B3 )
     => ( ( ord_le1552505484586773650_f_nat @ B3 @ C2 )
       => ( ord_le1552505484586773650_f_nat @ A6 @ C2 ) ) ) ).

% order.trans
thf(fact_449_order_Otrans,axiom,
    ! [A6: nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ord_less_eq_nat @ A6 @ C2 ) ) ) ).

% order.trans
thf(fact_450_order_Otrans,axiom,
    ! [A6: set_nat,B3: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C2 )
       => ( ord_less_eq_set_nat @ A6 @ C2 ) ) ) ).

% order.trans
thf(fact_451_order_Otrans,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f,C2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ B3 )
     => ( ( ord_le2735537439747282356term_f @ B3 @ C2 )
       => ( ord_le2735537439747282356term_f @ A6 @ C2 ) ) ) ).

% order.trans
thf(fact_452_order__antisym,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
     => ( ( ord_le8976984241387448984_f_nat @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_453_order__antisym,axiom,
    ! [X3: set_Pr989862937836626183term_f,Y: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ X3 @ Y )
     => ( ( ord_le263819222746101927term_f @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_454_order__antisym,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X3 @ Y )
     => ( ( ord_le1552505484586773650_f_nat @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_455_order__antisym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_456_order__antisym,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_457_order__antisym,axiom,
    ! [X3: set_Ground_gterm_f,Y: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ X3 @ Y )
     => ( ( ord_le2735537439747282356term_f @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_458_ord__le__eq__trans,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_le8976984241387448984_f_nat @ A6 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_459_ord__le__eq__trans,axiom,
    ! [A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f,C2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_le263819222746101927term_f @ A6 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_460_ord__le__eq__trans,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_le1552505484586773650_f_nat @ A6 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_461_ord__le__eq__trans,axiom,
    ! [A6: nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_less_eq_nat @ A6 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_462_ord__le__eq__trans,axiom,
    ! [A6: set_nat,B3: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_less_eq_set_nat @ A6 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_463_ord__le__eq__trans,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f,C2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_le2735537439747282356term_f @ A6 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_464_ord__eq__le__trans,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
      ( ( A6 = B3 )
     => ( ( ord_le8976984241387448984_f_nat @ B3 @ C2 )
       => ( ord_le8976984241387448984_f_nat @ A6 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_465_ord__eq__le__trans,axiom,
    ! [A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f,C2: set_Pr989862937836626183term_f] :
      ( ( A6 = B3 )
     => ( ( ord_le263819222746101927term_f @ B3 @ C2 )
       => ( ord_le263819222746101927term_f @ A6 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_466_ord__eq__le__trans,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( A6 = B3 )
     => ( ( ord_le1552505484586773650_f_nat @ B3 @ C2 )
       => ( ord_le1552505484586773650_f_nat @ A6 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_467_ord__eq__le__trans,axiom,
    ! [A6: nat,B3: nat,C2: nat] :
      ( ( A6 = B3 )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ord_less_eq_nat @ A6 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_468_ord__eq__le__trans,axiom,
    ! [A6: set_nat,B3: set_nat,C2: set_nat] :
      ( ( A6 = B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C2 )
       => ( ord_less_eq_set_nat @ A6 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_469_ord__eq__le__trans,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f,C2: set_Ground_gterm_f] :
      ( ( A6 = B3 )
     => ( ( ord_le2735537439747282356term_f @ B3 @ C2 )
       => ( ord_le2735537439747282356term_f @ A6 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_470_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_Pr5245412377734449720_f_nat,Z: set_Pr5245412377734449720_f_nat] : ( Y2 = Z ) )
    = ( ^ [X: set_Pr5245412377734449720_f_nat,Y4: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ X @ Y4 )
          & ( ord_le8976984241387448984_f_nat @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_471_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_Pr989862937836626183term_f,Z: set_Pr989862937836626183term_f] : ( Y2 = Z ) )
    = ( ^ [X: set_Pr989862937836626183term_f,Y4: set_Pr989862937836626183term_f] :
          ( ( ord_le263819222746101927term_f @ X @ Y4 )
          & ( ord_le263819222746101927term_f @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_472_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: fset_P6228066233360383026_f_nat,Z: fset_P6228066233360383026_f_nat] : ( Y2 = Z ) )
    = ( ^ [X: fset_P6228066233360383026_f_nat,Y4: fset_P6228066233360383026_f_nat] :
          ( ( ord_le1552505484586773650_f_nat @ X @ Y4 )
          & ( ord_le1552505484586773650_f_nat @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_473_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_474_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
    = ( ^ [X: set_nat,Y4: set_nat] :
          ( ( ord_less_eq_set_nat @ X @ Y4 )
          & ( ord_less_eq_set_nat @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_475_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_Ground_gterm_f,Z: set_Ground_gterm_f] : ( Y2 = Z ) )
    = ( ^ [X: set_Ground_gterm_f,Y4: set_Ground_gterm_f] :
          ( ( ord_le2735537439747282356term_f @ X @ Y4 )
          & ( ord_le2735537439747282356term_f @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_476_le__cases3,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_477_nle__le,axiom,
    ! [A6: nat,B3: nat] :
      ( ( ~ ( ord_less_eq_nat @ A6 @ B3 ) )
      = ( ( ord_less_eq_nat @ B3 @ A6 )
        & ( B3 != A6 ) ) ) ).

% nle_le
thf(fact_478_pred__equals__eq2,axiom,
    ! [R2: set_Pr989862937836626183term_f,S: set_Pr989862937836626183term_f] :
      ( ( ( ^ [X: ground_gterm_f,Y4: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y4 ) @ R2 ) )
        = ( ^ [X: ground_gterm_f,Y4: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y4 ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_479_pred__subset__eq,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,S: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le3748900571519617005_nat_o
        @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ R2 )
        @ ^ [X: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X @ S ) )
      = ( ord_le8976984241387448984_f_nat @ R2 @ S ) ) ).

% pred_subset_eq
thf(fact_480_pred__subset__eq,axiom,
    ! [R2: set_Pr989862937836626183term_f,S: set_Pr989862937836626183term_f] :
      ( ( ord_le1842562795780935414rm_f_o
        @ ^ [X: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X @ R2 )
        @ ^ [X: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X @ S ) )
      = ( ord_le263819222746101927term_f @ R2 @ S ) ) ).

% pred_subset_eq
thf(fact_481_pred__subset__eq,axiom,
    ! [R2: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R2 )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ord_less_eq_set_nat @ R2 @ S ) ) ).

% pred_subset_eq
thf(fact_482_pred__subset__eq,axiom,
    ! [R2: set_Ground_gterm_f,S: set_Ground_gterm_f] :
      ( ( ord_le4300966928821188649rm_f_o
        @ ^ [X: ground_gterm_f] : ( member5261315044688711901term_f @ X @ R2 )
        @ ^ [X: ground_gterm_f] : ( member5261315044688711901term_f @ X @ S ) )
      = ( ord_le2735537439747282356term_f @ R2 @ S ) ) ).

% pred_subset_eq
thf(fact_483_subrelI,axiom,
    ! [R4: set_Pr5245412377734449720_f_nat,S3: set_Pr5245412377734449720_f_nat] :
      ( ! [X2: f,Y3: nat] :
          ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X2 @ Y3 ) @ R4 )
         => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X2 @ Y3 ) @ S3 ) )
     => ( ord_le8976984241387448984_f_nat @ R4 @ S3 ) ) ).

% subrelI
thf(fact_484_subrelI,axiom,
    ! [R4: set_Pr989862937836626183term_f,S3: set_Pr989862937836626183term_f] :
      ( ! [X2: ground_gterm_f,Y3: ground_gterm_f] :
          ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X2 @ Y3 ) @ R4 )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X2 @ Y3 ) @ S3 ) )
     => ( ord_le263819222746101927term_f @ R4 @ S3 ) ) ).

% subrelI
thf(fact_485_option_Odistinct_I1_J,axiom,
    ! [X22: tree_r733329426570293750tion_f] :
      ( none_T5277256714714431317tion_f
     != ( some_T4055341017772447441tion_f @ X22 ) ) ).

% option.distinct(1)
thf(fact_486_option_OdiscI,axiom,
    ! [Option: option3789488934265196358tion_f,X22: tree_r733329426570293750tion_f] :
      ( ( Option
        = ( some_T4055341017772447441tion_f @ X22 ) )
     => ( Option != none_T5277256714714431317tion_f ) ) ).

% option.discI
thf(fact_487_option_Oexhaust,axiom,
    ! [Y: option3789488934265196358tion_f] :
      ( ( Y != none_T5277256714714431317tion_f )
     => ~ ! [X23: tree_r733329426570293750tion_f] :
            ( Y
           != ( some_T4055341017772447441tion_f @ X23 ) ) ) ).

% option.exhaust
thf(fact_488_split__option__ex,axiom,
    ( ( ^ [P3: option3789488934265196358tion_f > $o] :
        ? [X4: option3789488934265196358tion_f] : ( P3 @ X4 ) )
    = ( ^ [P4: option3789488934265196358tion_f > $o] :
          ( ( P4 @ none_T5277256714714431317tion_f )
          | ? [X: tree_r733329426570293750tion_f] : ( P4 @ ( some_T4055341017772447441tion_f @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_489_split__option__all,axiom,
    ( ( ^ [P3: option3789488934265196358tion_f > $o] :
        ! [X4: option3789488934265196358tion_f] : ( P3 @ X4 ) )
    = ( ^ [P4: option3789488934265196358tion_f > $o] :
          ( ( P4 @ none_T5277256714714431317tion_f )
          & ! [X: tree_r733329426570293750tion_f] : ( P4 @ ( some_T4055341017772447441tion_f @ X ) ) ) ) ) ).

% split_option_all
thf(fact_490_combine__options__cases,axiom,
    ! [X3: option3789488934265196358tion_f,P: option3789488934265196358tion_f > option3789488934265196358tion_f > $o,Y: option3789488934265196358tion_f] :
      ( ( ( X3 = none_T5277256714714431317tion_f )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_T5277256714714431317tion_f )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: tree_r733329426570293750tion_f,B5: tree_r733329426570293750tion_f] :
              ( ( X3
                = ( some_T4055341017772447441tion_f @ A3 ) )
             => ( ( Y
                  = ( some_T4055341017772447441tion_f @ B5 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_491_lift__less__eq__total_Ocases,axiom,
    ! [X3: produc2070832938513523962tion_f] :
      ( ! [F4: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f > $o,Uu2: option3789488934265196358tion_f] :
          ( X3
         != ( produc7984526239635384938tion_f @ F4 @ ( produc6849691629296390053tion_f @ Uu2 @ none_T5277256714714431317tion_f ) ) )
     => ( ! [F4: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f > $o,V: tree_r733329426570293750tion_f] :
            ( X3
           != ( produc7984526239635384938tion_f @ F4 @ ( produc6849691629296390053tion_f @ none_T5277256714714431317tion_f @ ( some_T4055341017772447441tion_f @ V ) ) ) )
       => ~ ! [F4: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f > $o,S4: tree_r733329426570293750tion_f,T4: tree_r733329426570293750tion_f] :
              ( X3
             != ( produc7984526239635384938tion_f @ F4 @ ( produc6849691629296390053tion_f @ ( some_T4055341017772447441tion_f @ S4 ) @ ( some_T4055341017772447441tion_f @ T4 ) ) ) ) ) ) ).

% lift_less_eq_total.cases
thf(fact_492_RR2__spec__eps__free__reg,axiom,
    ! [A8: tree_r733329426570293750tion_f,R2: set_Pr989862937836626183term_f] :
      ( ( rRn_RR2_spec_nat_f_f @ ( tree_e1535168946329719739tion_f @ A8 ) @ R2 )
      = ( rRn_RR2_spec_nat_f_f @ A8 @ R2 ) ) ).

% RR2_spec_eps_free_reg
thf(fact_493_gcomp__rel__sig,axiom,
    ! [R2: set_Pr989862937836626183term_f,F2: set_Pr5245412377734449720_f_nat,S: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ R2
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) )
     => ( ( ord_le263819222746101927term_f @ S
          @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
            @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) )
       => ( ord_le263819222746101927term_f @ ( ground_gcomp_rel_f @ F2 @ R2 @ S )
          @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
            @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) ) ) ) ).

% gcomp_rel_sig
thf(fact_494_assms,axiom,
    ! [X5: fset_P8018961893305114765rm_f_v] :
      ( ( member6790519936504491446rm_f_v @ X5 @ ( set_fs7270820277574336546rm_f_v @ rs ) )
     => ( ( lV_to_lv_trs_f_v @ ( fset_P4617584883882644886rm_f_v @ X5 ) )
        & ( ord_le1552505484586773650_f_nat @ ( lV_to_ffunas_trs_f_v @ X5 ) @ f2 ) ) ) ).

% assms
thf(fact_495_relabel__RR2__spec,axiom,
    ! [A: tree_r733329426570293750tion_f,T3: set_Pr989862937836626183term_f] :
      ( ( rRn_RR2_spec_nat_f_f @ ( tree_r1168672525086783368tion_f @ A ) @ T3 )
      = ( rRn_RR2_spec_nat_f_f @ A @ T3 ) ) ).

% relabel_RR2_spec
thf(fact_496_R1Fin,axiom,
    ! [F2: set_Pr5245412377734449720_f_nat,Rs: list_s8746099396510718605rm_f_v,R4: fOR_rr2_rel_FOR_ftrs] :
      ( ( fOR_eval_rr1_rel_f_v @ F2 @ Rs @ ( fOR_R1Fin_FOR_ftrs @ R4 ) )
      = ( collec3522850625759001119term_f
        @ ^ [T2: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ T2 @ ( ground_T_G_f @ F2 ) )
            & ( finite1923210418194103261term_f
              @ ( collec3522850625759001119term_f
                @ ^ [S2: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T2 @ S2 ) @ ( fOR_eval_rr2_rel_f_v @ F2 @ Rs @ R4 ) ) ) ) ) ) ) ).

% R1Fin
thf(fact_497_List_Ofinite__set,axiom,
    ! [Xs: list_f1824981274722084755rm_f_v] : ( finite5809181139301488822rm_f_v @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) ).

% List.finite_set
thf(fact_498_List_Ofinite__set,axiom,
    ! [Xs: list_Ground_gterm_f] : ( finite1923210418194103261term_f @ ( set_Ground_gterm_f2 @ Xs ) ) ).

% List.finite_set
thf(fact_499_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_500_List_Ofinite__set,axiom,
    ! [Xs: list_P3903862279629787026_f_nat] : ( finite3598747971073327385_f_nat @ ( set_Pr7102205019285007021_f_nat @ Xs ) ) ).

% List.finite_set
thf(fact_501_List_Ofinite__set,axiom,
    ! [Xs: list_s5016777017369925914term_f] : ( finite6824738741894949053term_f @ ( set_se8315813792620424745term_f @ Xs ) ) ).

% List.finite_set
thf(fact_502_List_Ofinite__set,axiom,
    ! [Xs: list_s8644638678082573581term_f] : ( finite5058142709724841776term_f @ ( set_se6127471701090326172term_f @ Xs ) ) ).

% List.finite_set
thf(fact_503_List_Ofinite__set,axiom,
    ! [Xs: list_s7480137856107671624_f_nat] : ( finite879454794548843727_f_nat @ ( set_se752066948783124067_f_nat @ Xs ) ) ).

% List.finite_set
thf(fact_504_List_Ofinite__set,axiom,
    ! [Xs: list_set_nat] : ( finite1152437895449049373et_nat @ ( set_set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_505_map__eq__conv,axiom,
    ! [F3: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,Xs: list_f1824981274722084755rm_f_v,G2: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ( ( map_fs8602507653405230974rm_f_v @ F3 @ Xs )
        = ( map_fs8602507653405230974rm_f_v @ G2 @ Xs ) )
      = ( ! [X: fset_P8018961893305114765rm_f_v] :
            ( ( member6790519936504491446rm_f_v @ X @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
           => ( ( F3 @ X )
              = ( G2 @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_506_subset__code_I1_J,axiom,
    ! [Xs: list_f1824981274722084755rm_f_v,B2: set_fs7307227306443116653rm_f_v] :
      ( ( ord_le2293696477246793741rm_f_v @ ( set_fs7270820277574336546rm_f_v @ Xs ) @ B2 )
      = ( ! [X: fset_P8018961893305114765rm_f_v] :
            ( ( member6790519936504491446rm_f_v @ X @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
           => ( member6790519936504491446rm_f_v @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_507_subset__code_I1_J,axiom,
    ! [Xs: list_P3903862279629787026_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ Xs ) @ B2 )
      = ( ! [X: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X @ ( set_Pr7102205019285007021_f_nat @ Xs ) )
           => ( member2678038196313615385_f_nat @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_508_subset__code_I1_J,axiom,
    ! [Xs: list_P4363786793477243949term_f,B2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ ( set_Pr1901606489578307004term_f @ Xs ) @ B2 )
      = ( ! [X: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X @ ( set_Pr1901606489578307004term_f @ Xs ) )
           => ( member848276444142703440term_f @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_509_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_510_subset__code_I1_J,axiom,
    ! [Xs: list_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ ( set_Ground_gterm_f2 @ Xs ) @ B2 )
      = ( ! [X: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X @ ( set_Ground_gterm_f2 @ Xs ) )
           => ( member5261315044688711901term_f @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_511_finite__list,axiom,
    ! [A: set_fs7307227306443116653rm_f_v] :
      ( ( finite5809181139301488822rm_f_v @ A )
     => ? [Xs2: list_f1824981274722084755rm_f_v] :
          ( ( set_fs7270820277574336546rm_f_v @ Xs2 )
          = A ) ) ).

% finite_list
thf(fact_512_finite__list,axiom,
    ! [A: set_Ground_gterm_f] :
      ( ( finite1923210418194103261term_f @ A )
     => ? [Xs2: list_Ground_gterm_f] :
          ( ( set_Ground_gterm_f2 @ Xs2 )
          = A ) ) ).

% finite_list
thf(fact_513_finite__list,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ? [Xs2: list_nat] :
          ( ( set_nat2 @ Xs2 )
          = A ) ) ).

% finite_list
thf(fact_514_finite__list,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ? [Xs2: list_P3903862279629787026_f_nat] :
          ( ( set_Pr7102205019285007021_f_nat @ Xs2 )
          = A ) ) ).

% finite_list
thf(fact_515_finite__list,axiom,
    ! [A: set_se421094335941524212term_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ? [Xs2: list_s5016777017369925914term_f] :
          ( ( set_se8315813792620424745term_f @ Xs2 )
          = A ) ) ).

% finite_list
thf(fact_516_finite__list,axiom,
    ! [A: set_se5570125628664547815term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ? [Xs2: list_s8644638678082573581term_f] :
          ( ( set_se6127471701090326172term_f @ Xs2 )
          = A ) ) ).

% finite_list
thf(fact_517_finite__list,axiom,
    ! [A: set_se1451291994713068270_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ? [Xs2: list_s7480137856107671624_f_nat] :
          ( ( set_se752066948783124067_f_nat @ Xs2 )
          = A ) ) ).

% finite_list
thf(fact_518_finite__list,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ? [Xs2: list_set_nat] :
          ( ( set_set_nat2 @ Xs2 )
          = A ) ) ).

% finite_list
thf(fact_519_list_Omap__cong,axiom,
    ! [X3: list_f1824981274722084755rm_f_v,Ya: list_f1824981274722084755rm_f_v,F3: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,G2: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ( X3 = Ya )
     => ( ! [Z3: fset_P8018961893305114765rm_f_v] :
            ( ( member6790519936504491446rm_f_v @ Z3 @ ( set_fs7270820277574336546rm_f_v @ Ya ) )
           => ( ( F3 @ Z3 )
              = ( G2 @ Z3 ) ) )
       => ( ( map_fs8602507653405230974rm_f_v @ F3 @ X3 )
          = ( map_fs8602507653405230974rm_f_v @ G2 @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_520_list_Omap__cong0,axiom,
    ! [X3: list_f1824981274722084755rm_f_v,F3: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,G2: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ! [Z3: fset_P8018961893305114765rm_f_v] :
          ( ( member6790519936504491446rm_f_v @ Z3 @ ( set_fs7270820277574336546rm_f_v @ X3 ) )
         => ( ( F3 @ Z3 )
            = ( G2 @ Z3 ) ) )
     => ( ( map_fs8602507653405230974rm_f_v @ F3 @ X3 )
        = ( map_fs8602507653405230974rm_f_v @ G2 @ X3 ) ) ) ).

% list.map_cong0
thf(fact_521_list_Oinj__map__strong,axiom,
    ! [X3: list_f1824981274722084755rm_f_v,Xa3: list_f1824981274722084755rm_f_v,F3: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,Fa: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ! [Z3: fset_P8018961893305114765rm_f_v,Za: fset_P8018961893305114765rm_f_v] :
          ( ( member6790519936504491446rm_f_v @ Z3 @ ( set_fs7270820277574336546rm_f_v @ X3 ) )
         => ( ( member6790519936504491446rm_f_v @ Za @ ( set_fs7270820277574336546rm_f_v @ Xa3 ) )
           => ( ( ( F3 @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_fs8602507653405230974rm_f_v @ F3 @ X3 )
          = ( map_fs8602507653405230974rm_f_v @ Fa @ Xa3 ) )
       => ( X3 = Xa3 ) ) ) ).

% list.inj_map_strong
thf(fact_522_list_Omap__ident__strong,axiom,
    ! [T: list_P4363786793477243949term_f,F3: produc7245736746747425831term_f > produc7245736746747425831term_f] :
      ( ! [Z3: produc7245736746747425831term_f] :
          ( ( member848276444142703440term_f @ Z3 @ ( set_Pr1901606489578307004term_f @ T ) )
         => ( ( F3 @ Z3 )
            = Z3 ) )
     => ( ( map_Pr2733089317190839300term_f @ F3 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_523_list_Omap__ident__strong,axiom,
    ! [T: list_nat,F3: nat > nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ T ) )
         => ( ( F3 @ Z3 )
            = Z3 ) )
     => ( ( map_nat_nat @ F3 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_524_list_Omap__ident__strong,axiom,
    ! [T: list_f1824981274722084755rm_f_v,F3: fset_P8018961893305114765rm_f_v > fset_P8018961893305114765rm_f_v] :
      ( ! [Z3: fset_P8018961893305114765rm_f_v] :
          ( ( member6790519936504491446rm_f_v @ Z3 @ ( set_fs7270820277574336546rm_f_v @ T ) )
         => ( ( F3 @ Z3 )
            = Z3 ) )
     => ( ( map_fs3095252337778551172rm_f_v @ F3 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_525_map__ext,axiom,
    ! [Xs: list_f1824981274722084755rm_f_v,F3: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,G2: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ! [X2: fset_P8018961893305114765rm_f_v] :
          ( ( member6790519936504491446rm_f_v @ X2 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
         => ( ( F3 @ X2 )
            = ( G2 @ X2 ) ) )
     => ( ( map_fs8602507653405230974rm_f_v @ F3 @ Xs )
        = ( map_fs8602507653405230974rm_f_v @ G2 @ Xs ) ) ) ).

% map_ext
thf(fact_526_map__idI,axiom,
    ! [Xs: list_P4363786793477243949term_f,F3: produc7245736746747425831term_f > produc7245736746747425831term_f] :
      ( ! [X2: produc7245736746747425831term_f] :
          ( ( member848276444142703440term_f @ X2 @ ( set_Pr1901606489578307004term_f @ Xs ) )
         => ( ( F3 @ X2 )
            = X2 ) )
     => ( ( map_Pr2733089317190839300term_f @ F3 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_527_map__idI,axiom,
    ! [Xs: list_nat,F3: nat > nat] :
      ( ! [X2: nat] :
          ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
         => ( ( F3 @ X2 )
            = X2 ) )
     => ( ( map_nat_nat @ F3 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_528_map__idI,axiom,
    ! [Xs: list_f1824981274722084755rm_f_v,F3: fset_P8018961893305114765rm_f_v > fset_P8018961893305114765rm_f_v] :
      ( ! [X2: fset_P8018961893305114765rm_f_v] :
          ( ( member6790519936504491446rm_f_v @ X2 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
         => ( ( F3 @ X2 )
            = X2 ) )
     => ( ( map_fs3095252337778551172rm_f_v @ F3 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_529_map__cong,axiom,
    ! [Xs: list_f1824981274722084755rm_f_v,Ys: list_f1824981274722084755rm_f_v,F3: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,G2: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ( Xs = Ys )
     => ( ! [X2: fset_P8018961893305114765rm_f_v] :
            ( ( member6790519936504491446rm_f_v @ X2 @ ( set_fs7270820277574336546rm_f_v @ Ys ) )
           => ( ( F3 @ X2 )
              = ( G2 @ X2 ) ) )
       => ( ( map_fs8602507653405230974rm_f_v @ F3 @ Xs )
          = ( map_fs8602507653405230974rm_f_v @ G2 @ Ys ) ) ) ) ).

% map_cong
thf(fact_530_ex__map__conv,axiom,
    ! [Ys: list_s8746099396510718605rm_f_v,F3: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ( ? [Xs3: list_f1824981274722084755rm_f_v] :
            ( Ys
            = ( map_fs8602507653405230974rm_f_v @ F3 @ Xs3 ) ) )
      = ( ! [X: set_Pr8827868859434726151rm_f_v] :
            ( ( member4041562125048333488rm_f_v @ X @ ( set_se722422988665441564rm_f_v @ Ys ) )
           => ? [Y4: fset_P8018961893305114765rm_f_v] :
                ( X
                = ( F3 @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_531_is__to__trs_H__props_I2_J,axiom,
    ! [Rs: list_f1824981274722084755rm_f_v,F2: fset_P6228066233360383026_f_nat,Is: list_FOR_ftrs,S: fset_P8018961893305114765rm_f_v] :
      ( ! [X2: fset_P8018961893305114765rm_f_v] :
          ( ( member6790519936504491446rm_f_v @ X2 @ ( set_fs7270820277574336546rm_f_v @ Rs ) )
         => ( ( lV_to_lv_trs_f_v @ ( fset_P4617584883882644886rm_f_v @ X2 ) )
            & ( ord_le1552505484586773650_f_nat @ ( lV_to_ffunas_trs_f_v @ X2 ) @ F2 ) ) )
     => ( ( ( fOR_is_to_trs_f_v @ Rs @ Is )
          = ( some_f4918923380007242610rm_f_v @ S ) )
       => ( lV_to_lv_trs_f_v @ ( fset_P4617584883882644886rm_f_v @ S ) ) ) ) ).

% is_to_trs'_props(2)
thf(fact_532_is__to__trs_H__props_I1_J,axiom,
    ! [Rs: list_f1824981274722084755rm_f_v,F2: fset_P6228066233360383026_f_nat,Is: list_FOR_ftrs,S: fset_P8018961893305114765rm_f_v] :
      ( ! [X2: fset_P8018961893305114765rm_f_v] :
          ( ( member6790519936504491446rm_f_v @ X2 @ ( set_fs7270820277574336546rm_f_v @ Rs ) )
         => ( ( lV_to_lv_trs_f_v @ ( fset_P4617584883882644886rm_f_v @ X2 ) )
            & ( ord_le1552505484586773650_f_nat @ ( lV_to_ffunas_trs_f_v @ X2 ) @ F2 ) ) )
     => ( ( ( fOR_is_to_trs_f_v @ Rs @ Is )
          = ( some_f4918923380007242610rm_f_v @ S ) )
       => ( ord_le1552505484586773650_f_nat @ ( lV_to_ffunas_trs_f_v @ S ) @ F2 ) ) ) ).

% is_to_trs'_props(1)
thf(fact_533_set__product,axiom,
    ! [Xs: list_f1824981274722084755rm_f_v,Ys: list_f1824981274722084755rm_f_v] :
      ( ( set_Pr7525886052349274108rm_f_v @ ( produc3634554572996078213rm_f_v @ Xs @ Ys ) )
      = ( produc6891897000406530696rm_f_v @ ( set_fs7270820277574336546rm_f_v @ Xs )
        @ ^ [Uu: fset_P8018961893305114765rm_f_v] : ( set_fs7270820277574336546rm_f_v @ Ys ) ) ) ).

% set_product
thf(fact_534_set__product,axiom,
    ! [Xs: list_Ground_gterm_f,Ys: list_Ground_gterm_f] :
      ( ( set_Pr1901606489578307004term_f @ ( produc6472808940620784069term_f @ Xs @ Ys ) )
      = ( produc6043514909933191368term_f @ ( set_Ground_gterm_f2 @ Xs )
        @ ^ [Uu: ground_gterm_f] : ( set_Ground_gterm_f2 @ Ys ) ) ) ).

% set_product
thf(fact_535_gtt__of__gtt__rel__correct,axiom,
    ! [Rs: list_f1824981274722084755rm_f_v,F2: fset_P6228066233360383026_f_nat,G2: fOR_gtt_rel_FOR_ftrs,G3: produc5774109276193113959_nat_f] :
      ( ! [X2: fset_P8018961893305114765rm_f_v] :
          ( ( member6790519936504491446rm_f_v @ X2 @ ( set_fs7270820277574336546rm_f_v @ Rs ) )
         => ( ( lV_to_lv_trs_f_v @ ( fset_P4617584883882644886rm_f_v @ X2 ) )
            & ( ord_le1552505484586773650_f_nat @ ( lV_to_ffunas_trs_f_v @ X2 ) @ F2 ) ) )
     => ( ( ( fOR_gt3598446537764997717el_f_v @ F2 @ Rs @ G2 )
          = ( some_P6781462078782365260_nat_f @ G3 ) )
       => ( ( agtt_lang_nat_f @ G3 )
          = ( fOR_eval_gtt_rel_f_v @ ( fset_P3576968334923099475_f_nat @ F2 ) @ ( map_fs8602507653405230974rm_f_v @ fset_P4617584883882644886rm_f_v @ Rs ) @ G2 ) ) ) ) ).

% gtt_of_gtt_rel_correct
thf(fact_536_gtrancl__rel__sig,axiom,
    ! [R2: set_Pr989862937836626183term_f,F2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le263819222746101927term_f @ R2
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) )
     => ( ord_le263819222746101927term_f @ ( ground_gtrancl_rel_f @ F2 @ R2 )
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) ) ) ).

% gtrancl_rel_sig
thf(fact_537_map__eq__map__tailrec,axiom,
    map_fs8602507653405230974rm_f_v = map_ta9055086340571775688rm_f_v ).

% map_eq_map_tailrec
thf(fact_538_R__in__gtrancl__rel,axiom,
    ! [R5: set_Pr989862937836626183term_f,F2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le263819222746101927term_f @ R5
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) )
     => ( ord_le263819222746101927term_f @ R5 @ ( ground_gtrancl_rel_f @ F2 @ R5 ) ) ) ).

% R_in_gtrancl_rel
thf(fact_539_gtrancl__rel__sound,axiom,
    ! [R5: set_Pr989862937836626183term_f,F2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le263819222746101927term_f @ R5
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) )
     => ( ord_le263819222746101927term_f @ ( ground_gtrancl_rel_f @ F2 @ R5 )
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) ) ) ).

% gtrancl_rel_sound
thf(fact_540_is__to__trs__conv,axiom,
    ! [Rs: list_f1824981274722084755rm_f_v,Is: list_FOR_ftrs,S: fset_P8018961893305114765rm_f_v] :
      ( ( ( fOR_is_to_trs_f_v @ Rs @ Is )
        = ( some_f4918923380007242610rm_f_v @ S ) )
     => ( ( fOR_is_to_trs_f_v2 @ ( map_fs8602507653405230974rm_f_v @ fset_P4617584883882644886rm_f_v @ Rs ) @ Is )
        = ( fset_P4617584883882644886rm_f_v @ S ) ) ) ).

% is_to_trs_conv
thf(fact_541_subset__Collect__iff,axiom,
    ! [B2: set_se5570125628664547815term_f,A: set_se5570125628664547815term_f,P: set_Pr989862937836626183term_f > $o] :
      ( ( ord_le5727726935103693191term_f @ B2 @ A )
     => ( ( ord_le5727726935103693191term_f @ B2
          @ ( collec4490547877585280114term_f
            @ ^ [X: set_Pr989862937836626183term_f] :
                ( ( member6919674941555055664term_f @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: set_Pr989862937836626183term_f] :
              ( ( member6919674941555055664term_f @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_542_subset__Collect__iff,axiom,
    ! [B2: set_se1451291994713068270_f_nat,A: set_se1451291994713068270_f_nat,P: set_Pr5245412377734449720_f_nat > $o] :
      ( ( ord_le5386115319958680398_f_nat @ B2 @ A )
     => ( ( ord_le5386115319958680398_f_nat @ B2
          @ ( collec6964094325902095629_f_nat
            @ ^ [X: set_Pr5245412377734449720_f_nat] :
                ( ( member8345558847462883279_f_nat @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: set_Pr5245412377734449720_f_nat] :
              ( ( member8345558847462883279_f_nat @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_543_subset__Collect__iff,axiom,
    ! [B2: set_set_nat,A: set_set_nat,P: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B2 @ A )
     => ( ( ord_le6893508408891458716et_nat @ B2
          @ ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( member_set_nat @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: set_nat] :
              ( ( member_set_nat @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_544_subset__Collect__iff,axiom,
    ! [B2: set_se421094335941524212term_f,A: set_se421094335941524212term_f,P: set_Ground_gterm_f > $o] :
      ( ( ord_le2325249104654787220term_f @ B2 @ A )
     => ( ( ord_le2325249104654787220term_f @ B2
          @ ( collec1979342819066322687term_f
            @ ^ [X: set_Ground_gterm_f] :
                ( ( member9219338147399679421term_f @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: set_Ground_gterm_f] :
              ( ( member9219338147399679421term_f @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_545_subset__Collect__iff,axiom,
    ! [B2: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,P: product_prod_f_nat > $o] :
      ( ( ord_le8976984241387448984_f_nat @ B2 @ A )
     => ( ( ord_le8976984241387448984_f_nat @ B2
          @ ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_546_subset__Collect__iff,axiom,
    ! [B2: set_Pr989862937836626183term_f,A: set_Pr989862937836626183term_f,P: produc7245736746747425831term_f > $o] :
      ( ( ord_le263819222746101927term_f @ B2 @ A )
     => ( ( ord_le263819222746101927term_f @ B2
          @ ( collec3209569126566747026term_f
            @ ^ [X: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: produc7245736746747425831term_f] :
              ( ( member848276444142703440term_f @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_547_subset__Collect__iff,axiom,
    ! [B2: set_nat,A: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B2 @ A )
     => ( ( ord_less_eq_set_nat @ B2
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_548_subset__Collect__iff,axiom,
    ! [B2: set_Ground_gterm_f,A: set_Ground_gterm_f,P: ground_gterm_f > $o] :
      ( ( ord_le2735537439747282356term_f @ B2 @ A )
     => ( ( ord_le2735537439747282356term_f @ B2
          @ ( collec3522850625759001119term_f
            @ ^ [X: ground_gterm_f] :
                ( ( member5261315044688711901term_f @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: ground_gterm_f] :
              ( ( member5261315044688711901term_f @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_549_subset__CollectI,axiom,
    ! [B2: set_se5570125628664547815term_f,A: set_se5570125628664547815term_f,Q: set_Pr989862937836626183term_f > $o,P: set_Pr989862937836626183term_f > $o] :
      ( ( ord_le5727726935103693191term_f @ B2 @ A )
     => ( ! [X2: set_Pr989862937836626183term_f] :
            ( ( member6919674941555055664term_f @ X2 @ B2 )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_le5727726935103693191term_f
          @ ( collec4490547877585280114term_f
            @ ^ [X: set_Pr989862937836626183term_f] :
                ( ( member6919674941555055664term_f @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collec4490547877585280114term_f
            @ ^ [X: set_Pr989862937836626183term_f] :
                ( ( member6919674941555055664term_f @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_550_subset__CollectI,axiom,
    ! [B2: set_se1451291994713068270_f_nat,A: set_se1451291994713068270_f_nat,Q: set_Pr5245412377734449720_f_nat > $o,P: set_Pr5245412377734449720_f_nat > $o] :
      ( ( ord_le5386115319958680398_f_nat @ B2 @ A )
     => ( ! [X2: set_Pr5245412377734449720_f_nat] :
            ( ( member8345558847462883279_f_nat @ X2 @ B2 )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_le5386115319958680398_f_nat
          @ ( collec6964094325902095629_f_nat
            @ ^ [X: set_Pr5245412377734449720_f_nat] :
                ( ( member8345558847462883279_f_nat @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collec6964094325902095629_f_nat
            @ ^ [X: set_Pr5245412377734449720_f_nat] :
                ( ( member8345558847462883279_f_nat @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_551_subset__CollectI,axiom,
    ! [B2: set_set_nat,A: set_set_nat,Q: set_nat > $o,P: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B2 @ A )
     => ( ! [X2: set_nat] :
            ( ( member_set_nat @ X2 @ B2 )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_le6893508408891458716et_nat
          @ ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( member_set_nat @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( member_set_nat @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_552_subset__CollectI,axiom,
    ! [B2: set_se421094335941524212term_f,A: set_se421094335941524212term_f,Q: set_Ground_gterm_f > $o,P: set_Ground_gterm_f > $o] :
      ( ( ord_le2325249104654787220term_f @ B2 @ A )
     => ( ! [X2: set_Ground_gterm_f] :
            ( ( member9219338147399679421term_f @ X2 @ B2 )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_le2325249104654787220term_f
          @ ( collec1979342819066322687term_f
            @ ^ [X: set_Ground_gterm_f] :
                ( ( member9219338147399679421term_f @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collec1979342819066322687term_f
            @ ^ [X: set_Ground_gterm_f] :
                ( ( member9219338147399679421term_f @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_553_subset__CollectI,axiom,
    ! [B2: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,Q: product_prod_f_nat > $o,P: product_prod_f_nat > $o] :
      ( ( ord_le8976984241387448984_f_nat @ B2 @ A )
     => ( ! [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ B2 )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_le8976984241387448984_f_nat
          @ ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_554_subset__CollectI,axiom,
    ! [B2: set_Pr989862937836626183term_f,A: set_Pr989862937836626183term_f,Q: produc7245736746747425831term_f > $o,P: produc7245736746747425831term_f > $o] :
      ( ( ord_le263819222746101927term_f @ B2 @ A )
     => ( ! [X2: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X2 @ B2 )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_le263819222746101927term_f
          @ ( collec3209569126566747026term_f
            @ ^ [X: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collec3209569126566747026term_f
            @ ^ [X: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_555_subset__CollectI,axiom,
    ! [B2: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B2 @ A )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ B2 )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_556_subset__CollectI,axiom,
    ! [B2: set_Ground_gterm_f,A: set_Ground_gterm_f,Q: ground_gterm_f > $o,P: ground_gterm_f > $o] :
      ( ( ord_le2735537439747282356term_f @ B2 @ A )
     => ( ! [X2: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X2 @ B2 )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_le2735537439747282356term_f
          @ ( collec3522850625759001119term_f
            @ ^ [X: ground_gterm_f] :
                ( ( member5261315044688711901term_f @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collec3522850625759001119term_f
            @ ^ [X: ground_gterm_f] :
                ( ( member5261315044688711901term_f @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_557_Collect__restrict,axiom,
    ! [X6: set_se5570125628664547815term_f,P: set_Pr989862937836626183term_f > $o] :
      ( ord_le5727726935103693191term_f
      @ ( collec4490547877585280114term_f
        @ ^ [X: set_Pr989862937836626183term_f] :
            ( ( member6919674941555055664term_f @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_558_Collect__restrict,axiom,
    ! [X6: set_se1451291994713068270_f_nat,P: set_Pr5245412377734449720_f_nat > $o] :
      ( ord_le5386115319958680398_f_nat
      @ ( collec6964094325902095629_f_nat
        @ ^ [X: set_Pr5245412377734449720_f_nat] :
            ( ( member8345558847462883279_f_nat @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_559_Collect__restrict,axiom,
    ! [X6: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( member_set_nat @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_560_Collect__restrict,axiom,
    ! [X6: set_se421094335941524212term_f,P: set_Ground_gterm_f > $o] :
      ( ord_le2325249104654787220term_f
      @ ( collec1979342819066322687term_f
        @ ^ [X: set_Ground_gterm_f] :
            ( ( member9219338147399679421term_f @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_561_Collect__restrict,axiom,
    ! [X6: set_Pr5245412377734449720_f_nat,P: product_prod_f_nat > $o] :
      ( ord_le8976984241387448984_f_nat
      @ ( collec1417984143148964183_f_nat
        @ ^ [X: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_562_Collect__restrict,axiom,
    ! [X6: set_Pr989862937836626183term_f,P: produc7245736746747425831term_f > $o] :
      ( ord_le263819222746101927term_f
      @ ( collec3209569126566747026term_f
        @ ^ [X: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_563_Collect__restrict,axiom,
    ! [X6: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_564_Collect__restrict,axiom,
    ! [X6: set_Ground_gterm_f,P: ground_gterm_f > $o] :
      ( ord_le2735537439747282356term_f
      @ ( collec3522850625759001119term_f
        @ ^ [X: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_565_prop__restrict,axiom,
    ! [X3: set_Pr989862937836626183term_f,Z4: set_se5570125628664547815term_f,X6: set_se5570125628664547815term_f,P: set_Pr989862937836626183term_f > $o] :
      ( ( member6919674941555055664term_f @ X3 @ Z4 )
     => ( ( ord_le5727726935103693191term_f @ Z4
          @ ( collec4490547877585280114term_f
            @ ^ [X: set_Pr989862937836626183term_f] :
                ( ( member6919674941555055664term_f @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_566_prop__restrict,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Z4: set_se1451291994713068270_f_nat,X6: set_se1451291994713068270_f_nat,P: set_Pr5245412377734449720_f_nat > $o] :
      ( ( member8345558847462883279_f_nat @ X3 @ Z4 )
     => ( ( ord_le5386115319958680398_f_nat @ Z4
          @ ( collec6964094325902095629_f_nat
            @ ^ [X: set_Pr5245412377734449720_f_nat] :
                ( ( member8345558847462883279_f_nat @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_567_prop__restrict,axiom,
    ! [X3: set_nat,Z4: set_set_nat,X6: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X3 @ Z4 )
     => ( ( ord_le6893508408891458716et_nat @ Z4
          @ ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( member_set_nat @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_568_prop__restrict,axiom,
    ! [X3: set_Ground_gterm_f,Z4: set_se421094335941524212term_f,X6: set_se421094335941524212term_f,P: set_Ground_gterm_f > $o] :
      ( ( member9219338147399679421term_f @ X3 @ Z4 )
     => ( ( ord_le2325249104654787220term_f @ Z4
          @ ( collec1979342819066322687term_f
            @ ^ [X: set_Ground_gterm_f] :
                ( ( member9219338147399679421term_f @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_569_prop__restrict,axiom,
    ! [X3: product_prod_f_nat,Z4: set_Pr5245412377734449720_f_nat,X6: set_Pr5245412377734449720_f_nat,P: product_prod_f_nat > $o] :
      ( ( member2678038196313615385_f_nat @ X3 @ Z4 )
     => ( ( ord_le8976984241387448984_f_nat @ Z4
          @ ( collec1417984143148964183_f_nat
            @ ^ [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_570_prop__restrict,axiom,
    ! [X3: produc7245736746747425831term_f,Z4: set_Pr989862937836626183term_f,X6: set_Pr989862937836626183term_f,P: produc7245736746747425831term_f > $o] :
      ( ( member848276444142703440term_f @ X3 @ Z4 )
     => ( ( ord_le263819222746101927term_f @ Z4
          @ ( collec3209569126566747026term_f
            @ ^ [X: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_571_prop__restrict,axiom,
    ! [X3: nat,Z4: set_nat,X6: set_nat,P: nat > $o] :
      ( ( member_nat @ X3 @ Z4 )
     => ( ( ord_less_eq_set_nat @ Z4
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_572_prop__restrict,axiom,
    ! [X3: ground_gterm_f,Z4: set_Ground_gterm_f,X6: set_Ground_gterm_f,P: ground_gterm_f > $o] :
      ( ( member5261315044688711901term_f @ X3 @ Z4 )
     => ( ( ord_le2735537439747282356term_f @ Z4
          @ ( collec3522850625759001119term_f
            @ ^ [X: ground_gterm_f] :
                ( ( member5261315044688711901term_f @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_573_ssubst__Pair__rhs,axiom,
    ! [R4: ground_gterm_f,S3: ground_gterm_f,R2: set_Pr989862937836626183term_f,S5: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ R4 @ S3 ) @ R2 )
     => ( ( S5 = S3 )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ R4 @ S5 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_574_conj__subset__def,axiom,
    ! [A: set_se5570125628664547815term_f,P: set_Pr989862937836626183term_f > $o,Q: set_Pr989862937836626183term_f > $o] :
      ( ( ord_le5727726935103693191term_f @ A
        @ ( collec4490547877585280114term_f
          @ ^ [X: set_Pr989862937836626183term_f] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le5727726935103693191term_f @ A @ ( collec4490547877585280114term_f @ P ) )
        & ( ord_le5727726935103693191term_f @ A @ ( collec4490547877585280114term_f @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_575_conj__subset__def,axiom,
    ! [A: set_se1451291994713068270_f_nat,P: set_Pr5245412377734449720_f_nat > $o,Q: set_Pr5245412377734449720_f_nat > $o] :
      ( ( ord_le5386115319958680398_f_nat @ A
        @ ( collec6964094325902095629_f_nat
          @ ^ [X: set_Pr5245412377734449720_f_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le5386115319958680398_f_nat @ A @ ( collec6964094325902095629_f_nat @ P ) )
        & ( ord_le5386115319958680398_f_nat @ A @ ( collec6964094325902095629_f_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_576_conj__subset__def,axiom,
    ! [A: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ A
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A @ ( collect_set_nat @ P ) )
        & ( ord_le6893508408891458716et_nat @ A @ ( collect_set_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_577_conj__subset__def,axiom,
    ! [A: set_se421094335941524212term_f,P: set_Ground_gterm_f > $o,Q: set_Ground_gterm_f > $o] :
      ( ( ord_le2325249104654787220term_f @ A
        @ ( collec1979342819066322687term_f
          @ ^ [X: set_Ground_gterm_f] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le2325249104654787220term_f @ A @ ( collec1979342819066322687term_f @ P ) )
        & ( ord_le2325249104654787220term_f @ A @ ( collec1979342819066322687term_f @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_578_conj__subset__def,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,P: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
      ( ( ord_le8976984241387448984_f_nat @ A
        @ ( collec1417984143148964183_f_nat
          @ ^ [X: product_prod_f_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le8976984241387448984_f_nat @ A @ ( collec1417984143148964183_f_nat @ P ) )
        & ( ord_le8976984241387448984_f_nat @ A @ ( collec1417984143148964183_f_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_579_conj__subset__def,axiom,
    ! [A: set_Pr989862937836626183term_f,P: produc7245736746747425831term_f > $o,Q: produc7245736746747425831term_f > $o] :
      ( ( ord_le263819222746101927term_f @ A
        @ ( collec3209569126566747026term_f
          @ ^ [X: produc7245736746747425831term_f] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le263819222746101927term_f @ A @ ( collec3209569126566747026term_f @ P ) )
        & ( ord_le263819222746101927term_f @ A @ ( collec3209569126566747026term_f @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_580_conj__subset__def,axiom,
    ! [A: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
        & ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_581_conj__subset__def,axiom,
    ! [A: set_Ground_gterm_f,P: ground_gterm_f > $o,Q: ground_gterm_f > $o] :
      ( ( ord_le2735537439747282356term_f @ A
        @ ( collec3522850625759001119term_f
          @ ^ [X: ground_gterm_f] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le2735537439747282356term_f @ A @ ( collec3522850625759001119term_f @ P ) )
        & ( ord_le2735537439747282356term_f @ A @ ( collec3522850625759001119term_f @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_582_lift__root__step__sig_H,axiom,
    ! [R5: set_Pr989862937836626183term_f,G: set_Pr5245412377734449720_f_nat,H: set_Pr5245412377734449720_f_nat,F2: set_Pr5245412377734449720_f_nat,W: fOR_pos_step,X6: fOR_ext_step] :
      ( ( ord_le263819222746101927term_f @ R5
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ G )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ H ) ) )
     => ( ( ord_le8976984241387448984_f_nat @ F2 @ G )
       => ( ( ord_le8976984241387448984_f_nat @ F2 @ H )
         => ( ord_le263819222746101927term_f @ ( lift_R8854095231239425060step_f @ F2 @ W @ X6 @ R5 )
            @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ G )
              @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ H ) ) ) ) ) ) ).

% lift_root_step_sig'
thf(fact_583_rr2__compositon,axiom,
    ! [R5: set_Pr989862937836626183term_f,F2: fset_P6228066233360383026_f_nat,LL: set_Pr989862937836626183term_f,A: tree_r733329426570293750tion_f,B2: tree_r733329426570293750tion_f] :
      ( ( ord_le263819222746101927term_f @ R5
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ ( fset_P3576968334923099475_f_nat @ F2 ) )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ ( fset_P3576968334923099475_f_nat @ F2 ) ) ) )
     => ( ( ord_le263819222746101927term_f @ LL
          @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ ( fset_P3576968334923099475_f_nat @ F2 ) )
            @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ ( fset_P3576968334923099475_f_nat @ F2 ) ) ) )
       => ( ( rRn_RR2_spec_nat_f_f @ A @ R5 )
         => ( ( rRn_RR2_spec_nat_f_f @ B2 @ LL )
           => ( rRn_RR7359223619187921291at_f_f @ ( rRn_rr8100768573751510567at_nat @ F2 @ A @ B2 ) @ ( relcom4354988593214781939term_f @ R5 @ LL ) ) ) ) ) ) ).

% rr2_compositon
thf(fact_584_fset__of__list__subset,axiom,
    ! [Xs: list_f1824981274722084755rm_f_v,Ys: list_f1824981274722084755rm_f_v] :
      ( ( ord_le2293696477246793741rm_f_v @ ( set_fs7270820277574336546rm_f_v @ Xs ) @ ( set_fs7270820277574336546rm_f_v @ Ys ) )
     => ( ord_le7606603380504677011rm_f_v @ ( fset_o5782745410740424019rm_f_v @ Xs ) @ ( fset_o5782745410740424019rm_f_v @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_585_fset__of__list__subset,axiom,
    ! [Xs: list_P3903862279629787026_f_nat,Ys: list_P3903862279629787026_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ Xs ) @ ( set_Pr7102205019285007021_f_nat @ Ys ) )
     => ( ord_le1552505484586773650_f_nat @ ( fset_o8009517685352940092_f_nat @ Xs ) @ ( fset_o8009517685352940092_f_nat @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_586_fset__of__list__subset,axiom,
    ! [Xs: list_P4363786793477243949term_f,Ys: list_P4363786793477243949term_f] :
      ( ( ord_le263819222746101927term_f @ ( set_Pr1901606489578307004term_f @ Xs ) @ ( set_Pr1901606489578307004term_f @ Ys ) )
     => ( ord_le8532657618882012973term_f @ ( fset_o6879037440527609069term_f @ Xs ) @ ( fset_o6879037440527609069term_f @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_587_fset__of__list__subset,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
     => ( ord_less_eq_fset_nat @ ( fset_of_list_nat @ Xs ) @ ( fset_of_list_nat @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_588_fset__of__list__subset,axiom,
    ! [Xs: list_Ground_gterm_f,Ys: list_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ ( set_Ground_gterm_f2 @ Xs ) @ ( set_Ground_gterm_f2 @ Ys ) )
     => ( ord_le8198718236663383866term_f @ ( fset_o1084718608627824890term_f @ Xs ) @ ( fset_o1084718608627824890term_f @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_589_eval__rr2__rel_Osimps_I8_J,axiom,
    ! [F2: set_Pr5245412377734449720_f_nat,Rs: list_s8746099396510718605rm_f_v,R2: fOR_rr1_rel_FOR_ftrs,S: fOR_rr1_rel_FOR_ftrs] :
      ( ( fOR_eval_rr2_rel_f_v @ F2 @ Rs @ ( fOR_rr1845157339234217046R_ftrs @ R2 @ S ) )
      = ( produc6043514909933191368term_f @ ( fOR_eval_rr1_rel_f_v @ F2 @ Rs @ R2 )
        @ ^ [Uu: ground_gterm_f] : ( fOR_eval_rr1_rel_f_v @ F2 @ Rs @ S ) ) ) ).

% eval_rr2_rel.simps(8)
thf(fact_590_trim__RR2__spec,axiom,
    ! [A: tree_r733329426570293750tion_f,T3: set_Pr989862937836626183term_f] :
      ( ( rRn_RR2_spec_nat_f_f @ ( tree_t6100411961293590077tion_f @ A ) @ T3 )
      = ( rRn_RR2_spec_nat_f_f @ A @ T3 ) ) ).

% trim_RR2_spec
thf(fact_591_relcompEpair,axiom,
    ! [A6: ground_gterm_f,C2: ground_gterm_f,R4: set_Pr989862937836626183term_f,S3: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A6 @ C2 ) @ ( relcom4354988593214781939term_f @ R4 @ S3 ) )
     => ~ ! [B5: ground_gterm_f] :
            ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A6 @ B5 ) @ R4 )
           => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B5 @ C2 ) @ S3 ) ) ) ).

% relcompEpair
thf(fact_592_relcompE,axiom,
    ! [Xz: produc7245736746747425831term_f,R4: set_Pr989862937836626183term_f,S3: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ Xz @ ( relcom4354988593214781939term_f @ R4 @ S3 ) )
     => ~ ! [X2: ground_gterm_f,Y3: ground_gterm_f,Z3: ground_gterm_f] :
            ( ( Xz
              = ( produc3560254623552331287term_f @ X2 @ Z3 ) )
           => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X2 @ Y3 ) @ R4 )
             => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ Y3 @ Z3 ) @ S3 ) ) ) ) ).

% relcompE
thf(fact_593_relcomp_OrelcompI,axiom,
    ! [A6: ground_gterm_f,B3: ground_gterm_f,R4: set_Pr989862937836626183term_f,C2: ground_gterm_f,S3: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A6 @ B3 ) @ R4 )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B3 @ C2 ) @ S3 )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A6 @ C2 ) @ ( relcom4354988593214781939term_f @ R4 @ S3 ) ) ) ) ).

% relcomp.relcompI
thf(fact_594_relcomp_Osimps,axiom,
    ! [A1: ground_gterm_f,A22: ground_gterm_f,R4: set_Pr989862937836626183term_f,S3: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ A22 ) @ ( relcom4354988593214781939term_f @ R4 @ S3 ) )
      = ( ? [A5: ground_gterm_f,B6: ground_gterm_f,C3: ground_gterm_f] :
            ( ( A1 = A5 )
            & ( A22 = C3 )
            & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A5 @ B6 ) @ R4 )
            & ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B6 @ C3 ) @ S3 ) ) ) ) ).

% relcomp.simps
thf(fact_595_relcomp_Ocases,axiom,
    ! [A1: ground_gterm_f,A22: ground_gterm_f,R4: set_Pr989862937836626183term_f,S3: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ A22 ) @ ( relcom4354988593214781939term_f @ R4 @ S3 ) )
     => ~ ! [B5: ground_gterm_f] :
            ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A1 @ B5 ) @ R4 )
           => ~ ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ B5 @ A22 ) @ S3 ) ) ) ).

% relcomp.cases
thf(fact_596_relcomp__mono,axiom,
    ! [R6: set_Product_prod_f_f,R4: set_Product_prod_f_f,S5: set_Pr5245412377734449720_f_nat,S3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le7147000192471188839od_f_f @ R6 @ R4 )
     => ( ( ord_le8976984241387448984_f_nat @ S5 @ S3 )
       => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ R6 @ S5 ) @ ( relcomp_f_f_nat @ R4 @ S3 ) ) ) ) ).

% relcomp_mono
thf(fact_597_relcomp__mono,axiom,
    ! [R6: set_Pr5245412377734449720_f_nat,R4: set_Pr5245412377734449720_f_nat,S5: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8976984241387448984_f_nat @ R6 @ R4 )
     => ( ( ord_le3146513528884898305at_nat @ S5 @ S3 )
       => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_nat_nat @ R6 @ S5 ) @ ( relcomp_f_nat_nat @ R4 @ S3 ) ) ) ) ).

% relcomp_mono
thf(fact_598_relcomp__mono,axiom,
    ! [R6: set_Pr989862937836626183term_f,R4: set_Pr989862937836626183term_f,S5: set_Pr989862937836626183term_f,S3: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ R6 @ R4 )
     => ( ( ord_le263819222746101927term_f @ S5 @ S3 )
       => ( ord_le263819222746101927term_f @ ( relcom4354988593214781939term_f @ R6 @ S5 ) @ ( relcom4354988593214781939term_f @ R4 @ S3 ) ) ) ) ).

% relcomp_mono
thf(fact_599_lift__root__step__incr,axiom,
    ! [R5: set_Pr989862937836626183term_f,S6: set_Pr989862937836626183term_f,F2: set_Pr5245412377734449720_f_nat,W: fOR_pos_step,X6: fOR_ext_step] :
      ( ( ord_le263819222746101927term_f @ R5 @ S6 )
     => ( ord_le263819222746101927term_f @ ( lift_R8854095231239425060step_f @ F2 @ W @ X6 @ R5 ) @ ( lift_R8854095231239425060step_f @ F2 @ W @ X6 @ S6 ) ) ) ).

% lift_root_step_incr
thf(fact_600_lift__root__step__mono,axiom,
    ! [F2: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,W: fOR_pos_step,X6: fOR_ext_step,R5: set_Pr989862937836626183term_f] :
      ( ( ord_le8976984241387448984_f_nat @ F2 @ G )
     => ( ord_le263819222746101927term_f @ ( lift_R8854095231239425060step_f @ F2 @ W @ X6 @ R5 ) @ ( lift_R8854095231239425060step_f @ G @ W @ X6 @ R5 ) ) ) ).

% lift_root_step_mono
thf(fact_601_fset__of__list_Orep__eq,axiom,
    ! [X3: list_f1824981274722084755rm_f_v] :
      ( ( fset_f1410452810862158076rm_f_v @ ( fset_o5782745410740424019rm_f_v @ X3 ) )
      = ( set_fs7270820277574336546rm_f_v @ X3 ) ) ).

% fset_of_list.rep_eq
thf(fact_602_fset__of__list_Orep__eq,axiom,
    ! [X3: list_P3903862279629787026_f_nat] :
      ( ( fset_P3576968334923099475_f_nat @ ( fset_o8009517685352940092_f_nat @ X3 ) )
      = ( set_Pr7102205019285007021_f_nat @ X3 ) ) ).

% fset_of_list.rep_eq
thf(fact_603_fset__of__list_Orep__eq,axiom,
    ! [X3: list_P4093298276913796397rm_f_v] :
      ( ( fset_P4617584883882644886rm_f_v @ ( fset_o4970844032613833069rm_f_v @ X3 ) )
      = ( set_Pr817814403484925884rm_f_v @ X3 ) ) ).

% fset_of_list.rep_eq
thf(fact_604_relcomp__subset__Sigma,axiom,
    ! [R4: set_Product_prod_f_f,A: set_f,B2: set_f,S3: set_Pr5245412377734449720_f_nat,C: set_nat] :
      ( ( ord_le7147000192471188839od_f_f @ R4
        @ ( product_Sigma_f_f @ A
          @ ^ [Uu: f] : B2 ) )
     => ( ( ord_le8976984241387448984_f_nat @ S3
          @ ( product_Sigma_f_nat @ B2
            @ ^ [Uu: f] : C ) )
       => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ R4 @ S3 )
          @ ( product_Sigma_f_nat @ A
            @ ^ [Uu: f] : C ) ) ) ) ).

% relcomp_subset_Sigma
thf(fact_605_relcomp__subset__Sigma,axiom,
    ! [R4: set_Pr5245412377734449720_f_nat,A: set_f,B2: set_nat,S3: set_Pr1261947904930325089at_nat,C: set_nat] :
      ( ( ord_le8976984241387448984_f_nat @ R4
        @ ( product_Sigma_f_nat @ A
          @ ^ [Uu: f] : B2 ) )
     => ( ( ord_le3146513528884898305at_nat @ S3
          @ ( produc457027306803732586at_nat @ B2
            @ ^ [Uu: nat] : C ) )
       => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_nat_nat @ R4 @ S3 )
          @ ( product_Sigma_f_nat @ A
            @ ^ [Uu: f] : C ) ) ) ) ).

% relcomp_subset_Sigma
thf(fact_606_relcomp__subset__Sigma,axiom,
    ! [R4: set_Pr989862937836626183term_f,A: set_Ground_gterm_f,B2: set_Ground_gterm_f,S3: set_Pr989862937836626183term_f,C: set_Ground_gterm_f] :
      ( ( ord_le263819222746101927term_f @ R4
        @ ( produc6043514909933191368term_f @ A
          @ ^ [Uu: ground_gterm_f] : B2 ) )
     => ( ( ord_le263819222746101927term_f @ S3
          @ ( produc6043514909933191368term_f @ B2
            @ ^ [Uu: ground_gterm_f] : C ) )
       => ( ord_le263819222746101927term_f @ ( relcom4354988593214781939term_f @ R4 @ S3 )
          @ ( produc6043514909933191368term_f @ A
            @ ^ [Uu: ground_gterm_f] : C ) ) ) ) ).

% relcomp_subset_Sigma
thf(fact_607_lift__root__step__sig,axiom,
    ! [R5: set_Pr989862937836626183term_f,F2: set_Pr5245412377734449720_f_nat,W: fOR_pos_step,X6: fOR_ext_step] :
      ( ( ord_le263819222746101927term_f @ R5
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) )
     => ( ord_le263819222746101927term_f @ ( lift_R8854095231239425060step_f @ F2 @ W @ X6 @ R5 )
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) ) ) ).

% lift_root_step_sig
thf(fact_608_finite__relcomp,axiom,
    ! [R2: set_Product_prod_f_f,S: set_Pr5245412377734449720_f_nat] :
      ( ( finite1830552743515526928od_f_f @ R2 )
     => ( ( finite3598747971073327385_f_nat @ S )
       => ( finite3598747971073327385_f_nat @ ( relcomp_f_f_nat @ R2 @ S ) ) ) ) ).

% finite_relcomp
thf(fact_609_finite__relcomp,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( finite3598747971073327385_f_nat @ R2 )
     => ( ( finite6177210948735845034at_nat @ S )
       => ( finite3598747971073327385_f_nat @ ( relcomp_f_nat_nat @ R2 @ S ) ) ) ) ).

% finite_relcomp
thf(fact_610_O__mono2,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,R7: set_Pr5245412377734449720_f_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8976984241387448984_f_nat @ R2 @ R7 )
     => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_nat_nat @ R2 @ T3 ) @ ( relcomp_f_nat_nat @ R7 @ T3 ) ) ) ).

% O_mono2
thf(fact_611_O__mono2,axiom,
    ! [R2: set_Pr989862937836626183term_f,R7: set_Pr989862937836626183term_f,T3: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ R2 @ R7 )
     => ( ord_le263819222746101927term_f @ ( relcom4354988593214781939term_f @ R2 @ T3 ) @ ( relcom4354988593214781939term_f @ R7 @ T3 ) ) ) ).

% O_mono2
thf(fact_612_O__mono1,axiom,
    ! [R2: set_Pr5245412377734449720_f_nat,R7: set_Pr5245412377734449720_f_nat,S: set_Product_prod_f_f] :
      ( ( ord_le8976984241387448984_f_nat @ R2 @ R7 )
     => ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ S @ R2 ) @ ( relcomp_f_f_nat @ S @ R7 ) ) ) ).

% O_mono1
thf(fact_613_O__mono1,axiom,
    ! [R2: set_Pr989862937836626183term_f,R7: set_Pr989862937836626183term_f,S: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ R2 @ R7 )
     => ( ord_le263819222746101927term_f @ ( relcom4354988593214781939term_f @ S @ R2 ) @ ( relcom4354988593214781939term_f @ S @ R7 ) ) ) ).

% O_mono1
thf(fact_614_relcomp3__I,axiom,
    ! [T: ground_gterm_f,U2: ground_gterm_f,A: set_Pr989862937836626183term_f,S3: ground_gterm_f,B2: set_Pr989862937836626183term_f,V2: ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ T @ U2 ) @ A )
     => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S3 @ T ) @ B2 )
       => ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ U2 @ V2 ) @ B2 )
         => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S3 @ V2 ) @ ( relcom4354988593214781939term_f @ B2 @ ( relcom4354988593214781939term_f @ A @ B2 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_615_ffunas__gterm_Orep__eq,axiom,
    ! [X3: ground_gterm_f] :
      ( ( fset_P3576968334923099475_f_nat @ ( fSet_ffunas_gterm_f @ X3 ) )
      = ( ground_funas_gterm_f @ X3 ) ) ).

% ffunas_gterm.rep_eq
thf(fact_616_Fpow__def,axiom,
    ( finite6503987499129345914term_f
    = ( ^ [A4: set_se421094335941524212term_f] :
          ( collec8921018771386908639term_f
          @ ^ [X7: set_se421094335941524212term_f] :
              ( ( ord_le2325249104654787220term_f @ X7 @ A4 )
              & ( finite6824738741894949053term_f @ X7 ) ) ) ) ) ).

% Fpow_def
thf(fact_617_Fpow__def,axiom,
    ( finite2263802542432393069term_f
    = ( ^ [A4: set_se5570125628664547815term_f] :
          ( collec6999422248803473746term_f
          @ ^ [X7: set_se5570125628664547815term_f] :
              ( ( ord_le5727726935103693191term_f @ X7 @ A4 )
              & ( finite5058142709724841776term_f @ X7 ) ) ) ) ) ).

% Fpow_def
thf(fact_618_Fpow__def,axiom,
    ( finite5541903527309716562_f_nat
    = ( ^ [A4: set_se1451291994713068270_f_nat] :
          ( collec2226960723385805251_f_nat
          @ ^ [X7: set_se1451291994713068270_f_nat] :
              ( ( ord_le5386115319958680398_f_nat @ X7 @ A4 )
              & ( finite879454794548843727_f_nat @ X7 ) ) ) ) ) ).

% Fpow_def
thf(fact_619_Fpow__def,axiom,
    ( finite_Fpow_set_nat
    = ( ^ [A4: set_set_nat] :
          ( collect_set_set_nat
          @ ^ [X7: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X7 @ A4 )
              & ( finite1152437895449049373et_nat @ X7 ) ) ) ) ) ).

% Fpow_def
thf(fact_620_Fpow__def,axiom,
    ( finite1077981519904105628_f_nat
    = ( ^ [A4: set_Pr5245412377734449720_f_nat] :
          ( collec6964094325902095629_f_nat
          @ ^ [X7: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X7 @ A4 )
              & ( finite3598747971073327385_f_nat @ X7 ) ) ) ) ) ).

% Fpow_def
thf(fact_621_Fpow__def,axiom,
    ( finite7689244754419647117term_f
    = ( ^ [A4: set_Pr989862937836626183term_f] :
          ( collec4490547877585280114term_f
          @ ^ [X7: set_Pr989862937836626183term_f] :
              ( ( ord_le263819222746101927term_f @ X7 @ A4 )
              & ( finite4071141763865312848term_f @ X7 ) ) ) ) ) ).

% Fpow_def
thf(fact_622_Fpow__def,axiom,
    ( finite_Fpow_nat
    = ( ^ [A4: set_nat] :
          ( collect_set_nat
          @ ^ [X7: set_nat] :
              ( ( ord_less_eq_set_nat @ X7 @ A4 )
              & ( finite_finite_nat @ X7 ) ) ) ) ) ).

% Fpow_def
thf(fact_623_Fpow__def,axiom,
    ( finite6612922506196852378term_f
    = ( ^ [A4: set_Ground_gterm_f] :
          ( collec1979342819066322687term_f
          @ ^ [X7: set_Ground_gterm_f] :
              ( ( ord_le2735537439747282356term_f @ X7 @ A4 )
              & ( finite1923210418194103261term_f @ X7 ) ) ) ) ) ).

% Fpow_def
thf(fact_624_Id__on__set,axiom,
    ! [Xs: list_f1824981274722084755rm_f_v] :
      ( ( id_on_4849849022345749802rm_f_v @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
      = ( set_Pr7525886052349274108rm_f_v
        @ ( map_fs3454058416658099038rm_f_v
          @ ^ [X: fset_P8018961893305114765rm_f_v] : ( produc989898760166608087rm_f_v @ X @ X )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_625_Id__on__set,axiom,
    ! [Xs: list_Ground_gterm_f] :
      ( ( id_on_Ground_gterm_f @ ( set_Ground_gterm_f2 @ Xs ) )
      = ( set_Pr1901606489578307004term_f
        @ ( map_Gr7565432863924955831term_f
          @ ^ [X: ground_gterm_f] : ( produc3560254623552331287term_f @ X @ X )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_626_Id__onI,axiom,
    ! [A6: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ A6 @ A )
     => ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ A6 @ A6 ) @ ( id_on_5951252762333000388term_f @ A ) ) ) ).

% Id_onI
thf(fact_627_Id__onI,axiom,
    ! [A6: nat,A: set_nat] :
      ( ( member_nat @ A6 @ A )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ A6 ) @ ( id_on_nat @ A ) ) ) ).

% Id_onI
thf(fact_628_Id__onI,axiom,
    ! [A6: ground_gterm_f,A: set_Ground_gterm_f] :
      ( ( member5261315044688711901term_f @ A6 @ A )
     => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A6 @ A6 ) @ ( id_on_Ground_gterm_f @ A ) ) ) ).

% Id_onI
thf(fact_629_Id__onE,axiom,
    ! [C2: produc8979644124618241255term_f,A: set_Pr989862937836626183term_f] :
      ( ( member3777300322277616400term_f @ C2 @ ( id_on_5951252762333000388term_f @ A ) )
     => ~ ! [X2: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X2 @ A )
           => ( C2
             != ( produc7348823797596265303term_f @ X2 @ X2 ) ) ) ) ).

% Id_onE
thf(fact_630_Id__onE,axiom,
    ! [C2: product_prod_nat_nat,A: set_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( id_on_nat @ A ) )
     => ~ ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ( C2
             != ( product_Pair_nat_nat @ X2 @ X2 ) ) ) ) ).

% Id_onE
thf(fact_631_Id__onE,axiom,
    ! [C2: produc7245736746747425831term_f,A: set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ C2 @ ( id_on_Ground_gterm_f @ A ) )
     => ~ ! [X2: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X2 @ A )
           => ( C2
             != ( produc3560254623552331287term_f @ X2 @ X2 ) ) ) ) ).

% Id_onE
thf(fact_632_Id__on__eqI,axiom,
    ! [A6: produc7245736746747425831term_f,B3: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f] :
      ( ( A6 = B3 )
     => ( ( member848276444142703440term_f @ A6 @ A )
       => ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ A6 @ B3 ) @ ( id_on_5951252762333000388term_f @ A ) ) ) ) ).

% Id_on_eqI
thf(fact_633_Id__on__eqI,axiom,
    ! [A6: nat,B3: nat,A: set_nat] :
      ( ( A6 = B3 )
     => ( ( member_nat @ A6 @ A )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B3 ) @ ( id_on_nat @ A ) ) ) ) ).

% Id_on_eqI
thf(fact_634_Id__on__eqI,axiom,
    ! [A6: ground_gterm_f,B3: ground_gterm_f,A: set_Ground_gterm_f] :
      ( ( A6 = B3 )
     => ( ( member5261315044688711901term_f @ A6 @ A )
       => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A6 @ B3 ) @ ( id_on_Ground_gterm_f @ A ) ) ) ) ).

% Id_on_eqI
thf(fact_635_Id__on__iff,axiom,
    ! [X3: produc7245736746747425831term_f,Y: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ X3 @ Y ) @ ( id_on_5951252762333000388term_f @ A ) )
      = ( ( X3 = Y )
        & ( member848276444142703440term_f @ X3 @ A ) ) ) ).

% Id_on_iff
thf(fact_636_Id__on__iff,axiom,
    ! [X3: nat,Y: nat,A: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( id_on_nat @ A ) )
      = ( ( X3 = Y )
        & ( member_nat @ X3 @ A ) ) ) ).

% Id_on_iff
thf(fact_637_Id__on__iff,axiom,
    ! [X3: ground_gterm_f,Y: ground_gterm_f,A: set_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y ) @ ( id_on_Ground_gterm_f @ A ) )
      = ( ( X3 = Y )
        & ( member5261315044688711901term_f @ X3 @ A ) ) ) ).

% Id_on_iff
thf(fact_638_Fpow__mono,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
     => ( ord_le5386115319958680398_f_nat @ ( finite1077981519904105628_f_nat @ A ) @ ( finite1077981519904105628_f_nat @ B2 ) ) ) ).

% Fpow_mono
thf(fact_639_Fpow__mono,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A @ B2 )
     => ( ord_le5727726935103693191term_f @ ( finite7689244754419647117term_f @ A ) @ ( finite7689244754419647117term_f @ B2 ) ) ) ).

% Fpow_mono
thf(fact_640_Fpow__mono,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( finite_Fpow_nat @ B2 ) ) ) ).

% Fpow_mono
thf(fact_641_Fpow__mono,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A @ B2 )
     => ( ord_le2325249104654787220term_f @ ( finite6612922506196852378term_f @ A ) @ ( finite6612922506196852378term_f @ B2 ) ) ) ).

% Fpow_mono
thf(fact_642_Id__on__subset__Times,axiom,
    ! [A: set_Ground_gterm_f] :
      ( ord_le263819222746101927term_f @ ( id_on_Ground_gterm_f @ A )
      @ ( produc6043514909933191368term_f @ A
        @ ^ [Uu: ground_gterm_f] : A ) ) ).

% Id_on_subset_Times
thf(fact_643_finite__finite__Fpow,axiom,
    ! [A: set_Pr989862937836626183term_f] :
      ( ( finite4071141763865312848term_f @ A )
     => ( finite5058142709724841776term_f @ ( finite7689244754419647117term_f @ A ) ) ) ).

% finite_finite_Fpow
thf(fact_644_finite__finite__Fpow,axiom,
    ! [A: set_Ground_gterm_f] :
      ( ( finite1923210418194103261term_f @ A )
     => ( finite6824738741894949053term_f @ ( finite6612922506196852378term_f @ A ) ) ) ).

% finite_finite_Fpow
thf(fact_645_finite__finite__Fpow,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite1152437895449049373et_nat @ ( finite_Fpow_nat @ A ) ) ) ).

% finite_finite_Fpow
thf(fact_646_finite__finite__Fpow,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( finite879454794548843727_f_nat @ ( finite1077981519904105628_f_nat @ A ) ) ) ).

% finite_finite_Fpow
thf(fact_647_finite__finite__Fpow,axiom,
    ! [A: set_se421094335941524212term_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( finite6023745176923099549term_f @ ( finite6503987499129345914term_f @ A ) ) ) ).

% finite_finite_Fpow
thf(fact_648_finite__finite__Fpow,axiom,
    ! [A: set_se5570125628664547815term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( finite9150278737398483984term_f @ ( finite2263802542432393069term_f @ A ) ) ) ).

% finite_finite_Fpow
thf(fact_649_finite__finite__Fpow,axiom,
    ! [A: set_se1451291994713068270_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( finite1774089887677170053_f_nat @ ( finite5541903527309716562_f_nat @ A ) ) ) ).

% finite_finite_Fpow
thf(fact_650_finite__finite__Fpow,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( finite6739761609112101331et_nat @ ( finite_Fpow_set_nat @ A ) ) ) ).

% finite_finite_Fpow
thf(fact_651_infinite__infinite__Fpow,axiom,
    ! [A: set_Pr989862937836626183term_f] :
      ( ~ ( finite4071141763865312848term_f @ A )
     => ~ ( finite5058142709724841776term_f @ ( finite7689244754419647117term_f @ A ) ) ) ).

% infinite_infinite_Fpow
thf(fact_652_infinite__infinite__Fpow,axiom,
    ! [A: set_Ground_gterm_f] :
      ( ~ ( finite1923210418194103261term_f @ A )
     => ~ ( finite6824738741894949053term_f @ ( finite6612922506196852378term_f @ A ) ) ) ).

% infinite_infinite_Fpow
thf(fact_653_infinite__infinite__Fpow,axiom,
    ! [A: set_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ~ ( finite1152437895449049373et_nat @ ( finite_Fpow_nat @ A ) ) ) ).

% infinite_infinite_Fpow
thf(fact_654_infinite__infinite__Fpow,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ~ ( finite3598747971073327385_f_nat @ A )
     => ~ ( finite879454794548843727_f_nat @ ( finite1077981519904105628_f_nat @ A ) ) ) ).

% infinite_infinite_Fpow
thf(fact_655_infinite__infinite__Fpow,axiom,
    ! [A: set_se421094335941524212term_f] :
      ( ~ ( finite6824738741894949053term_f @ A )
     => ~ ( finite6023745176923099549term_f @ ( finite6503987499129345914term_f @ A ) ) ) ).

% infinite_infinite_Fpow
thf(fact_656_infinite__infinite__Fpow,axiom,
    ! [A: set_se5570125628664547815term_f] :
      ( ~ ( finite5058142709724841776term_f @ A )
     => ~ ( finite9150278737398483984term_f @ ( finite2263802542432393069term_f @ A ) ) ) ).

% infinite_infinite_Fpow
thf(fact_657_infinite__infinite__Fpow,axiom,
    ! [A: set_se1451291994713068270_f_nat] :
      ( ~ ( finite879454794548843727_f_nat @ A )
     => ~ ( finite1774089887677170053_f_nat @ ( finite5541903527309716562_f_nat @ A ) ) ) ).

% infinite_infinite_Fpow
thf(fact_658_infinite__infinite__Fpow,axiom,
    ! [A: set_set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ A )
     => ~ ( finite6739761609112101331et_nat @ ( finite_Fpow_set_nat @ A ) ) ) ).

% infinite_infinite_Fpow
thf(fact_659_R2Eq,axiom,
    ! [F2: set_Pr5245412377734449720_f_nat,Rs: list_s8746099396510718605rm_f_v] :
      ( ( fOR_eval_rr2_rel_f_v @ F2 @ Rs @ fOR_R2Eq_FOR_ftrs )
      = ( id_on_Ground_gterm_f @ ( ground_T_G_f @ F2 ) ) ) ).

% R2Eq
thf(fact_660_fId__on_Orep__eq,axiom,
    ! [X3: fset_P6228066233360383026_f_nat] :
      ( ( fset_P604416012683968852_f_nat @ ( fSet_f5219732947226789882_f_nat @ X3 ) )
      = ( id_on_9063977485820054437_f_nat @ ( fset_P3576968334923099475_f_nat @ X3 ) ) ) ).

% fId_on.rep_eq
thf(fact_661_fId__on_Orep__eq,axiom,
    ! [X3: fset_P8018961893305114765rm_f_v] :
      ( ( fset_P8224949580060463702rm_f_v @ ( fSet_f4703993580257979951rm_f_v @ X3 ) )
      = ( id_on_7786124912184395972rm_f_v @ ( fset_P4617584883882644886rm_f_v @ X3 ) ) ) ).

% fId_on.rep_eq
thf(fact_662_fId__on_Orep__eq,axiom,
    ! [X3: fset_term_f_v] :
      ( ( fset_P4617584883882644886rm_f_v @ ( fSet_fId_on_term_f_v @ X3 ) )
      = ( id_on_term_f_v @ ( fset_term_f_v2 @ X3 ) ) ) ).

% fId_on.rep_eq
thf(fact_663_finite__SigmaI2,axiom,
    ! [A: set_f,B2: f > set_nat] :
      ( ( finite_finite_f
        @ ( collect_f
          @ ^ [X: f] :
              ( ( member_f @ X @ A )
              & ( ( B2 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A3: f] :
            ( ( member_f @ A3 @ A )
           => ( finite_finite_nat @ ( B2 @ A3 ) ) )
       => ( finite3598747971073327385_f_nat @ ( product_Sigma_f_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI2
thf(fact_664_finite__SigmaI2,axiom,
    ! [A: set_nat,B2: nat > set_nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( ( B2 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B2 @ A3 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI2
thf(fact_665_finite__SigmaI2,axiom,
    ! [A: set_Ground_gterm_f,B2: ground_gterm_f > set_nat] :
      ( ( finite1923210418194103261term_f
        @ ( collec3522850625759001119term_f
          @ ^ [X: ground_gterm_f] :
              ( ( member5261315044688711901term_f @ X @ A )
              & ( ( B2 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A3: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ A3 @ A )
           => ( finite_finite_nat @ ( B2 @ A3 ) ) )
       => ( finite311254710766229066_f_nat @ ( produc4746993288989285692_f_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI2
thf(fact_666_finite__SigmaI2,axiom,
    ! [A: set_nat,B2: nat > set_Ground_gterm_f] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( ( B2 @ X )
               != bot_bo3086195439153612416term_f ) ) ) )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite1923210418194103261term_f @ ( B2 @ A3 ) ) )
       => ( finite4078170753649457648term_f @ ( produc3100878011685873178term_f @ A @ B2 ) ) ) ) ).

% finite_SigmaI2
thf(fact_667_finite__SigmaI2,axiom,
    ! [A: set_nat,B2: nat > set_set_nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( ( B2 @ X )
               != bot_bot_set_set_nat ) ) ) )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite1152437895449049373et_nat @ ( B2 @ A3 ) ) )
       => ( finite962763233036973920et_nat @ ( produc8883945523214541856et_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI2
thf(fact_668_finite__SigmaI2,axiom,
    ! [A: set_set_nat,B2: set_nat > set_nat] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( member_set_nat @ X @ A )
              & ( ( B2 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A3: set_nat] :
            ( ( member_set_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B2 @ A3 ) ) )
       => ( finite6054027020396858976at_nat @ ( produc5318310618559172640at_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI2
thf(fact_669_finite__SigmaI2,axiom,
    ! [A: set_Ground_gterm_f,B2: ground_gterm_f > set_set_nat] :
      ( ( finite1923210418194103261term_f
        @ ( collec3522850625759001119term_f
          @ ^ [X: ground_gterm_f] :
              ( ( member5261315044688711901term_f @ X @ A )
              & ( ( B2 @ X )
               != bot_bot_set_set_nat ) ) ) )
     => ( ! [A3: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ A3 @ A )
           => ( finite1152437895449049373et_nat @ ( B2 @ A3 ) ) )
       => ( finite3836040682740435712et_nat @ ( produc2061566356152752370et_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI2
thf(fact_670_finite__SigmaI2,axiom,
    ! [A: set_nat,B2: nat > set_Pr5245412377734449720_f_nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( ( B2 @ X )
               != bot_bo136713897782694092_f_nat ) ) ) )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite3598747971073327385_f_nat @ ( B2 @ A3 ) ) )
       => ( finite5167703378860198236_f_nat @ ( produc1258315153622678812_f_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI2
thf(fact_671_finite__SigmaI2,axiom,
    ! [A: set_nat,B2: nat > set_se421094335941524212term_f] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( ( B2 @ X )
               != bot_bo6304853979754439520term_f ) ) ) )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite6824738741894949053term_f @ ( B2 @ A3 ) ) )
       => ( finite2583538073261184720term_f @ ( produc5271949834123747066term_f @ A @ B2 ) ) ) ) ).

% finite_SigmaI2
thf(fact_672_finite__SigmaI2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: product_prod_f_nat > set_nat] :
      ( ( finite3598747971073327385_f_nat
        @ ( collec1417984143148964183_f_nat
          @ ^ [X: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ X @ A )
              & ( ( B2 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A3: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B2 @ A3 ) ) )
       => ( finite1611537298927112028at_nat @ ( produc4600693851930136092at_nat @ A @ B2 ) ) ) ) ).

% finite_SigmaI2
thf(fact_673_finite__cartesian__productD1,axiom,
    ! [A: set_f,B2: set_nat] :
      ( ( finite3598747971073327385_f_nat
        @ ( product_Sigma_f_nat @ A
          @ ^ [Uu: f] : B2 ) )
     => ( ( B2 != bot_bot_set_nat )
       => ( finite_finite_f @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_674_finite__cartesian__productD1,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( finite4071141763865312848term_f
        @ ( produc6043514909933191368term_f @ A
          @ ^ [Uu: ground_gterm_f] : B2 ) )
     => ( ( B2 != bot_bo3086195439153612416term_f )
       => ( finite1923210418194103261term_f @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_675_finite__cartesian__productD2,axiom,
    ! [A: set_f,B2: set_nat] :
      ( ( finite3598747971073327385_f_nat
        @ ( product_Sigma_f_nat @ A
          @ ^ [Uu: f] : B2 ) )
     => ( ( A != bot_bot_set_f )
       => ( finite_finite_nat @ B2 ) ) ) ).

% finite_cartesian_productD2
thf(fact_676_finite__cartesian__productD2,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( finite4071141763865312848term_f
        @ ( produc6043514909933191368term_f @ A
          @ ^ [Uu: ground_gterm_f] : B2 ) )
     => ( ( A != bot_bo3086195439153612416term_f )
       => ( finite1923210418194103261term_f @ B2 ) ) ) ).

% finite_cartesian_productD2
thf(fact_677_empty__Collect__eq,axiom,
    ! [P: ground_gterm_f > $o] :
      ( ( bot_bo3086195439153612416term_f
        = ( collec3522850625759001119term_f @ P ) )
      = ( ! [X: ground_gterm_f] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_678_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_679_empty__Collect__eq,axiom,
    ! [P: set_Pr989862937836626183term_f > $o] :
      ( ( bot_bo5021337078593797203term_f
        = ( collec4490547877585280114term_f @ P ) )
      = ( ! [X: set_Pr989862937836626183term_f] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_680_empty__Collect__eq,axiom,
    ! [P: set_Pr5245412377734449720_f_nat > $o] :
      ( ( bot_bo6700018649133943170_f_nat
        = ( collec6964094325902095629_f_nat @ P ) )
      = ( ! [X: set_Pr5245412377734449720_f_nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_681_empty__Collect__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P ) )
      = ( ! [X: set_nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_682_empty__Collect__eq,axiom,
    ! [P: set_Ground_gterm_f > $o] :
      ( ( bot_bo6304853979754439520term_f
        = ( collec1979342819066322687term_f @ P ) )
      = ( ! [X: set_Ground_gterm_f] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_683_Collect__empty__eq,axiom,
    ! [P: ground_gterm_f > $o] :
      ( ( ( collec3522850625759001119term_f @ P )
        = bot_bo3086195439153612416term_f )
      = ( ! [X: ground_gterm_f] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_684_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_685_Collect__empty__eq,axiom,
    ! [P: set_Pr989862937836626183term_f > $o] :
      ( ( ( collec4490547877585280114term_f @ P )
        = bot_bo5021337078593797203term_f )
      = ( ! [X: set_Pr989862937836626183term_f] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_686_Collect__empty__eq,axiom,
    ! [P: set_Pr5245412377734449720_f_nat > $o] :
      ( ( ( collec6964094325902095629_f_nat @ P )
        = bot_bo6700018649133943170_f_nat )
      = ( ! [X: set_Pr5245412377734449720_f_nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_687_Collect__empty__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( ! [X: set_nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_688_Collect__empty__eq,axiom,
    ! [P: set_Ground_gterm_f > $o] :
      ( ( ( collec1979342819066322687term_f @ P )
        = bot_bo6304853979754439520term_f )
      = ( ! [X: set_Ground_gterm_f] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_689_all__not__in__conv,axiom,
    ! [A: set_Pr989862937836626183term_f] :
      ( ( ! [X: produc7245736746747425831term_f] :
            ~ ( member848276444142703440term_f @ X @ A ) )
      = ( A = bot_bo9223291547480392563term_f ) ) ).

% all_not_in_conv
thf(fact_690_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_691_empty__iff,axiom,
    ! [C2: produc7245736746747425831term_f] :
      ~ ( member848276444142703440term_f @ C2 @ bot_bo9223291547480392563term_f ) ).

% empty_iff
thf(fact_692_empty__iff,axiom,
    ! [C2: nat] :
      ~ ( member_nat @ C2 @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_693_empty__subsetI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ bot_bo136713897782694092_f_nat @ A ) ).

% empty_subsetI
thf(fact_694_empty__subsetI,axiom,
    ! [A: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ bot_bo9223291547480392563term_f @ A ) ).

% empty_subsetI
thf(fact_695_empty__subsetI,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% empty_subsetI
thf(fact_696_empty__subsetI,axiom,
    ! [A: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ bot_bo3086195439153612416term_f @ A ) ).

% empty_subsetI
thf(fact_697_subset__empty,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ bot_bo136713897782694092_f_nat )
      = ( A = bot_bo136713897782694092_f_nat ) ) ).

% subset_empty
thf(fact_698_subset__empty,axiom,
    ! [A: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A @ bot_bo9223291547480392563term_f )
      = ( A = bot_bo9223291547480392563term_f ) ) ).

% subset_empty
thf(fact_699_subset__empty,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_700_subset__empty,axiom,
    ! [A: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A @ bot_bo3086195439153612416term_f )
      = ( A = bot_bo3086195439153612416term_f ) ) ).

% subset_empty
thf(fact_701_Sigma__empty1,axiom,
    ! [B2: ground_gterm_f > set_Ground_gterm_f] :
      ( ( produc6043514909933191368term_f @ bot_bo3086195439153612416term_f @ B2 )
      = bot_bo9223291547480392563term_f ) ).

% Sigma_empty1
thf(fact_702_Times__empty,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( ( produc6043514909933191368term_f @ A
          @ ^ [Uu: ground_gterm_f] : B2 )
        = bot_bo9223291547480392563term_f )
      = ( ( A = bot_bo3086195439153612416term_f )
        | ( B2 = bot_bo3086195439153612416term_f ) ) ) ).

% Times_empty
thf(fact_703_Sigma__empty2,axiom,
    ! [A: set_Ground_gterm_f] :
      ( ( produc6043514909933191368term_f @ A
        @ ^ [Uu: ground_gterm_f] : bot_bo3086195439153612416term_f )
      = bot_bo9223291547480392563term_f ) ).

% Sigma_empty2
thf(fact_704_Sigma__empty__iff,axiom,
    ! [I2: set_Ground_gterm_f,X6: ground_gterm_f > set_Ground_gterm_f] :
      ( ( ( produc6043514909933191368term_f @ I2 @ X6 )
        = bot_bo9223291547480392563term_f )
      = ( ! [X: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X @ I2 )
           => ( ( X6 @ X )
              = bot_bo3086195439153612416term_f ) ) ) ) ).

% Sigma_empty_iff
thf(fact_705_Set_Oempty__def,axiom,
    ( bot_bo3086195439153612416term_f
    = ( collec3522850625759001119term_f
      @ ^ [X: ground_gterm_f] : $false ) ) ).

% Set.empty_def
thf(fact_706_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% Set.empty_def
thf(fact_707_Set_Oempty__def,axiom,
    ( bot_bo5021337078593797203term_f
    = ( collec4490547877585280114term_f
      @ ^ [X: set_Pr989862937836626183term_f] : $false ) ) ).

% Set.empty_def
thf(fact_708_Set_Oempty__def,axiom,
    ( bot_bo6700018649133943170_f_nat
    = ( collec6964094325902095629_f_nat
      @ ^ [X: set_Pr5245412377734449720_f_nat] : $false ) ) ).

% Set.empty_def
thf(fact_709_Set_Oempty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X: set_nat] : $false ) ) ).

% Set.empty_def
thf(fact_710_Set_Oempty__def,axiom,
    ( bot_bo6304853979754439520term_f
    = ( collec1979342819066322687term_f
      @ ^ [X: set_Ground_gterm_f] : $false ) ) ).

% Set.empty_def
thf(fact_711_ex__in__conv,axiom,
    ! [A: set_Pr989862937836626183term_f] :
      ( ( ? [X: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X @ A ) )
      = ( A != bot_bo9223291547480392563term_f ) ) ).

% ex_in_conv
thf(fact_712_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_713_equals0I,axiom,
    ! [A: set_Pr989862937836626183term_f] :
      ( ! [Y3: produc7245736746747425831term_f] :
          ~ ( member848276444142703440term_f @ Y3 @ A )
     => ( A = bot_bo9223291547480392563term_f ) ) ).

% equals0I
thf(fact_714_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_715_equals0D,axiom,
    ! [A: set_Pr989862937836626183term_f,A6: produc7245736746747425831term_f] :
      ( ( A = bot_bo9223291547480392563term_f )
     => ~ ( member848276444142703440term_f @ A6 @ A ) ) ).

% equals0D
thf(fact_716_equals0D,axiom,
    ! [A: set_nat,A6: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat @ A6 @ A ) ) ).

% equals0D
thf(fact_717_emptyE,axiom,
    ! [A6: produc7245736746747425831term_f] :
      ~ ( member848276444142703440term_f @ A6 @ bot_bo9223291547480392563term_f ) ).

% emptyE
thf(fact_718_emptyE,axiom,
    ! [A6: nat] :
      ~ ( member_nat @ A6 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_719_bot_Oextremum__uniqueI,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ bot_bo136713897782694092_f_nat )
     => ( A6 = bot_bo136713897782694092_f_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_720_bot_Oextremum__uniqueI,axiom,
    ! [A6: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ bot_bo9223291547480392563term_f )
     => ( A6 = bot_bo9223291547480392563term_f ) ) ).

% bot.extremum_uniqueI
thf(fact_721_bot_Oextremum__uniqueI,axiom,
    ! [A6: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ bot_bo3974441229639268550_f_nat )
     => ( A6 = bot_bo3974441229639268550_f_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_722_bot_Oextremum__uniqueI,axiom,
    ! [A6: nat] :
      ( ( ord_less_eq_nat @ A6 @ bot_bot_nat )
     => ( A6 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_723_bot_Oextremum__uniqueI,axiom,
    ! [A6: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ bot_bot_set_nat )
     => ( A6 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_724_bot_Oextremum__uniqueI,axiom,
    ! [A6: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ bot_bo3086195439153612416term_f )
     => ( A6 = bot_bo3086195439153612416term_f ) ) ).

% bot.extremum_uniqueI
thf(fact_725_bot_Oextremum__unique,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ bot_bo136713897782694092_f_nat )
      = ( A6 = bot_bo136713897782694092_f_nat ) ) ).

% bot.extremum_unique
thf(fact_726_bot_Oextremum__unique,axiom,
    ! [A6: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ bot_bo9223291547480392563term_f )
      = ( A6 = bot_bo9223291547480392563term_f ) ) ).

% bot.extremum_unique
thf(fact_727_bot_Oextremum__unique,axiom,
    ! [A6: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ bot_bo3974441229639268550_f_nat )
      = ( A6 = bot_bo3974441229639268550_f_nat ) ) ).

% bot.extremum_unique
thf(fact_728_bot_Oextremum__unique,axiom,
    ! [A6: nat] :
      ( ( ord_less_eq_nat @ A6 @ bot_bot_nat )
      = ( A6 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_729_bot_Oextremum__unique,axiom,
    ! [A6: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ bot_bot_set_nat )
      = ( A6 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_730_bot_Oextremum__unique,axiom,
    ! [A6: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ bot_bo3086195439153612416term_f )
      = ( A6 = bot_bo3086195439153612416term_f ) ) ).

% bot.extremum_unique
thf(fact_731_bot_Oextremum,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ bot_bo136713897782694092_f_nat @ A6 ) ).

% bot.extremum
thf(fact_732_bot_Oextremum,axiom,
    ! [A6: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ bot_bo9223291547480392563term_f @ A6 ) ).

% bot.extremum
thf(fact_733_bot_Oextremum,axiom,
    ! [A6: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ bot_bo3974441229639268550_f_nat @ A6 ) ).

% bot.extremum
thf(fact_734_bot_Oextremum,axiom,
    ! [A6: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A6 ) ).

% bot.extremum
thf(fact_735_bot_Oextremum,axiom,
    ! [A6: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A6 ) ).

% bot.extremum
thf(fact_736_bot_Oextremum,axiom,
    ! [A6: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ bot_bo3086195439153612416term_f @ A6 ) ).

% bot.extremum
thf(fact_737_subset__emptyI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ! [X2: product_prod_f_nat] :
          ~ ( member2678038196313615385_f_nat @ X2 @ A )
     => ( ord_le8976984241387448984_f_nat @ A @ bot_bo136713897782694092_f_nat ) ) ).

% subset_emptyI
thf(fact_738_subset__emptyI,axiom,
    ! [A: set_Pr989862937836626183term_f] :
      ( ! [X2: produc7245736746747425831term_f] :
          ~ ( member848276444142703440term_f @ X2 @ A )
     => ( ord_le263819222746101927term_f @ A @ bot_bo9223291547480392563term_f ) ) ).

% subset_emptyI
thf(fact_739_subset__emptyI,axiom,
    ! [A: set_nat] :
      ( ! [X2: nat] :
          ~ ( member_nat @ X2 @ A )
     => ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_740_subset__emptyI,axiom,
    ! [A: set_Ground_gterm_f] :
      ( ! [X2: ground_gterm_f] :
          ~ ( member5261315044688711901term_f @ X2 @ A )
     => ( ord_le2735537439747282356term_f @ A @ bot_bo3086195439153612416term_f ) ) ).

% subset_emptyI
thf(fact_741_infinite__imp__nonempty,axiom,
    ! [S: set_Ground_gterm_f] :
      ( ~ ( finite1923210418194103261term_f @ S )
     => ( S != bot_bo3086195439153612416term_f ) ) ).

% infinite_imp_nonempty
thf(fact_742_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_743_infinite__imp__nonempty,axiom,
    ! [S: set_Pr5245412377734449720_f_nat] :
      ( ~ ( finite3598747971073327385_f_nat @ S )
     => ( S != bot_bo136713897782694092_f_nat ) ) ).

% infinite_imp_nonempty
thf(fact_744_infinite__imp__nonempty,axiom,
    ! [S: set_se421094335941524212term_f] :
      ( ~ ( finite6824738741894949053term_f @ S )
     => ( S != bot_bo6304853979754439520term_f ) ) ).

% infinite_imp_nonempty
thf(fact_745_infinite__imp__nonempty,axiom,
    ! [S: set_se5570125628664547815term_f] :
      ( ~ ( finite5058142709724841776term_f @ S )
     => ( S != bot_bo5021337078593797203term_f ) ) ).

% infinite_imp_nonempty
thf(fact_746_infinite__imp__nonempty,axiom,
    ! [S: set_se1451291994713068270_f_nat] :
      ( ~ ( finite879454794548843727_f_nat @ S )
     => ( S != bot_bo6700018649133943170_f_nat ) ) ).

% infinite_imp_nonempty
thf(fact_747_infinite__imp__nonempty,axiom,
    ! [S: set_set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ S )
     => ( S != bot_bot_set_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_748_finite_OemptyI,axiom,
    finite1923210418194103261term_f @ bot_bo3086195439153612416term_f ).

% finite.emptyI
thf(fact_749_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_750_finite_OemptyI,axiom,
    finite3598747971073327385_f_nat @ bot_bo136713897782694092_f_nat ).

% finite.emptyI
thf(fact_751_finite_OemptyI,axiom,
    finite6824738741894949053term_f @ bot_bo6304853979754439520term_f ).

% finite.emptyI
thf(fact_752_finite_OemptyI,axiom,
    finite5058142709724841776term_f @ bot_bo5021337078593797203term_f ).

% finite.emptyI
thf(fact_753_finite_OemptyI,axiom,
    finite879454794548843727_f_nat @ bot_bo6700018649133943170_f_nat ).

% finite.emptyI
thf(fact_754_finite_OemptyI,axiom,
    finite1152437895449049373et_nat @ bot_bot_set_set_nat ).

% finite.emptyI
thf(fact_755_times__eq__iff,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f,C: set_Ground_gterm_f,D: set_Ground_gterm_f] :
      ( ( ( produc6043514909933191368term_f @ A
          @ ^ [Uu: ground_gterm_f] : B2 )
        = ( produc6043514909933191368term_f @ C
          @ ^ [Uu: ground_gterm_f] : D ) )
      = ( ( ( A = C )
          & ( B2 = D ) )
        | ( ( ( A = bot_bo3086195439153612416term_f )
            | ( B2 = bot_bo3086195439153612416term_f ) )
          & ( ( C = bot_bo3086195439153612416term_f )
            | ( D = bot_bo3086195439153612416term_f ) ) ) ) ) ).

% times_eq_iff
thf(fact_756_finite__has__minimal,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ? [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A )
            & ! [Xa: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ Xa @ A )
               => ( ( ord_le2234929214960958178_f_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_757_finite__has__minimal,axiom,
    ! [A: set_se1451291994713068270_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( A != bot_bo6700018649133943170_f_nat )
       => ? [X2: set_Pr5245412377734449720_f_nat] :
            ( ( member8345558847462883279_f_nat @ X2 @ A )
            & ! [Xa: set_Pr5245412377734449720_f_nat] :
                ( ( member8345558847462883279_f_nat @ Xa @ A )
               => ( ( ord_le8976984241387448984_f_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_758_finite__has__minimal,axiom,
    ! [A: set_se5570125628664547815term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( A != bot_bo5021337078593797203term_f )
       => ? [X2: set_Pr989862937836626183term_f] :
            ( ( member6919674941555055664term_f @ X2 @ A )
            & ! [Xa: set_Pr989862937836626183term_f] :
                ( ( member6919674941555055664term_f @ Xa @ A )
               => ( ( ord_le263819222746101927term_f @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_759_finite__has__minimal,axiom,
    ! [A: set_fs7237271039932420712_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( A != bot_bo2133226978013888252_f_nat )
       => ? [X2: fset_P6228066233360383026_f_nat] :
            ( ( member6677119207206147657_f_nat @ X2 @ A )
            & ! [Xa: fset_P6228066233360383026_f_nat] :
                ( ( member6677119207206147657_f_nat @ Xa @ A )
               => ( ( ord_le1552505484586773650_f_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_760_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_761_finite__has__minimal,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A )
               => ( ( ord_less_eq_set_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_762_finite__has__minimal,axiom,
    ! [A: set_se421094335941524212term_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( A != bot_bo6304853979754439520term_f )
       => ? [X2: set_Ground_gterm_f] :
            ( ( member9219338147399679421term_f @ X2 @ A )
            & ! [Xa: set_Ground_gterm_f] :
                ( ( member9219338147399679421term_f @ Xa @ A )
               => ( ( ord_le2735537439747282356term_f @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_763_finite__has__maximal,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ? [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A )
            & ! [Xa: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ Xa @ A )
               => ( ( ord_le2234929214960958178_f_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_764_finite__has__maximal,axiom,
    ! [A: set_se1451291994713068270_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( A != bot_bo6700018649133943170_f_nat )
       => ? [X2: set_Pr5245412377734449720_f_nat] :
            ( ( member8345558847462883279_f_nat @ X2 @ A )
            & ! [Xa: set_Pr5245412377734449720_f_nat] :
                ( ( member8345558847462883279_f_nat @ Xa @ A )
               => ( ( ord_le8976984241387448984_f_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_765_finite__has__maximal,axiom,
    ! [A: set_se5570125628664547815term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( A != bot_bo5021337078593797203term_f )
       => ? [X2: set_Pr989862937836626183term_f] :
            ( ( member6919674941555055664term_f @ X2 @ A )
            & ! [Xa: set_Pr989862937836626183term_f] :
                ( ( member6919674941555055664term_f @ Xa @ A )
               => ( ( ord_le263819222746101927term_f @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_766_finite__has__maximal,axiom,
    ! [A: set_fs7237271039932420712_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( A != bot_bo2133226978013888252_f_nat )
       => ? [X2: fset_P6228066233360383026_f_nat] :
            ( ( member6677119207206147657_f_nat @ X2 @ A )
            & ! [Xa: fset_P6228066233360383026_f_nat] :
                ( ( member6677119207206147657_f_nat @ Xa @ A )
               => ( ( ord_le1552505484586773650_f_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_767_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_768_finite__has__maximal,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A )
               => ( ( ord_less_eq_set_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_769_finite__has__maximal,axiom,
    ! [A: set_se421094335941524212term_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( A != bot_bo6304853979754439520term_f )
       => ? [X2: set_Ground_gterm_f] :
            ( ( member9219338147399679421term_f @ X2 @ A )
            & ! [Xa: set_Ground_gterm_f] :
                ( ( member9219338147399679421term_f @ Xa @ A )
               => ( ( ord_le2735537439747282356term_f @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_770_times__subset__iff,axiom,
    ! [A: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
      ( ( ord_le3146513528884898305at_nat
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : C )
        @ ( produc457027306803732586at_nat @ B2
          @ ^ [Uu: nat] : D ) )
      = ( ( A = bot_bot_set_nat )
        | ( C = bot_bot_set_nat )
        | ( ( ord_less_eq_set_nat @ A @ B2 )
          & ( ord_less_eq_set_nat @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_771_times__subset__iff,axiom,
    ! [A: set_f,C: set_nat,B2: set_f,D: set_nat] :
      ( ( ord_le8976984241387448984_f_nat
        @ ( product_Sigma_f_nat @ A
          @ ^ [Uu: f] : C )
        @ ( product_Sigma_f_nat @ B2
          @ ^ [Uu: f] : D ) )
      = ( ( A = bot_bot_set_f )
        | ( C = bot_bot_set_nat )
        | ( ( ord_less_eq_set_f @ A @ B2 )
          & ( ord_less_eq_set_nat @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_772_times__subset__iff,axiom,
    ! [A: set_nat,C: set_Ground_gterm_f,B2: set_nat,D: set_Ground_gterm_f] :
      ( ( ord_le4584778628501795311term_f
        @ ( produc3100878011685873178term_f @ A
          @ ^ [Uu: nat] : C )
        @ ( produc3100878011685873178term_f @ B2
          @ ^ [Uu: nat] : D ) )
      = ( ( A = bot_bot_set_nat )
        | ( C = bot_bo3086195439153612416term_f )
        | ( ( ord_less_eq_set_nat @ A @ B2 )
          & ( ord_le2735537439747282356term_f @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_773_times__subset__iff,axiom,
    ! [A: set_Ground_gterm_f,C: set_nat,B2: set_Ground_gterm_f,D: set_nat] :
      ( ( ord_le6666376546550546761_f_nat
        @ ( produc4746993288989285692_f_nat @ A
          @ ^ [Uu: ground_gterm_f] : C )
        @ ( produc4746993288989285692_f_nat @ B2
          @ ^ [Uu: ground_gterm_f] : D ) )
      = ( ( A = bot_bo3086195439153612416term_f )
        | ( C = bot_bot_set_nat )
        | ( ( ord_le2735537439747282356term_f @ A @ B2 )
          & ( ord_less_eq_set_nat @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_774_times__subset__iff,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,C: set_nat,B2: set_Pr5245412377734449720_f_nat,D: set_nat] :
      ( ( ord_le703013924939395507at_nat
        @ ( produc4600693851930136092at_nat @ A
          @ ^ [Uu: product_prod_f_nat] : C )
        @ ( produc4600693851930136092at_nat @ B2
          @ ^ [Uu: product_prod_f_nat] : D ) )
      = ( ( A = bot_bo136713897782694092_f_nat )
        | ( C = bot_bot_set_nat )
        | ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
          & ( ord_less_eq_set_nat @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_775_times__subset__iff,axiom,
    ! [A: set_nat,C: set_Pr5245412377734449720_f_nat,B2: set_nat,D: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le697507202650467251_f_nat
        @ ( produc1258315153622678812_f_nat @ A
          @ ^ [Uu: nat] : C )
        @ ( produc1258315153622678812_f_nat @ B2
          @ ^ [Uu: nat] : D ) )
      = ( ( A = bot_bot_set_nat )
        | ( C = bot_bo136713897782694092_f_nat )
        | ( ( ord_less_eq_set_nat @ A @ B2 )
          & ( ord_le8976984241387448984_f_nat @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_776_times__subset__iff,axiom,
    ! [A: set_Ground_gterm_f,C: set_Ground_gterm_f,B2: set_Ground_gterm_f,D: set_Ground_gterm_f] :
      ( ( ord_le263819222746101927term_f
        @ ( produc6043514909933191368term_f @ A
          @ ^ [Uu: ground_gterm_f] : C )
        @ ( produc6043514909933191368term_f @ B2
          @ ^ [Uu: ground_gterm_f] : D ) )
      = ( ( A = bot_bo3086195439153612416term_f )
        | ( C = bot_bo3086195439153612416term_f )
        | ( ( ord_le2735537439747282356term_f @ A @ B2 )
          & ( ord_le2735537439747282356term_f @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_777_times__subset__iff,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,C: set_Ground_gterm_f,B2: set_Pr5245412377734449720_f_nat,D: set_Ground_gterm_f] :
      ( ( ord_le6888376801088063933term_f
        @ ( produc2338249810576627688term_f @ A
          @ ^ [Uu: product_prod_f_nat] : C )
        @ ( produc2338249810576627688term_f @ B2
          @ ^ [Uu: product_prod_f_nat] : D ) )
      = ( ( A = bot_bo136713897782694092_f_nat )
        | ( C = bot_bo3086195439153612416term_f )
        | ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
          & ( ord_le2735537439747282356term_f @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_778_times__subset__iff,axiom,
    ! [A: set_Ground_gterm_f,C: set_Pr5245412377734449720_f_nat,B2: set_Ground_gterm_f,D: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le3590369893943491323_f_nat
        @ ( produc5546477881898244078_f_nat @ A
          @ ^ [Uu: ground_gterm_f] : C )
        @ ( produc5546477881898244078_f_nat @ B2
          @ ^ [Uu: ground_gterm_f] : D ) )
      = ( ( A = bot_bo3086195439153612416term_f )
        | ( C = bot_bo136713897782694092_f_nat )
        | ( ( ord_le2735537439747282356term_f @ A @ B2 )
          & ( ord_le8976984241387448984_f_nat @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_779_times__subset__iff,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat,D: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8946695678602162533_f_nat
        @ ( produc7667213020494768334_f_nat @ A
          @ ^ [Uu: product_prod_f_nat] : C )
        @ ( produc7667213020494768334_f_nat @ B2
          @ ^ [Uu: product_prod_f_nat] : D ) )
      = ( ( A = bot_bo136713897782694092_f_nat )
        | ( C = bot_bo136713897782694092_f_nat )
        | ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
          & ( ord_le8976984241387448984_f_nat @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_780_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B2 ) )
      = ( ( A = bot_bot_set_nat )
        | ( B2 = bot_bot_set_nat )
        | ( ( finite_finite_nat @ A )
          & ( finite_finite_nat @ B2 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_781_finite__cartesian__product__iff,axiom,
    ! [A: set_f,B2: set_nat] :
      ( ( finite3598747971073327385_f_nat
        @ ( product_Sigma_f_nat @ A
          @ ^ [Uu: f] : B2 ) )
      = ( ( A = bot_bot_set_f )
        | ( B2 = bot_bot_set_nat )
        | ( ( finite_finite_f @ A )
          & ( finite_finite_nat @ B2 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_782_finite__cartesian__product__iff,axiom,
    ! [A: set_Ground_gterm_f,B2: set_nat] :
      ( ( finite311254710766229066_f_nat
        @ ( produc4746993288989285692_f_nat @ A
          @ ^ [Uu: ground_gterm_f] : B2 ) )
      = ( ( A = bot_bo3086195439153612416term_f )
        | ( B2 = bot_bot_set_nat )
        | ( ( finite1923210418194103261term_f @ A )
          & ( finite_finite_nat @ B2 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_783_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B2: set_Ground_gterm_f] :
      ( ( finite4078170753649457648term_f
        @ ( produc3100878011685873178term_f @ A
          @ ^ [Uu: nat] : B2 ) )
      = ( ( A = bot_bot_set_nat )
        | ( B2 = bot_bo3086195439153612416term_f )
        | ( ( finite_finite_nat @ A )
          & ( finite1923210418194103261term_f @ B2 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_784_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B2: set_set_nat] :
      ( ( finite962763233036973920et_nat
        @ ( produc8883945523214541856et_nat @ A
          @ ^ [Uu: nat] : B2 ) )
      = ( ( A = bot_bot_set_nat )
        | ( B2 = bot_bot_set_set_nat )
        | ( ( finite_finite_nat @ A )
          & ( finite1152437895449049373et_nat @ B2 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_785_finite__cartesian__product__iff,axiom,
    ! [A: set_set_nat,B2: set_nat] :
      ( ( finite6054027020396858976at_nat
        @ ( produc5318310618559172640at_nat @ A
          @ ^ [Uu: set_nat] : B2 ) )
      = ( ( A = bot_bot_set_set_nat )
        | ( B2 = bot_bot_set_nat )
        | ( ( finite1152437895449049373et_nat @ A )
          & ( finite_finite_nat @ B2 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_786_finite__cartesian__product__iff,axiom,
    ! [A: set_Ground_gterm_f,B2: set_set_nat] :
      ( ( finite3836040682740435712et_nat
        @ ( produc2061566356152752370et_nat @ A
          @ ^ [Uu: ground_gterm_f] : B2 ) )
      = ( ( A = bot_bo3086195439153612416term_f )
        | ( B2 = bot_bot_set_set_nat )
        | ( ( finite1923210418194103261term_f @ A )
          & ( finite1152437895449049373et_nat @ B2 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_787_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( finite5167703378860198236_f_nat
        @ ( produc1258315153622678812_f_nat @ A
          @ ^ [Uu: nat] : B2 ) )
      = ( ( A = bot_bot_set_nat )
        | ( B2 = bot_bo136713897782694092_f_nat )
        | ( ( finite_finite_nat @ A )
          & ( finite3598747971073327385_f_nat @ B2 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_788_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B2: set_se421094335941524212term_f] :
      ( ( finite2583538073261184720term_f
        @ ( produc5271949834123747066term_f @ A
          @ ^ [Uu: nat] : B2 ) )
      = ( ( A = bot_bot_set_nat )
        | ( B2 = bot_bo6304853979754439520term_f )
        | ( ( finite_finite_nat @ A )
          & ( finite6824738741894949053term_f @ B2 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_789_finite__cartesian__product__iff,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_nat] :
      ( ( finite1611537298927112028at_nat
        @ ( produc4600693851930136092at_nat @ A
          @ ^ [Uu: product_prod_f_nat] : B2 ) )
      = ( ( A = bot_bo136713897782694092_f_nat )
        | ( B2 = bot_bot_set_nat )
        | ( ( finite3598747971073327385_f_nat @ A )
          & ( finite_finite_nat @ B2 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_790_finite__transitivity__chain,axiom,
    ! [A: set_Pr989862937836626183term_f,R2: produc7245736746747425831term_f > produc7245736746747425831term_f > $o] :
      ( ( finite4071141763865312848term_f @ A )
     => ( ! [X2: produc7245736746747425831term_f] :
            ~ ( R2 @ X2 @ X2 )
       => ( ! [X2: produc7245736746747425831term_f,Y3: produc7245736746747425831term_f,Z3: produc7245736746747425831term_f] :
              ( ( R2 @ X2 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X2 @ Z3 ) ) )
         => ( ! [X2: produc7245736746747425831term_f] :
                ( ( member848276444142703440term_f @ X2 @ A )
               => ? [Y5: produc7245736746747425831term_f] :
                    ( ( member848276444142703440term_f @ Y5 @ A )
                    & ( R2 @ X2 @ Y5 ) ) )
           => ( A = bot_bo9223291547480392563term_f ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_791_finite__transitivity__chain,axiom,
    ! [A: set_Ground_gterm_f,R2: ground_gterm_f > ground_gterm_f > $o] :
      ( ( finite1923210418194103261term_f @ A )
     => ( ! [X2: ground_gterm_f] :
            ~ ( R2 @ X2 @ X2 )
       => ( ! [X2: ground_gterm_f,Y3: ground_gterm_f,Z3: ground_gterm_f] :
              ( ( R2 @ X2 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X2 @ Z3 ) ) )
         => ( ! [X2: ground_gterm_f] :
                ( ( member5261315044688711901term_f @ X2 @ A )
               => ? [Y5: ground_gterm_f] :
                    ( ( member5261315044688711901term_f @ Y5 @ A )
                    & ( R2 @ X2 @ Y5 ) ) )
           => ( A = bot_bo3086195439153612416term_f ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_792_finite__transitivity__chain,axiom,
    ! [A: set_nat,R2: nat > nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ! [X2: nat] :
            ~ ( R2 @ X2 @ X2 )
       => ( ! [X2: nat,Y3: nat,Z3: nat] :
              ( ( R2 @ X2 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X2 @ Z3 ) ) )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A )
               => ? [Y5: nat] :
                    ( ( member_nat @ Y5 @ A )
                    & ( R2 @ X2 @ Y5 ) ) )
           => ( A = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_793_finite__transitivity__chain,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,R2: product_prod_f_nat > product_prod_f_nat > $o] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ! [X2: product_prod_f_nat] :
            ~ ( R2 @ X2 @ X2 )
       => ( ! [X2: product_prod_f_nat,Y3: product_prod_f_nat,Z3: product_prod_f_nat] :
              ( ( R2 @ X2 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X2 @ Z3 ) ) )
         => ( ! [X2: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X2 @ A )
               => ? [Y5: product_prod_f_nat] :
                    ( ( member2678038196313615385_f_nat @ Y5 @ A )
                    & ( R2 @ X2 @ Y5 ) ) )
           => ( A = bot_bo136713897782694092_f_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_794_finite__transitivity__chain,axiom,
    ! [A: set_se421094335941524212term_f,R2: set_Ground_gterm_f > set_Ground_gterm_f > $o] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ! [X2: set_Ground_gterm_f] :
            ~ ( R2 @ X2 @ X2 )
       => ( ! [X2: set_Ground_gterm_f,Y3: set_Ground_gterm_f,Z3: set_Ground_gterm_f] :
              ( ( R2 @ X2 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X2 @ Z3 ) ) )
         => ( ! [X2: set_Ground_gterm_f] :
                ( ( member9219338147399679421term_f @ X2 @ A )
               => ? [Y5: set_Ground_gterm_f] :
                    ( ( member9219338147399679421term_f @ Y5 @ A )
                    & ( R2 @ X2 @ Y5 ) ) )
           => ( A = bot_bo6304853979754439520term_f ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_795_finite__transitivity__chain,axiom,
    ! [A: set_se5570125628664547815term_f,R2: set_Pr989862937836626183term_f > set_Pr989862937836626183term_f > $o] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ! [X2: set_Pr989862937836626183term_f] :
            ~ ( R2 @ X2 @ X2 )
       => ( ! [X2: set_Pr989862937836626183term_f,Y3: set_Pr989862937836626183term_f,Z3: set_Pr989862937836626183term_f] :
              ( ( R2 @ X2 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X2 @ Z3 ) ) )
         => ( ! [X2: set_Pr989862937836626183term_f] :
                ( ( member6919674941555055664term_f @ X2 @ A )
               => ? [Y5: set_Pr989862937836626183term_f] :
                    ( ( member6919674941555055664term_f @ Y5 @ A )
                    & ( R2 @ X2 @ Y5 ) ) )
           => ( A = bot_bo5021337078593797203term_f ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_796_finite__transitivity__chain,axiom,
    ! [A: set_se1451291994713068270_f_nat,R2: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat > $o] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ! [X2: set_Pr5245412377734449720_f_nat] :
            ~ ( R2 @ X2 @ X2 )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat,Z3: set_Pr5245412377734449720_f_nat] :
              ( ( R2 @ X2 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X2 @ Z3 ) ) )
         => ( ! [X2: set_Pr5245412377734449720_f_nat] :
                ( ( member8345558847462883279_f_nat @ X2 @ A )
               => ? [Y5: set_Pr5245412377734449720_f_nat] :
                    ( ( member8345558847462883279_f_nat @ Y5 @ A )
                    & ( R2 @ X2 @ Y5 ) ) )
           => ( A = bot_bo6700018649133943170_f_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_797_finite__transitivity__chain,axiom,
    ! [A: set_set_nat,R2: set_nat > set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ! [X2: set_nat] :
            ~ ( R2 @ X2 @ X2 )
       => ( ! [X2: set_nat,Y3: set_nat,Z3: set_nat] :
              ( ( R2 @ X2 @ Y3 )
             => ( ( R2 @ Y3 @ Z3 )
               => ( R2 @ X2 @ Z3 ) ) )
         => ( ! [X2: set_nat] :
                ( ( member_set_nat @ X2 @ A )
               => ? [Y5: set_nat] :
                    ( ( member_set_nat @ Y5 @ A )
                    & ( R2 @ X2 @ Y5 ) ) )
           => ( A = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_798_eval__rr2__rel_Osimps_I7_J,axiom,
    ! [F2: set_Pr5245412377734449720_f_nat,Rs: list_s8746099396510718605rm_f_v,R2: fOR_rr1_rel_FOR_ftrs] :
      ( ( fOR_eval_rr2_rel_f_v @ F2 @ Rs @ ( fOR_rr6911657030925137002R_ftrs @ R2 ) )
      = ( id_on_Ground_gterm_f @ ( fOR_eval_rr1_rel_f_v @ F2 @ Rs @ R2 ) ) ) ).

% eval_rr2_rel.simps(7)
thf(fact_799_map__of__map__Pair__key,axiom,
    ! [X3: ground_gterm_f,Xs: list_Ground_gterm_f,F3: ground_gterm_f > ground_gterm_f] :
      ( ( ( member5261315044688711901term_f @ X3 @ ( set_Ground_gterm_f2 @ Xs ) )
       => ( ( map_of953081065202312803term_f
            @ ( map_Gr7565432863924955831term_f
              @ ^ [K: ground_gterm_f] : ( produc3560254623552331287term_f @ K @ ( F3 @ K ) )
              @ Xs )
            @ X3 )
          = ( some_Ground_gterm_f @ ( F3 @ X3 ) ) ) )
      & ( ~ ( member5261315044688711901term_f @ X3 @ ( set_Ground_gterm_f2 @ Xs ) )
       => ( ( map_of953081065202312803term_f
            @ ( map_Gr7565432863924955831term_f
              @ ^ [K: ground_gterm_f] : ( produc3560254623552331287term_f @ K @ ( F3 @ K ) )
              @ Xs )
            @ X3 )
          = none_Ground_gterm_f ) ) ) ).

% map_of_map_Pair_key
thf(fact_800_map__of__map__Pair__key,axiom,
    ! [X3: produc7245736746747425831term_f,Xs: list_P4363786793477243949term_f,F3: produc7245736746747425831term_f > tree_r733329426570293750tion_f] :
      ( ( ( member848276444142703440term_f @ X3 @ ( set_Pr1901606489578307004term_f @ Xs ) )
       => ( ( map_of8294215395849376634tion_f
            @ ( map_Pr7132808054754551983tion_f
              @ ^ [K: produc7245736746747425831term_f] : ( produc1542041472756485830tion_f @ K @ ( F3 @ K ) )
              @ Xs )
            @ X3 )
          = ( some_T4055341017772447441tion_f @ ( F3 @ X3 ) ) ) )
      & ( ~ ( member848276444142703440term_f @ X3 @ ( set_Pr1901606489578307004term_f @ Xs ) )
       => ( ( map_of8294215395849376634tion_f
            @ ( map_Pr7132808054754551983tion_f
              @ ^ [K: produc7245736746747425831term_f] : ( produc1542041472756485830tion_f @ K @ ( F3 @ K ) )
              @ Xs )
            @ X3 )
          = none_T5277256714714431317tion_f ) ) ) ).

% map_of_map_Pair_key
thf(fact_801_map__of__map__Pair__key,axiom,
    ! [X3: nat,Xs: list_nat,F3: nat > tree_r733329426570293750tion_f] :
      ( ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
       => ( ( map_of4331776557111929259tion_f
            @ ( map_na5319156817545281297tion_f
              @ ^ [K: nat] : ( produc7427067040957846367tion_f @ K @ ( F3 @ K ) )
              @ Xs )
            @ X3 )
          = ( some_T4055341017772447441tion_f @ ( F3 @ X3 ) ) ) )
      & ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
       => ( ( map_of4331776557111929259tion_f
            @ ( map_na5319156817545281297tion_f
              @ ^ [K: nat] : ( produc7427067040957846367tion_f @ K @ ( F3 @ K ) )
              @ Xs )
            @ X3 )
          = none_T5277256714714431317tion_f ) ) ) ).

% map_of_map_Pair_key
thf(fact_802_map__of__map__Pair__key,axiom,
    ! [X3: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v,F3: fset_P8018961893305114765rm_f_v > tree_r733329426570293750tion_f] :
      ( ( ( member6790519936504491446rm_f_v @ X3 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
       => ( ( map_of7630570576832274912tion_f
            @ ( map_fs5340168066763766139tion_f
              @ ^ [K: fset_P8018961893305114765rm_f_v] : ( produc605741161548973356tion_f @ K @ ( F3 @ K ) )
              @ Xs )
            @ X3 )
          = ( some_T4055341017772447441tion_f @ ( F3 @ X3 ) ) ) )
      & ( ~ ( member6790519936504491446rm_f_v @ X3 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
       => ( ( map_of7630570576832274912tion_f
            @ ( map_fs5340168066763766139tion_f
              @ ^ [K: fset_P8018961893305114765rm_f_v] : ( produc605741161548973356tion_f @ K @ ( F3 @ K ) )
              @ Xs )
            @ X3 )
          = none_T5277256714714431317tion_f ) ) ) ).

% map_of_map_Pair_key
thf(fact_803_map__of__map__Pair__const,axiom,
    ! [V2: ground_gterm_f,Xs: list_Ground_gterm_f] :
      ( ( map_of953081065202312803term_f
        @ ( map_Gr7565432863924955831term_f
          @ ^ [X: ground_gterm_f] : ( produc3560254623552331287term_f @ X @ V2 )
          @ Xs ) )
      = ( ^ [X: ground_gterm_f] : ( if_opt1517256166862243008term_f @ ( member5261315044688711901term_f @ X @ ( set_Ground_gterm_f2 @ Xs ) ) @ ( some_Ground_gterm_f @ V2 ) @ none_Ground_gterm_f ) ) ) ).

% map_of_map_Pair_const
thf(fact_804_map__of__map__Pair__const,axiom,
    ! [V2: tree_r733329426570293750tion_f,Xs: list_P4363786793477243949term_f] :
      ( ( map_of8294215395849376634tion_f
        @ ( map_Pr7132808054754551983tion_f
          @ ^ [X: produc7245736746747425831term_f] : ( produc1542041472756485830tion_f @ X @ V2 )
          @ Xs ) )
      = ( ^ [X: produc7245736746747425831term_f] : ( if_opt8252839400935543808tion_f @ ( member848276444142703440term_f @ X @ ( set_Pr1901606489578307004term_f @ Xs ) ) @ ( some_T4055341017772447441tion_f @ V2 ) @ none_T5277256714714431317tion_f ) ) ) ).

% map_of_map_Pair_const
thf(fact_805_map__of__map__Pair__const,axiom,
    ! [V2: tree_r733329426570293750tion_f,Xs: list_nat] :
      ( ( map_of4331776557111929259tion_f
        @ ( map_na5319156817545281297tion_f
          @ ^ [X: nat] : ( produc7427067040957846367tion_f @ X @ V2 )
          @ Xs ) )
      = ( ^ [X: nat] : ( if_opt8252839400935543808tion_f @ ( member_nat @ X @ ( set_nat2 @ Xs ) ) @ ( some_T4055341017772447441tion_f @ V2 ) @ none_T5277256714714431317tion_f ) ) ) ).

% map_of_map_Pair_const
thf(fact_806_map__of__map__Pair__const,axiom,
    ! [V2: tree_r733329426570293750tion_f,Xs: list_f1824981274722084755rm_f_v] :
      ( ( map_of7630570576832274912tion_f
        @ ( map_fs5340168066763766139tion_f
          @ ^ [X: fset_P8018961893305114765rm_f_v] : ( produc605741161548973356tion_f @ X @ V2 )
          @ Xs ) )
      = ( ^ [X: fset_P8018961893305114765rm_f_v] : ( if_opt8252839400935543808tion_f @ ( member6790519936504491446rm_f_v @ X @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) @ ( some_T4055341017772447441tion_f @ V2 ) @ none_T5277256714714431317tion_f ) ) ) ).

% map_of_map_Pair_const
thf(fact_807_arg__min__least,axiom,
    ! [S: set_Pr989862937836626183term_f,Y: produc7245736746747425831term_f,F3: produc7245736746747425831term_f > nat] :
      ( ( finite4071141763865312848term_f @ S )
     => ( ( S != bot_bo9223291547480392563term_f )
       => ( ( member848276444142703440term_f @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic7026156477881839006_f_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_808_arg__min__least,axiom,
    ! [S: set_Ground_gterm_f,Y: ground_gterm_f,F3: ground_gterm_f > nat] :
      ( ( finite1923210418194103261term_f @ S )
     => ( ( S != bot_bo3086195439153612416term_f )
       => ( ( member5261315044688711901term_f @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic132315326754989483_f_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_809_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F3: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic7446932960582359483at_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_810_arg__min__least,axiom,
    ! [S: set_Pr5245412377734449720_f_nat,Y: product_prod_f_nat,F3: product_prod_f_nat > nat] :
      ( ( finite3598747971073327385_f_nat @ S )
     => ( ( S != bot_bo136713897782694092_f_nat )
       => ( ( member2678038196313615385_f_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic1542944825072176749at_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_811_arg__min__least,axiom,
    ! [S: set_se421094335941524212term_f,Y: set_Ground_gterm_f,F3: set_Ground_gterm_f > nat] :
      ( ( finite6824738741894949053term_f @ S )
     => ( ( S != bot_bo6304853979754439520term_f )
       => ( ( member9219338147399679421term_f @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic9034591371731082891_f_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_812_arg__min__least,axiom,
    ! [S: set_se5570125628664547815term_f,Y: set_Pr989862937836626183term_f,F3: set_Pr989862937836626183term_f > nat] :
      ( ( finite5058142709724841776term_f @ S )
     => ( ( S != bot_bo5021337078593797203term_f )
       => ( ( member6919674941555055664term_f @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic436307122699764350_f_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_813_arg__min__least,axiom,
    ! [S: set_se1451291994713068270_f_nat,Y: set_Pr5245412377734449720_f_nat,F3: set_Pr5245412377734449720_f_nat > nat] :
      ( ( finite879454794548843727_f_nat @ S )
     => ( ( S != bot_bo6700018649133943170_f_nat )
       => ( ( member8345558847462883279_f_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic3209461880563875363at_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_814_arg__min__least,axiom,
    ! [S: set_set_nat,Y: set_nat,F3: set_nat > nat] :
      ( ( finite1152437895449049373et_nat @ S )
     => ( ( S != bot_bot_set_set_nat )
       => ( ( member_set_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic7132588981422310769at_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_815_ffunas__trs_Orep__eq,axiom,
    ! [X3: fset_P8018961893305114765rm_f_v] :
      ( ( fset_P3576968334923099475_f_nat @ ( lV_to_ffunas_trs_f_v @ X3 ) )
      = ( funas_trs_f_v_v @ ( fset_P4617584883882644886rm_f_v @ X3 ) ) ) ).

% ffunas_trs.rep_eq
thf(fact_816_fempty__fsubsetI,axiom,
    ! [X3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ bot_bo3974441229639268550_f_nat @ X3 ) ).

% fempty_fsubsetI
thf(fact_817_fsubset__fempty,axiom,
    ! [A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ bot_bo3974441229639268550_f_nat )
      = ( A = bot_bo3974441229639268550_f_nat ) ) ).

% fsubset_fempty
thf(fact_818_bot__fset_Orep__eq,axiom,
    ( ( fset_P3576968334923099475_f_nat @ bot_bo3974441229639268550_f_nat )
    = bot_bo136713897782694092_f_nat ) ).

% bot_fset.rep_eq
thf(fact_819_bot__fset_Orep__eq,axiom,
    ( ( fset_P4617584883882644886rm_f_v @ bot_bo4614643899460950777rm_f_v )
    = bot_bo4638812505780724595rm_f_v ) ).

% bot_fset.rep_eq
thf(fact_820_bot__empty__eq,axiom,
    ( bot_bo6175410582289378090rm_f_o
    = ( ^ [X: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X @ bot_bo9223291547480392563term_f ) ) ) ).

% bot_empty_eq
thf(fact_821_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_822_bot__set__def,axiom,
    ( bot_bo3086195439153612416term_f
    = ( collec3522850625759001119term_f @ bot_bo6690664261123731037rm_f_o ) ) ).

% bot_set_def
thf(fact_823_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_824_bot__set__def,axiom,
    ( bot_bo5021337078593797203term_f
    = ( collec4490547877585280114term_f @ bot_bo7729305435646466634rm_f_o ) ) ).

% bot_set_def
thf(fact_825_bot__set__def,axiom,
    ( bot_bo6700018649133943170_f_nat
    = ( collec6964094325902095629_f_nat @ bot_bo7832956155940689283_nat_o ) ) ).

% bot_set_def
thf(fact_826_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_827_bot__set__def,axiom,
    ( bot_bo6304853979754439520term_f
    = ( collec1979342819066322687term_f @ bot_bo3747463815411251581rm_f_o ) ) ).

% bot_set_def
thf(fact_828_bot__empty__eq2,axiom,
    ( bot_bo3990891926076045402rm_f_o
    = ( ^ [X: ground_gterm_f,Y4: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y4 ) @ bot_bo9223291547480392563term_f ) ) ) ).

% bot_empty_eq2
thf(fact_829_map__of__SomeD,axiom,
    ! [Xs: list_P4363786793477243949term_f,K2: ground_gterm_f,Y: ground_gterm_f] :
      ( ( ( map_of953081065202312803term_f @ Xs @ K2 )
        = ( some_Ground_gterm_f @ Y ) )
     => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ K2 @ Y ) @ ( set_Pr1901606489578307004term_f @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_830_weak__map__of__SomeI,axiom,
    ! [K2: ground_gterm_f,X3: ground_gterm_f,L: list_P4363786793477243949term_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ K2 @ X3 ) @ ( set_Pr1901606489578307004term_f @ L ) )
     => ? [X2: ground_gterm_f] :
          ( ( map_of953081065202312803term_f @ L @ K2 )
          = ( some_Ground_gterm_f @ X2 ) ) ) ).

% weak_map_of_SomeI
thf(fact_831_less__by__empty,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( A = bot_bo9223291547480392563term_f )
     => ( ord_le263819222746101927term_f @ A @ B2 ) ) ).

% less_by_empty
thf(fact_832_Inf__fin_Osubset__imp,axiom,
    ! [A: set_se1451291994713068270_f_nat,B2: set_se1451291994713068270_f_nat] :
      ( ( ord_le5386115319958680398_f_nat @ A @ B2 )
     => ( ( A != bot_bo6700018649133943170_f_nat )
       => ( ( finite879454794548843727_f_nat @ B2 )
         => ( ord_le8976984241387448984_f_nat @ ( lattic837720719766856155_f_nat @ B2 ) @ ( lattic837720719766856155_f_nat @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_833_Inf__fin_Osubset__imp,axiom,
    ! [A: set_se5570125628664547815term_f,B2: set_se5570125628664547815term_f] :
      ( ( ord_le5727726935103693191term_f @ A @ B2 )
     => ( ( A != bot_bo5021337078593797203term_f )
       => ( ( finite5058142709724841776term_f @ B2 )
         => ( ord_le263819222746101927term_f @ ( lattic5334677991264475428term_f @ B2 ) @ ( lattic5334677991264475428term_f @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_834_Inf__fin_Osubset__imp,axiom,
    ! [A: set_fs7237271039932420712_f_nat,B2: set_fs7237271039932420712_f_nat] :
      ( ( ord_le6859684168973566664_f_nat @ A @ B2 )
     => ( ( A != bot_bo2133226978013888252_f_nat )
       => ( ( finite910431038387235657_f_nat @ B2 )
         => ( ord_le1552505484586773650_f_nat @ ( lattic2851005640092480085_f_nat @ B2 ) @ ( lattic2851005640092480085_f_nat @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_835_Inf__fin_Osubset__imp,axiom,
    ! [A: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B2 )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( finite1152437895449049373et_nat @ B2 )
         => ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B2 ) @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_836_Inf__fin_Osubset__imp,axiom,
    ! [A: set_se421094335941524212term_f,B2: set_se421094335941524212term_f] :
      ( ( ord_le2325249104654787220term_f @ A @ B2 )
     => ( ( A != bot_bo6304853979754439520term_f )
       => ( ( finite6824738741894949053term_f @ B2 )
         => ( ord_le2735537439747282356term_f @ ( lattic7387541653291857073term_f @ B2 ) @ ( lattic7387541653291857073term_f @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_837_Inf__fin_Osubset__imp,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ( finite3598747971073327385_f_nat @ B2 )
         => ( ord_le2234929214960958178_f_nat @ ( lattic6562015589159468581_f_nat @ B2 ) @ ( lattic6562015589159468581_f_nat @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_838_Inf__fin_Osubset__imp,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B2 )
         => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B2 ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_839_Inf__fin_OcoboundedI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,A6: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( member2678038196313615385_f_nat @ A6 @ A )
       => ( ord_le2234929214960958178_f_nat @ ( lattic6562015589159468581_f_nat @ A ) @ A6 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_840_Inf__fin_OcoboundedI,axiom,
    ! [A: set_se1451291994713068270_f_nat,A6: set_Pr5245412377734449720_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( member8345558847462883279_f_nat @ A6 @ A )
       => ( ord_le8976984241387448984_f_nat @ ( lattic837720719766856155_f_nat @ A ) @ A6 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_841_Inf__fin_OcoboundedI,axiom,
    ! [A: set_se5570125628664547815term_f,A6: set_Pr989862937836626183term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( member6919674941555055664term_f @ A6 @ A )
       => ( ord_le263819222746101927term_f @ ( lattic5334677991264475428term_f @ A ) @ A6 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_842_Inf__fin_OcoboundedI,axiom,
    ! [A: set_fs7237271039932420712_f_nat,A6: fset_P6228066233360383026_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( member6677119207206147657_f_nat @ A6 @ A )
       => ( ord_le1552505484586773650_f_nat @ ( lattic2851005640092480085_f_nat @ A ) @ A6 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_843_Inf__fin_OcoboundedI,axiom,
    ! [A: set_nat,A6: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A6 @ A )
       => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ A6 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_844_Inf__fin_OcoboundedI,axiom,
    ! [A: set_set_nat,A6: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ A6 @ A )
       => ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ A6 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_845_Inf__fin_OcoboundedI,axiom,
    ! [A: set_se421094335941524212term_f,A6: set_Ground_gterm_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( member9219338147399679421term_f @ A6 @ A )
       => ( ord_le2735537439747282356term_f @ ( lattic7387541653291857073term_f @ A ) @ A6 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_846_Inf__fin_OboundedE,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ( ord_le2234929214960958178_f_nat @ X3 @ ( lattic6562015589159468581_f_nat @ A ) )
         => ! [A9: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ A9 @ A )
             => ( ord_le2234929214960958178_f_nat @ X3 @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_847_Inf__fin_OboundedE,axiom,
    ! [A: set_se1451291994713068270_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( A != bot_bo6700018649133943170_f_nat )
       => ( ( ord_le8976984241387448984_f_nat @ X3 @ ( lattic837720719766856155_f_nat @ A ) )
         => ! [A9: set_Pr5245412377734449720_f_nat] :
              ( ( member8345558847462883279_f_nat @ A9 @ A )
             => ( ord_le8976984241387448984_f_nat @ X3 @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_848_Inf__fin_OboundedE,axiom,
    ! [A: set_se5570125628664547815term_f,X3: set_Pr989862937836626183term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( A != bot_bo5021337078593797203term_f )
       => ( ( ord_le263819222746101927term_f @ X3 @ ( lattic5334677991264475428term_f @ A ) )
         => ! [A9: set_Pr989862937836626183term_f] :
              ( ( member6919674941555055664term_f @ A9 @ A )
             => ( ord_le263819222746101927term_f @ X3 @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_849_Inf__fin_OboundedE,axiom,
    ! [A: set_fs7237271039932420712_f_nat,X3: fset_P6228066233360383026_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( A != bot_bo2133226978013888252_f_nat )
       => ( ( ord_le1552505484586773650_f_nat @ X3 @ ( lattic2851005640092480085_f_nat @ A ) )
         => ! [A9: fset_P6228066233360383026_f_nat] :
              ( ( member6677119207206147657_f_nat @ A9 @ A )
             => ( ord_le1552505484586773650_f_nat @ X3 @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_850_Inf__fin_OboundedE,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) )
         => ! [A9: nat] :
              ( ( member_nat @ A9 @ A )
             => ( ord_less_eq_nat @ X3 @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_851_Inf__fin_OboundedE,axiom,
    ! [A: set_set_nat,X3: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( ord_less_eq_set_nat @ X3 @ ( lattic3014633134055518761et_nat @ A ) )
         => ! [A9: set_nat] :
              ( ( member_set_nat @ A9 @ A )
             => ( ord_less_eq_set_nat @ X3 @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_852_Inf__fin_OboundedE,axiom,
    ! [A: set_se421094335941524212term_f,X3: set_Ground_gterm_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( A != bot_bo6304853979754439520term_f )
       => ( ( ord_le2735537439747282356term_f @ X3 @ ( lattic7387541653291857073term_f @ A ) )
         => ! [A9: set_Ground_gterm_f] :
              ( ( member9219338147399679421term_f @ A9 @ A )
             => ( ord_le2735537439747282356term_f @ X3 @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_853_Inf__fin_OboundedI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ! [A3: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ A3 @ A )
             => ( ord_le2234929214960958178_f_nat @ X3 @ A3 ) )
         => ( ord_le2234929214960958178_f_nat @ X3 @ ( lattic6562015589159468581_f_nat @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_854_Inf__fin_OboundedI,axiom,
    ! [A: set_se1451291994713068270_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( A != bot_bo6700018649133943170_f_nat )
       => ( ! [A3: set_Pr5245412377734449720_f_nat] :
              ( ( member8345558847462883279_f_nat @ A3 @ A )
             => ( ord_le8976984241387448984_f_nat @ X3 @ A3 ) )
         => ( ord_le8976984241387448984_f_nat @ X3 @ ( lattic837720719766856155_f_nat @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_855_Inf__fin_OboundedI,axiom,
    ! [A: set_se5570125628664547815term_f,X3: set_Pr989862937836626183term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( A != bot_bo5021337078593797203term_f )
       => ( ! [A3: set_Pr989862937836626183term_f] :
              ( ( member6919674941555055664term_f @ A3 @ A )
             => ( ord_le263819222746101927term_f @ X3 @ A3 ) )
         => ( ord_le263819222746101927term_f @ X3 @ ( lattic5334677991264475428term_f @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_856_Inf__fin_OboundedI,axiom,
    ! [A: set_fs7237271039932420712_f_nat,X3: fset_P6228066233360383026_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( A != bot_bo2133226978013888252_f_nat )
       => ( ! [A3: fset_P6228066233360383026_f_nat] :
              ( ( member6677119207206147657_f_nat @ A3 @ A )
             => ( ord_le1552505484586773650_f_nat @ X3 @ A3 ) )
         => ( ord_le1552505484586773650_f_nat @ X3 @ ( lattic2851005640092480085_f_nat @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_857_Inf__fin_OboundedI,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A )
             => ( ord_less_eq_nat @ X3 @ A3 ) )
         => ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_858_Inf__fin_OboundedI,axiom,
    ! [A: set_set_nat,X3: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ! [A3: set_nat] :
              ( ( member_set_nat @ A3 @ A )
             => ( ord_less_eq_set_nat @ X3 @ A3 ) )
         => ( ord_less_eq_set_nat @ X3 @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_859_Inf__fin_OboundedI,axiom,
    ! [A: set_se421094335941524212term_f,X3: set_Ground_gterm_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( A != bot_bo6304853979754439520term_f )
       => ( ! [A3: set_Ground_gterm_f] :
              ( ( member9219338147399679421term_f @ A3 @ A )
             => ( ord_le2735537439747282356term_f @ X3 @ A3 ) )
         => ( ord_le2735537439747282356term_f @ X3 @ ( lattic7387541653291857073term_f @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_860_Inf__fin_Obounded__iff,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ( ord_le2234929214960958178_f_nat @ X3 @ ( lattic6562015589159468581_f_nat @ A ) )
          = ( ! [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ A )
               => ( ord_le2234929214960958178_f_nat @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_861_Inf__fin_Obounded__iff,axiom,
    ! [A: set_se1451291994713068270_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( A != bot_bo6700018649133943170_f_nat )
       => ( ( ord_le8976984241387448984_f_nat @ X3 @ ( lattic837720719766856155_f_nat @ A ) )
          = ( ! [X: set_Pr5245412377734449720_f_nat] :
                ( ( member8345558847462883279_f_nat @ X @ A )
               => ( ord_le8976984241387448984_f_nat @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_862_Inf__fin_Obounded__iff,axiom,
    ! [A: set_se5570125628664547815term_f,X3: set_Pr989862937836626183term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( A != bot_bo5021337078593797203term_f )
       => ( ( ord_le263819222746101927term_f @ X3 @ ( lattic5334677991264475428term_f @ A ) )
          = ( ! [X: set_Pr989862937836626183term_f] :
                ( ( member6919674941555055664term_f @ X @ A )
               => ( ord_le263819222746101927term_f @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_863_Inf__fin_Obounded__iff,axiom,
    ! [A: set_fs7237271039932420712_f_nat,X3: fset_P6228066233360383026_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( A != bot_bo2133226978013888252_f_nat )
       => ( ( ord_le1552505484586773650_f_nat @ X3 @ ( lattic2851005640092480085_f_nat @ A ) )
          = ( ! [X: fset_P6228066233360383026_f_nat] :
                ( ( member6677119207206147657_f_nat @ X @ A )
               => ( ord_le1552505484586773650_f_nat @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_864_Inf__fin_Obounded__iff,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_865_Inf__fin_Obounded__iff,axiom,
    ! [A: set_set_nat,X3: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( ord_less_eq_set_nat @ X3 @ ( lattic3014633134055518761et_nat @ A ) )
          = ( ! [X: set_nat] :
                ( ( member_set_nat @ X @ A )
               => ( ord_less_eq_set_nat @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_866_Inf__fin_Obounded__iff,axiom,
    ! [A: set_se421094335941524212term_f,X3: set_Ground_gterm_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( A != bot_bo6304853979754439520term_f )
       => ( ( ord_le2735537439747282356term_f @ X3 @ ( lattic7387541653291857073term_f @ A ) )
          = ( ! [X: set_Ground_gterm_f] :
                ( ( member9219338147399679421term_f @ X @ A )
               => ( ord_le2735537439747282356term_f @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_867_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ord_le2234929214960958178_f_nat @ ( lattic6562015589159468581_f_nat @ A ) @ ( lattic7432751783671504395_f_nat @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_868_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_se1451291994713068270_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( A != bot_bo6700018649133943170_f_nat )
       => ( ord_le8976984241387448984_f_nat @ ( lattic837720719766856155_f_nat @ A ) @ ( lattic2791705783812008385_f_nat @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_869_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_se5570125628664547815term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( A != bot_bo5021337078593797203term_f )
       => ( ord_le263819222746101927term_f @ ( lattic5334677991264475428term_f @ A ) @ ( lattic4254796103941830846term_f @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_870_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_fs7237271039932420712_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( A != bot_bo2133226978013888252_f_nat )
       => ( ord_le1552505484586773650_f_nat @ ( lattic2851005640092480085_f_nat @ A ) @ ( lattic6606183005006063419_f_nat @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_871_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_872_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_873_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_se421094335941524212term_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( A != bot_bo6304853979754439520term_f )
       => ( ord_le2735537439747282356term_f @ ( lattic7387541653291857073term_f @ A ) @ ( lattic2636499749083849035term_f @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_874_map__of__zip__map,axiom,
    ! [Xs: list_f1824981274722084755rm_f_v,F3: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
      ( ( map_of7671611574113310365rm_f_v @ ( zip_fs4618486450128382833rm_f_v @ Xs @ ( map_fs8602507653405230974rm_f_v @ F3 @ Xs ) ) )
      = ( ^ [X: fset_P8018961893305114765rm_f_v] : ( if_opt1261302912030445331rm_f_v @ ( member6790519936504491446rm_f_v @ X @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) @ ( some_s6421656870786367596rm_f_v @ ( F3 @ X ) ) @ none_s1889411087592137448rm_f_v ) ) ) ).

% map_of_zip_map
thf(fact_875_map__of__zip__map,axiom,
    ! [Xs: list_P4363786793477243949term_f,F3: produc7245736746747425831term_f > tree_r733329426570293750tion_f] :
      ( ( map_of8294215395849376634tion_f @ ( zip_Pr1362359762346110886tion_f @ Xs @ ( map_Pr5969741078461996505tion_f @ F3 @ Xs ) ) )
      = ( ^ [X: produc7245736746747425831term_f] : ( if_opt8252839400935543808tion_f @ ( member848276444142703440term_f @ X @ ( set_Pr1901606489578307004term_f @ Xs ) ) @ ( some_T4055341017772447441tion_f @ ( F3 @ X ) ) @ none_T5277256714714431317tion_f ) ) ) ).

% map_of_zip_map
thf(fact_876_map__of__zip__map,axiom,
    ! [Xs: list_nat,F3: nat > tree_r733329426570293750tion_f] :
      ( ( map_of4331776557111929259tion_f @ ( zip_na9062919652488870527tion_f @ Xs @ ( map_na2027997827885215116tion_f @ F3 @ Xs ) ) )
      = ( ^ [X: nat] : ( if_opt8252839400935543808tion_f @ ( member_nat @ X @ ( set_nat2 @ Xs ) ) @ ( some_T4055341017772447441tion_f @ ( F3 @ X ) ) @ none_T5277256714714431317tion_f ) ) ) ).

% map_of_zip_map
thf(fact_877_map__of__zip__map,axiom,
    ! [Xs: list_f1824981274722084755rm_f_v,F3: fset_P8018961893305114765rm_f_v > tree_r733329426570293750tion_f] :
      ( ( map_of7630570576832274912tion_f @ ( zip_fs8106067013599143436tion_f @ Xs @ ( map_fs7137784065334668863tion_f @ F3 @ Xs ) ) )
      = ( ^ [X: fset_P8018961893305114765rm_f_v] : ( if_opt8252839400935543808tion_f @ ( member6790519936504491446rm_f_v @ X @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) @ ( some_T4055341017772447441tion_f @ ( F3 @ X ) ) @ none_T5277256714714431317tion_f ) ) ) ).

% map_of_zip_map
thf(fact_878_Sup__fin_Osubset__imp,axiom,
    ! [A: set_se1451291994713068270_f_nat,B2: set_se1451291994713068270_f_nat] :
      ( ( ord_le5386115319958680398_f_nat @ A @ B2 )
     => ( ( A != bot_bo6700018649133943170_f_nat )
       => ( ( finite879454794548843727_f_nat @ B2 )
         => ( ord_le8976984241387448984_f_nat @ ( lattic2791705783812008385_f_nat @ A ) @ ( lattic2791705783812008385_f_nat @ B2 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_879_Sup__fin_Osubset__imp,axiom,
    ! [A: set_se5570125628664547815term_f,B2: set_se5570125628664547815term_f] :
      ( ( ord_le5727726935103693191term_f @ A @ B2 )
     => ( ( A != bot_bo5021337078593797203term_f )
       => ( ( finite5058142709724841776term_f @ B2 )
         => ( ord_le263819222746101927term_f @ ( lattic4254796103941830846term_f @ A ) @ ( lattic4254796103941830846term_f @ B2 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_880_Sup__fin_Osubset__imp,axiom,
    ! [A: set_fs7237271039932420712_f_nat,B2: set_fs7237271039932420712_f_nat] :
      ( ( ord_le6859684168973566664_f_nat @ A @ B2 )
     => ( ( A != bot_bo2133226978013888252_f_nat )
       => ( ( finite910431038387235657_f_nat @ B2 )
         => ( ord_le1552505484586773650_f_nat @ ( lattic6606183005006063419_f_nat @ A ) @ ( lattic6606183005006063419_f_nat @ B2 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_881_Sup__fin_Osubset__imp,axiom,
    ! [A: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B2 )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( finite1152437895449049373et_nat @ B2 )
         => ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ ( lattic3835124923745554447et_nat @ B2 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_882_Sup__fin_Osubset__imp,axiom,
    ! [A: set_se421094335941524212term_f,B2: set_se421094335941524212term_f] :
      ( ( ord_le2325249104654787220term_f @ A @ B2 )
     => ( ( A != bot_bo6304853979754439520term_f )
       => ( ( finite6824738741894949053term_f @ B2 )
         => ( ord_le2735537439747282356term_f @ ( lattic2636499749083849035term_f @ A ) @ ( lattic2636499749083849035term_f @ B2 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_883_Sup__fin_Osubset__imp,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ( finite3598747971073327385_f_nat @ B2 )
         => ( ord_le2234929214960958178_f_nat @ ( lattic7432751783671504395_f_nat @ A ) @ ( lattic7432751783671504395_f_nat @ B2 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_884_Sup__fin_Osubset__imp,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B2 )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B2 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_885_Inf__fin_Osubset,axiom,
    ! [A: set_se421094335941524212term_f,B2: set_se421094335941524212term_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( B2 != bot_bo6304853979754439520term_f )
       => ( ( ord_le2325249104654787220term_f @ B2 @ A )
         => ( ( inf_in2342200634362942210term_f @ ( lattic7387541653291857073term_f @ B2 ) @ ( lattic7387541653291857073term_f @ A ) )
            = ( lattic7387541653291857073term_f @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_886_Inf__fin_Osubset,axiom,
    ! [A: set_se5570125628664547815term_f,B2: set_se5570125628664547815term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( B2 != bot_bo5021337078593797203term_f )
       => ( ( ord_le5727726935103693191term_f @ B2 @ A )
         => ( ( inf_in6246359444536921845term_f @ ( lattic5334677991264475428term_f @ B2 ) @ ( lattic5334677991264475428term_f @ A ) )
            = ( lattic5334677991264475428term_f @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_887_Inf__fin_Osubset,axiom,
    ! [A: set_se1451291994713068270_f_nat,B2: set_se1451291994713068270_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( B2 != bot_bo6700018649133943170_f_nat )
       => ( ( ord_le5386115319958680398_f_nat @ B2 @ A )
         => ( ( inf_in4124010953442585290_f_nat @ ( lattic837720719766856155_f_nat @ B2 ) @ ( lattic837720719766856155_f_nat @ A ) )
            = ( lattic837720719766856155_f_nat @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_888_Inf__fin_Osubset,axiom,
    ! [A: set_set_nat,B2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( B2 != bot_bot_set_set_nat )
       => ( ( ord_le6893508408891458716et_nat @ B2 @ A )
         => ( ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ B2 ) @ ( lattic3014633134055518761et_nat @ A ) )
            = ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_889_Inf__fin_Osubset,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( B2 != bot_bo136713897782694092_f_nat )
       => ( ( ord_le8976984241387448984_f_nat @ B2 @ A )
         => ( ( inf_in1797683127523738388_f_nat @ ( lattic6562015589159468581_f_nat @ B2 ) @ ( lattic6562015589159468581_f_nat @ A ) )
            = ( lattic6562015589159468581_f_nat @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_890_Inf__fin_Osubset,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( B2 != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B2 @ A )
         => ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B2 ) @ ( lattic5238388535129920115in_nat @ A ) )
            = ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_891_Max_Osubset__imp,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ( finite3598747971073327385_f_nat @ B2 )
         => ( ord_le2234929214960958178_f_nat @ ( lattic2060086870095978963_f_nat @ A ) @ ( lattic2060086870095978963_f_nat @ B2 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_892_Max_Osubset__imp,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B2 )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ ( lattic8265883725875713057ax_nat @ B2 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_893_Int__subset__iff,axiom,
    ! [C: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ C @ ( inf_in4124010953442585290_f_nat @ A @ B2 ) )
      = ( ( ord_le8976984241387448984_f_nat @ C @ A )
        & ( ord_le8976984241387448984_f_nat @ C @ B2 ) ) ) ).

% Int_subset_iff
thf(fact_894_Int__subset__iff,axiom,
    ! [C: set_Pr989862937836626183term_f,A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ C @ ( inf_in6246359444536921845term_f @ A @ B2 ) )
      = ( ( ord_le263819222746101927term_f @ C @ A )
        & ( ord_le263819222746101927term_f @ C @ B2 ) ) ) ).

% Int_subset_iff
thf(fact_895_Int__subset__iff,axiom,
    ! [C: set_nat,A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
      = ( ( ord_less_eq_set_nat @ C @ A )
        & ( ord_less_eq_set_nat @ C @ B2 ) ) ) ).

% Int_subset_iff
thf(fact_896_Int__subset__iff,axiom,
    ! [C: set_Ground_gterm_f,A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ C @ ( inf_in2342200634362942210term_f @ A @ B2 ) )
      = ( ( ord_le2735537439747282356term_f @ C @ A )
        & ( ord_le2735537439747282356term_f @ C @ B2 ) ) ) ).

% Int_subset_iff
thf(fact_897_finite__Int,axiom,
    ! [F5: set_Ground_gterm_f,G4: set_Ground_gterm_f] :
      ( ( ( finite1923210418194103261term_f @ F5 )
        | ( finite1923210418194103261term_f @ G4 ) )
     => ( finite1923210418194103261term_f @ ( inf_in2342200634362942210term_f @ F5 @ G4 ) ) ) ).

% finite_Int
thf(fact_898_finite__Int,axiom,
    ! [F5: set_nat,G4: set_nat] :
      ( ( ( finite_finite_nat @ F5 )
        | ( finite_finite_nat @ G4 ) )
     => ( finite_finite_nat @ ( inf_inf_set_nat @ F5 @ G4 ) ) ) ).

% finite_Int
thf(fact_899_finite__Int,axiom,
    ! [F5: set_Pr5245412377734449720_f_nat,G4: set_Pr5245412377734449720_f_nat] :
      ( ( ( finite3598747971073327385_f_nat @ F5 )
        | ( finite3598747971073327385_f_nat @ G4 ) )
     => ( finite3598747971073327385_f_nat @ ( inf_in4124010953442585290_f_nat @ F5 @ G4 ) ) ) ).

% finite_Int
thf(fact_900_finite__Int,axiom,
    ! [F5: set_se421094335941524212term_f,G4: set_se421094335941524212term_f] :
      ( ( ( finite6824738741894949053term_f @ F5 )
        | ( finite6824738741894949053term_f @ G4 ) )
     => ( finite6824738741894949053term_f @ ( inf_in6321094595705628642term_f @ F5 @ G4 ) ) ) ).

% finite_Int
thf(fact_901_finite__Int,axiom,
    ! [F5: set_se5570125628664547815term_f,G4: set_se5570125628664547815term_f] :
      ( ( ( finite5058142709724841776term_f @ F5 )
        | ( finite5058142709724841776term_f @ G4 ) )
     => ( finite5058142709724841776term_f @ ( inf_in2982042037914615765term_f @ F5 @ G4 ) ) ) ).

% finite_Int
thf(fact_902_finite__Int,axiom,
    ! [F5: set_se1451291994713068270_f_nat,G4: set_se1451291994713068270_f_nat] :
      ( ( ( finite879454794548843727_f_nat @ F5 )
        | ( finite879454794548843727_f_nat @ G4 ) )
     => ( finite879454794548843727_f_nat @ ( inf_in5161999638925519744_f_nat @ F5 @ G4 ) ) ) ).

% finite_Int
thf(fact_903_finite__Int,axiom,
    ! [F5: set_set_nat,G4: set_set_nat] :
      ( ( ( finite1152437895449049373et_nat @ F5 )
        | ( finite1152437895449049373et_nat @ G4 ) )
     => ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ F5 @ G4 ) ) ) ).

% finite_Int
thf(fact_904_inter__fset,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Xa3: fset_P6228066233360383026_f_nat] :
      ( ( fset_P3576968334923099475_f_nat @ ( inf_in6532419608991314884_f_nat @ X3 @ Xa3 ) )
      = ( inf_in4124010953442585290_f_nat @ ( fset_P3576968334923099475_f_nat @ X3 ) @ ( fset_P3576968334923099475_f_nat @ Xa3 ) ) ) ).

% inter_fset
thf(fact_905_inter__fset,axiom,
    ! [X3: fset_P8018961893305114765rm_f_v,Xa3: fset_P8018961893305114765rm_f_v] :
      ( ( fset_P4617584883882644886rm_f_v @ ( inf_in6563521360087198331rm_f_v @ X3 @ Xa3 ) )
      = ( inf_in6135112614888068085rm_f_v @ ( fset_P4617584883882644886rm_f_v @ X3 ) @ ( fset_P4617584883882644886rm_f_v @ Xa3 ) ) ) ).

% inter_fset
thf(fact_906_inf__Sup__absorb,axiom,
    ! [A: set_nat,A6: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A6 @ A )
       => ( ( inf_inf_nat @ A6 @ ( lattic1093996805478795353in_nat @ A ) )
          = A6 ) ) ) ).

% inf_Sup_absorb
thf(fact_907_inf__Sup__absorb,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,A6: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( member2678038196313615385_f_nat @ A6 @ A )
       => ( ( inf_in1797683127523738388_f_nat @ A6 @ ( lattic7432751783671504395_f_nat @ A ) )
          = A6 ) ) ) ).

% inf_Sup_absorb
thf(fact_908_inf__Sup__absorb,axiom,
    ! [A: set_se421094335941524212term_f,A6: set_Ground_gterm_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( member9219338147399679421term_f @ A6 @ A )
       => ( ( inf_in2342200634362942210term_f @ A6 @ ( lattic2636499749083849035term_f @ A ) )
          = A6 ) ) ) ).

% inf_Sup_absorb
thf(fact_909_inf__Sup__absorb,axiom,
    ! [A: set_se5570125628664547815term_f,A6: set_Pr989862937836626183term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( member6919674941555055664term_f @ A6 @ A )
       => ( ( inf_in6246359444536921845term_f @ A6 @ ( lattic4254796103941830846term_f @ A ) )
          = A6 ) ) ) ).

% inf_Sup_absorb
thf(fact_910_inf__Sup__absorb,axiom,
    ! [A: set_se1451291994713068270_f_nat,A6: set_Pr5245412377734449720_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( member8345558847462883279_f_nat @ A6 @ A )
       => ( ( inf_in4124010953442585290_f_nat @ A6 @ ( lattic2791705783812008385_f_nat @ A ) )
          = A6 ) ) ) ).

% inf_Sup_absorb
thf(fact_911_inf__Sup__absorb,axiom,
    ! [A: set_set_nat,A6: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ A6 @ A )
       => ( ( inf_inf_set_nat @ A6 @ ( lattic3835124923745554447et_nat @ A ) )
          = A6 ) ) ) ).

% inf_Sup_absorb
thf(fact_912_Max_Obounded__iff,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ( ord_le2234929214960958178_f_nat @ ( lattic2060086870095978963_f_nat @ A ) @ X3 )
          = ( ! [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ A )
               => ( ord_le2234929214960958178_f_nat @ X @ X3 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_913_Max_Obounded__iff,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X3 )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_914_Sigma__Int__distrib1,axiom,
    ! [I2: set_Ground_gterm_f,J2: set_Ground_gterm_f,C: ground_gterm_f > set_Ground_gterm_f] :
      ( ( produc6043514909933191368term_f @ ( inf_in2342200634362942210term_f @ I2 @ J2 ) @ C )
      = ( inf_in6246359444536921845term_f @ ( produc6043514909933191368term_f @ I2 @ C ) @ ( produc6043514909933191368term_f @ J2 @ C ) ) ) ).

% Sigma_Int_distrib1
thf(fact_915_inf__Int__eq2,axiom,
    ! [R2: set_Pr989862937836626183term_f,S: set_Pr989862937836626183term_f] :
      ( ( inf_in6121564803225898972rm_f_o
        @ ^ [X: ground_gterm_f,Y4: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y4 ) @ R2 )
        @ ^ [X: ground_gterm_f,Y4: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y4 ) @ S ) )
      = ( ^ [X: ground_gterm_f,Y4: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X @ Y4 ) @ ( inf_in6246359444536921845term_f @ R2 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_916_Times__Int__Times,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f,C: set_Ground_gterm_f,D: set_Ground_gterm_f] :
      ( ( inf_in6246359444536921845term_f
        @ ( produc6043514909933191368term_f @ A
          @ ^ [Uu: ground_gterm_f] : B2 )
        @ ( produc6043514909933191368term_f @ C
          @ ^ [Uu: ground_gterm_f] : D ) )
      = ( produc6043514909933191368term_f @ ( inf_in2342200634362942210term_f @ A @ C )
        @ ^ [Uu: ground_gterm_f] : ( inf_in2342200634362942210term_f @ B2 @ D ) ) ) ).

% Times_Int_Times
thf(fact_917_Sigma__Int__distrib2,axiom,
    ! [I2: set_Ground_gterm_f,A: ground_gterm_f > set_Ground_gterm_f,B2: ground_gterm_f > set_Ground_gterm_f] :
      ( ( produc6043514909933191368term_f @ I2
        @ ^ [I: ground_gterm_f] : ( inf_in2342200634362942210term_f @ ( A @ I ) @ ( B2 @ I ) ) )
      = ( inf_in6246359444536921845term_f @ ( produc6043514909933191368term_f @ I2 @ A ) @ ( produc6043514909933191368term_f @ I2 @ B2 ) ) ) ).

% Sigma_Int_distrib2
thf(fact_918_Times__Int__distrib1,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f,C: set_Ground_gterm_f] :
      ( ( produc6043514909933191368term_f @ ( inf_in2342200634362942210term_f @ A @ B2 )
        @ ^ [Uu: ground_gterm_f] : C )
      = ( inf_in6246359444536921845term_f
        @ ( produc6043514909933191368term_f @ A
          @ ^ [Uu: ground_gterm_f] : C )
        @ ( produc6043514909933191368term_f @ B2
          @ ^ [Uu: ground_gterm_f] : C ) ) ) ).

% Times_Int_distrib1
thf(fact_919_Int__mono,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat,D: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ C )
     => ( ( ord_le8976984241387448984_f_nat @ B2 @ D )
       => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B2 ) @ ( inf_in4124010953442585290_f_nat @ C @ D ) ) ) ) ).

% Int_mono
thf(fact_920_Int__mono,axiom,
    ! [A: set_Pr989862937836626183term_f,C: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f,D: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A @ C )
     => ( ( ord_le263819222746101927term_f @ B2 @ D )
       => ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ A @ B2 ) @ ( inf_in6246359444536921845term_f @ C @ D ) ) ) ) ).

% Int_mono
thf(fact_921_Int__mono,axiom,
    ! [A: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ( ord_less_eq_set_nat @ B2 @ D )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).

% Int_mono
thf(fact_922_Int__mono,axiom,
    ! [A: set_Ground_gterm_f,C: set_Ground_gterm_f,B2: set_Ground_gterm_f,D: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A @ C )
     => ( ( ord_le2735537439747282356term_f @ B2 @ D )
       => ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ A @ B2 ) @ ( inf_in2342200634362942210term_f @ C @ D ) ) ) ) ).

% Int_mono
thf(fact_923_Int__lower1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B2 ) @ A ) ).

% Int_lower1
thf(fact_924_Int__lower1,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ A @ B2 ) @ A ) ).

% Int_lower1
thf(fact_925_Int__lower1,axiom,
    ! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ A ) ).

% Int_lower1
thf(fact_926_Int__lower1,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ A @ B2 ) @ A ) ).

% Int_lower1
thf(fact_927_Int__lower2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B2 ) @ B2 ) ).

% Int_lower2
thf(fact_928_Int__lower2,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ A @ B2 ) @ B2 ) ).

% Int_lower2
thf(fact_929_Int__lower2,axiom,
    ! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ B2 ) ).

% Int_lower2
thf(fact_930_Int__lower2,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ A @ B2 ) @ B2 ) ).

% Int_lower2
thf(fact_931_Int__absorb1,axiom,
    ! [B2: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B2 @ A )
     => ( ( inf_in4124010953442585290_f_nat @ A @ B2 )
        = B2 ) ) ).

% Int_absorb1
thf(fact_932_Int__absorb1,axiom,
    ! [B2: set_Pr989862937836626183term_f,A: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ B2 @ A )
     => ( ( inf_in6246359444536921845term_f @ A @ B2 )
        = B2 ) ) ).

% Int_absorb1
thf(fact_933_Int__absorb1,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A )
     => ( ( inf_inf_set_nat @ A @ B2 )
        = B2 ) ) ).

% Int_absorb1
thf(fact_934_Int__absorb1,axiom,
    ! [B2: set_Ground_gterm_f,A: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ B2 @ A )
     => ( ( inf_in2342200634362942210term_f @ A @ B2 )
        = B2 ) ) ).

% Int_absorb1
thf(fact_935_Int__absorb2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
     => ( ( inf_in4124010953442585290_f_nat @ A @ B2 )
        = A ) ) ).

% Int_absorb2
thf(fact_936_Int__absorb2,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A @ B2 )
     => ( ( inf_in6246359444536921845term_f @ A @ B2 )
        = A ) ) ).

% Int_absorb2
thf(fact_937_Int__absorb2,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( inf_inf_set_nat @ A @ B2 )
        = A ) ) ).

% Int_absorb2
thf(fact_938_Int__absorb2,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A @ B2 )
     => ( ( inf_in2342200634362942210term_f @ A @ B2 )
        = A ) ) ).

% Int_absorb2
thf(fact_939_Int__greatest,axiom,
    ! [C: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ C @ A )
     => ( ( ord_le8976984241387448984_f_nat @ C @ B2 )
       => ( ord_le8976984241387448984_f_nat @ C @ ( inf_in4124010953442585290_f_nat @ A @ B2 ) ) ) ) ).

% Int_greatest
thf(fact_940_Int__greatest,axiom,
    ! [C: set_Pr989862937836626183term_f,A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ C @ A )
     => ( ( ord_le263819222746101927term_f @ C @ B2 )
       => ( ord_le263819222746101927term_f @ C @ ( inf_in6246359444536921845term_f @ A @ B2 ) ) ) ) ).

% Int_greatest
thf(fact_941_Int__greatest,axiom,
    ! [C: set_nat,A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A )
     => ( ( ord_less_eq_set_nat @ C @ B2 )
       => ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).

% Int_greatest
thf(fact_942_Int__greatest,axiom,
    ! [C: set_Ground_gterm_f,A: set_Ground_gterm_f,B2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ C @ A )
     => ( ( ord_le2735537439747282356term_f @ C @ B2 )
       => ( ord_le2735537439747282356term_f @ C @ ( inf_in2342200634362942210term_f @ A @ B2 ) ) ) ) ).

% Int_greatest
thf(fact_943_Int__Collect__mono,axiom,
    ! [A: set_se5570125628664547815term_f,B2: set_se5570125628664547815term_f,P: set_Pr989862937836626183term_f > $o,Q: set_Pr989862937836626183term_f > $o] :
      ( ( ord_le5727726935103693191term_f @ A @ B2 )
     => ( ! [X2: set_Pr989862937836626183term_f] :
            ( ( member6919674941555055664term_f @ X2 @ A )
           => ( ( P @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_le5727726935103693191term_f @ ( inf_in2982042037914615765term_f @ A @ ( collec4490547877585280114term_f @ P ) ) @ ( inf_in2982042037914615765term_f @ B2 @ ( collec4490547877585280114term_f @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_944_Int__Collect__mono,axiom,
    ! [A: set_se1451291994713068270_f_nat,B2: set_se1451291994713068270_f_nat,P: set_Pr5245412377734449720_f_nat > $o,Q: set_Pr5245412377734449720_f_nat > $o] :
      ( ( ord_le5386115319958680398_f_nat @ A @ B2 )
     => ( ! [X2: set_Pr5245412377734449720_f_nat] :
            ( ( member8345558847462883279_f_nat @ X2 @ A )
           => ( ( P @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_le5386115319958680398_f_nat @ ( inf_in5161999638925519744_f_nat @ A @ ( collec6964094325902095629_f_nat @ P ) ) @ ( inf_in5161999638925519744_f_nat @ B2 @ ( collec6964094325902095629_f_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_945_Int__Collect__mono,axiom,
    ! [A: set_set_nat,B2: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ A @ B2 )
     => ( ! [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A )
           => ( ( P @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B2 @ ( collect_set_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_946_Int__Collect__mono,axiom,
    ! [A: set_se421094335941524212term_f,B2: set_se421094335941524212term_f,P: set_Ground_gterm_f > $o,Q: set_Ground_gterm_f > $o] :
      ( ( ord_le2325249104654787220term_f @ A @ B2 )
     => ( ! [X2: set_Ground_gterm_f] :
            ( ( member9219338147399679421term_f @ X2 @ A )
           => ( ( P @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_le2325249104654787220term_f @ ( inf_in6321094595705628642term_f @ A @ ( collec1979342819066322687term_f @ P ) ) @ ( inf_in6321094595705628642term_f @ B2 @ ( collec1979342819066322687term_f @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_947_Int__Collect__mono,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat,P: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
     => ( ! [X2: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X2 @ A )
           => ( ( P @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ ( collec1417984143148964183_f_nat @ P ) ) @ ( inf_in4124010953442585290_f_nat @ B2 @ ( collec1417984143148964183_f_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_948_Int__Collect__mono,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f,P: produc7245736746747425831term_f > $o,Q: produc7245736746747425831term_f > $o] :
      ( ( ord_le263819222746101927term_f @ A @ B2 )
     => ( ! [X2: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X2 @ A )
           => ( ( P @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ A @ ( collec3209569126566747026term_f @ P ) ) @ ( inf_in6246359444536921845term_f @ B2 @ ( collec3209569126566747026term_f @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_949_Int__Collect__mono,axiom,
    ! [A: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ( ( P @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_950_Int__Collect__mono,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f,P: ground_gterm_f > $o,Q: ground_gterm_f > $o] :
      ( ( ord_le2735537439747282356term_f @ A @ B2 )
     => ( ! [X2: ground_gterm_f] :
            ( ( member5261315044688711901term_f @ X2 @ A )
           => ( ( P @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ A @ ( collec3522850625759001119term_f @ P ) ) @ ( inf_in2342200634362942210term_f @ B2 @ ( collec3522850625759001119term_f @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_951_Int__emptyI,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ! [X2: produc7245736746747425831term_f] :
          ( ( member848276444142703440term_f @ X2 @ A )
         => ~ ( member848276444142703440term_f @ X2 @ B2 ) )
     => ( ( inf_in6246359444536921845term_f @ A @ B2 )
        = bot_bo9223291547480392563term_f ) ) ).

% Int_emptyI
thf(fact_952_Int__emptyI,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ! [X2: nat] :
          ( ( member_nat @ X2 @ A )
         => ~ ( member_nat @ X2 @ B2 ) )
     => ( ( inf_inf_set_nat @ A @ B2 )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_953_disjoint__iff,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( ( inf_in6246359444536921845term_f @ A @ B2 )
        = bot_bo9223291547480392563term_f )
      = ( ! [X: produc7245736746747425831term_f] :
            ( ( member848276444142703440term_f @ X @ A )
           => ~ ( member848276444142703440term_f @ X @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_954_disjoint__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B2 )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A )
           => ~ ( member_nat @ X @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_955_finter__greatest,axiom,
    ! [C: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat,B2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ C @ A )
     => ( ( ord_le1552505484586773650_f_nat @ C @ B2 )
       => ( ord_le1552505484586773650_f_nat @ C @ ( inf_in6532419608991314884_f_nat @ A @ B2 ) ) ) ) ).

% finter_greatest
thf(fact_956_finter__absorb2,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ B2 )
     => ( ( inf_in6532419608991314884_f_nat @ A @ B2 )
        = A ) ) ).

% finter_absorb2
thf(fact_957_finter__absorb1,axiom,
    ! [B2: fset_P6228066233360383026_f_nat,A: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B2 @ A )
     => ( ( inf_in6532419608991314884_f_nat @ A @ B2 )
        = B2 ) ) ).

% finter_absorb1
thf(fact_958_finter__lower2,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B2: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A @ B2 ) @ B2 ) ).

% finter_lower2
thf(fact_959_finter__lower1,axiom,
    ! [A: fset_P6228066233360383026_f_nat,B2: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A @ B2 ) @ A ) ).

% finter_lower1
thf(fact_960_finter__mono,axiom,
    ! [A: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat,B2: fset_P6228066233360383026_f_nat,D: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A @ C )
     => ( ( ord_le1552505484586773650_f_nat @ B2 @ D )
       => ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A @ B2 ) @ ( inf_in6532419608991314884_f_nat @ C @ D ) ) ) ) ).

% finter_mono
thf(fact_961__092_060T_062_092_060_094sub_062G__intersection,axiom,
    ! [S3: ground_gterm_f,F2: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat] :
      ( ( member5261315044688711901term_f @ S3 @ ( ground_T_G_f @ F2 ) )
     => ( ( member5261315044688711901term_f @ S3 @ ( ground_T_G_f @ G ) )
       => ( member5261315044688711901term_f @ S3 @ ( ground_T_G_f @ ( inf_in4124010953442585290_f_nat @ F2 @ G ) ) ) ) ) ).

% \<T>\<^sub>G_intersection
thf(fact_962_Max__ge,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( member2678038196313615385_f_nat @ X3 @ A )
       => ( ord_le2234929214960958178_f_nat @ X3 @ ( lattic2060086870095978963_f_nat @ A ) ) ) ) ).

% Max_ge
thf(fact_963_Max__ge,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X3 @ A )
       => ( ord_less_eq_nat @ X3 @ ( lattic8265883725875713057ax_nat @ A ) ) ) ) ).

% Max_ge
thf(fact_964_Max__eqI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ! [Y3: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ Y3 @ A )
           => ( ord_le2234929214960958178_f_nat @ Y3 @ X3 ) )
       => ( ( member2678038196313615385_f_nat @ X3 @ A )
         => ( ( lattic2060086870095978963_f_nat @ A )
            = X3 ) ) ) ) ).

% Max_eqI
thf(fact_965_Max__eqI,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A )
           => ( ord_less_eq_nat @ Y3 @ X3 ) )
       => ( ( member_nat @ X3 @ A )
         => ( ( lattic8265883725875713057ax_nat @ A )
            = X3 ) ) ) ) ).

% Max_eqI
thf(fact_966_Max__eq__if,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( finite3598747971073327385_f_nat @ B2 )
       => ( ! [X2: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ X2 @ A )
             => ? [Xa: product_prod_f_nat] :
                  ( ( member2678038196313615385_f_nat @ Xa @ B2 )
                  & ( ord_le2234929214960958178_f_nat @ X2 @ Xa ) ) )
         => ( ! [X2: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X2 @ B2 )
               => ? [Xa: product_prod_f_nat] :
                    ( ( member2678038196313615385_f_nat @ Xa @ A )
                    & ( ord_le2234929214960958178_f_nat @ X2 @ Xa ) ) )
           => ( ( lattic2060086870095978963_f_nat @ A )
              = ( lattic2060086870095978963_f_nat @ B2 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_967_Max__eq__if,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( ord_less_eq_nat @ X2 @ Xa ) ) )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ B2 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ A )
                    & ( ord_less_eq_nat @ X2 @ Xa ) ) )
           => ( ( lattic8265883725875713057ax_nat @ A )
              = ( lattic8265883725875713057ax_nat @ B2 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_968_Max_OcoboundedI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,A6: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( member2678038196313615385_f_nat @ A6 @ A )
       => ( ord_le2234929214960958178_f_nat @ A6 @ ( lattic2060086870095978963_f_nat @ A ) ) ) ) ).

% Max.coboundedI
thf(fact_969_Max_OcoboundedI,axiom,
    ! [A: set_nat,A6: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A6 @ A )
       => ( ord_less_eq_nat @ A6 @ ( lattic8265883725875713057ax_nat @ A ) ) ) ) ).

% Max.coboundedI
thf(fact_970_Max__in,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( member_nat @ ( lattic8265883725875713057ax_nat @ A ) @ A ) ) ) ).

% Max_in
thf(fact_971_Max__in,axiom,
    ! [A: set_Pr5245412377734449720_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( member2678038196313615385_f_nat @ ( lattic2060086870095978963_f_nat @ A ) @ A ) ) ) ).

% Max_in
thf(fact_972_Inf__fin_Oin__idem,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X3 @ A )
       => ( ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) )
          = ( lattic5238388535129920115in_nat @ A ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_973_Inf__fin_Oin__idem,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( member2678038196313615385_f_nat @ X3 @ A )
       => ( ( inf_in1797683127523738388_f_nat @ X3 @ ( lattic6562015589159468581_f_nat @ A ) )
          = ( lattic6562015589159468581_f_nat @ A ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_974_Inf__fin_Oin__idem,axiom,
    ! [A: set_se421094335941524212term_f,X3: set_Ground_gterm_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( member9219338147399679421term_f @ X3 @ A )
       => ( ( inf_in2342200634362942210term_f @ X3 @ ( lattic7387541653291857073term_f @ A ) )
          = ( lattic7387541653291857073term_f @ A ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_975_Inf__fin_Oin__idem,axiom,
    ! [A: set_se5570125628664547815term_f,X3: set_Pr989862937836626183term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( member6919674941555055664term_f @ X3 @ A )
       => ( ( inf_in6246359444536921845term_f @ X3 @ ( lattic5334677991264475428term_f @ A ) )
          = ( lattic5334677991264475428term_f @ A ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_976_Inf__fin_Oin__idem,axiom,
    ! [A: set_se1451291994713068270_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( member8345558847462883279_f_nat @ X3 @ A )
       => ( ( inf_in4124010953442585290_f_nat @ X3 @ ( lattic837720719766856155_f_nat @ A ) )
          = ( lattic837720719766856155_f_nat @ A ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_977_Inf__fin_Oin__idem,axiom,
    ! [A: set_set_nat,X3: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ X3 @ A )
       => ( ( inf_inf_set_nat @ X3 @ ( lattic3014633134055518761et_nat @ A ) )
          = ( lattic3014633134055518761et_nat @ A ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_978_Sup__fin_OcoboundedI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,A6: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( member2678038196313615385_f_nat @ A6 @ A )
       => ( ord_le2234929214960958178_f_nat @ A6 @ ( lattic7432751783671504395_f_nat @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_979_Sup__fin_OcoboundedI,axiom,
    ! [A: set_se1451291994713068270_f_nat,A6: set_Pr5245412377734449720_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( member8345558847462883279_f_nat @ A6 @ A )
       => ( ord_le8976984241387448984_f_nat @ A6 @ ( lattic2791705783812008385_f_nat @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_980_Sup__fin_OcoboundedI,axiom,
    ! [A: set_se5570125628664547815term_f,A6: set_Pr989862937836626183term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( member6919674941555055664term_f @ A6 @ A )
       => ( ord_le263819222746101927term_f @ A6 @ ( lattic4254796103941830846term_f @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_981_Sup__fin_OcoboundedI,axiom,
    ! [A: set_fs7237271039932420712_f_nat,A6: fset_P6228066233360383026_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( member6677119207206147657_f_nat @ A6 @ A )
       => ( ord_le1552505484586773650_f_nat @ A6 @ ( lattic6606183005006063419_f_nat @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_982_Sup__fin_OcoboundedI,axiom,
    ! [A: set_nat,A6: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A6 @ A )
       => ( ord_less_eq_nat @ A6 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_983_Sup__fin_OcoboundedI,axiom,
    ! [A: set_set_nat,A6: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ A6 @ A )
       => ( ord_less_eq_set_nat @ A6 @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_984_Sup__fin_OcoboundedI,axiom,
    ! [A: set_se421094335941524212term_f,A6: set_Ground_gterm_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( member9219338147399679421term_f @ A6 @ A )
       => ( ord_le2735537439747282356term_f @ A6 @ ( lattic2636499749083849035term_f @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_985_zip__same__conv__map,axiom,
    ! [Xs: list_Ground_gterm_f] :
      ( ( zip_Gr5577144313533543735term_f @ Xs @ Xs )
      = ( map_Gr7565432863924955831term_f
        @ ^ [X: ground_gterm_f] : ( produc3560254623552331287term_f @ X @ X )
        @ Xs ) ) ).

% zip_same_conv_map
thf(fact_986_set__zip__rightD,axiom,
    ! [X3: ground_gterm_f,Y: ground_gterm_f,Xs: list_Ground_gterm_f,Ys: list_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y ) @ ( set_Pr1901606489578307004term_f @ ( zip_Gr5577144313533543735term_f @ Xs @ Ys ) ) )
     => ( member5261315044688711901term_f @ Y @ ( set_Ground_gterm_f2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_987_set__zip__leftD,axiom,
    ! [X3: ground_gterm_f,Y: ground_gterm_f,Xs: list_Ground_gterm_f,Ys: list_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y ) @ ( set_Pr1901606489578307004term_f @ ( zip_Gr5577144313533543735term_f @ Xs @ Ys ) ) )
     => ( member5261315044688711901term_f @ X3 @ ( set_Ground_gterm_f2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_988_in__set__zipE,axiom,
    ! [X3: produc7245736746747425831term_f,Y: produc7245736746747425831term_f,Xs: list_P4363786793477243949term_f,Ys: list_P4363786793477243949term_f] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ X3 @ Y ) @ ( set_Pr7884216185503574396term_f @ ( zip_Pr6240786406182277751term_f @ Xs @ Ys ) ) )
     => ~ ( ( member848276444142703440term_f @ X3 @ ( set_Pr1901606489578307004term_f @ Xs ) )
         => ~ ( member848276444142703440term_f @ Y @ ( set_Pr1901606489578307004term_f @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_989_in__set__zipE,axiom,
    ! [X3: produc7245736746747425831term_f,Y: nat,Xs: list_P4363786793477243949term_f,Ys: list_nat] :
      ( ( member3662771866509663549_f_nat @ ( produc5217865480721399968_f_nat @ X3 @ Y ) @ ( set_Pr795494564038687697_f_nat @ ( zip_Pr2102475093691875712_f_nat @ Xs @ Ys ) ) )
     => ~ ( ( member848276444142703440term_f @ X3 @ ( set_Pr1901606489578307004term_f @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_990_in__set__zipE,axiom,
    ! [X3: nat,Y: produc7245736746747425831term_f,Xs: list_nat,Ys: list_P4363786793477243949term_f] :
      ( ( member4003629742723934691term_f @ ( produc6263668818416315902term_f @ X3 @ Y ) @ ( set_Pr1136352440252958839term_f @ ( zip_na3148278431386791646term_f @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( member848276444142703440term_f @ Y @ ( set_Pr1901606489578307004term_f @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_991_in__set__zipE,axiom,
    ! [X3: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_992_in__set__zipE,axiom,
    ! [X3: produc7245736746747425831term_f,Y: fset_P8018961893305114765rm_f_v,Xs: list_P4363786793477243949term_f,Ys: list_f1824981274722084755rm_f_v] :
      ( ( member2505966486127315062rm_f_v @ ( produc1765471333835581373rm_f_v @ X3 @ Y ) @ ( set_Pr3149645275019554530rm_f_v @ ( zip_Pr1896772707301498589rm_f_v @ Xs @ Ys ) ) )
     => ~ ( ( member848276444142703440term_f @ X3 @ ( set_Pr1901606489578307004term_f @ Xs ) )
         => ~ ( member6790519936504491446rm_f_v @ Y @ ( set_fs7270820277574336546rm_f_v @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_993_in__set__zipE,axiom,
    ! [X3: nat,Y: fset_P8018961893305114765rm_f_v,Xs: list_nat,Ys: list_f1824981274722084755rm_f_v] :
      ( ( member7904933363068100937rm_f_v @ ( produc1420575044300057700rm_f_v @ X3 @ Y ) @ ( set_Pr1814501619541350365rm_f_v @ ( zip_na383336925718687044rm_f_v @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( member6790519936504491446rm_f_v @ Y @ ( set_fs7270820277574336546rm_f_v @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_994_in__set__zipE,axiom,
    ! [X3: fset_P8018961893305114765rm_f_v,Y: produc7245736746747425831term_f,Xs: list_f1824981274722084755rm_f_v,Ys: list_P4363786793477243949term_f] :
      ( ( member6363105357409103914term_f @ ( produc4505198872610157681term_f @ X3 @ Y ) @ ( set_Pr7006784146301343382term_f @ ( zip_fs4636500246076074897term_f @ Xs @ Ys ) ) )
     => ~ ( ( member6790519936504491446rm_f_v @ X3 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
         => ~ ( member848276444142703440term_f @ Y @ ( set_Pr1901606489578307004term_f @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_995_in__set__zipE,axiom,
    ! [X3: fset_P8018961893305114765rm_f_v,Y: nat,Xs: list_f1824981274722084755rm_f_v,Ys: list_nat] :
      ( ( member8567861511764899235_v_nat @ ( produc261146399008998662_v_nat @ X3 @ Y ) @ ( set_Pr2477429768238148663_v_nat @ ( zip_fs8447280317282403814_v_nat @ Xs @ Ys ) ) )
     => ~ ( ( member6790519936504491446rm_f_v @ X3 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_996_in__set__zipE,axiom,
    ! [X3: fset_P8018961893305114765rm_f_v,Y: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v,Ys: list_f1824981274722084755rm_f_v] :
      ( ( member4344164543411788176rm_f_v @ ( produc989898760166608087rm_f_v @ X3 @ Y ) @ ( set_Pr7525886052349274108rm_f_v @ ( zip_fs8516469554775540727rm_f_v @ Xs @ Ys ) ) )
     => ~ ( ( member6790519936504491446rm_f_v @ X3 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
         => ~ ( member6790519936504491446rm_f_v @ Y @ ( set_fs7270820277574336546rm_f_v @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_997_in__set__zipE,axiom,
    ! [X3: ground_gterm_f,Y: ground_gterm_f,Xs: list_Ground_gterm_f,Ys: list_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ X3 @ Y ) @ ( set_Pr1901606489578307004term_f @ ( zip_Gr5577144313533543735term_f @ Xs @ Ys ) ) )
     => ~ ( ( member5261315044688711901term_f @ X3 @ ( set_Ground_gterm_f2 @ Xs ) )
         => ~ ( member5261315044688711901term_f @ Y @ ( set_Ground_gterm_f2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_998_zip__same,axiom,
    ! [A6: produc7245736746747425831term_f,B3: produc7245736746747425831term_f,Xs: list_P4363786793477243949term_f] :
      ( ( member3777300322277616400term_f @ ( produc7348823797596265303term_f @ A6 @ B3 ) @ ( set_Pr7884216185503574396term_f @ ( zip_Pr6240786406182277751term_f @ Xs @ Xs ) ) )
      = ( ( member848276444142703440term_f @ A6 @ ( set_Pr1901606489578307004term_f @ Xs ) )
        & ( A6 = B3 ) ) ) ).

% zip_same
thf(fact_999_zip__same,axiom,
    ! [A6: nat,B3: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat @ A6 @ ( set_nat2 @ Xs ) )
        & ( A6 = B3 ) ) ) ).

% zip_same
thf(fact_1000_zip__same,axiom,
    ! [A6: fset_P8018961893305114765rm_f_v,B3: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
      ( ( member4344164543411788176rm_f_v @ ( produc989898760166608087rm_f_v @ A6 @ B3 ) @ ( set_Pr7525886052349274108rm_f_v @ ( zip_fs8516469554775540727rm_f_v @ Xs @ Xs ) ) )
      = ( ( member6790519936504491446rm_f_v @ A6 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
        & ( A6 = B3 ) ) ) ).

% zip_same
thf(fact_1001_zip__same,axiom,
    ! [A6: ground_gterm_f,B3: ground_gterm_f,Xs: list_Ground_gterm_f] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ A6 @ B3 ) @ ( set_Pr1901606489578307004term_f @ ( zip_Gr5577144313533543735term_f @ Xs @ Xs ) ) )
      = ( ( member5261315044688711901term_f @ A6 @ ( set_Ground_gterm_f2 @ Xs ) )
        & ( A6 = B3 ) ) ) ).

% zip_same
thf(fact_1002_Max_OboundedI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ! [A3: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ A3 @ A )
             => ( ord_le2234929214960958178_f_nat @ A3 @ X3 ) )
         => ( ord_le2234929214960958178_f_nat @ ( lattic2060086870095978963_f_nat @ A ) @ X3 ) ) ) ) ).

% Max.boundedI
thf(fact_1003_Max_OboundedI,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A )
             => ( ord_less_eq_nat @ A3 @ X3 ) )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X3 ) ) ) ) ).

% Max.boundedI
thf(fact_1004_Max_OboundedE,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ( ord_le2234929214960958178_f_nat @ ( lattic2060086870095978963_f_nat @ A ) @ X3 )
         => ! [A9: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ A9 @ A )
             => ( ord_le2234929214960958178_f_nat @ A9 @ X3 ) ) ) ) ) ).

% Max.boundedE
thf(fact_1005_Max_OboundedE,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X3 )
         => ! [A9: nat] :
              ( ( member_nat @ A9 @ A )
             => ( ord_less_eq_nat @ A9 @ X3 ) ) ) ) ) ).

% Max.boundedE
thf(fact_1006_eq__Max__iff,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,M: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ( M
            = ( lattic2060086870095978963_f_nat @ A ) )
          = ( ( member2678038196313615385_f_nat @ M @ A )
            & ! [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ A )
               => ( ord_le2234929214960958178_f_nat @ X @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_1007_eq__Max__iff,axiom,
    ! [A: set_nat,M: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( M
            = ( lattic8265883725875713057ax_nat @ A ) )
          = ( ( member_nat @ M @ A )
            & ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_eq_nat @ X @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_1008_Max__ge__iff,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ( ord_le2234929214960958178_f_nat @ X3 @ ( lattic2060086870095978963_f_nat @ A ) )
          = ( ? [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ A )
                & ( ord_le2234929214960958178_f_nat @ X3 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_1009_Max__ge__iff,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X3 @ ( lattic8265883725875713057ax_nat @ A ) )
          = ( ? [X: nat] :
                ( ( member_nat @ X @ A )
                & ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_1010_Max__eq__iff,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,M: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ( ( lattic2060086870095978963_f_nat @ A )
            = M )
          = ( ( member2678038196313615385_f_nat @ M @ A )
            & ! [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ A )
               => ( ord_le2234929214960958178_f_nat @ X @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_1011_Max__eq__iff,axiom,
    ! [A: set_nat,M: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ( lattic8265883725875713057ax_nat @ A )
            = M )
          = ( ( member_nat @ M @ A )
            & ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_eq_nat @ X @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_1012_Sup__fin_Obounded__iff,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ( ord_le2234929214960958178_f_nat @ ( lattic7432751783671504395_f_nat @ A ) @ X3 )
          = ( ! [X: product_prod_f_nat] :
                ( ( member2678038196313615385_f_nat @ X @ A )
               => ( ord_le2234929214960958178_f_nat @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1013_Sup__fin_Obounded__iff,axiom,
    ! [A: set_se1451291994713068270_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( A != bot_bo6700018649133943170_f_nat )
       => ( ( ord_le8976984241387448984_f_nat @ ( lattic2791705783812008385_f_nat @ A ) @ X3 )
          = ( ! [X: set_Pr5245412377734449720_f_nat] :
                ( ( member8345558847462883279_f_nat @ X @ A )
               => ( ord_le8976984241387448984_f_nat @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1014_Sup__fin_Obounded__iff,axiom,
    ! [A: set_se5570125628664547815term_f,X3: set_Pr989862937836626183term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( A != bot_bo5021337078593797203term_f )
       => ( ( ord_le263819222746101927term_f @ ( lattic4254796103941830846term_f @ A ) @ X3 )
          = ( ! [X: set_Pr989862937836626183term_f] :
                ( ( member6919674941555055664term_f @ X @ A )
               => ( ord_le263819222746101927term_f @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1015_Sup__fin_Obounded__iff,axiom,
    ! [A: set_fs7237271039932420712_f_nat,X3: fset_P6228066233360383026_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( A != bot_bo2133226978013888252_f_nat )
       => ( ( ord_le1552505484586773650_f_nat @ ( lattic6606183005006063419_f_nat @ A ) @ X3 )
          = ( ! [X: fset_P6228066233360383026_f_nat] :
                ( ( member6677119207206147657_f_nat @ X @ A )
               => ( ord_le1552505484586773650_f_nat @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1016_Sup__fin_Obounded__iff,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X3 )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1017_Sup__fin_Obounded__iff,axiom,
    ! [A: set_set_nat,X3: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X3 )
          = ( ! [X: set_nat] :
                ( ( member_set_nat @ X @ A )
               => ( ord_less_eq_set_nat @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1018_Sup__fin_Obounded__iff,axiom,
    ! [A: set_se421094335941524212term_f,X3: set_Ground_gterm_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( A != bot_bo6304853979754439520term_f )
       => ( ( ord_le2735537439747282356term_f @ ( lattic2636499749083849035term_f @ A ) @ X3 )
          = ( ! [X: set_Ground_gterm_f] :
                ( ( member9219338147399679421term_f @ X @ A )
               => ( ord_le2735537439747282356term_f @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1019_Sup__fin_OboundedI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ! [A3: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ A3 @ A )
             => ( ord_le2234929214960958178_f_nat @ A3 @ X3 ) )
         => ( ord_le2234929214960958178_f_nat @ ( lattic7432751783671504395_f_nat @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1020_Sup__fin_OboundedI,axiom,
    ! [A: set_se1451291994713068270_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( A != bot_bo6700018649133943170_f_nat )
       => ( ! [A3: set_Pr5245412377734449720_f_nat] :
              ( ( member8345558847462883279_f_nat @ A3 @ A )
             => ( ord_le8976984241387448984_f_nat @ A3 @ X3 ) )
         => ( ord_le8976984241387448984_f_nat @ ( lattic2791705783812008385_f_nat @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1021_Sup__fin_OboundedI,axiom,
    ! [A: set_se5570125628664547815term_f,X3: set_Pr989862937836626183term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( A != bot_bo5021337078593797203term_f )
       => ( ! [A3: set_Pr989862937836626183term_f] :
              ( ( member6919674941555055664term_f @ A3 @ A )
             => ( ord_le263819222746101927term_f @ A3 @ X3 ) )
         => ( ord_le263819222746101927term_f @ ( lattic4254796103941830846term_f @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1022_Sup__fin_OboundedI,axiom,
    ! [A: set_fs7237271039932420712_f_nat,X3: fset_P6228066233360383026_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( A != bot_bo2133226978013888252_f_nat )
       => ( ! [A3: fset_P6228066233360383026_f_nat] :
              ( ( member6677119207206147657_f_nat @ A3 @ A )
             => ( ord_le1552505484586773650_f_nat @ A3 @ X3 ) )
         => ( ord_le1552505484586773650_f_nat @ ( lattic6606183005006063419_f_nat @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1023_Sup__fin_OboundedI,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A )
             => ( ord_less_eq_nat @ A3 @ X3 ) )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1024_Sup__fin_OboundedI,axiom,
    ! [A: set_set_nat,X3: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ! [A3: set_nat] :
              ( ( member_set_nat @ A3 @ A )
             => ( ord_less_eq_set_nat @ A3 @ X3 ) )
         => ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1025_Sup__fin_OboundedI,axiom,
    ! [A: set_se421094335941524212term_f,X3: set_Ground_gterm_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( A != bot_bo6304853979754439520term_f )
       => ( ! [A3: set_Ground_gterm_f] :
              ( ( member9219338147399679421term_f @ A3 @ A )
             => ( ord_le2735537439747282356term_f @ A3 @ X3 ) )
         => ( ord_le2735537439747282356term_f @ ( lattic2636499749083849035term_f @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1026_Sup__fin_OboundedE,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A != bot_bo136713897782694092_f_nat )
       => ( ( ord_le2234929214960958178_f_nat @ ( lattic7432751783671504395_f_nat @ A ) @ X3 )
         => ! [A9: product_prod_f_nat] :
              ( ( member2678038196313615385_f_nat @ A9 @ A )
             => ( ord_le2234929214960958178_f_nat @ A9 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1027_Sup__fin_OboundedE,axiom,
    ! [A: set_se1451291994713068270_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( finite879454794548843727_f_nat @ A )
     => ( ( A != bot_bo6700018649133943170_f_nat )
       => ( ( ord_le8976984241387448984_f_nat @ ( lattic2791705783812008385_f_nat @ A ) @ X3 )
         => ! [A9: set_Pr5245412377734449720_f_nat] :
              ( ( member8345558847462883279_f_nat @ A9 @ A )
             => ( ord_le8976984241387448984_f_nat @ A9 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1028_Sup__fin_OboundedE,axiom,
    ! [A: set_se5570125628664547815term_f,X3: set_Pr989862937836626183term_f] :
      ( ( finite5058142709724841776term_f @ A )
     => ( ( A != bot_bo5021337078593797203term_f )
       => ( ( ord_le263819222746101927term_f @ ( lattic4254796103941830846term_f @ A ) @ X3 )
         => ! [A9: set_Pr989862937836626183term_f] :
              ( ( member6919674941555055664term_f @ A9 @ A )
             => ( ord_le263819222746101927term_f @ A9 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1029_Sup__fin_OboundedE,axiom,
    ! [A: set_fs7237271039932420712_f_nat,X3: fset_P6228066233360383026_f_nat] :
      ( ( finite910431038387235657_f_nat @ A )
     => ( ( A != bot_bo2133226978013888252_f_nat )
       => ( ( ord_le1552505484586773650_f_nat @ ( lattic6606183005006063419_f_nat @ A ) @ X3 )
         => ! [A9: fset_P6228066233360383026_f_nat] :
              ( ( member6677119207206147657_f_nat @ A9 @ A )
             => ( ord_le1552505484586773650_f_nat @ A9 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1030_Sup__fin_OboundedE,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X3 )
         => ! [A9: nat] :
              ( ( member_nat @ A9 @ A )
             => ( ord_less_eq_nat @ A9 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1031_Sup__fin_OboundedE,axiom,
    ! [A: set_set_nat,X3: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X3 )
         => ! [A9: set_nat] :
              ( ( member_set_nat @ A9 @ A )
             => ( ord_less_eq_set_nat @ A9 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1032_Sup__fin_OboundedE,axiom,
    ! [A: set_se421094335941524212term_f,X3: set_Ground_gterm_f] :
      ( ( finite6824738741894949053term_f @ A )
     => ( ( A != bot_bo6304853979754439520term_f )
       => ( ( ord_le2735537439747282356term_f @ ( lattic2636499749083849035term_f @ A ) @ X3 )
         => ! [A9: set_Ground_gterm_f] :
              ( ( member9219338147399679421term_f @ A9 @ A )
             => ( ord_le2735537439747282356term_f @ A9 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1033_Restr__simps_I4_J,axiom,
    ! [R2: set_Pr989862937836626183term_f,X6: set_Ground_gterm_f,S: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ R2
        @ ( produc6043514909933191368term_f @ X6
          @ ^ [Uu: ground_gterm_f] : X6 ) )
     => ( ( ord_le263819222746101927term_f @ S
          @ ( produc6043514909933191368term_f @ X6
            @ ^ [Uu: ground_gterm_f] : X6 ) )
       => ( ( inf_in6246359444536921845term_f @ ( relcom4354988593214781939term_f @ R2 @ S )
            @ ( produc6043514909933191368term_f @ X6
              @ ^ [Uu: ground_gterm_f] : X6 ) )
          = ( relcom4354988593214781939term_f @ R2 @ S ) ) ) ) ).

% Restr_simps(4)
thf(fact_1034_Max__mono,axiom,
    ! [M2: set_Pr5245412377734449720_f_nat,N: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ M2 @ N )
     => ( ( M2 != bot_bo136713897782694092_f_nat )
       => ( ( finite3598747971073327385_f_nat @ N )
         => ( ord_le2234929214960958178_f_nat @ ( lattic2060086870095978963_f_nat @ M2 ) @ ( lattic2060086870095978963_f_nat @ N ) ) ) ) ) ).

% Max_mono
thf(fact_1035_Max__mono,axiom,
    ! [M2: set_nat,N: set_nat] :
      ( ( ord_less_eq_set_nat @ M2 @ N )
     => ( ( M2 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ N )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M2 ) @ ( lattic8265883725875713057ax_nat @ N ) ) ) ) ) ).

% Max_mono
thf(fact_1036_inf_Obounded__iff,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ ( inf_in4124010953442585290_f_nat @ B3 @ C2 ) )
      = ( ( ord_le8976984241387448984_f_nat @ A6 @ B3 )
        & ( ord_le8976984241387448984_f_nat @ A6 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1037_inf_Obounded__iff,axiom,
    ! [A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f,C2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ ( inf_in6246359444536921845term_f @ B3 @ C2 ) )
      = ( ( ord_le263819222746101927term_f @ A6 @ B3 )
        & ( ord_le263819222746101927term_f @ A6 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1038_inf_Obounded__iff,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ ( inf_in6532419608991314884_f_nat @ B3 @ C2 ) )
      = ( ( ord_le1552505484586773650_f_nat @ A6 @ B3 )
        & ( ord_le1552505484586773650_f_nat @ A6 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1039_inf_Obounded__iff,axiom,
    ! [A6: nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A6 @ ( inf_inf_nat @ B3 @ C2 ) )
      = ( ( ord_less_eq_nat @ A6 @ B3 )
        & ( ord_less_eq_nat @ A6 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1040_inf_Obounded__iff,axiom,
    ! [A6: set_nat,B3: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ ( inf_inf_set_nat @ B3 @ C2 ) )
      = ( ( ord_less_eq_set_nat @ A6 @ B3 )
        & ( ord_less_eq_set_nat @ A6 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1041_inf_Obounded__iff,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f,C2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ ( inf_in2342200634362942210term_f @ B3 @ C2 ) )
      = ( ( ord_le2735537439747282356term_f @ A6 @ B3 )
        & ( ord_le2735537439747282356term_f @ A6 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1042_le__inf__iff,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ Y @ Z2 ) )
      = ( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
        & ( ord_le8976984241387448984_f_nat @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_1043_le__inf__iff,axiom,
    ! [X3: set_Pr989862937836626183term_f,Y: set_Pr989862937836626183term_f,Z2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ X3 @ ( inf_in6246359444536921845term_f @ Y @ Z2 ) )
      = ( ( ord_le263819222746101927term_f @ X3 @ Y )
        & ( ord_le263819222746101927term_f @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_1044_le__inf__iff,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat,Z2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X3 @ ( inf_in6532419608991314884_f_nat @ Y @ Z2 ) )
      = ( ( ord_le1552505484586773650_f_nat @ X3 @ Y )
        & ( ord_le1552505484586773650_f_nat @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_1045_le__inf__iff,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) )
      = ( ( ord_less_eq_nat @ X3 @ Y )
        & ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_1046_le__inf__iff,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
      = ( ( ord_less_eq_set_nat @ X3 @ Y )
        & ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_1047_le__inf__iff,axiom,
    ! [X3: set_Ground_gterm_f,Y: set_Ground_gterm_f,Z2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ X3 @ ( inf_in2342200634362942210term_f @ Y @ Z2 ) )
      = ( ( ord_le2735537439747282356term_f @ X3 @ Y )
        & ( ord_le2735537439747282356term_f @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_1048_map__eq__set__zipD,axiom,
    ! [F3: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,Xs: list_f1824981274722084755rm_f_v,Ys: list_f1824981274722084755rm_f_v,X3: fset_P8018961893305114765rm_f_v,Y: fset_P8018961893305114765rm_f_v] :
      ( ( ( map_fs8602507653405230974rm_f_v @ F3 @ Xs )
        = ( map_fs8602507653405230974rm_f_v @ F3 @ Ys ) )
     => ( ( member4344164543411788176rm_f_v @ ( produc989898760166608087rm_f_v @ X3 @ Y ) @ ( set_Pr7525886052349274108rm_f_v @ ( zip_fs8516469554775540727rm_f_v @ Xs @ Ys ) ) )
       => ( ( F3 @ X3 )
          = ( F3 @ Y ) ) ) ) ).

% map_eq_set_zipD
thf(fact_1049_Restr__gnrrstepD__dist,axiom,
    ! [F2: set_Pr5245412377734449720_f_nat,R5: set_Pr989862937836626183term_f,G: set_Pr5245412377734449720_f_nat] :
      ( ( inf_in6246359444536921845term_f @ ( lift_R6632485188928184278tepD_f @ F2 @ R5 )
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ G )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ G ) ) )
      = ( lift_R6632485188928184278tepD_f @ ( inf_in4124010953442585290_f_nat @ F2 @ G )
        @ ( inf_in6246359444536921845term_f @ R5
          @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ G )
            @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ G ) ) ) ) ) ).

% Restr_gnrrstepD_dist
thf(fact_1050_Restr__gpar__rstepD__dist,axiom,
    ! [F2: set_Pr5245412377734449720_f_nat,R5: set_Pr989862937836626183term_f,G: set_Pr5245412377734449720_f_nat] :
      ( ( inf_in6246359444536921845term_f @ ( lift_R6706595793745066653tepD_f @ F2 @ R5 )
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ G )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ G ) ) )
      = ( lift_R6706595793745066653tepD_f @ ( inf_in4124010953442585290_f_nat @ F2 @ G )
        @ ( inf_in6246359444536921845term_f @ R5
          @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ G )
            @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ G ) ) ) ) ) ).

% Restr_gpar_rstepD_dist
thf(fact_1051_Restr__subset,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat,R4: set_Pr516226060960033221_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B2 )
     => ( ( inf_in7383377402786884787_f_nat
          @ ( inf_in7383377402786884787_f_nat @ R4
            @ ( produc7667213020494768334_f_nat @ B2
              @ ^ [Uu: product_prod_f_nat] : B2 ) )
          @ ( produc7667213020494768334_f_nat @ A
            @ ^ [Uu: product_prod_f_nat] : A ) )
        = ( inf_in7383377402786884787_f_nat @ R4
          @ ( produc7667213020494768334_f_nat @ A
            @ ^ [Uu: product_prod_f_nat] : A ) ) ) ) ).

% Restr_subset
thf(fact_1052_Restr__subset,axiom,
    ! [A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f,R4: set_Pr1766941801908210887term_f] :
      ( ( ord_le263819222746101927term_f @ A @ B2 )
     => ( ( inf_in6212978850389331125term_f
          @ ( inf_in6212978850389331125term_f @ R4
            @ ( produc4509621232857779464term_f @ B2
              @ ^ [Uu: produc7245736746747425831term_f] : B2 ) )
          @ ( produc4509621232857779464term_f @ A
            @ ^ [Uu: produc7245736746747425831term_f] : A ) )
        = ( inf_in6212978850389331125term_f @ R4
          @ ( produc4509621232857779464term_f @ A
            @ ^ [Uu: produc7245736746747425831term_f] : A ) ) ) ) ).

% Restr_subset
thf(fact_1053_Restr__subset,axiom,
    ! [A: set_nat,B2: set_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( inf_in2572325071724192079at_nat
          @ ( inf_in2572325071724192079at_nat @ R4
            @ ( produc457027306803732586at_nat @ B2
              @ ^ [Uu: nat] : B2 ) )
          @ ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : A ) )
        = ( inf_in2572325071724192079at_nat @ R4
          @ ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : A ) ) ) ) ).

% Restr_subset
thf(fact_1054_Restr__subset,axiom,
    ! [A: set_Ground_gterm_f,B2: set_Ground_gterm_f,R4: set_Pr989862937836626183term_f] :
      ( ( ord_le2735537439747282356term_f @ A @ B2 )
     => ( ( inf_in6246359444536921845term_f
          @ ( inf_in6246359444536921845term_f @ R4
            @ ( produc6043514909933191368term_f @ B2
              @ ^ [Uu: ground_gterm_f] : B2 ) )
          @ ( produc6043514909933191368term_f @ A
            @ ^ [Uu: ground_gterm_f] : A ) )
        = ( inf_in6246359444536921845term_f @ R4
          @ ( produc6043514909933191368term_f @ A
            @ ^ [Uu: ground_gterm_f] : A ) ) ) ) ).

% Restr_subset
thf(fact_1055_IntI,axiom,
    ! [C2: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ C2 @ A )
     => ( ( member848276444142703440term_f @ C2 @ B2 )
       => ( member848276444142703440term_f @ C2 @ ( inf_in6246359444536921845term_f @ A @ B2 ) ) ) ) ).

% IntI
thf(fact_1056_IntI,axiom,
    ! [C2: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat @ C2 @ A )
     => ( ( member_nat @ C2 @ B2 )
       => ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).

% IntI
thf(fact_1057_Int__iff,axiom,
    ! [C2: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ C2 @ ( inf_in6246359444536921845term_f @ A @ B2 ) )
      = ( ( member848276444142703440term_f @ C2 @ A )
        & ( member848276444142703440term_f @ C2 @ B2 ) ) ) ).

% Int_iff
thf(fact_1058_Int__iff,axiom,
    ! [C2: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
      = ( ( member_nat @ C2 @ A )
        & ( member_nat @ C2 @ B2 ) ) ) ).

% Int_iff
thf(fact_1059_inf__Int__eq,axiom,
    ! [R2: set_Pr989862937836626183term_f,S: set_Pr989862937836626183term_f] :
      ( ( inf_in1406940722187435560rm_f_o
        @ ^ [X: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X @ R2 )
        @ ^ [X: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X @ S ) )
      = ( ^ [X: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X @ ( inf_in6246359444536921845term_f @ R2 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1060_inf__Int__eq,axiom,
    ! [R2: set_nat,S: set_nat] :
      ( ( inf_inf_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R2 )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( inf_inf_set_nat @ R2 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1061_IntE,axiom,
    ! [C2: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ C2 @ ( inf_in6246359444536921845term_f @ A @ B2 ) )
     => ~ ( ( member848276444142703440term_f @ C2 @ A )
         => ~ ( member848276444142703440term_f @ C2 @ B2 ) ) ) ).

% IntE
thf(fact_1062_IntE,axiom,
    ! [C2: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
     => ~ ( ( member_nat @ C2 @ A )
         => ~ ( member_nat @ C2 @ B2 ) ) ) ).

% IntE
thf(fact_1063_IntD1,axiom,
    ! [C2: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ C2 @ ( inf_in6246359444536921845term_f @ A @ B2 ) )
     => ( member848276444142703440term_f @ C2 @ A ) ) ).

% IntD1
thf(fact_1064_IntD1,axiom,
    ! [C2: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
     => ( member_nat @ C2 @ A ) ) ).

% IntD1
thf(fact_1065_IntD2,axiom,
    ! [C2: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f,B2: set_Pr989862937836626183term_f] :
      ( ( member848276444142703440term_f @ C2 @ ( inf_in6246359444536921845term_f @ A @ B2 ) )
     => ( member848276444142703440term_f @ C2 @ B2 ) ) ).

% IntD2
thf(fact_1066_IntD2,axiom,
    ! [C2: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
     => ( member_nat @ C2 @ B2 ) ) ).

% IntD2
thf(fact_1067_inf__set__def,axiom,
    ( inf_in6246359444536921845term_f
    = ( ^ [A4: set_Pr989862937836626183term_f,B: set_Pr989862937836626183term_f] :
          ( collec3209569126566747026term_f
          @ ( inf_in1406940722187435560rm_f_o
            @ ^ [X: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X @ A4 )
            @ ^ [X: produc7245736746747425831term_f] : ( member848276444142703440term_f @ X @ B ) ) ) ) ) ).

% inf_set_def
thf(fact_1068_inf__set__def,axiom,
    ( inf_in2342200634362942210term_f
    = ( ^ [A4: set_Ground_gterm_f,B: set_Ground_gterm_f] :
          ( collec3522850625759001119term_f
          @ ( inf_in3863720841383968859rm_f_o
            @ ^ [X: ground_gterm_f] : ( member5261315044688711901term_f @ X @ A4 )
            @ ^ [X: ground_gterm_f] : ( member5261315044688711901term_f @ X @ B ) ) ) ) ) ).

% inf_set_def
thf(fact_1069_inf__set__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A4: set_nat,B: set_nat] :
          ( collect_nat
          @ ( inf_inf_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A4 )
            @ ^ [X: nat] : ( member_nat @ X @ B ) ) ) ) ) ).

% inf_set_def
thf(fact_1070_inf__set__def,axiom,
    ( inf_in2982042037914615765term_f
    = ( ^ [A4: set_se5570125628664547815term_f,B: set_se5570125628664547815term_f] :
          ( collec4490547877585280114term_f
          @ ( inf_in6097971200903918920rm_f_o
            @ ^ [X: set_Pr989862937836626183term_f] : ( member6919674941555055664term_f @ X @ A4 )
            @ ^ [X: set_Pr989862937836626183term_f] : ( member6919674941555055664term_f @ X @ B ) ) ) ) ) ).

% inf_set_def
thf(fact_1071_inf__set__def,axiom,
    ( inf_in5161999638925519744_f_nat
    = ( ^ [A4: set_se1451291994713068270_f_nat,B: set_se1451291994713068270_f_nat] :
          ( collec6964094325902095629_f_nat
          @ ( inf_in825707178096620805_nat_o
            @ ^ [X: set_Pr5245412377734449720_f_nat] : ( member8345558847462883279_f_nat @ X @ A4 )
            @ ^ [X: set_Pr5245412377734449720_f_nat] : ( member8345558847462883279_f_nat @ X @ B ) ) ) ) ) ).

% inf_set_def
thf(fact_1072_inf__set__def,axiom,
    ( inf_inf_set_set_nat
    = ( ^ [A4: set_set_nat,B: set_set_nat] :
          ( collect_set_nat
          @ ( inf_inf_set_nat_o
            @ ^ [X: set_nat] : ( member_set_nat @ X @ A4 )
            @ ^ [X: set_nat] : ( member_set_nat @ X @ B ) ) ) ) ) ).

% inf_set_def
thf(fact_1073_inf__set__def,axiom,
    ( inf_in6321094595705628642term_f
    = ( ^ [A4: set_se421094335941524212term_f,B: set_se421094335941524212term_f] :
          ( collec1979342819066322687term_f
          @ ( inf_in7734760871071142779rm_f_o
            @ ^ [X: set_Ground_gterm_f] : ( member9219338147399679421term_f @ X @ A4 )
            @ ^ [X: set_Ground_gterm_f] : ( member9219338147399679421term_f @ X @ B ) ) ) ) ) ).

% inf_set_def
thf(fact_1074_Int__def,axiom,
    ( inf_in6246359444536921845term_f
    = ( ^ [A4: set_Pr989862937836626183term_f,B: set_Pr989862937836626183term_f] :
          ( collec3209569126566747026term_f
          @ ^ [X: produc7245736746747425831term_f] :
              ( ( member848276444142703440term_f @ X @ A4 )
              & ( member848276444142703440term_f @ X @ B ) ) ) ) ) ).

% Int_def
thf(fact_1075_Int__def,axiom,
    ( inf_in2342200634362942210term_f
    = ( ^ [A4: set_Ground_gterm_f,B: set_Ground_gterm_f] :
          ( collec3522850625759001119term_f
          @ ^ [X: ground_gterm_f] :
              ( ( member5261315044688711901term_f @ X @ A4 )
              & ( member5261315044688711901term_f @ X @ B ) ) ) ) ) ).

% Int_def
thf(fact_1076_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A4: set_nat,B: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A4 )
              & ( member_nat @ X @ B ) ) ) ) ) ).

% Int_def
thf(fact_1077_Int__def,axiom,
    ( inf_in2982042037914615765term_f
    = ( ^ [A4: set_se5570125628664547815term_f,B: set_se5570125628664547815term_f] :
          ( collec4490547877585280114term_f
          @ ^ [X: set_Pr989862937836626183term_f] :
              ( ( member6919674941555055664term_f @ X @ A4 )
              & ( member6919674941555055664term_f @ X @ B ) ) ) ) ) ).

% Int_def
thf(fact_1078_Int__def,axiom,
    ( inf_in5161999638925519744_f_nat
    = ( ^ [A4: set_se1451291994713068270_f_nat,B: set_se1451291994713068270_f_nat] :
          ( collec6964094325902095629_f_nat
          @ ^ [X: set_Pr5245412377734449720_f_nat] :
              ( ( member8345558847462883279_f_nat @ X @ A4 )
              & ( member8345558847462883279_f_nat @ X @ B ) ) ) ) ) ).

% Int_def
thf(fact_1079_Int__def,axiom,
    ( inf_inf_set_set_nat
    = ( ^ [A4: set_set_nat,B: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( member_set_nat @ X @ A4 )
              & ( member_set_nat @ X @ B ) ) ) ) ) ).

% Int_def
thf(fact_1080_Int__def,axiom,
    ( inf_in6321094595705628642term_f
    = ( ^ [A4: set_se421094335941524212term_f,B: set_se421094335941524212term_f] :
          ( collec1979342819066322687term_f
          @ ^ [X: set_Ground_gterm_f] :
              ( ( member9219338147399679421term_f @ X @ A4 )
              & ( member9219338147399679421term_f @ X @ B ) ) ) ) ) ).

% Int_def
thf(fact_1081_Int__Collect,axiom,
    ! [X3: produc7245736746747425831term_f,A: set_Pr989862937836626183term_f,P: produc7245736746747425831term_f > $o] :
      ( ( member848276444142703440term_f @ X3 @ ( inf_in6246359444536921845term_f @ A @ ( collec3209569126566747026term_f @ P ) ) )
      = ( ( member848276444142703440term_f @ X3 @ A )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1082_Int__Collect,axiom,
    ! [X3: ground_gterm_f,A: set_Ground_gterm_f,P: ground_gterm_f > $o] :
      ( ( member5261315044688711901term_f @ X3 @ ( inf_in2342200634362942210term_f @ A @ ( collec3522850625759001119term_f @ P ) ) )
      = ( ( member5261315044688711901term_f @ X3 @ A )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1083_Int__Collect,axiom,
    ! [X3: nat,A: set_nat,P: nat > $o] :
      ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
      = ( ( member_nat @ X3 @ A )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1084_Int__Collect,axiom,
    ! [X3: set_Pr989862937836626183term_f,A: set_se5570125628664547815term_f,P: set_Pr989862937836626183term_f > $o] :
      ( ( member6919674941555055664term_f @ X3 @ ( inf_in2982042037914615765term_f @ A @ ( collec4490547877585280114term_f @ P ) ) )
      = ( ( member6919674941555055664term_f @ X3 @ A )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1085_Int__Collect,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,A: set_se1451291994713068270_f_nat,P: set_Pr5245412377734449720_f_nat > $o] :
      ( ( member8345558847462883279_f_nat @ X3 @ ( inf_in5161999638925519744_f_nat @ A @ ( collec6964094325902095629_f_nat @ P ) ) )
      = ( ( member8345558847462883279_f_nat @ X3 @ A )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1086_Int__Collect,axiom,
    ! [X3: set_nat,A: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X3 @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P ) ) )
      = ( ( member_set_nat @ X3 @ A )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1087_Int__Collect,axiom,
    ! [X3: set_Ground_gterm_f,A: set_se421094335941524212term_f,P: set_Ground_gterm_f > $o] :
      ( ( member9219338147399679421term_f @ X3 @ ( inf_in6321094595705628642term_f @ A @ ( collec1979342819066322687term_f @ P ) ) )
      = ( ( member9219338147399679421term_f @ X3 @ A )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1088_Collect__conj__eq,axiom,
    ! [P: ground_gterm_f > $o,Q: ground_gterm_f > $o] :
      ( ( collec3522850625759001119term_f
        @ ^ [X: ground_gterm_f] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_in2342200634362942210term_f @ ( collec3522850625759001119term_f @ P ) @ ( collec3522850625759001119term_f @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1089_Collect__conj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1090_Collect__conj__eq,axiom,
    ! [P: set_Pr989862937836626183term_f > $o,Q: set_Pr989862937836626183term_f > $o] :
      ( ( collec4490547877585280114term_f
        @ ^ [X: set_Pr989862937836626183term_f] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_in2982042037914615765term_f @ ( collec4490547877585280114term_f @ P ) @ ( collec4490547877585280114term_f @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1091_Collect__conj__eq,axiom,
    ! [P: set_Pr5245412377734449720_f_nat > $o,Q: set_Pr5245412377734449720_f_nat > $o] :
      ( ( collec6964094325902095629_f_nat
        @ ^ [X: set_Pr5245412377734449720_f_nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_in5161999638925519744_f_nat @ ( collec6964094325902095629_f_nat @ P ) @ ( collec6964094325902095629_f_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1092_Collect__conj__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1093_Collect__conj__eq,axiom,
    ! [P: set_Ground_gterm_f > $o,Q: set_Ground_gterm_f > $o] :
      ( ( collec1979342819066322687term_f
        @ ^ [X: set_Ground_gterm_f] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_in6321094595705628642term_f @ ( collec1979342819066322687term_f @ P ) @ ( collec1979342819066322687term_f @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1094_inf_OcoboundedI2,axiom,
    ! [B3: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat,A6: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B3 @ C2 )
     => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A6 @ B3 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_1095_inf_OcoboundedI2,axiom,
    ! [B3: set_Pr989862937836626183term_f,C2: set_Pr989862937836626183term_f,A6: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ B3 @ C2 )
     => ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ A6 @ B3 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_1096_inf_OcoboundedI2,axiom,
    ! [B3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat,A6: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B3 @ C2 )
     => ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A6 @ B3 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_1097_inf_OcoboundedI2,axiom,
    ! [B3: nat,C2: nat,A6: nat] :
      ( ( ord_less_eq_nat @ B3 @ C2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A6 @ B3 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_1098_inf_OcoboundedI2,axiom,
    ! [B3: set_nat,C2: set_nat,A6: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ C2 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A6 @ B3 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_1099_inf_OcoboundedI2,axiom,
    ! [B3: set_Ground_gterm_f,C2: set_Ground_gterm_f,A6: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ B3 @ C2 )
     => ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ A6 @ B3 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_1100_inf_OcoboundedI1,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ C2 )
     => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A6 @ B3 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_1101_inf_OcoboundedI1,axiom,
    ! [A6: set_Pr989862937836626183term_f,C2: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ C2 )
     => ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ A6 @ B3 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_1102_inf_OcoboundedI1,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ C2 )
     => ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A6 @ B3 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_1103_inf_OcoboundedI1,axiom,
    ! [A6: nat,C2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A6 @ C2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A6 @ B3 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_1104_inf_OcoboundedI1,axiom,
    ! [A6: set_nat,C2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ C2 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A6 @ B3 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_1105_inf_OcoboundedI1,axiom,
    ! [A6: set_Ground_gterm_f,C2: set_Ground_gterm_f,B3: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ C2 )
     => ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ A6 @ B3 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_1106_inf_Oabsorb__iff2,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [B6: set_Pr5245412377734449720_f_nat,A5: set_Pr5245412377734449720_f_nat] :
          ( ( inf_in4124010953442585290_f_nat @ A5 @ B6 )
          = B6 ) ) ) ).

% inf.absorb_iff2
thf(fact_1107_inf_Oabsorb__iff2,axiom,
    ( ord_le263819222746101927term_f
    = ( ^ [B6: set_Pr989862937836626183term_f,A5: set_Pr989862937836626183term_f] :
          ( ( inf_in6246359444536921845term_f @ A5 @ B6 )
          = B6 ) ) ) ).

% inf.absorb_iff2
thf(fact_1108_inf_Oabsorb__iff2,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [B6: fset_P6228066233360383026_f_nat,A5: fset_P6228066233360383026_f_nat] :
          ( ( inf_in6532419608991314884_f_nat @ A5 @ B6 )
          = B6 ) ) ) ).

% inf.absorb_iff2
thf(fact_1109_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B6: nat,A5: nat] :
          ( ( inf_inf_nat @ A5 @ B6 )
          = B6 ) ) ) ).

% inf.absorb_iff2
thf(fact_1110_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B6: set_nat,A5: set_nat] :
          ( ( inf_inf_set_nat @ A5 @ B6 )
          = B6 ) ) ) ).

% inf.absorb_iff2
thf(fact_1111_inf_Oabsorb__iff2,axiom,
    ( ord_le2735537439747282356term_f
    = ( ^ [B6: set_Ground_gterm_f,A5: set_Ground_gterm_f] :
          ( ( inf_in2342200634362942210term_f @ A5 @ B6 )
          = B6 ) ) ) ).

% inf.absorb_iff2
thf(fact_1112_inf_Oabsorb__iff1,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B6: set_Pr5245412377734449720_f_nat] :
          ( ( inf_in4124010953442585290_f_nat @ A5 @ B6 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_1113_inf_Oabsorb__iff1,axiom,
    ( ord_le263819222746101927term_f
    = ( ^ [A5: set_Pr989862937836626183term_f,B6: set_Pr989862937836626183term_f] :
          ( ( inf_in6246359444536921845term_f @ A5 @ B6 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_1114_inf_Oabsorb__iff1,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [A5: fset_P6228066233360383026_f_nat,B6: fset_P6228066233360383026_f_nat] :
          ( ( inf_in6532419608991314884_f_nat @ A5 @ B6 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_1115_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B6: nat] :
          ( ( inf_inf_nat @ A5 @ B6 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_1116_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B6: set_nat] :
          ( ( inf_inf_set_nat @ A5 @ B6 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_1117_inf_Oabsorb__iff1,axiom,
    ( ord_le2735537439747282356term_f
    = ( ^ [A5: set_Ground_gterm_f,B6: set_Ground_gterm_f] :
          ( ( inf_in2342200634362942210term_f @ A5 @ B6 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_1118_inf_Ocobounded2,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A6 @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_1119_inf_Ocobounded2,axiom,
    ! [A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ A6 @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_1120_inf_Ocobounded2,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A6 @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_1121_inf_Ocobounded2,axiom,
    ! [A6: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A6 @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_1122_inf_Ocobounded2,axiom,
    ! [A6: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A6 @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_1123_inf_Ocobounded2,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ A6 @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_1124_inf_Ocobounded1,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A6 @ B3 ) @ A6 ) ).

% inf.cobounded1
thf(fact_1125_inf_Ocobounded1,axiom,
    ! [A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ A6 @ B3 ) @ A6 ) ).

% inf.cobounded1
thf(fact_1126_inf_Ocobounded1,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A6 @ B3 ) @ A6 ) ).

% inf.cobounded1
thf(fact_1127_inf_Ocobounded1,axiom,
    ! [A6: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A6 @ B3 ) @ A6 ) ).

% inf.cobounded1
thf(fact_1128_inf_Ocobounded1,axiom,
    ! [A6: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A6 @ B3 ) @ A6 ) ).

% inf.cobounded1
thf(fact_1129_inf_Ocobounded1,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ A6 @ B3 ) @ A6 ) ).

% inf.cobounded1
thf(fact_1130_inf_Oorder__iff,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B6: set_Pr5245412377734449720_f_nat] :
          ( A5
          = ( inf_in4124010953442585290_f_nat @ A5 @ B6 ) ) ) ) ).

% inf.order_iff
thf(fact_1131_inf_Oorder__iff,axiom,
    ( ord_le263819222746101927term_f
    = ( ^ [A5: set_Pr989862937836626183term_f,B6: set_Pr989862937836626183term_f] :
          ( A5
          = ( inf_in6246359444536921845term_f @ A5 @ B6 ) ) ) ) ).

% inf.order_iff
thf(fact_1132_inf_Oorder__iff,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [A5: fset_P6228066233360383026_f_nat,B6: fset_P6228066233360383026_f_nat] :
          ( A5
          = ( inf_in6532419608991314884_f_nat @ A5 @ B6 ) ) ) ) ).

% inf.order_iff
thf(fact_1133_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B6: nat] :
          ( A5
          = ( inf_inf_nat @ A5 @ B6 ) ) ) ) ).

% inf.order_iff
thf(fact_1134_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B6: set_nat] :
          ( A5
          = ( inf_inf_set_nat @ A5 @ B6 ) ) ) ) ).

% inf.order_iff
thf(fact_1135_inf_Oorder__iff,axiom,
    ( ord_le2735537439747282356term_f
    = ( ^ [A5: set_Ground_gterm_f,B6: set_Ground_gterm_f] :
          ( A5
          = ( inf_in2342200634362942210term_f @ A5 @ B6 ) ) ) ) ).

% inf.order_iff
thf(fact_1136_inf__greatest,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
     => ( ( ord_le8976984241387448984_f_nat @ X3 @ Z2 )
       => ( ord_le8976984241387448984_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_1137_inf__greatest,axiom,
    ! [X3: set_Pr989862937836626183term_f,Y: set_Pr989862937836626183term_f,Z2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ X3 @ Y )
     => ( ( ord_le263819222746101927term_f @ X3 @ Z2 )
       => ( ord_le263819222746101927term_f @ X3 @ ( inf_in6246359444536921845term_f @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_1138_inf__greatest,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat,Z2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X3 @ Y )
     => ( ( ord_le1552505484586773650_f_nat @ X3 @ Z2 )
       => ( ord_le1552505484586773650_f_nat @ X3 @ ( inf_in6532419608991314884_f_nat @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_1139_inf__greatest,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ X3 @ Z2 )
       => ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_1140_inf__greatest,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( ord_less_eq_set_nat @ X3 @ Z2 )
       => ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_1141_inf__greatest,axiom,
    ! [X3: set_Ground_gterm_f,Y: set_Ground_gterm_f,Z2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ X3 @ Y )
     => ( ( ord_le2735537439747282356term_f @ X3 @ Z2 )
       => ( ord_le2735537439747282356term_f @ X3 @ ( inf_in2342200634362942210term_f @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_1142_inf_OboundedI,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ B3 )
     => ( ( ord_le8976984241387448984_f_nat @ A6 @ C2 )
       => ( ord_le8976984241387448984_f_nat @ A6 @ ( inf_in4124010953442585290_f_nat @ B3 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_1143_inf_OboundedI,axiom,
    ! [A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f,C2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ B3 )
     => ( ( ord_le263819222746101927term_f @ A6 @ C2 )
       => ( ord_le263819222746101927term_f @ A6 @ ( inf_in6246359444536921845term_f @ B3 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_1144_inf_OboundedI,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ B3 )
     => ( ( ord_le1552505484586773650_f_nat @ A6 @ C2 )
       => ( ord_le1552505484586773650_f_nat @ A6 @ ( inf_in6532419608991314884_f_nat @ B3 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_1145_inf_OboundedI,axiom,
    ! [A6: nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( ord_less_eq_nat @ A6 @ C2 )
       => ( ord_less_eq_nat @ A6 @ ( inf_inf_nat @ B3 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_1146_inf_OboundedI,axiom,
    ! [A6: set_nat,B3: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ B3 )
     => ( ( ord_less_eq_set_nat @ A6 @ C2 )
       => ( ord_less_eq_set_nat @ A6 @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_1147_inf_OboundedI,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f,C2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ B3 )
     => ( ( ord_le2735537439747282356term_f @ A6 @ C2 )
       => ( ord_le2735537439747282356term_f @ A6 @ ( inf_in2342200634362942210term_f @ B3 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_1148_inf_OboundedE,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ ( inf_in4124010953442585290_f_nat @ B3 @ C2 ) )
     => ~ ( ( ord_le8976984241387448984_f_nat @ A6 @ B3 )
         => ~ ( ord_le8976984241387448984_f_nat @ A6 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_1149_inf_OboundedE,axiom,
    ! [A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f,C2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ ( inf_in6246359444536921845term_f @ B3 @ C2 ) )
     => ~ ( ( ord_le263819222746101927term_f @ A6 @ B3 )
         => ~ ( ord_le263819222746101927term_f @ A6 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_1150_inf_OboundedE,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ ( inf_in6532419608991314884_f_nat @ B3 @ C2 ) )
     => ~ ( ( ord_le1552505484586773650_f_nat @ A6 @ B3 )
         => ~ ( ord_le1552505484586773650_f_nat @ A6 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_1151_inf_OboundedE,axiom,
    ! [A6: nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A6 @ ( inf_inf_nat @ B3 @ C2 ) )
     => ~ ( ( ord_less_eq_nat @ A6 @ B3 )
         => ~ ( ord_less_eq_nat @ A6 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_1152_inf_OboundedE,axiom,
    ! [A6: set_nat,B3: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ ( inf_inf_set_nat @ B3 @ C2 ) )
     => ~ ( ( ord_less_eq_set_nat @ A6 @ B3 )
         => ~ ( ord_less_eq_set_nat @ A6 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_1153_inf_OboundedE,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f,C2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ ( inf_in2342200634362942210term_f @ B3 @ C2 ) )
     => ~ ( ( ord_le2735537439747282356term_f @ A6 @ B3 )
         => ~ ( ord_le2735537439747282356term_f @ A6 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_1154_inf__absorb2,axiom,
    ! [Y: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ Y @ X3 )
     => ( ( inf_in4124010953442585290_f_nat @ X3 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1155_inf__absorb2,axiom,
    ! [Y: set_Pr989862937836626183term_f,X3: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ Y @ X3 )
     => ( ( inf_in6246359444536921845term_f @ X3 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1156_inf__absorb2,axiom,
    ! [Y: fset_P6228066233360383026_f_nat,X3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ Y @ X3 )
     => ( ( inf_in6532419608991314884_f_nat @ X3 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1157_inf__absorb2,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ( ( inf_inf_nat @ X3 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1158_inf__absorb2,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X3 )
     => ( ( inf_inf_set_nat @ X3 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1159_inf__absorb2,axiom,
    ! [Y: set_Ground_gterm_f,X3: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ Y @ X3 )
     => ( ( inf_in2342200634362942210term_f @ X3 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1160_inf__absorb1,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
     => ( ( inf_in4124010953442585290_f_nat @ X3 @ Y )
        = X3 ) ) ).

% inf_absorb1
thf(fact_1161_inf__absorb1,axiom,
    ! [X3: set_Pr989862937836626183term_f,Y: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ X3 @ Y )
     => ( ( inf_in6246359444536921845term_f @ X3 @ Y )
        = X3 ) ) ).

% inf_absorb1
thf(fact_1162_inf__absorb1,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X3 @ Y )
     => ( ( inf_in6532419608991314884_f_nat @ X3 @ Y )
        = X3 ) ) ).

% inf_absorb1
thf(fact_1163_inf__absorb1,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( inf_inf_nat @ X3 @ Y )
        = X3 ) ) ).

% inf_absorb1
thf(fact_1164_inf__absorb1,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( inf_inf_set_nat @ X3 @ Y )
        = X3 ) ) ).

% inf_absorb1
thf(fact_1165_inf__absorb1,axiom,
    ! [X3: set_Ground_gterm_f,Y: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ X3 @ Y )
     => ( ( inf_in2342200634362942210term_f @ X3 @ Y )
        = X3 ) ) ).

% inf_absorb1
thf(fact_1166_inf_Oabsorb2,axiom,
    ! [B3: set_Pr5245412377734449720_f_nat,A6: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B3 @ A6 )
     => ( ( inf_in4124010953442585290_f_nat @ A6 @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_1167_inf_Oabsorb2,axiom,
    ! [B3: set_Pr989862937836626183term_f,A6: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ B3 @ A6 )
     => ( ( inf_in6246359444536921845term_f @ A6 @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_1168_inf_Oabsorb2,axiom,
    ! [B3: fset_P6228066233360383026_f_nat,A6: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B3 @ A6 )
     => ( ( inf_in6532419608991314884_f_nat @ A6 @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_1169_inf_Oabsorb2,axiom,
    ! [B3: nat,A6: nat] :
      ( ( ord_less_eq_nat @ B3 @ A6 )
     => ( ( inf_inf_nat @ A6 @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_1170_inf_Oabsorb2,axiom,
    ! [B3: set_nat,A6: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A6 )
     => ( ( inf_inf_set_nat @ A6 @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_1171_inf_Oabsorb2,axiom,
    ! [B3: set_Ground_gterm_f,A6: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ B3 @ A6 )
     => ( ( inf_in2342200634362942210term_f @ A6 @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_1172_inf_Oabsorb1,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ B3 )
     => ( ( inf_in4124010953442585290_f_nat @ A6 @ B3 )
        = A6 ) ) ).

% inf.absorb1
thf(fact_1173_inf_Oabsorb1,axiom,
    ! [A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ B3 )
     => ( ( inf_in6246359444536921845term_f @ A6 @ B3 )
        = A6 ) ) ).

% inf.absorb1
thf(fact_1174_inf_Oabsorb1,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ B3 )
     => ( ( inf_in6532419608991314884_f_nat @ A6 @ B3 )
        = A6 ) ) ).

% inf.absorb1
thf(fact_1175_inf_Oabsorb1,axiom,
    ! [A6: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( ( inf_inf_nat @ A6 @ B3 )
        = A6 ) ) ).

% inf.absorb1
thf(fact_1176_inf_Oabsorb1,axiom,
    ! [A6: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ B3 )
     => ( ( inf_inf_set_nat @ A6 @ B3 )
        = A6 ) ) ).

% inf.absorb1
thf(fact_1177_inf_Oabsorb1,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ B3 )
     => ( ( inf_in2342200634362942210term_f @ A6 @ B3 )
        = A6 ) ) ).

% inf.absorb1
thf(fact_1178_le__iff__inf,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [X: set_Pr5245412377734449720_f_nat,Y4: set_Pr5245412377734449720_f_nat] :
          ( ( inf_in4124010953442585290_f_nat @ X @ Y4 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_1179_le__iff__inf,axiom,
    ( ord_le263819222746101927term_f
    = ( ^ [X: set_Pr989862937836626183term_f,Y4: set_Pr989862937836626183term_f] :
          ( ( inf_in6246359444536921845term_f @ X @ Y4 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_1180_le__iff__inf,axiom,
    ( ord_le1552505484586773650_f_nat
    = ( ^ [X: fset_P6228066233360383026_f_nat,Y4: fset_P6228066233360383026_f_nat] :
          ( ( inf_in6532419608991314884_f_nat @ X @ Y4 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_1181_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( inf_inf_nat @ X @ Y4 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_1182_le__iff__inf,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X: set_nat,Y4: set_nat] :
          ( ( inf_inf_set_nat @ X @ Y4 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_1183_le__iff__inf,axiom,
    ( ord_le2735537439747282356term_f
    = ( ^ [X: set_Ground_gterm_f,Y4: set_Ground_gterm_f] :
          ( ( inf_in2342200634362942210term_f @ X @ Y4 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_1184_inf__unique,axiom,
    ! [F3: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
      ( ! [X2: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 @ Y3 ) @ X2 )
     => ( ! [X2: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( F3 @ X2 @ Y3 ) @ Y3 )
       => ( ! [X2: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat,Z3: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X2 @ Y3 )
             => ( ( ord_le8976984241387448984_f_nat @ X2 @ Z3 )
               => ( ord_le8976984241387448984_f_nat @ X2 @ ( F3 @ Y3 @ Z3 ) ) ) )
         => ( ( inf_in4124010953442585290_f_nat @ X3 @ Y )
            = ( F3 @ X3 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1185_inf__unique,axiom,
    ! [F3: set_Pr989862937836626183term_f > set_Pr989862937836626183term_f > set_Pr989862937836626183term_f,X3: set_Pr989862937836626183term_f,Y: set_Pr989862937836626183term_f] :
      ( ! [X2: set_Pr989862937836626183term_f,Y3: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ ( F3 @ X2 @ Y3 ) @ X2 )
     => ( ! [X2: set_Pr989862937836626183term_f,Y3: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ ( F3 @ X2 @ Y3 ) @ Y3 )
       => ( ! [X2: set_Pr989862937836626183term_f,Y3: set_Pr989862937836626183term_f,Z3: set_Pr989862937836626183term_f] :
              ( ( ord_le263819222746101927term_f @ X2 @ Y3 )
             => ( ( ord_le263819222746101927term_f @ X2 @ Z3 )
               => ( ord_le263819222746101927term_f @ X2 @ ( F3 @ Y3 @ Z3 ) ) ) )
         => ( ( inf_in6246359444536921845term_f @ X3 @ Y )
            = ( F3 @ X3 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1186_inf__unique,axiom,
    ! [F3: fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat,X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
      ( ! [X2: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( F3 @ X2 @ Y3 ) @ X2 )
     => ( ! [X2: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( F3 @ X2 @ Y3 ) @ Y3 )
       => ( ! [X2: fset_P6228066233360383026_f_nat,Y3: fset_P6228066233360383026_f_nat,Z3: fset_P6228066233360383026_f_nat] :
              ( ( ord_le1552505484586773650_f_nat @ X2 @ Y3 )
             => ( ( ord_le1552505484586773650_f_nat @ X2 @ Z3 )
               => ( ord_le1552505484586773650_f_nat @ X2 @ ( F3 @ Y3 @ Z3 ) ) ) )
         => ( ( inf_in6532419608991314884_f_nat @ X3 @ Y )
            = ( F3 @ X3 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1187_inf__unique,axiom,
    ! [F3: nat > nat > nat,X3: nat,Y: nat] :
      ( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( F3 @ X2 @ Y3 ) @ X2 )
     => ( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( F3 @ X2 @ Y3 ) @ Y3 )
       => ( ! [X2: nat,Y3: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ( ord_less_eq_nat @ X2 @ Z3 )
               => ( ord_less_eq_nat @ X2 @ ( F3 @ Y3 @ Z3 ) ) ) )
         => ( ( inf_inf_nat @ X3 @ Y )
            = ( F3 @ X3 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1188_inf__unique,axiom,
    ! [F3: set_nat > set_nat > set_nat,X3: set_nat,Y: set_nat] :
      ( ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F3 @ X2 @ Y3 ) @ X2 )
     => ( ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F3 @ X2 @ Y3 ) @ Y3 )
       => ( ! [X2: set_nat,Y3: set_nat,Z3: set_nat] :
              ( ( ord_less_eq_set_nat @ X2 @ Y3 )
             => ( ( ord_less_eq_set_nat @ X2 @ Z3 )
               => ( ord_less_eq_set_nat @ X2 @ ( F3 @ Y3 @ Z3 ) ) ) )
         => ( ( inf_inf_set_nat @ X3 @ Y )
            = ( F3 @ X3 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1189_inf__unique,axiom,
    ! [F3: set_Ground_gterm_f > set_Ground_gterm_f > set_Ground_gterm_f,X3: set_Ground_gterm_f,Y: set_Ground_gterm_f] :
      ( ! [X2: set_Ground_gterm_f,Y3: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ ( F3 @ X2 @ Y3 ) @ X2 )
     => ( ! [X2: set_Ground_gterm_f,Y3: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ ( F3 @ X2 @ Y3 ) @ Y3 )
       => ( ! [X2: set_Ground_gterm_f,Y3: set_Ground_gterm_f,Z3: set_Ground_gterm_f] :
              ( ( ord_le2735537439747282356term_f @ X2 @ Y3 )
             => ( ( ord_le2735537439747282356term_f @ X2 @ Z3 )
               => ( ord_le2735537439747282356term_f @ X2 @ ( F3 @ Y3 @ Z3 ) ) ) )
         => ( ( inf_in2342200634362942210term_f @ X3 @ Y )
            = ( F3 @ X3 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1190_inf_OorderI,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
      ( ( A6
        = ( inf_in4124010953442585290_f_nat @ A6 @ B3 ) )
     => ( ord_le8976984241387448984_f_nat @ A6 @ B3 ) ) ).

% inf.orderI
thf(fact_1191_inf_OorderI,axiom,
    ! [A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f] :
      ( ( A6
        = ( inf_in6246359444536921845term_f @ A6 @ B3 ) )
     => ( ord_le263819222746101927term_f @ A6 @ B3 ) ) ).

% inf.orderI
thf(fact_1192_inf_OorderI,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
      ( ( A6
        = ( inf_in6532419608991314884_f_nat @ A6 @ B3 ) )
     => ( ord_le1552505484586773650_f_nat @ A6 @ B3 ) ) ).

% inf.orderI
thf(fact_1193_inf_OorderI,axiom,
    ! [A6: nat,B3: nat] :
      ( ( A6
        = ( inf_inf_nat @ A6 @ B3 ) )
     => ( ord_less_eq_nat @ A6 @ B3 ) ) ).

% inf.orderI
thf(fact_1194_inf_OorderI,axiom,
    ! [A6: set_nat,B3: set_nat] :
      ( ( A6
        = ( inf_inf_set_nat @ A6 @ B3 ) )
     => ( ord_less_eq_set_nat @ A6 @ B3 ) ) ).

% inf.orderI
thf(fact_1195_inf_OorderI,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f] :
      ( ( A6
        = ( inf_in2342200634362942210term_f @ A6 @ B3 ) )
     => ( ord_le2735537439747282356term_f @ A6 @ B3 ) ) ).

% inf.orderI
thf(fact_1196_inf_OorderE,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ B3 )
     => ( A6
        = ( inf_in4124010953442585290_f_nat @ A6 @ B3 ) ) ) ).

% inf.orderE
thf(fact_1197_inf_OorderE,axiom,
    ! [A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ B3 )
     => ( A6
        = ( inf_in6246359444536921845term_f @ A6 @ B3 ) ) ) ).

% inf.orderE
thf(fact_1198_inf_OorderE,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ B3 )
     => ( A6
        = ( inf_in6532419608991314884_f_nat @ A6 @ B3 ) ) ) ).

% inf.orderE
thf(fact_1199_inf_OorderE,axiom,
    ! [A6: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A6 @ B3 )
     => ( A6
        = ( inf_inf_nat @ A6 @ B3 ) ) ) ).

% inf.orderE
thf(fact_1200_inf_OorderE,axiom,
    ! [A6: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ B3 )
     => ( A6
        = ( inf_inf_set_nat @ A6 @ B3 ) ) ) ).

% inf.orderE
thf(fact_1201_inf_OorderE,axiom,
    ! [A6: set_Ground_gterm_f,B3: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ B3 )
     => ( A6
        = ( inf_in2342200634362942210term_f @ A6 @ B3 ) ) ) ).

% inf.orderE
thf(fact_1202_le__infI2,axiom,
    ! [B3: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,A6: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B3 @ X3 )
     => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A6 @ B3 ) @ X3 ) ) ).

% le_infI2
thf(fact_1203_le__infI2,axiom,
    ! [B3: set_Pr989862937836626183term_f,X3: set_Pr989862937836626183term_f,A6: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ B3 @ X3 )
     => ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ A6 @ B3 ) @ X3 ) ) ).

% le_infI2
thf(fact_1204_le__infI2,axiom,
    ! [B3: fset_P6228066233360383026_f_nat,X3: fset_P6228066233360383026_f_nat,A6: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ B3 @ X3 )
     => ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A6 @ B3 ) @ X3 ) ) ).

% le_infI2
thf(fact_1205_le__infI2,axiom,
    ! [B3: nat,X3: nat,A6: nat] :
      ( ( ord_less_eq_nat @ B3 @ X3 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A6 @ B3 ) @ X3 ) ) ).

% le_infI2
thf(fact_1206_le__infI2,axiom,
    ! [B3: set_nat,X3: set_nat,A6: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ X3 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A6 @ B3 ) @ X3 ) ) ).

% le_infI2
thf(fact_1207_le__infI2,axiom,
    ! [B3: set_Ground_gterm_f,X3: set_Ground_gterm_f,A6: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ B3 @ X3 )
     => ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ A6 @ B3 ) @ X3 ) ) ).

% le_infI2
thf(fact_1208_le__infI1,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ X3 )
     => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A6 @ B3 ) @ X3 ) ) ).

% le_infI1
thf(fact_1209_le__infI1,axiom,
    ! [A6: set_Pr989862937836626183term_f,X3: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ X3 )
     => ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ A6 @ B3 ) @ X3 ) ) ).

% le_infI1
thf(fact_1210_le__infI1,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,X3: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ X3 )
     => ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A6 @ B3 ) @ X3 ) ) ).

% le_infI1
thf(fact_1211_le__infI1,axiom,
    ! [A6: nat,X3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A6 @ X3 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A6 @ B3 ) @ X3 ) ) ).

% le_infI1
thf(fact_1212_le__infI1,axiom,
    ! [A6: set_nat,X3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ X3 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A6 @ B3 ) @ X3 ) ) ).

% le_infI1
thf(fact_1213_le__infI1,axiom,
    ! [A6: set_Ground_gterm_f,X3: set_Ground_gterm_f,B3: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ X3 )
     => ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ A6 @ B3 ) @ X3 ) ) ).

% le_infI1
thf(fact_1214_inf__mono,axiom,
    ! [A6: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat,D2: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A6 @ C2 )
     => ( ( ord_le8976984241387448984_f_nat @ B3 @ D2 )
       => ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A6 @ B3 ) @ ( inf_in4124010953442585290_f_nat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1215_inf__mono,axiom,
    ! [A6: set_Pr989862937836626183term_f,C2: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f,D2: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ A6 @ C2 )
     => ( ( ord_le263819222746101927term_f @ B3 @ D2 )
       => ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ A6 @ B3 ) @ ( inf_in6246359444536921845term_f @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1216_inf__mono,axiom,
    ! [A6: fset_P6228066233360383026_f_nat,C2: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,D2: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ A6 @ C2 )
     => ( ( ord_le1552505484586773650_f_nat @ B3 @ D2 )
       => ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ A6 @ B3 ) @ ( inf_in6532419608991314884_f_nat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1217_inf__mono,axiom,
    ! [A6: nat,C2: nat,B3: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A6 @ C2 )
     => ( ( ord_less_eq_nat @ B3 @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A6 @ B3 ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1218_inf__mono,axiom,
    ! [A6: set_nat,C2: set_nat,B3: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ C2 )
     => ( ( ord_less_eq_set_nat @ B3 @ D2 )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A6 @ B3 ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1219_inf__mono,axiom,
    ! [A6: set_Ground_gterm_f,C2: set_Ground_gterm_f,B3: set_Ground_gterm_f,D2: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ A6 @ C2 )
     => ( ( ord_le2735537439747282356term_f @ B3 @ D2 )
       => ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ A6 @ B3 ) @ ( inf_in2342200634362942210term_f @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1220_le__infI,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ A6 )
     => ( ( ord_le8976984241387448984_f_nat @ X3 @ B3 )
       => ( ord_le8976984241387448984_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ A6 @ B3 ) ) ) ) ).

% le_infI
thf(fact_1221_le__infI,axiom,
    ! [X3: set_Pr989862937836626183term_f,A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ X3 @ A6 )
     => ( ( ord_le263819222746101927term_f @ X3 @ B3 )
       => ( ord_le263819222746101927term_f @ X3 @ ( inf_in6246359444536921845term_f @ A6 @ B3 ) ) ) ) ).

% le_infI
thf(fact_1222_le__infI,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X3 @ A6 )
     => ( ( ord_le1552505484586773650_f_nat @ X3 @ B3 )
       => ( ord_le1552505484586773650_f_nat @ X3 @ ( inf_in6532419608991314884_f_nat @ A6 @ B3 ) ) ) ) ).

% le_infI
thf(fact_1223_le__infI,axiom,
    ! [X3: nat,A6: nat,B3: nat] :
      ( ( ord_less_eq_nat @ X3 @ A6 )
     => ( ( ord_less_eq_nat @ X3 @ B3 )
       => ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A6 @ B3 ) ) ) ) ).

% le_infI
thf(fact_1224_le__infI,axiom,
    ! [X3: set_nat,A6: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ A6 )
     => ( ( ord_less_eq_set_nat @ X3 @ B3 )
       => ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A6 @ B3 ) ) ) ) ).

% le_infI
thf(fact_1225_le__infI,axiom,
    ! [X3: set_Ground_gterm_f,A6: set_Ground_gterm_f,B3: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ X3 @ A6 )
     => ( ( ord_le2735537439747282356term_f @ X3 @ B3 )
       => ( ord_le2735537439747282356term_f @ X3 @ ( inf_in2342200634362942210term_f @ A6 @ B3 ) ) ) ) ).

% le_infI
thf(fact_1226_le__infE,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,A6: set_Pr5245412377734449720_f_nat,B3: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ A6 @ B3 ) )
     => ~ ( ( ord_le8976984241387448984_f_nat @ X3 @ A6 )
         => ~ ( ord_le8976984241387448984_f_nat @ X3 @ B3 ) ) ) ).

% le_infE
thf(fact_1227_le__infE,axiom,
    ! [X3: set_Pr989862937836626183term_f,A6: set_Pr989862937836626183term_f,B3: set_Pr989862937836626183term_f] :
      ( ( ord_le263819222746101927term_f @ X3 @ ( inf_in6246359444536921845term_f @ A6 @ B3 ) )
     => ~ ( ( ord_le263819222746101927term_f @ X3 @ A6 )
         => ~ ( ord_le263819222746101927term_f @ X3 @ B3 ) ) ) ).

% le_infE
thf(fact_1228_le__infE,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,A6: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
      ( ( ord_le1552505484586773650_f_nat @ X3 @ ( inf_in6532419608991314884_f_nat @ A6 @ B3 ) )
     => ~ ( ( ord_le1552505484586773650_f_nat @ X3 @ A6 )
         => ~ ( ord_le1552505484586773650_f_nat @ X3 @ B3 ) ) ) ).

% le_infE
thf(fact_1229_le__infE,axiom,
    ! [X3: nat,A6: nat,B3: nat] :
      ( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A6 @ B3 ) )
     => ~ ( ( ord_less_eq_nat @ X3 @ A6 )
         => ~ ( ord_less_eq_nat @ X3 @ B3 ) ) ) ).

% le_infE
thf(fact_1230_le__infE,axiom,
    ! [X3: set_nat,A6: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A6 @ B3 ) )
     => ~ ( ( ord_less_eq_set_nat @ X3 @ A6 )
         => ~ ( ord_less_eq_set_nat @ X3 @ B3 ) ) ) ).

% le_infE
thf(fact_1231_le__infE,axiom,
    ! [X3: set_Ground_gterm_f,A6: set_Ground_gterm_f,B3: set_Ground_gterm_f] :
      ( ( ord_le2735537439747282356term_f @ X3 @ ( inf_in2342200634362942210term_f @ A6 @ B3 ) )
     => ~ ( ( ord_le2735537439747282356term_f @ X3 @ A6 )
         => ~ ( ord_le2735537439747282356term_f @ X3 @ B3 ) ) ) ).

% le_infE
thf(fact_1232_inf__le2,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y ) @ Y ) ).

% inf_le2
thf(fact_1233_inf__le2,axiom,
    ! [X3: set_Pr989862937836626183term_f,Y: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ X3 @ Y ) @ Y ) ).

% inf_le2
thf(fact_1234_inf__le2,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ X3 @ Y ) @ Y ) ).

% inf_le2
thf(fact_1235_inf__le2,axiom,
    ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).

% inf_le2
thf(fact_1236_inf__le2,axiom,
    ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y ) ).

% inf_le2
thf(fact_1237_inf__le2,axiom,
    ! [X3: set_Ground_gterm_f,Y: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ X3 @ Y ) @ Y ) ).

% inf_le2
thf(fact_1238_inf__le1,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y ) @ X3 ) ).

% inf_le1
thf(fact_1239_inf__le1,axiom,
    ! [X3: set_Pr989862937836626183term_f,Y: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ X3 @ Y ) @ X3 ) ).

% inf_le1
thf(fact_1240_inf__le1,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ X3 @ Y ) @ X3 ) ).

% inf_le1
thf(fact_1241_inf__le1,axiom,
    ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).

% inf_le1
thf(fact_1242_inf__le1,axiom,
    ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ X3 ) ).

% inf_le1
thf(fact_1243_inf__le1,axiom,
    ! [X3: set_Ground_gterm_f,Y: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ X3 @ Y ) @ X3 ) ).

% inf_le1
thf(fact_1244_inf__sup__ord_I1_J,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_1245_inf__sup__ord_I1_J,axiom,
    ! [X3: set_Pr989862937836626183term_f,Y: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ X3 @ Y ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_1246_inf__sup__ord_I1_J,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ X3 @ Y ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_1247_inf__sup__ord_I1_J,axiom,
    ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_1248_inf__sup__ord_I1_J,axiom,
    ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_1249_inf__sup__ord_I1_J,axiom,
    ! [X3: set_Ground_gterm_f,Y: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ X3 @ Y ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_1250_inf__sup__ord_I2_J,axiom,
    ! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1251_inf__sup__ord_I2_J,axiom,
    ! [X3: set_Pr989862937836626183term_f,Y: set_Pr989862937836626183term_f] : ( ord_le263819222746101927term_f @ ( inf_in6246359444536921845term_f @ X3 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1252_inf__sup__ord_I2_J,axiom,
    ! [X3: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ ( inf_in6532419608991314884_f_nat @ X3 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1253_inf__sup__ord_I2_J,axiom,
    ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1254_inf__sup__ord_I2_J,axiom,
    ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1255_inf__sup__ord_I2_J,axiom,
    ! [X3: set_Ground_gterm_f,Y: set_Ground_gterm_f] : ( ord_le2735537439747282356term_f @ ( inf_in2342200634362942210term_f @ X3 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1256_Restr__grstepD__dist,axiom,
    ! [F2: set_Pr5245412377734449720_f_nat,R5: set_Pr989862937836626183term_f,G: set_Pr5245412377734449720_f_nat] :
      ( ( inf_in6246359444536921845term_f @ ( lift_Root_grstepD_f @ F2 @ R5 )
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ G )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ G ) ) )
      = ( lift_Root_grstepD_f @ ( inf_in4124010953442585290_f_nat @ F2 @ G )
        @ ( inf_in6246359444536921845term_f @ R5
          @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ G )
            @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ G ) ) ) ) ) ).

% Restr_grstepD_dist
thf(fact_1257_R2ParStep,axiom,
    ! [F2: set_Pr5245412377734449720_f_nat,Rs: list_s8746099396510718605rm_f_v,Ts: list_FOR_ftrs] :
      ( ( fOR_eval_rr2_rel_f_v @ F2 @ Rs @ ( fOR_R27104206116318976481R_ftrs @ Ts ) )
      = ( inf_in6246359444536921845term_f @ ( gpar_rstep_f_v @ ( fOR_is_to_trs_f_v2 @ Rs @ Ts ) )
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) ) ) ).

% R2ParStep
thf(fact_1258_agtt__grrstep,axiom,
    ! [R2: fset_P8018961893305114765rm_f_v,F2: fset_P6228066233360383026_f_nat] :
      ( ( lV_to_lv_trs_f_v @ ( fset_P4617584883882644886rm_f_v @ R2 ) )
     => ( ( ord_le1552505484586773650_f_nat @ ( lV_to_ffunas_trs_f_v @ R2 ) @ F2 )
       => ( ( agtt_l8776386655999288343rm_f_f @ ( lV_to_5725947919369604231ep_f_v @ R2 @ F2 ) )
          = ( inf_in6246359444536921845term_f @ ( grrstep_f_v @ ( fset_P4617584883882644886rm_f_v @ R2 ) )
            @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ ( fset_P3576968334923099475_f_nat @ F2 ) )
              @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ ( fset_P3576968334923099475_f_nat @ F2 ) ) ) ) ) ) ) ).

% agtt_grrstep
thf(fact_1259_dual__Min,axiom,
    ( ( lattices_Min_nat
      @ ^ [X: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X ) )
    = lattic8265883725875713057ax_nat ) ).

% dual_Min
thf(fact_1260_R2NonRootStep,axiom,
    ! [F2: set_Pr5245412377734449720_f_nat,Rs: list_s8746099396510718605rm_f_v,Ts: list_FOR_ftrs] :
      ( ( fOR_eval_rr2_rel_f_v @ F2 @ Rs @ ( fOR_R22350627647438995247R_ftrs @ Ts ) )
      = ( inf_in6246359444536921845term_f @ ( gnrrstep_f_v @ ( fOR_is_to_trs_f_v2 @ Rs @ Ts ) )
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) ) ) ).

% R2NonRootStep
thf(fact_1261_grstepD,axiom,
    ! [S3: ground_gterm_f,T: ground_gterm_f,R5: set_Pr989862937836626183term_f,F2: set_Pr5245412377734449720_f_nat] :
      ( ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S3 @ T ) @ R5 )
     => ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ S3 @ T ) @ ( lift_Root_grstepD_f @ F2 @ R5 ) ) ) ).

% grstepD
thf(fact_1262_R2RootStep,axiom,
    ! [F2: set_Pr5245412377734449720_f_nat,Rs: list_s8746099396510718605rm_f_v,Ts: list_FOR_ftrs] :
      ( ( fOR_eval_rr2_rel_f_v @ F2 @ Rs @ ( fOR_R25033456731906645142R_ftrs @ Ts ) )
      = ( inf_in6246359444536921845term_f @ ( grrstep_f_v @ ( fOR_is_to_trs_f_v2 @ Rs @ Ts ) )
        @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
          @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) ) ) ).

% R2RootStep
thf(fact_1263_Max__suffI,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A = B2 )
       => ( ( lattic8265883725875713057ax_nat @ A )
          = ( lattic8265883725875713057ax_nat @ B2 ) ) ) ) ).

% Max_suffI
thf(fact_1264_Max__suffI,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B2: set_Pr5245412377734449720_f_nat] :
      ( ( finite3598747971073327385_f_nat @ A )
     => ( ( A = B2 )
       => ( ( lattic2060086870095978963_f_nat @ A )
          = ( lattic2060086870095978963_f_nat @ B2 ) ) ) ) ).

% Max_suffI
thf(fact_1265_root__pair__automaton__grrstep,axiom,
    ! [R2: fset_P8018961893305114765rm_f_v,F2: fset_P6228066233360383026_f_nat] :
      ( ( lV_to_lv_trs_f_v @ ( fset_P4617584883882644886rm_f_v @ R2 ) )
     => ( ( ord_le1552505484586773650_f_nat @ ( lV_to_ffunas_trs_f_v @ R2 ) @ F2 )
       => ( ( pair_p5686396426817080910rm_f_f @ ( lV_to_8065481040984897572_f_v_v @ F2 @ R2 ) @ ( lV_to_Rel_f_f_v @ R2 ) )
          = ( inf_in6246359444536921845term_f @ ( grrstep_f_v @ ( fset_P4617584883882644886rm_f_v @ R2 ) )
            @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ ( fset_P3576968334923099475_f_nat @ F2 ) )
              @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ ( fset_P3576968334923099475_f_nat @ F2 ) ) ) ) ) ) ) ).

% root_pair_automaton_grrstep
thf(fact_1266_agtt__grrstep__set,axiom,
    ! [R2: set_Pr8827868859434726151rm_f_v,F2: set_Pr5245412377734449720_f_nat] :
      ( ( finite6110293842680836688rm_f_v @ R2 )
     => ( ( finite3598747971073327385_f_nat @ F2 )
       => ( ( lV_to_lv_trs_f_v @ R2 )
         => ( ( ord_le8976984241387448984_f_nat @ ( funas_trs_f_v_v @ R2 ) @ F2 )
           => ( ( agtt_l8776386655999288343rm_f_f @ ( lV_to_5725947919369604231ep_f_v @ ( abs_fs251882036029440684rm_f_v @ R2 ) @ ( abs_fs3093487306503459773_f_nat @ F2 ) ) )
              = ( inf_in6246359444536921845term_f @ ( grrstep_f_v @ R2 )
                @ ( produc6043514909933191368term_f @ ( ground_T_G_f @ F2 )
                  @ ^ [Uu: ground_gterm_f] : ( ground_T_G_f @ F2 ) ) ) ) ) ) ) ) ).

% agtt_grrstep_set
thf(fact_1267_fset__inverse,axiom,
    ! [X3: fset_P6228066233360383026_f_nat] :
      ( ( abs_fs3093487306503459773_f_nat @ ( fset_P3576968334923099475_f_nat @ X3 ) )
      = X3 ) ).

% fset_inverse
thf(fact_1268_fset__inverse,axiom,
    ! [X3: fset_P8018961893305114765rm_f_v] :
      ( ( abs_fs251882036029440684rm_f_v @ ( fset_P4617584883882644886rm_f_v @ X3 ) )
      = X3 ) ).

% fset_inverse
thf(fact_1269_Abs__fset__inject,axiom,
    ! [X3: set_set_nat,Y: set_set_nat] :
      ( ( member_set_set_nat @ X3 @ ( collect_set_set_nat @ finite1152437895449049373et_nat ) )
     => ( ( member_set_set_nat @ Y @ ( collect_set_set_nat @ finite1152437895449049373et_nat ) )
       => ( ( ( abs_fset_set_nat @ X3 )
            = ( abs_fset_set_nat @ Y ) )
          = ( X3 = Y ) ) ) ) ).

% Abs_fset_inject
thf(fact_1270_finite__Collect__le__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K2 ) ) ) ).

% finite_Collect_le_nat
thf(fact_1271_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M3: nat] :
          ? [N2: nat] :
            ( ( ord_less_eq_nat @ M3 @ N2 )
            & ( member_nat @ N2 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_1272_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat

% Helper facts (7)
thf(help_If_2_1_If_001t__Option__Ooption_It__Ground____Terms__Ogterm_Itf__f_J_J_T,axiom,
    ! [X3: option3580545445573873146term_f,Y: option3580545445573873146term_f] :
      ( ( if_opt1517256166862243008term_f @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Ground____Terms__Ogterm_Itf__f_J_J_T,axiom,
    ! [X3: option3580545445573873146term_f,Y: option3580545445573873146term_f] :
      ( ( if_opt1517256166862243008term_f @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J_T,axiom,
    ! [X3: option3296083141436081229rm_f_v,Y: option3296083141436081229rm_f_v] :
      ( ( if_opt1261302912030445331rm_f_v @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J_T,axiom,
    ! [X3: option3296083141436081229rm_f_v,Y: option3296083141436081229rm_f_v] :
      ( ( if_opt1261302912030445331rm_f_v @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_3_1_If_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_T,axiom,
    ! [X3: option3789488934265196358tion_f,Y: option3789488934265196358tion_f] :
      ( ( if_opt8252839400935543808tion_f @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_T,axiom,
    ! [X3: option3789488934265196358tion_f,Y: option3789488934265196358tion_f] :
      ( ( if_opt8252839400935543808tion_f @ $true @ X3 @ Y )
      = X3 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ~ ( finite1923210418194103261term_f
      @ ( collec3522850625759001119term_f
        @ ^ [U: ground_gterm_f] : ( member848276444142703440term_f @ ( produc3560254623552331287term_f @ t @ U ) @ ( fOR_eval_rr2_rel_f_v @ ( fset_P3576968334923099475_f_nat @ f2 ) @ ( map_fs8602507653405230974rm_f_v @ fset_P4617584883882644886rm_f_v @ rs ) @ r ) ) ) ) ).

thf(conj_1,conjecture,
    ord_le8976984241387448984_f_nat @ ( ground_funas_gterm_f @ t ) @ ( fset_P3576968334923099475_f_nat @ f2 ) ).

%------------------------------------------------------------------------------