TPTP Problem File: ITP028^1.p

View Solutions - Solve Problem

%------------------------------------------------------------------------------
% File     : ITP028^1 : TPTP v9.0.0. Released v7.5.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer Algebra8 problem prob_962__6421504_1
% Version  : Especial.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source   : [Des21]
% Names    : Algebra8/prob_962__6421504_1 [Des21]

% Status   : Theorem
% Rating   : 0.12 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax   : Number of formulae    :  456 ( 120 unt; 102 typ;   0 def)
%            Number of atoms       : 1002 ( 288 equ;   0 cnn)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives : 2874 (  78   ~;  10   |;  54   &;2255   @)
%                                         (   0 <=>; 477  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   7 avg)
%            Number of types       :   16 (  15 usr)
%            Number of type conns  :  618 ( 618   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   90 (  87 usr;  12 con; 0-5 aty)
%            Number of variables   : 1103 (  97   ^; 962   !;  44   ?;1103   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Sledgehammer 2021-02-23 15:45:55.322
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
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% Explicit typings (87)
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001tf__a,type,
    image_set_set_a_a: ( set_set_a > a ) > set_set_set_a > set_a ).

thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_062_Itf__a_Mtf__e_J,type,
    image_set_a_a_e: ( set_a > a > e ) > set_set_a > set_a_e ).

thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J,type,
    image_210900495_a_a_e: ( set_a > produc715398134_a_a_e ) > set_set_a > set_Pr204296108_a_a_e ).

thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    image_1930509541_set_a: ( set_a > set_set_a ) > set_set_a > set_set_set_a ).

thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
    image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).

thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
    image_set_a_a: ( set_a > a ) > set_set_a > set_a ).

thf(sy_c_Set_Oimage_001tf__a_001_062_Itf__a_Mtf__e_J,type,
    image_a_a_e: ( a > a > e ) > set_a > set_a_e ).

thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    image_a_set_set_a: ( a > set_set_a ) > set_a > set_set_set_a ).

thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
    image_a_set_a: ( a > set_a ) > set_a > set_set_a ).

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

thf(sy_c_member_001_062_Itf__a_Mtf__e_J,type,
    member_a_e: ( a > e ) > set_a_e > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_Itf__a_Mtf__e_J_Mt__Set__Oset_Itf__a_J_J,type,
    member360729153_set_a: produc1965228714_set_a > set_Pr305929312_set_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J,type,
    member1258382221_a_a_e: produc715398134_a_a_e > set_Pr204296108_a_a_e > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_M_062_Itf__a_Mtf__e_J_J_J,type,
    member1437121347_a_a_e: set_Pr204296108_a_a_e > set_se821189730_a_a_e > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    member_set_set_a: set_set_a > set_set_set_a > $o ).

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

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

thf(sy_v_C,type,
    c: set_Pr204296108_a_a_e ).

thf(sy_v_a1,type,
    a1: set_a ).

thf(sy_v_a2,type,
    a2: set_a ).

thf(sy_v_a3,type,
    a3: set_a ).

thf(sy_v_b1,type,
    b1: a > e ).

thf(sy_v_b2,type,
    b2: a > e ).

thf(sy_v_b3,type,
    b3: a > e ).

thf(sy_v_thesis____,type,
    thesis: $o ).

% Relevant facts (352)
thf(fact_0__092_060open_062_092_060forall_062a_092_060in_062C_O_A_092_060forall_062b_092_060in_062C_O_Afst_Aa_A_092_060subseteq_062_Afst_Ab_A_092_060or_062_Afst_Ab_A_092_060subseteq_062_Afst_Aa_092_060close_062,axiom,
    ! [X: produc715398134_a_a_e] :
      ( ( member1258382221_a_a_e @ X @ c )
     => ! [Xa: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ Xa @ c )
         => ( ( ord_less_eq_set_a @ ( produc1040928388_a_a_e @ X ) @ ( produc1040928388_a_a_e @ Xa ) )
            | ( ord_less_eq_set_a @ ( produc1040928388_a_a_e @ Xa ) @ ( produc1040928388_a_a_e @ X ) ) ) ) ) ).

% \<open>\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a\<close>
thf(fact_1__092_060open_062_092_060exists_062a_092_060in_062fst_A_096_AC_O_Aa1_A_092_060subseteq_062_Aa_A_092_060and_062_Aa2_A_092_060subseteq_062_Aa_A_092_060and_062_Aa3_A_092_060subseteq_062_Aa_092_060close_062,axiom,
    ? [X2: set_a] :
      ( ( member_set_a @ X2 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
      & ( ord_less_eq_set_a @ a1 @ X2 )
      & ( ord_less_eq_set_a @ a2 @ X2 )
      & ( ord_less_eq_set_a @ a3 @ X2 ) ) ).

% \<open>\<exists>a\<in>fst ` C. a1 \<subseteq> a \<and> a2 \<subseteq> a \<and> a3 \<subseteq> a\<close>
thf(fact_2__092_060open_062a1_A_092_060subseteq_062_Aa2_A_092_060or_062_Aa2_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
    ( ( ord_less_eq_set_a @ a1 @ a2 )
    | ( ord_less_eq_set_a @ a2 @ a1 ) ) ).

% \<open>a1 \<subseteq> a2 \<or> a2 \<subseteq> a1\<close>
thf(fact_3__092_060open_062a1_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
    ( ( ord_less_eq_set_a @ a1 @ a3 )
    | ( ord_less_eq_set_a @ a3 @ a1 ) ) ).

% \<open>a1 \<subseteq> a3 \<or> a3 \<subseteq> a1\<close>
thf(fact_4__092_060open_062a2_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa2_092_060close_062,axiom,
    ( ( ord_less_eq_set_a @ a2 @ a3 )
    | ( ord_less_eq_set_a @ a3 @ a2 ) ) ).

% \<open>a2 \<subseteq> a3 \<or> a3 \<subseteq> a2\<close>
thf(fact_5_A_I3_J,axiom,
    member_set_a @ a3 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) ).

% A(3)
thf(fact_6_A_I2_J,axiom,
    member_set_a @ a2 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) ).

% A(2)
thf(fact_7_A_I1_J,axiom,
    member_set_a @ a1 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) ).

% A(1)
thf(fact_8__092_060open_062_Ia1_M_Ab1_J_A_092_060in_062_AC_092_060close_062,axiom,
    member1258382221_a_a_e @ ( produc1134456688_a_a_e @ a1 @ b1 ) @ c ).

% \<open>(a1, b1) \<in> C\<close>
thf(fact_9__092_060open_062_Ia2_M_Ab2_J_A_092_060in_062_AC_092_060close_062,axiom,
    member1258382221_a_a_e @ ( produc1134456688_a_a_e @ a2 @ b2 ) @ c ).

% \<open>(a2, b2) \<in> C\<close>
thf(fact_10__092_060open_062_Ia3_M_Ab3_J_A_092_060in_062_AC_092_060close_062,axiom,
    member1258382221_a_a_e @ ( produc1134456688_a_a_e @ a3 @ b3 ) @ c ).

% \<open>(a3, b3) \<in> C\<close>
thf(fact_11_subsetI,axiom,
    ! [A: set_a_e,B: set_a_e] :
      ( ! [X2: a > e] :
          ( ( member_a_e @ X2 @ A )
         => ( member_a_e @ X2 @ B ) )
     => ( ord_less_eq_set_a_e @ A @ B ) ) ).

% subsetI
thf(fact_12_subsetI,axiom,
    ! [A: set_Pr305929312_set_a,B: set_Pr305929312_set_a] :
      ( ! [X2: produc1965228714_set_a] :
          ( ( member360729153_set_a @ X2 @ A )
         => ( member360729153_set_a @ X2 @ B ) )
     => ( ord_le802356416_set_a @ A @ B ) ) ).

% subsetI
thf(fact_13_subsetI,axiom,
    ! [A: set_set_set_a,B: set_set_set_a] :
      ( ! [X2: set_set_a] :
          ( ( member_set_set_a @ X2 @ A )
         => ( member_set_set_a @ X2 @ B ) )
     => ( ord_le2103863998_set_a @ A @ B ) ) ).

% subsetI
thf(fact_14_subsetI,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ! [X2: set_a] :
          ( ( member_set_a @ X2 @ A )
         => ( member_set_a @ X2 @ B ) )
     => ( ord_le318720350_set_a @ A @ B ) ) ).

% subsetI
thf(fact_15_subsetI,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
      ( ! [X2: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ X2 @ A )
         => ( member1258382221_a_a_e @ X2 @ B ) )
     => ( ord_le700723212_a_a_e @ A @ B ) ) ).

% subsetI
thf(fact_16_subsetI,axiom,
    ! [A: set_a,B: set_a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A )
         => ( member_a @ X2 @ B ) )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% subsetI
thf(fact_17_subset__antisym,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
      ( ( ord_le700723212_a_a_e @ A @ B )
     => ( ( ord_le700723212_a_a_e @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_18_subset__antisym,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( ord_le318720350_set_a @ A @ B )
     => ( ( ord_le318720350_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_19_subset__antisym,axiom,
    ! [A: set_set_set_a,B: set_set_set_a] :
      ( ( ord_le2103863998_set_a @ A @ B )
     => ( ( ord_le2103863998_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_20_subset__antisym,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_21_image__eqI,axiom,
    ! [B2: set_a,F: set_a > set_a,X3: set_a,A: set_set_a] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_set_a @ X3 @ A )
       => ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_22_image__eqI,axiom,
    ! [B2: produc715398134_a_a_e,F: set_a > produc715398134_a_a_e,X3: set_a,A: set_set_a] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_set_a @ X3 @ A )
       => ( member1258382221_a_a_e @ B2 @ ( image_210900495_a_a_e @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_23_image__eqI,axiom,
    ! [B2: set_a,F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member1258382221_a_a_e @ X3 @ A )
       => ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_24_image__eqI,axiom,
    ! [B2: produc715398134_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e,X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member1258382221_a_a_e @ X3 @ A )
       => ( member1258382221_a_a_e @ B2 @ ( image_625620211_a_a_e @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_25_image__eqI,axiom,
    ! [B2: a,F: a > a,X3: a,A: set_a] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_a @ X3 @ A )
       => ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_26_image__eqI,axiom,
    ! [B2: a,F: set_a > a,X3: set_a,A: set_set_a] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_set_a @ X3 @ A )
       => ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_27_image__eqI,axiom,
    ! [B2: set_a,F: a > set_a,X3: a,A: set_a] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_a @ X3 @ A )
       => ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_28_image__eqI,axiom,
    ! [B2: a > e,F: a > a > e,X3: a,A: set_a] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_a @ X3 @ A )
       => ( member_a_e @ B2 @ ( image_a_a_e @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_29_image__eqI,axiom,
    ! [B2: a,F: ( a > e ) > a,X3: a > e,A: set_a_e] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_a_e @ X3 @ A )
       => ( member_a @ B2 @ ( image_a_e_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_30_image__eqI,axiom,
    ! [B2: a > e,F: set_a > a > e,X3: set_a,A: set_set_a] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_set_a @ X3 @ A )
       => ( member_a_e @ B2 @ ( image_set_a_a_e @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_31_order__refl,axiom,
    ! [X3: set_Pr204296108_a_a_e] : ( ord_le700723212_a_a_e @ X3 @ X3 ) ).

% order_refl
thf(fact_32_order__refl,axiom,
    ! [X3: set_set_a] : ( ord_le318720350_set_a @ X3 @ X3 ) ).

% order_refl
thf(fact_33_order__refl,axiom,
    ! [X3: set_set_set_a] : ( ord_le2103863998_set_a @ X3 @ X3 ) ).

% order_refl
thf(fact_34_order__refl,axiom,
    ! [X3: $o > set_a] : ( ord_less_eq_o_set_a @ X3 @ X3 ) ).

% order_refl
thf(fact_35_order__refl,axiom,
    ! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).

% order_refl
thf(fact_36_image__mono,axiom,
    ! [A: set_a,B: set_a,F: a > a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).

% image_mono
thf(fact_37_image__mono,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a] :
      ( ( ord_le700723212_a_a_e @ A @ B )
     => ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ ( image_64240033_set_a @ F @ B ) ) ) ).

% image_mono
thf(fact_38_image__mono,axiom,
    ! [A: set_a,B: set_a,F: a > set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_le318720350_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).

% image_mono
thf(fact_39_image__mono,axiom,
    ! [A: set_set_a,B: set_set_a,F: set_a > a] :
      ( ( ord_le318720350_set_a @ A @ B )
     => ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ ( image_set_a_a @ F @ B ) ) ) ).

% image_mono
thf(fact_40_image__mono,axiom,
    ! [A: set_a,B: set_a,F: a > set_set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_le2103863998_set_a @ ( image_a_set_set_a @ F @ A ) @ ( image_a_set_set_a @ F @ B ) ) ) ).

% image_mono
thf(fact_41_image__mono,axiom,
    ! [A: set_set_a,B: set_set_a,F: set_a > set_a] :
      ( ( ord_le318720350_set_a @ A @ B )
     => ( ord_le318720350_set_a @ ( image_set_a_set_a @ F @ A ) @ ( image_set_a_set_a @ F @ B ) ) ) ).

% image_mono
thf(fact_42_image__mono,axiom,
    ! [A: set_set_set_a,B: set_set_set_a,F: set_set_a > a] :
      ( ( ord_le2103863998_set_a @ A @ B )
     => ( ord_less_eq_set_a @ ( image_set_set_a_a @ F @ A ) @ ( image_set_set_a_a @ F @ B ) ) ) ).

% image_mono
thf(fact_43_image__mono,axiom,
    ! [A: set_set_a,B: set_set_a,F: set_a > set_set_a] :
      ( ( ord_le318720350_set_a @ A @ B )
     => ( ord_le2103863998_set_a @ ( image_1930509541_set_a @ F @ A ) @ ( image_1930509541_set_a @ F @ B ) ) ) ).

% image_mono
thf(fact_44_image__mono,axiom,
    ! [A: set_set_set_a,B: set_set_set_a,F: set_set_a > set_a] :
      ( ( ord_le2103863998_set_a @ A @ B )
     => ( ord_le318720350_set_a @ ( image_402591269_set_a @ F @ A ) @ ( image_402591269_set_a @ F @ B ) ) ) ).

% image_mono
thf(fact_45_image__mono,axiom,
    ! [A: set_set_set_a,B: set_set_set_a,F: set_set_a > set_set_a] :
      ( ( ord_le2103863998_set_a @ A @ B )
     => ( ord_le2103863998_set_a @ ( image_1956910981_set_a @ F @ A ) @ ( image_1956910981_set_a @ F @ B ) ) ) ).

% image_mono
thf(fact_46_image__subsetI,axiom,
    ! [A: set_set_a,F: set_a > a,B: set_a] :
      ( ! [X2: set_a] :
          ( ( member_set_a @ X2 @ A )
         => ( member_a @ ( F @ X2 ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_47_image__subsetI,axiom,
    ! [A: set_set_a,F: set_a > set_a,B: set_set_a] :
      ( ! [X2: set_a] :
          ( ( member_set_a @ X2 @ A )
         => ( member_set_a @ ( F @ X2 ) @ B ) )
     => ( ord_le318720350_set_a @ ( image_set_a_set_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_48_image__subsetI,axiom,
    ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > a,B: set_a] :
      ( ! [X2: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ X2 @ A )
         => ( member_a @ ( F @ X2 ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_2116492353_a_e_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_49_image__subsetI,axiom,
    ! [A: set_set_a,F: set_a > produc715398134_a_a_e,B: set_Pr204296108_a_a_e] :
      ( ! [X2: set_a] :
          ( ( member_set_a @ X2 @ A )
         => ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
     => ( ord_le700723212_a_a_e @ ( image_210900495_a_a_e @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_50_image__subsetI,axiom,
    ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a,B: set_set_a] :
      ( ! [X2: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ X2 @ A )
         => ( member_set_a @ ( F @ X2 ) @ B ) )
     => ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_51_image__subsetI,axiom,
    ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e,B: set_Pr204296108_a_a_e] :
      ( ! [X2: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ X2 @ A )
         => ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
     => ( ord_le700723212_a_a_e @ ( image_625620211_a_a_e @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_52_image__subsetI,axiom,
    ! [A: set_a,F: a > a,B: set_a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A )
         => ( member_a @ ( F @ X2 ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_53_image__subsetI,axiom,
    ! [A: set_a,F: a > set_a,B: set_set_a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A )
         => ( member_set_a @ ( F @ X2 ) @ B ) )
     => ( ord_le318720350_set_a @ ( image_a_set_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_54_image__subsetI,axiom,
    ! [A: set_a,F: a > a > e,B: set_a_e] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A )
         => ( member_a_e @ ( F @ X2 ) @ B ) )
     => ( ord_less_eq_set_a_e @ ( image_a_a_e @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_55_image__subsetI,axiom,
    ! [A: set_a_e,F: ( a > e ) > a,B: set_a] :
      ( ! [X2: a > e] :
          ( ( member_a_e @ X2 @ A )
         => ( member_a @ ( F @ X2 ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_a_e_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_56_subset__imageE,axiom,
    ! [B: set_a,F: a > a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
     => ~ ! [C: set_a] :
            ( ( ord_less_eq_set_a @ C @ A )
           => ( B
             != ( image_a_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_57_subset__imageE,axiom,
    ! [B: set_set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
      ( ( ord_le318720350_set_a @ B @ ( image_64240033_set_a @ F @ A ) )
     => ~ ! [C: set_Pr204296108_a_a_e] :
            ( ( ord_le700723212_a_a_e @ C @ A )
           => ( B
             != ( image_64240033_set_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_58_subset__imageE,axiom,
    ! [B: set_a,F: set_a > a,A: set_set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
     => ~ ! [C: set_set_a] :
            ( ( ord_le318720350_set_a @ C @ A )
           => ( B
             != ( image_set_a_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_59_subset__imageE,axiom,
    ! [B: set_set_a,F: a > set_a,A: set_a] :
      ( ( ord_le318720350_set_a @ B @ ( image_a_set_a @ F @ A ) )
     => ~ ! [C: set_a] :
            ( ( ord_less_eq_set_a @ C @ A )
           => ( B
             != ( image_a_set_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_60_subset__imageE,axiom,
    ! [B: set_a,F: set_set_a > a,A: set_set_set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_set_set_a_a @ F @ A ) )
     => ~ ! [C: set_set_set_a] :
            ( ( ord_le2103863998_set_a @ C @ A )
           => ( B
             != ( image_set_set_a_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_61_subset__imageE,axiom,
    ! [B: set_set_a,F: set_a > set_a,A: set_set_a] :
      ( ( ord_le318720350_set_a @ B @ ( image_set_a_set_a @ F @ A ) )
     => ~ ! [C: set_set_a] :
            ( ( ord_le318720350_set_a @ C @ A )
           => ( B
             != ( image_set_a_set_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_62_subset__imageE,axiom,
    ! [B: set_set_set_a,F: a > set_set_a,A: set_a] :
      ( ( ord_le2103863998_set_a @ B @ ( image_a_set_set_a @ F @ A ) )
     => ~ ! [C: set_a] :
            ( ( ord_less_eq_set_a @ C @ A )
           => ( B
             != ( image_a_set_set_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_63_subset__imageE,axiom,
    ! [B: set_set_a,F: set_set_a > set_a,A: set_set_set_a] :
      ( ( ord_le318720350_set_a @ B @ ( image_402591269_set_a @ F @ A ) )
     => ~ ! [C: set_set_set_a] :
            ( ( ord_le2103863998_set_a @ C @ A )
           => ( B
             != ( image_402591269_set_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_64_subset__imageE,axiom,
    ! [B: set_set_set_a,F: set_a > set_set_a,A: set_set_a] :
      ( ( ord_le2103863998_set_a @ B @ ( image_1930509541_set_a @ F @ A ) )
     => ~ ! [C: set_set_a] :
            ( ( ord_le318720350_set_a @ C @ A )
           => ( B
             != ( image_1930509541_set_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_65_subset__imageE,axiom,
    ! [B: set_set_set_a,F: set_set_a > set_set_a,A: set_set_set_a] :
      ( ( ord_le2103863998_set_a @ B @ ( image_1956910981_set_a @ F @ A ) )
     => ~ ! [C: set_set_set_a] :
            ( ( ord_le2103863998_set_a @ C @ A )
           => ( B
             != ( image_1956910981_set_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_66_image__subset__iff,axiom,
    ! [F: set_a > a,A: set_set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B )
      = ( ! [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
           => ( member_a @ ( F @ X4 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_67_image__subset__iff,axiom,
    ! [F: a > a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A )
           => ( member_a @ ( F @ X4 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_68_image__subset__iff,axiom,
    ! [F: set_a > produc715398134_a_a_e,A: set_set_a,B: set_Pr204296108_a_a_e] :
      ( ( ord_le700723212_a_a_e @ ( image_210900495_a_a_e @ F @ A ) @ B )
      = ( ! [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
           => ( member1258382221_a_a_e @ ( F @ X4 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_69_image__subset__iff,axiom,
    ! [F: set_a > set_a,A: set_set_a,B: set_set_a] :
      ( ( ord_le318720350_set_a @ ( image_set_a_set_a @ F @ A ) @ B )
      = ( ! [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
           => ( member_set_a @ ( F @ X4 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_70_image__subset__iff,axiom,
    ! [F: set_Pr204296108_a_a_e > set_set_a,A: set_se821189730_a_a_e,B: set_set_set_a] :
      ( ( ord_le2103863998_set_a @ ( image_1441218763_set_a @ F @ A ) @ B )
      = ( ! [X4: set_Pr204296108_a_a_e] :
            ( ( member1437121347_a_a_e @ X4 @ A )
           => ( member_set_set_a @ ( F @ X4 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_71_image__subset__iff,axiom,
    ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
      ( ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B )
      = ( ! [X4: produc715398134_a_a_e] :
            ( ( member1258382221_a_a_e @ X4 @ A )
           => ( member_set_a @ ( F @ X4 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_72_subset__image__iff,axiom,
    ! [B: set_a,F: a > a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A )
            & ( B
              = ( image_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_73_subset__image__iff,axiom,
    ! [B: set_set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
      ( ( ord_le318720350_set_a @ B @ ( image_64240033_set_a @ F @ A ) )
      = ( ? [AA: set_Pr204296108_a_a_e] :
            ( ( ord_le700723212_a_a_e @ AA @ A )
            & ( B
              = ( image_64240033_set_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_74_subset__image__iff,axiom,
    ! [B: set_a,F: set_a > a,A: set_set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
      = ( ? [AA: set_set_a] :
            ( ( ord_le318720350_set_a @ AA @ A )
            & ( B
              = ( image_set_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_75_subset__image__iff,axiom,
    ! [B: set_set_a,F: a > set_a,A: set_a] :
      ( ( ord_le318720350_set_a @ B @ ( image_a_set_a @ F @ A ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A )
            & ( B
              = ( image_a_set_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_76_subset__image__iff,axiom,
    ! [B: set_a,F: set_set_a > a,A: set_set_set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_set_set_a_a @ F @ A ) )
      = ( ? [AA: set_set_set_a] :
            ( ( ord_le2103863998_set_a @ AA @ A )
            & ( B
              = ( image_set_set_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_77_subset__image__iff,axiom,
    ! [B: set_set_a,F: set_a > set_a,A: set_set_a] :
      ( ( ord_le318720350_set_a @ B @ ( image_set_a_set_a @ F @ A ) )
      = ( ? [AA: set_set_a] :
            ( ( ord_le318720350_set_a @ AA @ A )
            & ( B
              = ( image_set_a_set_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_78_subset__image__iff,axiom,
    ! [B: set_set_set_a,F: a > set_set_a,A: set_a] :
      ( ( ord_le2103863998_set_a @ B @ ( image_a_set_set_a @ F @ A ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A )
            & ( B
              = ( image_a_set_set_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_79_subset__image__iff,axiom,
    ! [B: set_set_a,F: set_set_a > set_a,A: set_set_set_a] :
      ( ( ord_le318720350_set_a @ B @ ( image_402591269_set_a @ F @ A ) )
      = ( ? [AA: set_set_set_a] :
            ( ( ord_le2103863998_set_a @ AA @ A )
            & ( B
              = ( image_402591269_set_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_80_subset__image__iff,axiom,
    ! [B: set_set_set_a,F: set_a > set_set_a,A: set_set_a] :
      ( ( ord_le2103863998_set_a @ B @ ( image_1930509541_set_a @ F @ A ) )
      = ( ? [AA: set_set_a] :
            ( ( ord_le318720350_set_a @ AA @ A )
            & ( B
              = ( image_1930509541_set_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_81_subset__image__iff,axiom,
    ! [B: set_set_set_a,F: set_set_a > set_set_a,A: set_set_set_a] :
      ( ( ord_le2103863998_set_a @ B @ ( image_1956910981_set_a @ F @ A ) )
      = ( ? [AA: set_set_set_a] :
            ( ( ord_le2103863998_set_a @ AA @ A )
            & ( B
              = ( image_1956910981_set_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_82_PairE__lemma,axiom,
    ! [P: produc1965228714_set_a] :
    ? [X2: a > e,Y: set_a] :
      ( P
      = ( produc479206876_set_a @ X2 @ Y ) ) ).

% PairE_lemma
thf(fact_83_PairE__lemma,axiom,
    ! [P: produc715398134_a_a_e] :
    ? [X2: set_a,Y: a > e] :
      ( P
      = ( produc1134456688_a_a_e @ X2 @ Y ) ) ).

% PairE_lemma
thf(fact_84_dual__order_Oantisym,axiom,
    ! [B2: set_Pr204296108_a_a_e,A2: set_Pr204296108_a_a_e] :
      ( ( ord_le700723212_a_a_e @ B2 @ A2 )
     => ( ( ord_le700723212_a_a_e @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_85_dual__order_Oantisym,axiom,
    ! [B2: set_set_a,A2: set_set_a] :
      ( ( ord_le318720350_set_a @ B2 @ A2 )
     => ( ( ord_le318720350_set_a @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_86_dual__order_Oantisym,axiom,
    ! [B2: set_set_set_a,A2: set_set_set_a] :
      ( ( ord_le2103863998_set_a @ B2 @ A2 )
     => ( ( ord_le2103863998_set_a @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_87_dual__order_Oantisym,axiom,
    ! [B2: $o > set_a,A2: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_o_set_a @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_88_dual__order_Oantisym,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_set_a @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_89_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_Pr204296108_a_a_e,Z: set_Pr204296108_a_a_e] : ( Y2 = Z ) )
    = ( ^ [A3: set_Pr204296108_a_a_e,B3: set_Pr204296108_a_a_e] :
          ( ( ord_le700723212_a_a_e @ B3 @ A3 )
          & ( ord_le700723212_a_a_e @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_90_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_set_a,Z: set_set_a] : ( Y2 = Z ) )
    = ( ^ [A3: set_set_a,B3: set_set_a] :
          ( ( ord_le318720350_set_a @ B3 @ A3 )
          & ( ord_le318720350_set_a @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_91_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_set_set_a,Z: set_set_set_a] : ( Y2 = Z ) )
    = ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
          ( ( ord_le2103863998_set_a @ B3 @ A3 )
          & ( ord_le2103863998_set_a @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_92_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: $o > set_a,Z: $o > set_a] : ( Y2 = Z ) )
    = ( ^ [A3: $o > set_a,B3: $o > set_a] :
          ( ( ord_less_eq_o_set_a @ B3 @ A3 )
          & ( ord_less_eq_o_set_a @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_93_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
    = ( ^ [A3: set_a,B3: set_a] :
          ( ( ord_less_eq_set_a @ B3 @ A3 )
          & ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_94_dual__order_Otrans,axiom,
    ! [B2: set_Pr204296108_a_a_e,A2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
      ( ( ord_le700723212_a_a_e @ B2 @ A2 )
     => ( ( ord_le700723212_a_a_e @ C2 @ B2 )
       => ( ord_le700723212_a_a_e @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_95_dual__order_Otrans,axiom,
    ! [B2: set_set_a,A2: set_set_a,C2: set_set_a] :
      ( ( ord_le318720350_set_a @ B2 @ A2 )
     => ( ( ord_le318720350_set_a @ C2 @ B2 )
       => ( ord_le318720350_set_a @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_96_dual__order_Otrans,axiom,
    ! [B2: set_set_set_a,A2: set_set_set_a,C2: set_set_set_a] :
      ( ( ord_le2103863998_set_a @ B2 @ A2 )
     => ( ( ord_le2103863998_set_a @ C2 @ B2 )
       => ( ord_le2103863998_set_a @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_97_dual__order_Otrans,axiom,
    ! [B2: $o > set_a,A2: $o > set_a,C2: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_o_set_a @ C2 @ B2 )
       => ( ord_less_eq_o_set_a @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_98_dual__order_Otrans,axiom,
    ! [B2: set_a,A2: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_set_a @ C2 @ B2 )
       => ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_99_dual__order_Orefl,axiom,
    ! [A2: set_Pr204296108_a_a_e] : ( ord_le700723212_a_a_e @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_100_dual__order_Orefl,axiom,
    ! [A2: set_set_a] : ( ord_le318720350_set_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_101_dual__order_Orefl,axiom,
    ! [A2: set_set_set_a] : ( ord_le2103863998_set_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_102_dual__order_Orefl,axiom,
    ! [A2: $o > set_a] : ( ord_less_eq_o_set_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_103_dual__order_Orefl,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_104_order__trans,axiom,
    ! [X3: set_Pr204296108_a_a_e,Y3: set_Pr204296108_a_a_e,Z2: set_Pr204296108_a_a_e] :
      ( ( ord_le700723212_a_a_e @ X3 @ Y3 )
     => ( ( ord_le700723212_a_a_e @ Y3 @ Z2 )
       => ( ord_le700723212_a_a_e @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_105_order__trans,axiom,
    ! [X3: set_set_a,Y3: set_set_a,Z2: set_set_a] :
      ( ( ord_le318720350_set_a @ X3 @ Y3 )
     => ( ( ord_le318720350_set_a @ Y3 @ Z2 )
       => ( ord_le318720350_set_a @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_106_order__trans,axiom,
    ! [X3: set_set_set_a,Y3: set_set_set_a,Z2: set_set_set_a] :
      ( ( ord_le2103863998_set_a @ X3 @ Y3 )
     => ( ( ord_le2103863998_set_a @ Y3 @ Z2 )
       => ( ord_le2103863998_set_a @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_107_order__trans,axiom,
    ! [X3: $o > set_a,Y3: $o > set_a,Z2: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ X3 @ Y3 )
     => ( ( ord_less_eq_o_set_a @ Y3 @ Z2 )
       => ( ord_less_eq_o_set_a @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_108_order__trans,axiom,
    ! [X3: set_a,Y3: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ X3 @ Y3 )
     => ( ( ord_less_eq_set_a @ Y3 @ Z2 )
       => ( ord_less_eq_set_a @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_109_order__class_Oorder_Oantisym,axiom,
    ! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e] :
      ( ( ord_le700723212_a_a_e @ A2 @ B2 )
     => ( ( ord_le700723212_a_a_e @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% order_class.order.antisym
thf(fact_110_order__class_Oorder_Oantisym,axiom,
    ! [A2: set_set_a,B2: set_set_a] :
      ( ( ord_le318720350_set_a @ A2 @ B2 )
     => ( ( ord_le318720350_set_a @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% order_class.order.antisym
thf(fact_111_order__class_Oorder_Oantisym,axiom,
    ! [A2: set_set_set_a,B2: set_set_set_a] :
      ( ( ord_le2103863998_set_a @ A2 @ B2 )
     => ( ( ord_le2103863998_set_a @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% order_class.order.antisym
thf(fact_112_order__class_Oorder_Oantisym,axiom,
    ! [A2: $o > set_a,B2: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_o_set_a @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% order_class.order.antisym
thf(fact_113_order__class_Oorder_Oantisym,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% order_class.order.antisym
thf(fact_114_ord__le__eq__trans,axiom,
    ! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
      ( ( ord_le700723212_a_a_e @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le700723212_a_a_e @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_115_ord__le__eq__trans,axiom,
    ! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
      ( ( ord_le318720350_set_a @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le318720350_set_a @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_116_ord__le__eq__trans,axiom,
    ! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
      ( ( ord_le2103863998_set_a @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le2103863998_set_a @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_117_ord__le__eq__trans,axiom,
    ! [A2: $o > set_a,B2: $o > set_a,C2: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_o_set_a @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_118_ord__le__eq__trans,axiom,
    ! [A2: set_a,B2: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_119_ord__eq__le__trans,axiom,
    ! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
      ( ( A2 = B2 )
     => ( ( ord_le700723212_a_a_e @ B2 @ C2 )
       => ( ord_le700723212_a_a_e @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_120_ord__eq__le__trans,axiom,
    ! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
      ( ( A2 = B2 )
     => ( ( ord_le318720350_set_a @ B2 @ C2 )
       => ( ord_le318720350_set_a @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_121_ord__eq__le__trans,axiom,
    ! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
      ( ( A2 = B2 )
     => ( ( ord_le2103863998_set_a @ B2 @ C2 )
       => ( ord_le2103863998_set_a @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_122_ord__eq__le__trans,axiom,
    ! [A2: $o > set_a,B2: $o > set_a,C2: $o > set_a] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
       => ( ord_less_eq_o_set_a @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_123_ord__eq__le__trans,axiom,
    ! [A2: set_a,B2: set_a,C2: set_a] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_124_order__class_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y2: set_Pr204296108_a_a_e,Z: set_Pr204296108_a_a_e] : ( Y2 = Z ) )
    = ( ^ [A3: set_Pr204296108_a_a_e,B3: set_Pr204296108_a_a_e] :
          ( ( ord_le700723212_a_a_e @ A3 @ B3 )
          & ( ord_le700723212_a_a_e @ B3 @ A3 ) ) ) ) ).

% order_class.order.eq_iff
thf(fact_125_order__class_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y2: set_set_a,Z: set_set_a] : ( Y2 = Z ) )
    = ( ^ [A3: set_set_a,B3: set_set_a] :
          ( ( ord_le318720350_set_a @ A3 @ B3 )
          & ( ord_le318720350_set_a @ B3 @ A3 ) ) ) ) ).

% order_class.order.eq_iff
thf(fact_126_order__class_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y2: set_set_set_a,Z: set_set_set_a] : ( Y2 = Z ) )
    = ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
          ( ( ord_le2103863998_set_a @ A3 @ B3 )
          & ( ord_le2103863998_set_a @ B3 @ A3 ) ) ) ) ).

% order_class.order.eq_iff
thf(fact_127_order__class_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y2: $o > set_a,Z: $o > set_a] : ( Y2 = Z ) )
    = ( ^ [A3: $o > set_a,B3: $o > set_a] :
          ( ( ord_less_eq_o_set_a @ A3 @ B3 )
          & ( ord_less_eq_o_set_a @ B3 @ A3 ) ) ) ) ).

% order_class.order.eq_iff
thf(fact_128_order__class_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
    = ( ^ [A3: set_a,B3: set_a] :
          ( ( ord_less_eq_set_a @ A3 @ B3 )
          & ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).

% order_class.order.eq_iff
thf(fact_129_antisym__conv,axiom,
    ! [Y3: set_Pr204296108_a_a_e,X3: set_Pr204296108_a_a_e] :
      ( ( ord_le700723212_a_a_e @ Y3 @ X3 )
     => ( ( ord_le700723212_a_a_e @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv
thf(fact_130_antisym__conv,axiom,
    ! [Y3: set_set_a,X3: set_set_a] :
      ( ( ord_le318720350_set_a @ Y3 @ X3 )
     => ( ( ord_le318720350_set_a @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv
thf(fact_131_antisym__conv,axiom,
    ! [Y3: set_set_set_a,X3: set_set_set_a] :
      ( ( ord_le2103863998_set_a @ Y3 @ X3 )
     => ( ( ord_le2103863998_set_a @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv
thf(fact_132_antisym__conv,axiom,
    ! [Y3: $o > set_a,X3: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ Y3 @ X3 )
     => ( ( ord_less_eq_o_set_a @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv
thf(fact_133_antisym__conv,axiom,
    ! [Y3: set_a,X3: set_a] :
      ( ( ord_less_eq_set_a @ Y3 @ X3 )
     => ( ( ord_less_eq_set_a @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv
thf(fact_134_order_Otrans,axiom,
    ! [A2: set_Pr204296108_a_a_e,B2: set_Pr204296108_a_a_e,C2: set_Pr204296108_a_a_e] :
      ( ( ord_le700723212_a_a_e @ A2 @ B2 )
     => ( ( ord_le700723212_a_a_e @ B2 @ C2 )
       => ( ord_le700723212_a_a_e @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_135_order_Otrans,axiom,
    ! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
      ( ( ord_le318720350_set_a @ A2 @ B2 )
     => ( ( ord_le318720350_set_a @ B2 @ C2 )
       => ( ord_le318720350_set_a @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_136_order_Otrans,axiom,
    ! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
      ( ( ord_le2103863998_set_a @ A2 @ B2 )
     => ( ( ord_le2103863998_set_a @ B2 @ C2 )
       => ( ord_le2103863998_set_a @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_137_order_Otrans,axiom,
    ! [A2: $o > set_a,B2: $o > set_a,C2: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
       => ( ord_less_eq_o_set_a @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_138_order_Otrans,axiom,
    ! [A2: set_a,B2: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_139_eq__refl,axiom,
    ! [X3: set_Pr204296108_a_a_e,Y3: set_Pr204296108_a_a_e] :
      ( ( X3 = Y3 )
     => ( ord_le700723212_a_a_e @ X3 @ Y3 ) ) ).

% eq_refl
thf(fact_140_eq__refl,axiom,
    ! [X3: set_set_a,Y3: set_set_a] :
      ( ( X3 = Y3 )
     => ( ord_le318720350_set_a @ X3 @ Y3 ) ) ).

% eq_refl
thf(fact_141_eq__refl,axiom,
    ! [X3: set_set_set_a,Y3: set_set_set_a] :
      ( ( X3 = Y3 )
     => ( ord_le2103863998_set_a @ X3 @ Y3 ) ) ).

% eq_refl
thf(fact_142_eq__refl,axiom,
    ! [X3: $o > set_a,Y3: $o > set_a] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_o_set_a @ X3 @ Y3 ) ) ).

% eq_refl
thf(fact_143_eq__refl,axiom,
    ! [X3: set_a,Y3: set_a] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_set_a @ X3 @ Y3 ) ) ).

% eq_refl
thf(fact_144_antisym,axiom,
    ! [X3: set_Pr204296108_a_a_e,Y3: set_Pr204296108_a_a_e] :
      ( ( ord_le700723212_a_a_e @ X3 @ Y3 )
     => ( ( ord_le700723212_a_a_e @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% antisym
thf(fact_145_antisym,axiom,
    ! [X3: set_set_a,Y3: set_set_a] :
      ( ( ord_le318720350_set_a @ X3 @ Y3 )
     => ( ( ord_le318720350_set_a @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% antisym
thf(fact_146_antisym,axiom,
    ! [X3: set_set_set_a,Y3: set_set_set_a] :
      ( ( ord_le2103863998_set_a @ X3 @ Y3 )
     => ( ( ord_le2103863998_set_a @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% antisym
thf(fact_147_antisym,axiom,
    ! [X3: $o > set_a,Y3: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ X3 @ Y3 )
     => ( ( ord_less_eq_o_set_a @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% antisym
thf(fact_148_antisym,axiom,
    ! [X3: set_a,Y3: set_a] :
      ( ( ord_less_eq_set_a @ X3 @ Y3 )
     => ( ( ord_less_eq_set_a @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% antisym
thf(fact_149_eq__iff,axiom,
    ( ( ^ [Y2: set_Pr204296108_a_a_e,Z: set_Pr204296108_a_a_e] : ( Y2 = Z ) )
    = ( ^ [X4: set_Pr204296108_a_a_e,Y4: set_Pr204296108_a_a_e] :
          ( ( ord_le700723212_a_a_e @ X4 @ Y4 )
          & ( ord_le700723212_a_a_e @ Y4 @ X4 ) ) ) ) ).

% eq_iff
thf(fact_150_eq__iff,axiom,
    ( ( ^ [Y2: set_set_a,Z: set_set_a] : ( Y2 = Z ) )
    = ( ^ [X4: set_set_a,Y4: set_set_a] :
          ( ( ord_le318720350_set_a @ X4 @ Y4 )
          & ( ord_le318720350_set_a @ Y4 @ X4 ) ) ) ) ).

% eq_iff
thf(fact_151_eq__iff,axiom,
    ( ( ^ [Y2: set_set_set_a,Z: set_set_set_a] : ( Y2 = Z ) )
    = ( ^ [X4: set_set_set_a,Y4: set_set_set_a] :
          ( ( ord_le2103863998_set_a @ X4 @ Y4 )
          & ( ord_le2103863998_set_a @ Y4 @ X4 ) ) ) ) ).

% eq_iff
thf(fact_152_eq__iff,axiom,
    ( ( ^ [Y2: $o > set_a,Z: $o > set_a] : ( Y2 = Z ) )
    = ( ^ [X4: $o > set_a,Y4: $o > set_a] :
          ( ( ord_less_eq_o_set_a @ X4 @ Y4 )
          & ( ord_less_eq_o_set_a @ Y4 @ X4 ) ) ) ) ).

% eq_iff
thf(fact_153_eq__iff,axiom,
    ( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
    = ( ^ [X4: set_a,Y4: set_a] :
          ( ( ord_less_eq_set_a @ X4 @ Y4 )
          & ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).

% eq_iff
thf(fact_154_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_a,Y: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_155_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_set_a,C2: set_set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_a,Y: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y )
             => ( ord_le318720350_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_le318720350_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_156_ord__le__eq__subst,axiom,
    ! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C2: set_a] :
      ( ( ord_le318720350_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_set_a,Y: set_set_a] :
              ( ( ord_le318720350_set_a @ X2 @ Y )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_157_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_set_set_a,C2: set_set_set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_a,Y: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y )
             => ( ord_le2103863998_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_le2103863998_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_158_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > $o > set_a,C2: $o > set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_a,Y: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y )
             => ( ord_less_eq_o_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_o_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_159_ord__le__eq__subst,axiom,
    ! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_a,C2: set_set_a] :
      ( ( ord_le318720350_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_set_a,Y: set_set_a] :
              ( ( ord_le318720350_set_a @ X2 @ Y )
             => ( ord_le318720350_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_le318720350_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_160_ord__le__eq__subst,axiom,
    ! [A2: set_set_set_a,B2: set_set_set_a,F: set_set_set_a > set_a,C2: set_a] :
      ( ( ord_le2103863998_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_set_set_a,Y: set_set_set_a] :
              ( ( ord_le2103863998_set_a @ X2 @ Y )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_161_ord__le__eq__subst,axiom,
    ! [A2: $o > set_a,B2: $o > set_a,F: ( $o > set_a ) > set_a,C2: set_a] :
      ( ( ord_less_eq_o_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: $o > set_a,Y: $o > set_a] :
              ( ( ord_less_eq_o_set_a @ X2 @ Y )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_162_ord__le__eq__subst,axiom,
    ! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_set_a,C2: set_set_set_a] :
      ( ( ord_le318720350_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_set_a,Y: set_set_a] :
              ( ( ord_le318720350_set_a @ X2 @ Y )
             => ( ord_le2103863998_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_le2103863998_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_163_ord__le__eq__subst,axiom,
    ! [A2: set_set_a,B2: set_set_a,F: set_set_a > $o > set_a,C2: $o > set_a] :
      ( ( ord_le318720350_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_set_a,Y: set_set_a] :
              ( ( ord_le318720350_set_a @ X2 @ Y )
             => ( ord_less_eq_o_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_o_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_164_ord__eq__le__subst,axiom,
    ! [A2: set_Pr204296108_a_a_e,F: ( $o > set_a ) > set_Pr204296108_a_a_e,B2: $o > set_a,C2: $o > set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
       => ( ! [X2: $o > set_a,Y: $o > set_a] :
              ( ( ord_less_eq_o_set_a @ X2 @ Y )
             => ( ord_le700723212_a_a_e @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_le700723212_a_a_e @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_165_ord__eq__le__subst,axiom,
    ! [A2: set_set_a,F: ( $o > set_a ) > set_set_a,B2: $o > set_a,C2: $o > set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
       => ( ! [X2: $o > set_a,Y: $o > set_a] :
              ( ( ord_less_eq_o_set_a @ X2 @ Y )
             => ( ord_le318720350_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_le318720350_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_166_ord__eq__le__subst,axiom,
    ! [A2: set_set_set_a,F: ( $o > set_a ) > set_set_set_a,B2: $o > set_a,C2: $o > set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
       => ( ! [X2: $o > set_a,Y: $o > set_a] :
              ( ( ord_less_eq_o_set_a @ X2 @ Y )
             => ( ord_le2103863998_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_le2103863998_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_167_ord__eq__le__subst,axiom,
    ! [A2: $o > set_a,F: ( $o > set_a ) > $o > set_a,B2: $o > set_a,C2: $o > set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_o_set_a @ B2 @ C2 )
       => ( ! [X2: $o > set_a,Y: $o > set_a] :
              ( ( ord_less_eq_o_set_a @ X2 @ Y )
             => ( ord_less_eq_o_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_o_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_168_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ! [X2: set_a,Y: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_169_order__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
       => ( ! [X2: set_a,Y: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_170_order__subst1,axiom,
    ! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ! [X2: set_a,Y: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_171_mem__Collect__eq,axiom,
    ! [A2: set_a,P2: set_a > $o] :
      ( ( member_set_a @ A2 @ ( collect_set_a @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_172_mem__Collect__eq,axiom,
    ! [A2: produc715398134_a_a_e,P2: produc715398134_a_a_e > $o] :
      ( ( member1258382221_a_a_e @ A2 @ ( collec1369693387_a_a_e @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_173_Collect__mem__eq,axiom,
    ! [A: set_set_a] :
      ( ( collect_set_a
        @ ^ [X4: set_a] : ( member_set_a @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_174_Collect__mem__eq,axiom,
    ! [A: set_Pr204296108_a_a_e] :
      ( ( collec1369693387_a_a_e
        @ ^ [X4: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_175_rev__image__eqI,axiom,
    ! [X3: set_a,A: set_set_a,B2: set_a,F: set_a > set_a] :
      ( ( member_set_a @ X3 @ A )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_176_rev__image__eqI,axiom,
    ! [X3: set_a,A: set_set_a,B2: produc715398134_a_a_e,F: set_a > produc715398134_a_a_e] :
      ( ( member_set_a @ X3 @ A )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member1258382221_a_a_e @ B2 @ ( image_210900495_a_a_e @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_177_rev__image__eqI,axiom,
    ! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,B2: set_a,F: produc715398134_a_a_e > set_a] :
      ( ( member1258382221_a_a_e @ X3 @ A )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_178_rev__image__eqI,axiom,
    ! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,B2: produc715398134_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e] :
      ( ( member1258382221_a_a_e @ X3 @ A )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member1258382221_a_a_e @ B2 @ ( image_625620211_a_a_e @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_179_ball__imageD,axiom,
    ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,P2: set_a > $o] :
      ( ! [X2: set_a] :
          ( ( member_set_a @ X2 @ ( image_64240033_set_a @ F @ A ) )
         => ( P2 @ X2 ) )
     => ! [X: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ X @ A )
         => ( P2 @ ( F @ X ) ) ) ) ).

% ball_imageD
thf(fact_180_image__cong,axiom,
    ! [M: set_Pr204296108_a_a_e,N: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a,G: produc715398134_a_a_e > set_a] :
      ( ( M = N )
     => ( ! [X2: produc715398134_a_a_e] :
            ( ( member1258382221_a_a_e @ X2 @ N )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_64240033_set_a @ F @ M )
          = ( image_64240033_set_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_181_bex__imageD,axiom,
    ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,P2: set_a > $o] :
      ( ? [X: set_a] :
          ( ( member_set_a @ X @ ( image_64240033_set_a @ F @ A ) )
          & ( P2 @ X ) )
     => ? [X2: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ X2 @ A )
          & ( P2 @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_182_image__iff,axiom,
    ! [Z2: set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
      ( ( member_set_a @ Z2 @ ( image_64240033_set_a @ F @ A ) )
      = ( ? [X4: produc715398134_a_a_e] :
            ( ( member1258382221_a_a_e @ X4 @ A )
            & ( Z2
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_183_imageI,axiom,
    ! [X3: set_a,A: set_set_a,F: set_a > set_a] :
      ( ( member_set_a @ X3 @ A )
     => ( member_set_a @ ( F @ X3 ) @ ( image_set_a_set_a @ F @ A ) ) ) ).

% imageI
thf(fact_184_imageI,axiom,
    ! [X3: set_a,A: set_set_a,F: set_a > produc715398134_a_a_e] :
      ( ( member_set_a @ X3 @ A )
     => ( member1258382221_a_a_e @ ( F @ X3 ) @ ( image_210900495_a_a_e @ F @ A ) ) ) ).

% imageI
thf(fact_185_imageI,axiom,
    ! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a] :
      ( ( member1258382221_a_a_e @ X3 @ A )
     => ( member_set_a @ ( F @ X3 ) @ ( image_64240033_set_a @ F @ A ) ) ) ).

% imageI
thf(fact_186_imageI,axiom,
    ! [X3: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e] :
      ( ( member1258382221_a_a_e @ X3 @ A )
     => ( member1258382221_a_a_e @ ( F @ X3 ) @ ( image_625620211_a_a_e @ F @ A ) ) ) ).

% imageI
thf(fact_187_Collect__mono__iff,axiom,
    ! [P2: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
      = ( ! [X4: a] :
            ( ( P2 @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_188_set__eq__subset,axiom,
    ( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ A4 @ B4 )
          & ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_189_subset__trans,axiom,
    ! [A: set_a,B: set_a,C3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ C3 )
       => ( ord_less_eq_set_a @ A @ C3 ) ) ) ).

% subset_trans
thf(fact_190_Collect__mono,axiom,
    ! [P2: a > $o,Q: a > $o] :
      ( ! [X2: a] :
          ( ( P2 @ X2 )
         => ( Q @ X2 ) )
     => ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).

% Collect_mono
thf(fact_191_subset__iff,axiom,
    ( ord_le318720350_set_a
    = ( ^ [A4: set_set_a,B4: set_set_a] :
        ! [T: set_a] :
          ( ( member_set_a @ T @ A4 )
         => ( member_set_a @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_192_subset__iff,axiom,
    ( ord_le700723212_a_a_e
    = ( ^ [A4: set_Pr204296108_a_a_e,B4: set_Pr204296108_a_a_e] :
        ! [T: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ T @ A4 )
         => ( member1258382221_a_a_e @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_193_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
        ! [T: a] :
          ( ( member_a @ T @ A4 )
         => ( member_a @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_194_equalityD2,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ( ord_less_eq_set_a @ B @ A ) ) ).

% equalityD2
thf(fact_195_subset__eq,axiom,
    ( ord_le318720350_set_a
    = ( ^ [A4: set_set_a,B4: set_set_a] :
        ! [X4: set_a] :
          ( ( member_set_a @ X4 @ A4 )
         => ( member_set_a @ X4 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_196_subset__eq,axiom,
    ( ord_le700723212_a_a_e
    = ( ^ [A4: set_Pr204296108_a_a_e,B4: set_Pr204296108_a_a_e] :
        ! [X4: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ X4 @ A4 )
         => ( member1258382221_a_a_e @ X4 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_197_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
        ! [X4: a] :
          ( ( member_a @ X4 @ A4 )
         => ( member_a @ X4 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_198_equalityE,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_a @ A @ B )
         => ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).

% equalityE
thf(fact_199_subsetD,axiom,
    ! [A: set_set_a,B: set_set_a,C2: set_a] :
      ( ( ord_le318720350_set_a @ A @ B )
     => ( ( member_set_a @ C2 @ A )
       => ( member_set_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_200_subsetD,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,C2: produc715398134_a_a_e] :
      ( ( ord_le700723212_a_a_e @ A @ B )
     => ( ( member1258382221_a_a_e @ C2 @ A )
       => ( member1258382221_a_a_e @ C2 @ B ) ) ) ).

% subsetD
thf(fact_201_subsetD,axiom,
    ! [A: set_a,B: set_a,C2: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ C2 @ A )
       => ( member_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_202_in__mono,axiom,
    ! [A: set_set_a,B: set_set_a,X3: set_a] :
      ( ( ord_le318720350_set_a @ A @ B )
     => ( ( member_set_a @ X3 @ A )
       => ( member_set_a @ X3 @ B ) ) ) ).

% in_mono
thf(fact_203_in__mono,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
      ( ( ord_le700723212_a_a_e @ A @ B )
     => ( ( member1258382221_a_a_e @ X3 @ A )
       => ( member1258382221_a_a_e @ X3 @ B ) ) ) ).

% in_mono
thf(fact_204_in__mono,axiom,
    ! [A: set_a,B: set_a,X3: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ X3 @ A )
       => ( member_a @ X3 @ B ) ) ) ).

% in_mono
thf(fact_205_prod_Oinject,axiom,
    ! [X1: set_a,X22: a > e,Y1: set_a,Y22: a > e] :
      ( ( ( produc1134456688_a_a_e @ X1 @ X22 )
        = ( produc1134456688_a_a_e @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_206_old_Oprod_Oinject,axiom,
    ! [A2: set_a,B2: a > e,A5: set_a,B5: a > e] :
      ( ( ( produc1134456688_a_a_e @ A2 @ B2 )
        = ( produc1134456688_a_a_e @ A5 @ B5 ) )
      = ( ( A2 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
thf(fact_207_eq__fst__iff,axiom,
    ! [A2: set_a,P: produc715398134_a_a_e] :
      ( ( A2
        = ( produc1040928388_a_a_e @ P ) )
      = ( ? [B3: a > e] :
            ( P
            = ( produc1134456688_a_a_e @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_208_fst__conv,axiom,
    ! [X1: set_a,X22: a > e] :
      ( ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_209_fst__eqD,axiom,
    ! [X3: set_a,Y3: a > e,A2: set_a] :
      ( ( ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) )
        = A2 )
     => ( X3 = A2 ) ) ).

% fst_eqD
thf(fact_210_fstI,axiom,
    ! [X3: produc715398134_a_a_e,Y3: set_a,Z2: a > e] :
      ( ( X3
        = ( produc1134456688_a_a_e @ Y3 @ Z2 ) )
     => ( ( produc1040928388_a_a_e @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_211_all__subset__image,axiom,
    ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,P2: set_set_a > $o] :
      ( ( ! [B4: set_set_a] :
            ( ( ord_le318720350_set_a @ B4 @ ( image_64240033_set_a @ F @ A ) )
           => ( P2 @ B4 ) ) )
      = ( ! [B4: set_Pr204296108_a_a_e] :
            ( ( ord_le700723212_a_a_e @ B4 @ A )
           => ( P2 @ ( image_64240033_set_a @ F @ B4 ) ) ) ) ) ).

% all_subset_image
thf(fact_212_all__subset__image,axiom,
    ! [F: a > a,A: set_a,P2: set_a > $o] :
      ( ( ! [B4: set_a] :
            ( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A ) )
           => ( P2 @ B4 ) ) )
      = ( ! [B4: set_a] :
            ( ( ord_less_eq_set_a @ B4 @ A )
           => ( P2 @ ( image_a_a @ F @ B4 ) ) ) ) ) ).

% all_subset_image
thf(fact_213_Greatest__equality,axiom,
    ! [P2: set_a > $o,X3: set_a] :
      ( ( P2 @ X3 )
     => ( ! [Y: set_a] :
            ( ( P2 @ Y )
           => ( ord_less_eq_set_a @ Y @ X3 ) )
       => ( ( order_Greatest_set_a @ P2 )
          = X3 ) ) ) ).

% Greatest_equality
thf(fact_214_GreatestI2__order,axiom,
    ! [P2: set_a > $o,X3: set_a,Q: set_a > $o] :
      ( ( P2 @ X3 )
     => ( ! [Y: set_a] :
            ( ( P2 @ Y )
           => ( ord_less_eq_set_a @ Y @ X3 ) )
       => ( ! [X2: set_a] :
              ( ( P2 @ X2 )
             => ( ! [Y5: set_a] :
                    ( ( P2 @ Y5 )
                   => ( ord_less_eq_set_a @ Y5 @ X2 ) )
               => ( Q @ X2 ) ) )
         => ( Q @ ( order_Greatest_set_a @ P2 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_215_old_Oprod_Oinducts,axiom,
    ! [P2: produc715398134_a_a_e > $o,Prod: produc715398134_a_a_e] :
      ( ! [A6: set_a,B6: a > e] : ( P2 @ ( produc1134456688_a_a_e @ A6 @ B6 ) )
     => ( P2 @ Prod ) ) ).

% old.prod.inducts
thf(fact_216_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc715398134_a_a_e] :
      ~ ! [A6: set_a,B6: a > e] :
          ( Y3
         != ( produc1134456688_a_a_e @ A6 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_217_Pair__inject,axiom,
    ! [A2: set_a,B2: a > e,A5: set_a,B5: a > e] :
      ( ( ( produc1134456688_a_a_e @ A2 @ B2 )
        = ( produc1134456688_a_a_e @ A5 @ B5 ) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
thf(fact_218_prod__cases,axiom,
    ! [P2: produc715398134_a_a_e > $o,P: produc715398134_a_a_e] :
      ( ! [A6: set_a,B6: a > e] : ( P2 @ ( produc1134456688_a_a_e @ A6 @ B6 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_219_image__Fpow__mono,axiom,
    ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
      ( ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B )
     => ( ord_le2103863998_set_a @ ( image_1441218763_set_a @ ( image_64240033_set_a @ F ) @ ( finite33065488_a_a_e @ A ) ) @ ( finite_Fpow_set_a @ B ) ) ) ).

% image_Fpow_mono
thf(fact_220_not__sub,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ~ ( ord_le318720350_set_a @ A @ B )
     => ? [A6: set_a] :
          ( ( member_set_a @ A6 @ A )
          & ~ ( member_set_a @ A6 @ B ) ) ) ).

% not_sub
thf(fact_221_not__sub,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
      ( ~ ( ord_le700723212_a_a_e @ A @ B )
     => ? [A6: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ A6 @ A )
          & ~ ( member1258382221_a_a_e @ A6 @ B ) ) ) ).

% not_sub
thf(fact_222_not__sub,axiom,
    ! [A: set_a,B: set_a] :
      ( ~ ( ord_less_eq_set_a @ A @ B )
     => ? [A6: a] :
          ( ( member_a @ A6 @ A )
          & ~ ( member_a @ A6 @ B ) ) ) ).

% not_sub
thf(fact_223_eqsets__sub,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% eqsets_sub
thf(fact_224_sub__which1,axiom,
    ! [A: set_set_a,B: set_set_a,X3: set_a] :
      ( ( ( ord_le318720350_set_a @ A @ B )
        | ( ord_le318720350_set_a @ B @ A ) )
     => ( ( member_set_a @ X3 @ A )
       => ( ~ ( member_set_a @ X3 @ B )
         => ( ord_le318720350_set_a @ B @ A ) ) ) ) ).

% sub_which1
thf(fact_225_sub__which1,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
      ( ( ( ord_le700723212_a_a_e @ A @ B )
        | ( ord_le700723212_a_a_e @ B @ A ) )
     => ( ( member1258382221_a_a_e @ X3 @ A )
       => ( ~ ( member1258382221_a_a_e @ X3 @ B )
         => ( ord_le700723212_a_a_e @ B @ A ) ) ) ) ).

% sub_which1
thf(fact_226_sub__which1,axiom,
    ! [A: set_a,B: set_a,X3: a] :
      ( ( ( ord_less_eq_set_a @ A @ B )
        | ( ord_less_eq_set_a @ B @ A ) )
     => ( ( member_a @ X3 @ A )
       => ( ~ ( member_a @ X3 @ B )
         => ( ord_less_eq_set_a @ B @ A ) ) ) ) ).

% sub_which1
thf(fact_227_Fpow__mono,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_le318720350_set_a @ ( finite_Fpow_a @ A ) @ ( finite_Fpow_a @ B ) ) ) ).

% Fpow_mono
thf(fact_228_le__convert,axiom,
    ! [A2: set_a,B2: set_a,C2: set_a] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_a @ A2 @ C2 )
       => ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).

% le_convert
thf(fact_229_ge__convert,axiom,
    ! [A2: set_a,B2: set_a,C2: set_a] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_a @ C2 @ A2 )
       => ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).

% ge_convert
thf(fact_230_mem__in__image3,axiom,
    ! [B2: set_a,F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e] :
      ( ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ A ) )
     => ? [X2: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ X2 @ A )
          & ( B2
            = ( F @ X2 ) ) ) ) ).

% mem_in_image3
thf(fact_231_mem__in__image2,axiom,
    ! [A2: set_a,A: set_set_a,F: set_a > set_a] :
      ( ( member_set_a @ A2 @ A )
     => ( member_set_a @ ( F @ A2 ) @ ( image_set_a_set_a @ F @ A ) ) ) ).

% mem_in_image2
thf(fact_232_mem__in__image2,axiom,
    ! [A2: set_a,A: set_set_a,F: set_a > produc715398134_a_a_e] :
      ( ( member_set_a @ A2 @ A )
     => ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_210900495_a_a_e @ F @ A ) ) ) ).

% mem_in_image2
thf(fact_233_mem__in__image2,axiom,
    ! [A2: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a] :
      ( ( member1258382221_a_a_e @ A2 @ A )
     => ( member_set_a @ ( F @ A2 ) @ ( image_64240033_set_a @ F @ A ) ) ) ).

% mem_in_image2
thf(fact_234_mem__in__image2,axiom,
    ! [A2: produc715398134_a_a_e,A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e] :
      ( ( member1258382221_a_a_e @ A2 @ A )
     => ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_625620211_a_a_e @ F @ A ) ) ) ).

% mem_in_image2
thf(fact_235_mem__in__image1,axiom,
    ! [A: set_set_a,F: set_a > set_a,B: set_set_a,A2: set_a] :
      ( ! [X2: set_a] :
          ( ( member_set_a @ X2 @ A )
         => ( member_set_a @ ( F @ X2 ) @ B ) )
     => ( ( member_set_a @ A2 @ A )
       => ( member_set_a @ ( F @ A2 ) @ ( image_set_a_set_a @ F @ A ) ) ) ) ).

% mem_in_image1
thf(fact_236_mem__in__image1,axiom,
    ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > set_a,B: set_set_a,A2: produc715398134_a_a_e] :
      ( ! [X2: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ X2 @ A )
         => ( member_set_a @ ( F @ X2 ) @ B ) )
     => ( ( member1258382221_a_a_e @ A2 @ A )
       => ( member_set_a @ ( F @ A2 ) @ ( image_64240033_set_a @ F @ A ) ) ) ) ).

% mem_in_image1
thf(fact_237_mem__in__image1,axiom,
    ! [A: set_set_a,F: set_a > produc715398134_a_a_e,B: set_Pr204296108_a_a_e,A2: set_a] :
      ( ! [X2: set_a] :
          ( ( member_set_a @ X2 @ A )
         => ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
     => ( ( member_set_a @ A2 @ A )
       => ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_210900495_a_a_e @ F @ A ) ) ) ) ).

% mem_in_image1
thf(fact_238_mem__in__image1,axiom,
    ! [A: set_Pr204296108_a_a_e,F: produc715398134_a_a_e > produc715398134_a_a_e,B: set_Pr204296108_a_a_e,A2: produc715398134_a_a_e] :
      ( ! [X2: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ X2 @ A )
         => ( member1258382221_a_a_e @ ( F @ X2 ) @ B ) )
     => ( ( member1258382221_a_a_e @ A2 @ A )
       => ( member1258382221_a_a_e @ ( F @ A2 ) @ ( image_625620211_a_a_e @ F @ A ) ) ) ) ).

% mem_in_image1
thf(fact_239_proper__subset,axiom,
    ! [A: set_set_a,B: set_set_a,X3: set_a] :
      ( ( ord_le318720350_set_a @ A @ B )
     => ( ~ ( member_set_a @ X3 @ A )
       => ( ( member_set_a @ X3 @ B )
         => ( A != B ) ) ) ) ).

% proper_subset
thf(fact_240_proper__subset,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
      ( ( ord_le700723212_a_a_e @ A @ B )
     => ( ~ ( member1258382221_a_a_e @ X3 @ A )
       => ( ( member1258382221_a_a_e @ X3 @ B )
         => ( A != B ) ) ) ) ).

% proper_subset
thf(fact_241_proper__subset,axiom,
    ! [A: set_a,B: set_a,X3: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ~ ( member_a @ X3 @ A )
       => ( ( member_a @ X3 @ B )
         => ( A != B ) ) ) ) ).

% proper_subset
thf(fact_242_not__subseteq,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ~ ( ord_le318720350_set_a @ A @ B )
     => ? [X2: set_a] :
          ( ( member_set_a @ X2 @ A )
          & ~ ( member_set_a @ X2 @ B ) ) ) ).

% not_subseteq
thf(fact_243_not__subseteq,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
      ( ~ ( ord_le700723212_a_a_e @ A @ B )
     => ? [X2: produc715398134_a_a_e] :
          ( ( member1258382221_a_a_e @ X2 @ A )
          & ~ ( member1258382221_a_a_e @ X2 @ B ) ) ) ).

% not_subseteq
thf(fact_244_not__subseteq,axiom,
    ! [A: set_a,B: set_a] :
      ( ~ ( ord_less_eq_set_a @ A @ B )
     => ? [X2: a] :
          ( ( member_a @ X2 @ A )
          & ~ ( member_a @ X2 @ B ) ) ) ).

% not_subseteq
thf(fact_245_subset__self,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).

% subset_self
thf(fact_246_sets__not__eq,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( A != B )
     => ( ( ord_le318720350_set_a @ B @ A )
       => ? [X2: set_a] :
            ( ( member_set_a @ X2 @ A )
            & ~ ( member_set_a @ X2 @ B ) ) ) ) ).

% sets_not_eq
thf(fact_247_sets__not__eq,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e] :
      ( ( A != B )
     => ( ( ord_le700723212_a_a_e @ B @ A )
       => ? [X2: produc715398134_a_a_e] :
            ( ( member1258382221_a_a_e @ X2 @ A )
            & ~ ( member1258382221_a_a_e @ X2 @ B ) ) ) ) ).

% sets_not_eq
thf(fact_248_sets__not__eq,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A != B )
     => ( ( ord_less_eq_set_a @ B @ A )
       => ? [X2: a] :
            ( ( member_a @ X2 @ A )
            & ~ ( member_a @ X2 @ B ) ) ) ) ).

% sets_not_eq
thf(fact_249_sub__which2,axiom,
    ! [A: set_set_a,B: set_set_a,X3: set_a] :
      ( ( ( ord_le318720350_set_a @ A @ B )
        | ( ord_le318720350_set_a @ B @ A ) )
     => ( ~ ( member_set_a @ X3 @ A )
       => ( ( member_set_a @ X3 @ B )
         => ( ord_le318720350_set_a @ A @ B ) ) ) ) ).

% sub_which2
thf(fact_250_sub__which2,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,X3: produc715398134_a_a_e] :
      ( ( ( ord_le700723212_a_a_e @ A @ B )
        | ( ord_le700723212_a_a_e @ B @ A ) )
     => ( ~ ( member1258382221_a_a_e @ X3 @ A )
       => ( ( member1258382221_a_a_e @ X3 @ B )
         => ( ord_le700723212_a_a_e @ A @ B ) ) ) ) ).

% sub_which2
thf(fact_251_sub__which2,axiom,
    ! [A: set_a,B: set_a,X3: a] :
      ( ( ( ord_less_eq_set_a @ A @ B )
        | ( ord_less_eq_set_a @ B @ A ) )
     => ( ~ ( member_a @ X3 @ A )
       => ( ( member_a @ X3 @ B )
         => ( ord_less_eq_set_a @ A @ B ) ) ) ) ).

% sub_which2
thf(fact_252_image__Pow__mono,axiom,
    ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
      ( ( ord_le318720350_set_a @ ( image_64240033_set_a @ F @ A ) @ B )
     => ( ord_le2103863998_set_a @ ( image_1441218763_set_a @ ( image_64240033_set_a @ F ) @ ( pow_Pr328302361_a_a_e @ A ) ) @ ( pow_set_a @ B ) ) ) ).

% image_Pow_mono
thf(fact_253_Sup_OSUP__cong,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,C3: produc715398134_a_a_e > set_a,D: produc715398134_a_a_e > set_a,Sup: set_set_a > set_a] :
      ( ( A = B )
     => ( ! [X2: produc715398134_a_a_e] :
            ( ( member1258382221_a_a_e @ X2 @ B )
           => ( ( C3 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Sup @ ( image_64240033_set_a @ C3 @ A ) )
          = ( Sup @ ( image_64240033_set_a @ D @ B ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_254_Inf_OINF__cong,axiom,
    ! [A: set_Pr204296108_a_a_e,B: set_Pr204296108_a_a_e,C3: produc715398134_a_a_e > set_a,D: produc715398134_a_a_e > set_a,Inf: set_set_a > set_a] :
      ( ( A = B )
     => ( ! [X2: produc715398134_a_a_e] :
            ( ( member1258382221_a_a_e @ X2 @ B )
           => ( ( C3 @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Inf @ ( image_64240033_set_a @ C3 @ A ) )
          = ( Inf @ ( image_64240033_set_a @ D @ B ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_255_subrelI,axiom,
    ! [R: set_Pr204296108_a_a_e,S: set_Pr204296108_a_a_e] :
      ( ! [X2: set_a,Y: a > e] :
          ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ R )
         => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ S ) )
     => ( ord_le700723212_a_a_e @ R @ S ) ) ).

% subrelI
thf(fact_256_ssubst__Pair__rhs,axiom,
    ! [R: set_a,S: a > e,R2: set_Pr204296108_a_a_e,S2: a > e] :
      ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_257_PowI,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( member_set_a @ A @ ( pow_a @ B ) ) ) ).

% PowI
thf(fact_258_Pow__iff,axiom,
    ! [A: set_a,B: set_a] :
      ( ( member_set_a @ A @ ( pow_a @ B ) )
      = ( ord_less_eq_set_a @ A @ B ) ) ).

% Pow_iff
thf(fact_259_Pow__top,axiom,
    ! [A: set_a] : ( member_set_a @ A @ ( pow_a @ A ) ) ).

% Pow_top
thf(fact_260_PowD,axiom,
    ! [A: set_a,B: set_a] :
      ( ( member_set_a @ A @ ( pow_a @ B ) )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% PowD
thf(fact_261_Pow__mono,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_le318720350_set_a @ ( pow_a @ A ) @ ( pow_a @ B ) ) ) ).

% Pow_mono
thf(fact_262_image__Pow__surj,axiom,
    ! [F: produc715398134_a_a_e > set_a,A: set_Pr204296108_a_a_e,B: set_set_a] :
      ( ( ( image_64240033_set_a @ F @ A )
        = B )
     => ( ( image_1441218763_set_a @ ( image_64240033_set_a @ F ) @ ( pow_Pr328302361_a_a_e @ A ) )
        = ( pow_set_a @ B ) ) ) ).

% image_Pow_surj
thf(fact_263_le__rel__bool__arg__iff,axiom,
    ( ord_less_eq_o_set_a
    = ( ^ [X5: $o > set_a,Y6: $o > set_a] :
          ( ( ord_less_eq_set_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
          & ( ord_less_eq_set_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_264_curryI,axiom,
    ! [F: produc715398134_a_a_e > $o,A2: set_a,B2: a > e] :
      ( ( F @ ( produc1134456688_a_a_e @ A2 @ B2 ) )
     => ( produc952037978_a_e_o @ F @ A2 @ B2 ) ) ).

% curryI
thf(fact_265_curryD,axiom,
    ! [F: produc715398134_a_a_e > $o,A2: set_a,B2: a > e] :
      ( ( produc952037978_a_e_o @ F @ A2 @ B2 )
     => ( F @ ( produc1134456688_a_a_e @ A2 @ B2 ) ) ) ).

% curryD
thf(fact_266_curryE,axiom,
    ! [F: produc715398134_a_a_e > $o,A2: set_a,B2: a > e] :
      ( ( produc952037978_a_e_o @ F @ A2 @ B2 )
     => ( F @ ( produc1134456688_a_a_e @ A2 @ B2 ) ) ) ).

% curryE
thf(fact_267_fst__eq__Domain,axiom,
    ! [R2: set_Pr204296108_a_a_e] :
      ( ( image_64240033_set_a @ produc1040928388_a_a_e @ R2 )
      = ( domain_set_a_a_e @ R2 ) ) ).

% fst_eq_Domain
thf(fact_268_Domain__fst,axiom,
    ( domain_set_a_a_e
    = ( image_64240033_set_a @ produc1040928388_a_a_e ) ) ).

% Domain_fst
thf(fact_269_RangeE,axiom,
    ! [B2: a > e,R: set_Pr204296108_a_a_e] :
      ( ( member_a_e @ B2 @ ( range_set_a_a_e @ R ) )
     => ~ ! [A6: set_a] :
            ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B2 ) @ R ) ) ).

% RangeE
thf(fact_270_Range__iff,axiom,
    ! [A2: a > e,R: set_Pr204296108_a_a_e] :
      ( ( member_a_e @ A2 @ ( range_set_a_a_e @ R ) )
      = ( ? [Y4: set_a] : ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ Y4 @ A2 ) @ R ) ) ) ).

% Range_iff
thf(fact_271_Range_Ocases,axiom,
    ! [A2: a > e,R: set_Pr204296108_a_a_e] :
      ( ( member_a_e @ A2 @ ( range_set_a_a_e @ R ) )
     => ~ ! [A6: set_a] :
            ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ A2 ) @ R ) ) ).

% Range.cases
thf(fact_272_Range_Osimps,axiom,
    ! [A2: a > e,R: set_Pr204296108_a_a_e] :
      ( ( member_a_e @ A2 @ ( range_set_a_a_e @ R ) )
      = ( ? [A3: set_a,B3: a > e] :
            ( ( A2 = B3 )
            & ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A3 @ B3 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_273_Range_Ointros,axiom,
    ! [A2: set_a,B2: a > e,R: set_Pr204296108_a_a_e] :
      ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R )
     => ( member_a_e @ B2 @ ( range_set_a_a_e @ R ) ) ) ).

% Range.intros
thf(fact_274_Range_Oinducts,axiom,
    ! [X3: a > e,R: set_Pr204296108_a_a_e,P2: ( a > e ) > $o] :
      ( ( member_a_e @ X3 @ ( range_set_a_a_e @ R ) )
     => ( ! [A6: set_a,B6: a > e] :
            ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B6 ) @ R )
           => ( P2 @ B6 ) )
       => ( P2 @ X3 ) ) ) ).

% Range.inducts
thf(fact_275_DomainE,axiom,
    ! [A2: set_a,R: set_Pr204296108_a_a_e] :
      ( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
     => ~ ! [B6: a > e] :
            ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B6 ) @ R ) ) ).

% DomainE
thf(fact_276_Domain__iff,axiom,
    ! [A2: set_a,R: set_Pr204296108_a_a_e] :
      ( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
      = ( ? [Y4: a > e] : ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ Y4 ) @ R ) ) ) ).

% Domain_iff
thf(fact_277_Domain_Ocases,axiom,
    ! [A2: set_a,R: set_Pr204296108_a_a_e] :
      ( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
     => ~ ! [B6: a > e] :
            ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B6 ) @ R ) ) ).

% Domain.cases
thf(fact_278_Domain_Osimps,axiom,
    ! [A2: set_a,R: set_Pr204296108_a_a_e] :
      ( ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) )
      = ( ? [A3: set_a,B3: a > e] :
            ( ( A2 = A3 )
            & ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A3 @ B3 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_279_Domain_ODomainI,axiom,
    ! [A2: set_a,B2: a > e,R: set_Pr204296108_a_a_e] :
      ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R )
     => ( member_set_a @ A2 @ ( domain_set_a_a_e @ R ) ) ) ).

% Domain.DomainI
thf(fact_280_Domain_Oinducts,axiom,
    ! [X3: set_a,R: set_Pr204296108_a_a_e,P2: set_a > $o] :
      ( ( member_set_a @ X3 @ ( domain_set_a_a_e @ R ) )
     => ( ! [A6: set_a,B6: a > e] :
            ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B6 ) @ R )
           => ( P2 @ A6 ) )
       => ( P2 @ X3 ) ) ) ).

% Domain.inducts
thf(fact_281_pair__in__swap__image,axiom,
    ! [Y3: a > e,X3: set_a,A: set_Pr204296108_a_a_e] :
      ( ( member360729153_set_a @ ( produc479206876_set_a @ Y3 @ X3 ) @ ( image_1875450791_set_a @ produc755714512_a_a_e @ A ) )
      = ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) @ A ) ) ).

% pair_in_swap_image
thf(fact_282_pair__in__swap__image,axiom,
    ! [Y3: set_a,X3: a > e,A: set_Pr305929312_set_a] :
      ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ Y3 @ X3 ) @ ( image_376979367_a_a_e @ produc100464700_set_a @ A ) )
      = ( member360729153_set_a @ ( produc479206876_set_a @ X3 @ Y3 ) @ A ) ) ).

% pair_in_swap_image
thf(fact_283_swap__simp,axiom,
    ! [X3: a > e,Y3: set_a] :
      ( ( produc100464700_set_a @ ( produc479206876_set_a @ X3 @ Y3 ) )
      = ( produc1134456688_a_a_e @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_284_swap__simp,axiom,
    ! [X3: set_a,Y3: a > e] :
      ( ( produc755714512_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) )
      = ( produc479206876_set_a @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_285_converse__iff,axiom,
    ! [A2: a > e,B2: set_a,R: set_Pr204296108_a_a_e] :
      ( ( member360729153_set_a @ ( produc479206876_set_a @ A2 @ B2 ) @ ( converse_set_a_a_e @ R ) )
      = ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ A2 ) @ R ) ) ).

% converse_iff
thf(fact_286_converse__iff,axiom,
    ! [A2: set_a,B2: a > e,R: set_Pr305929312_set_a] :
      ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ ( converse_a_e_set_a @ R ) )
      = ( member360729153_set_a @ ( produc479206876_set_a @ B2 @ A2 ) @ R ) ) ).

% converse_iff
thf(fact_287_converse_Oinducts,axiom,
    ! [X1: a > e,X22: set_a,R: set_Pr204296108_a_a_e,P2: ( a > e ) > set_a > $o] :
      ( ( member360729153_set_a @ ( produc479206876_set_a @ X1 @ X22 ) @ ( converse_set_a_a_e @ R ) )
     => ( ! [A6: set_a,B6: a > e] :
            ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A6 @ B6 ) @ R )
           => ( P2 @ B6 @ A6 ) )
       => ( P2 @ X1 @ X22 ) ) ) ).

% converse.inducts
thf(fact_288_converse_Oinducts,axiom,
    ! [X1: set_a,X22: a > e,R: set_Pr305929312_set_a,P2: set_a > ( a > e ) > $o] :
      ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X1 @ X22 ) @ ( converse_a_e_set_a @ R ) )
     => ( ! [A6: a > e,B6: set_a] :
            ( ( member360729153_set_a @ ( produc479206876_set_a @ A6 @ B6 ) @ R )
           => ( P2 @ B6 @ A6 ) )
       => ( P2 @ X1 @ X22 ) ) ) ).

% converse.inducts
thf(fact_289_converse_Ointros,axiom,
    ! [A2: a > e,B2: set_a,R: set_Pr305929312_set_a] :
      ( ( member360729153_set_a @ ( produc479206876_set_a @ A2 @ B2 ) @ R )
     => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ A2 ) @ ( converse_a_e_set_a @ R ) ) ) ).

% converse.intros
thf(fact_290_converse_Ointros,axiom,
    ! [A2: set_a,B2: a > e,R: set_Pr204296108_a_a_e] :
      ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R )
     => ( member360729153_set_a @ ( produc479206876_set_a @ B2 @ A2 ) @ ( converse_set_a_a_e @ R ) ) ) ).

% converse.intros
thf(fact_291_converse_Osimps,axiom,
    ! [A1: a > e,A22: set_a,R: set_Pr204296108_a_a_e] :
      ( ( member360729153_set_a @ ( produc479206876_set_a @ A1 @ A22 ) @ ( converse_set_a_a_e @ R ) )
      = ( ? [A3: set_a,B3: a > e] :
            ( ( A1 = B3 )
            & ( A22 = A3 )
            & ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A3 @ B3 ) @ R ) ) ) ) ).

% converse.simps
thf(fact_292_converse_Osimps,axiom,
    ! [A1: set_a,A22: a > e,R: set_Pr305929312_set_a] :
      ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A1 @ A22 ) @ ( converse_a_e_set_a @ R ) )
      = ( ? [A3: a > e,B3: set_a] :
            ( ( A1 = B3 )
            & ( A22 = A3 )
            & ( member360729153_set_a @ ( produc479206876_set_a @ A3 @ B3 ) @ R ) ) ) ) ).

% converse.simps
thf(fact_293_converse_Ocases,axiom,
    ! [A1: a > e,A22: set_a,R: set_Pr204296108_a_a_e] :
      ( ( member360729153_set_a @ ( produc479206876_set_a @ A1 @ A22 ) @ ( converse_set_a_a_e @ R ) )
     => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A22 @ A1 ) @ R ) ) ).

% converse.cases
thf(fact_294_converse_Ocases,axiom,
    ! [A1: set_a,A22: a > e,R: set_Pr305929312_set_a] :
      ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A1 @ A22 ) @ ( converse_a_e_set_a @ R ) )
     => ( member360729153_set_a @ ( produc479206876_set_a @ A22 @ A1 ) @ R ) ) ).

% converse.cases
thf(fact_295_converseE,axiom,
    ! [Yx: produc1965228714_set_a,R: set_Pr204296108_a_a_e] :
      ( ( member360729153_set_a @ Yx @ ( converse_set_a_a_e @ R ) )
     => ~ ! [X2: set_a,Y: a > e] :
            ( ( Yx
              = ( produc479206876_set_a @ Y @ X2 ) )
           => ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ R ) ) ) ).

% converseE
thf(fact_296_converseE,axiom,
    ! [Yx: produc715398134_a_a_e,R: set_Pr305929312_set_a] :
      ( ( member1258382221_a_a_e @ Yx @ ( converse_a_e_set_a @ R ) )
     => ~ ! [X2: a > e,Y: set_a] :
            ( ( Yx
              = ( produc1134456688_a_a_e @ Y @ X2 ) )
           => ~ ( member360729153_set_a @ ( produc479206876_set_a @ X2 @ Y ) @ R ) ) ) ).

% converseE
thf(fact_297_converseD,axiom,
    ! [A2: a > e,B2: set_a,R: set_Pr204296108_a_a_e] :
      ( ( member360729153_set_a @ ( produc479206876_set_a @ A2 @ B2 ) @ ( converse_set_a_a_e @ R ) )
     => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ A2 ) @ R ) ) ).

% converseD
thf(fact_298_converseD,axiom,
    ! [A2: set_a,B2: a > e,R: set_Pr305929312_set_a] :
      ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ ( converse_a_e_set_a @ R ) )
     => ( member360729153_set_a @ ( produc479206876_set_a @ B2 @ A2 ) @ R ) ) ).

% converseD
thf(fact_299_prod_Oswap__def,axiom,
    ( produc100464700_set_a
    = ( ^ [P3: produc1965228714_set_a] : ( produc1134456688_a_a_e @ ( produc1809800494_set_a @ P3 ) @ ( produc385678576_set_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_300_prod_Oswap__def,axiom,
    ( produc755714512_a_a_e
    = ( ^ [P3: produc715398134_a_a_e] : ( produc479206876_set_a @ ( produc317566658_a_a_e @ P3 ) @ ( produc1040928388_a_a_e @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_301_map__prod__imageI,axiom,
    ! [A2: set_a,B2: a > e,R2: set_Pr204296108_a_a_e,F: set_a > set_a,G: ( a > e ) > a > e] :
      ( ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ A2 @ B2 ) @ R2 )
     => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ ( F @ A2 ) @ ( G @ B2 ) ) @ ( image_625620211_a_a_e @ ( produc1443219402_e_a_e @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_302_UNIV__I,axiom,
    ! [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ).

% UNIV_I
thf(fact_303_UNIV__I,axiom,
    ! [X3: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X3 @ top_to1601855068_a_a_e ) ).

% UNIV_I
thf(fact_304_map__prod__simp,axiom,
    ! [F: set_a > set_a,G: ( a > e ) > a > e,A2: set_a,B2: a > e] :
      ( ( produc1443219402_e_a_e @ F @ G @ ( produc1134456688_a_a_e @ A2 @ B2 ) )
      = ( produc1134456688_a_a_e @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_305_fst__map__prod,axiom,
    ! [F: set_a > set_a,G: ( a > e ) > a > e,X3: produc715398134_a_a_e] :
      ( ( produc1040928388_a_a_e @ ( produc1443219402_e_a_e @ F @ G @ X3 ) )
      = ( F @ ( produc1040928388_a_a_e @ X3 ) ) ) ).

% fst_map_prod
thf(fact_306_prod_Ocollapse,axiom,
    ! [Prod: produc715398134_a_a_e] :
      ( ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ Prod ) @ ( produc317566658_a_a_e @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_307_fst__swap,axiom,
    ! [X3: produc1965228714_set_a] :
      ( ( produc1040928388_a_a_e @ ( produc100464700_set_a @ X3 ) )
      = ( produc1809800494_set_a @ X3 ) ) ).

% fst_swap
thf(fact_308_snd__swap,axiom,
    ! [X3: produc715398134_a_a_e] :
      ( ( produc1809800494_set_a @ ( produc755714512_a_a_e @ X3 ) )
      = ( produc1040928388_a_a_e @ X3 ) ) ).

% snd_swap
thf(fact_309_range__fst,axiom,
    ( ( image_64240033_set_a @ produc1040928388_a_a_e @ top_to1601855068_a_a_e )
    = top_top_set_set_a ) ).

% range_fst
thf(fact_310_rangeI,axiom,
    ! [F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e] : ( member_set_a @ ( F @ X3 ) @ ( image_64240033_set_a @ F @ top_to1601855068_a_a_e ) ) ).

% rangeI
thf(fact_311_range__eqI,axiom,
    ! [B2: set_a,F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e] :
      ( ( B2
        = ( F @ X3 ) )
     => ( member_set_a @ B2 @ ( image_64240033_set_a @ F @ top_to1601855068_a_a_e ) ) ) ).

% range_eqI
thf(fact_312_surj__def,axiom,
    ! [F: produc715398134_a_a_e > set_a] :
      ( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
        = top_top_set_set_a )
      = ( ! [Y4: set_a] :
          ? [X4: produc715398134_a_a_e] :
            ( Y4
            = ( F @ X4 ) ) ) ) ).

% surj_def
thf(fact_313_surjI,axiom,
    ! [G: produc715398134_a_a_e > set_a,F: set_a > produc715398134_a_a_e] :
      ( ! [X2: set_a] :
          ( ( G @ ( F @ X2 ) )
          = X2 )
     => ( ( image_64240033_set_a @ G @ top_to1601855068_a_a_e )
        = top_top_set_set_a ) ) ).

% surjI
thf(fact_314_surjE,axiom,
    ! [F: produc715398134_a_a_e > set_a,Y3: set_a] :
      ( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
        = top_top_set_set_a )
     => ~ ! [X2: produc715398134_a_a_e] :
            ( Y3
           != ( F @ X2 ) ) ) ).

% surjE
thf(fact_315_surjD,axiom,
    ! [F: produc715398134_a_a_e > set_a,Y3: set_a] :
      ( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
        = top_top_set_set_a )
     => ? [X2: produc715398134_a_a_e] :
          ( Y3
          = ( F @ X2 ) ) ) ).

% surjD
thf(fact_316_map__prod__surj,axiom,
    ! [F: produc715398134_a_a_e > set_a,G: produc715398134_a_a_e > set_a] :
      ( ( ( image_64240033_set_a @ F @ top_to1601855068_a_a_e )
        = top_top_set_set_a )
     => ( ( ( image_64240033_set_a @ G @ top_to1601855068_a_a_e )
          = top_top_set_set_a )
       => ( ( image_2101504031_set_a @ ( produc387505142_set_a @ F @ G ) @ top_to970499581_a_a_e )
          = top_to144004695_set_a ) ) ) ).

% map_prod_surj
thf(fact_317_top_Oextremum__uniqueI,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
     => ( A2 = top_top_set_a ) ) ).

% top.extremum_uniqueI
thf(fact_318_top_Oextremum__unique,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
      = ( A2 = top_top_set_a ) ) ).

% top.extremum_unique
thf(fact_319_top__greatest,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).

% top_greatest
thf(fact_320_subset__UNIV,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).

% subset_UNIV
thf(fact_321_sndI,axiom,
    ! [X3: produc715398134_a_a_e,Y3: set_a,Z2: a > e] :
      ( ( X3
        = ( produc1134456688_a_a_e @ Y3 @ Z2 ) )
     => ( ( produc317566658_a_a_e @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_322_snd__eqD,axiom,
    ! [X3: set_a,Y3: a > e,A2: a > e] :
      ( ( ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) )
        = A2 )
     => ( Y3 = A2 ) ) ).

% snd_eqD
thf(fact_323_snd__conv,axiom,
    ! [X1: set_a,X22: a > e] :
      ( ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_324_eq__snd__iff,axiom,
    ! [B2: a > e,P: produc715398134_a_a_e] :
      ( ( B2
        = ( produc317566658_a_a_e @ P ) )
      = ( ? [A3: set_a] :
            ( P
            = ( produc1134456688_a_a_e @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_325_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc715398134_a_a_e,Z: produc715398134_a_a_e] : ( Y2 = Z ) )
    = ( ^ [S3: produc715398134_a_a_e,T: produc715398134_a_a_e] :
          ( ( ( produc1040928388_a_a_e @ S3 )
            = ( produc1040928388_a_a_e @ T ) )
          & ( ( produc317566658_a_a_e @ S3 )
            = ( produc317566658_a_a_e @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_326_prod_Oexpand,axiom,
    ! [Prod: produc715398134_a_a_e,Prod2: produc715398134_a_a_e] :
      ( ( ( ( produc1040928388_a_a_e @ Prod )
          = ( produc1040928388_a_a_e @ Prod2 ) )
        & ( ( produc317566658_a_a_e @ Prod )
          = ( produc317566658_a_a_e @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_327_prod__eqI,axiom,
    ! [P: produc715398134_a_a_e,Q2: produc715398134_a_a_e] :
      ( ( ( produc1040928388_a_a_e @ P )
        = ( produc1040928388_a_a_e @ Q2 ) )
     => ( ( ( produc317566658_a_a_e @ P )
          = ( produc317566658_a_a_e @ Q2 ) )
       => ( P = Q2 ) ) ) ).

% prod_eqI
thf(fact_328_UNIV__eq__I,axiom,
    ! [A: set_set_a] :
      ( ! [X2: set_a] : ( member_set_a @ X2 @ A )
     => ( top_top_set_set_a = A ) ) ).

% UNIV_eq_I
thf(fact_329_UNIV__eq__I,axiom,
    ! [A: set_Pr204296108_a_a_e] :
      ( ! [X2: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X2 @ A )
     => ( top_to1601855068_a_a_e = A ) ) ).

% UNIV_eq_I
thf(fact_330_UNIV__witness,axiom,
    ? [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).

% UNIV_witness
thf(fact_331_UNIV__witness,axiom,
    ? [X2: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X2 @ top_to1601855068_a_a_e ) ).

% UNIV_witness
thf(fact_332_surjective__pairing,axiom,
    ! [T2: produc715398134_a_a_e] :
      ( T2
      = ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ T2 ) @ ( produc317566658_a_a_e @ T2 ) ) ) ).

% surjective_pairing
thf(fact_333_prod_Oexhaust__sel,axiom,
    ! [Prod: produc715398134_a_a_e] :
      ( Prod
      = ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ Prod ) @ ( produc317566658_a_a_e @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_334_prod__fun__imageE,axiom,
    ! [C2: produc715398134_a_a_e,F: set_a > set_a,G: ( a > e ) > a > e,R2: set_Pr204296108_a_a_e] :
      ( ( member1258382221_a_a_e @ C2 @ ( image_625620211_a_a_e @ ( produc1443219402_e_a_e @ F @ G ) @ R2 ) )
     => ~ ! [X2: set_a,Y: a > e] :
            ( ( C2
              = ( produc1134456688_a_a_e @ ( F @ X2 ) @ ( G @ Y ) ) )
           => ~ ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X2 @ Y ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_335_exI__realizer,axiom,
    ! [P2: ( a > e ) > set_a > $o,Y3: a > e,X3: set_a] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) ) @ ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_336_conjI__realizer,axiom,
    ! [P2: set_a > $o,P: set_a,Q: ( a > e ) > $o,Q2: a > e] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc1040928388_a_a_e @ ( produc1134456688_a_a_e @ P @ Q2 ) ) )
          & ( Q @ ( produc317566658_a_a_e @ ( produc1134456688_a_a_e @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_337_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: set_a > ( a > e ) > $o,X3: set_a,Y3: a > e,A2: produc715398134_a_a_e] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A2
          = ( produc1134456688_a_a_e @ X3 @ Y3 ) )
       => ( P2 @ ( produc1040928388_a_a_e @ A2 ) @ ( produc317566658_a_a_e @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_338_top__empty__eq,axiom,
    ( top_top_set_a_o
    = ( ^ [X4: set_a] : ( member_set_a @ X4 @ top_top_set_set_a ) ) ) ).

% top_empty_eq
thf(fact_339_top__empty__eq,axiom,
    ( top_to2000253225_a_e_o
    = ( ^ [X4: produc715398134_a_a_e] : ( member1258382221_a_a_e @ X4 @ top_to1601855068_a_a_e ) ) ) ).

% top_empty_eq
thf(fact_340_exE__realizer_H,axiom,
    ! [P2: ( a > e ) > set_a > $o,P: produc715398134_a_a_e] :
      ( ( P2 @ ( produc317566658_a_a_e @ P ) @ ( produc1040928388_a_a_e @ P ) )
     => ~ ! [X2: set_a,Y: a > e] :
            ~ ( P2 @ Y @ X2 ) ) ).

% exE_realizer'
thf(fact_341_Collect__split__mono__strong,axiom,
    ! [X6: set_set_a,A: set_Pr204296108_a_a_e,Y7: set_a_e,P2: set_a > ( a > e ) > $o,Q: set_a > ( a > e ) > $o] :
      ( ( X6
        = ( image_64240033_set_a @ produc1040928388_a_a_e @ A ) )
     => ( ( Y7
          = ( image_1149879910_e_a_e @ produc317566658_a_a_e @ A ) )
       => ( ! [X2: set_a] :
              ( ( member_set_a @ X2 @ X6 )
             => ! [Xa2: a > e] :
                  ( ( member_a_e @ Xa2 @ Y7 )
                 => ( ( P2 @ X2 @ Xa2 )
                   => ( Q @ X2 @ Xa2 ) ) ) )
         => ( ( ord_le700723212_a_a_e @ A @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ P2 ) ) )
           => ( ord_le700723212_a_a_e @ A @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_342_sndOp__def,axiom,
    ( bNF_sn375433437_a_a_e
    = ( ^ [P4: set_a > set_a > $o,Q3: set_a > ( a > e ) > $o,Ac: produc715398134_a_a_e] : ( produc1134456688_a_a_e @ ( bNF_pi929741261_a_a_e @ P4 @ Q3 @ ( produc1040928388_a_a_e @ Ac ) @ ( produc317566658_a_a_e @ Ac ) ) @ ( produc317566658_a_a_e @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_343_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc715398134_a_a_e,A: set_a > ( a > e ) > $o] :
      ( ( member1258382221_a_a_e @ X3 @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ A ) ) )
     => ( A @ ( produc1040928388_a_a_e @ X3 ) @ ( produc317566658_a_a_e @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_344_fstOp__def,axiom,
    ( bNF_fs1024561536_e_a_e
    = ( ^ [P4: set_a > ( a > e ) > $o,Q3: ( a > e ) > ( a > e ) > $o,Ac: produc715398134_a_a_e] : ( produc1134456688_a_a_e @ ( produc1040928388_a_a_e @ Ac ) @ ( bNF_pi1175744974_e_a_e @ P4 @ Q3 @ ( produc1040928388_a_a_e @ Ac ) @ ( produc317566658_a_a_e @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_345_mem__case__prodE,axiom,
    ! [Z2: set_a,C2: set_a > ( a > e ) > set_set_a,P: produc715398134_a_a_e] :
      ( ( member_set_a @ Z2 @ ( produc1620938593_set_a @ C2 @ P ) )
     => ~ ! [X2: set_a,Y: a > e] :
            ( ( P
              = ( produc1134456688_a_a_e @ X2 @ Y ) )
           => ~ ( member_set_a @ Z2 @ ( C2 @ X2 @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_346_mem__case__prodE,axiom,
    ! [Z2: produc715398134_a_a_e,C2: set_a > ( a > e ) > set_Pr204296108_a_a_e,P: produc715398134_a_a_e] :
      ( ( member1258382221_a_a_e @ Z2 @ ( produc1096319049_a_a_e @ C2 @ P ) )
     => ~ ! [X2: set_a,Y: a > e] :
            ( ( P
              = ( produc1134456688_a_a_e @ X2 @ Y ) )
           => ~ ( member1258382221_a_a_e @ Z2 @ ( C2 @ X2 @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_347_in__rel__def,axiom,
    ( fun_in_rel_set_a_a_e
    = ( ^ [R3: set_Pr204296108_a_a_e,X4: set_a,Y4: a > e] : ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ X4 @ Y4 ) @ R3 ) ) ) ).

% in_rel_def
thf(fact_348_fstOp__in,axiom,
    ! [Ac2: produc715398134_a_a_e,P2: set_a > ( a > e ) > $o,Q: ( a > e ) > ( a > e ) > $o] :
      ( ( member1258382221_a_a_e @ Ac2 @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ ( relcom658603713_e_a_e @ P2 @ Q ) ) ) )
     => ( member1258382221_a_a_e @ ( bNF_fs1024561536_e_a_e @ P2 @ Q @ Ac2 ) @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_349_sndOp__in,axiom,
    ! [Ac2: produc715398134_a_a_e,P2: set_a > set_a > $o,Q: set_a > ( a > e ) > $o] :
      ( ( member1258382221_a_a_e @ Ac2 @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ ( relcom779348378_a_a_e @ P2 @ Q ) ) ) )
     => ( member1258382221_a_a_e @ ( bNF_sn375433437_a_a_e @ P2 @ Q @ Ac2 ) @ ( collec1369693387_a_a_e @ ( produc1341862587_a_e_o @ Q ) ) ) ) ).

% sndOp_in
thf(fact_350_image2__eqI,axiom,
    ! [B2: set_a,F: set_a > set_a,X3: set_a,C2: a > e,G: set_a > a > e,A: set_set_a] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member_set_a @ X3 @ A )
         => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ C2 ) @ ( bNF_Gr1872575811_a_a_e @ A @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_351_image2__eqI,axiom,
    ! [B2: set_a,F: produc715398134_a_a_e > set_a,X3: produc715398134_a_a_e,C2: a > e,G: produc715398134_a_a_e > a > e,A: set_Pr204296108_a_a_e] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member1258382221_a_a_e @ X3 @ A )
         => ( member1258382221_a_a_e @ ( produc1134456688_a_a_e @ B2 @ C2 ) @ ( bNF_Gr2072183031_a_a_e @ A @ F @ G ) ) ) ) ) ).

% image2_eqI

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [A7: set_a] :
      ( ( member_set_a @ A7 @ ( image_64240033_set_a @ produc1040928388_a_a_e @ c ) )
     => ( ( ord_less_eq_set_a @ a1 @ A7 )
       => ( ( ord_less_eq_set_a @ a2 @ A7 )
         => ( ( ord_less_eq_set_a @ a3 @ A7 )
           => thesis ) ) ) ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------