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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Real_Time_Deque/0026_RealTimeDeque_Dequeue_Proof/prob_00385_013031__7211920_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1741 ( 507 unt; 463 typ;   0 def)
%            Number of atoms       : 3740 (1252 equ;   0 cnn)
%            Maximal formula atoms :    9 (   2 avg)
%            Number of connectives : 11862 ( 308   ~;  32   |; 185   &;9634   @)
%                                         (   0 <=>;1703  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   7 avg)
%            Number of types       :   48 (  47 usr)
%            Number of type conns  : 2555 (2555   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  419 ( 416 usr;  21 con; 0-5 aty)
%            Number of variables   : 4050 ( 237   ^;3686   !; 127   ?;4050   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 15:53:15.171
%------------------------------------------------------------------------------
% Could-be-implicit typings (47)
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thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_It__Big__Ostate_Itf__a_J_M_Eo_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Big__Ostate_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(ty_n_t__Big__Ostate_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (416)
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Big__Ostate_Itf__a_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_J,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_J,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001tf__a_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_J,type,
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thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Big__Ostate_Itf__a_J_Mt__Big__Ostate_Itf__a_J_J_001t__Product____Type__Oprod_It__Big__Ostate_Itf__a_J_Mtf__a_J_001tf__a_001t__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J,type,
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thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_001t__Big__Ostate_Itf__a_J_001t__Product____Type__Oprod_It__Big__Ostate_Itf__a_J_Mtf__a_J,type,
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thf(sy_c_BNF__Def_OfstOp_001t__Big__Ostate_Itf__a_J_001t__Big__Ostate_Itf__a_J_001t__Big__Ostate_Itf__a_J,type,
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thf(sy_c_BNF__Def_OfstOp_001t__Big__Ostate_Itf__a_J_001t__Big__Ostate_Itf__a_J_001tf__a,type,
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thf(sy_c_BNF__Def_OfstOp_001t__Big__Ostate_Itf__a_J_001tf__a_001t__Big__Ostate_Itf__a_J,type,
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thf(sy_c_BNF__Def_OfstOp_001t__Big__Ostate_Itf__a_J_001tf__a_001tf__a,type,
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thf(sy_c_BNF__Def_OfstOp_001t__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_001t__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_001t__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J,type,
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thf(sy_c_BNF__Def_OfstOp_001tf__a_001t__Big__Ostate_Itf__a_J_001t__Big__Ostate_Itf__a_J,type,
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thf(sy_c_BNF__Def_OfstOp_001tf__a_001t__Big__Ostate_Itf__a_J_001tf__a,type,
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thf(sy_c_BNF__Def_OfstOp_001tf__a_001tf__a_001t__Big__Ostate_Itf__a_J,type,
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thf(sy_c_BNF__Def_OfstOp_001tf__a_001tf__a_001tf__a,type,
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thf(sy_c_BNF__Def_Opick__middlep_001t__Big__Ostate_Itf__a_J_001t__Big__Ostate_Itf__a_J_001tf__a,type,
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thf(sy_c_BNF__Def_Opick__middlep_001tf__a_001t__Big__Ostate_Itf__a_J_001t__Big__Ostate_Itf__a_J,type,
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thf(sy_c_BNF__Def_Opick__middlep_001tf__a_001tf__a_001t__Big__Ostate_Itf__a_J,type,
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thf(sy_c_BNF__Def_OsndOp_001t__Big__Ostate_Itf__a_J_001t__Big__Ostate_Itf__a_J_001t__Big__Ostate_Itf__a_J,type,
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thf(sy_c_BNF__Def_OsndOp_001t__Big__Ostate_Itf__a_J_001t__Big__Ostate_Itf__a_J_001tf__a,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Big__Ostate_Itf__a_J_Mtf__a_J,type,
    member1329951143264900189te_a_a: produc5126262593522865844te_a_a > set_Pr2749968112124914068te_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Big__Ostate_Itf__a_J_Mt__Big__Ostate_Itf__a_J_J_Mt__Product____Type__Oprod_It__Big__Ostate_Itf__a_J_Mt__Big__Ostate_Itf__a_J_J_J,type,
    member7923945571256933136tate_a: produc4620573296977836263tate_a > set_Pr2656942177312230599tate_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Big__Ostate_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_J_Mt__Product____Type__Oprod_It__Big__Ostate_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_J_J,type,
    member761508830101964240tate_a: produc8214081372919072167tate_a > set_Pr996548553778265991tate_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Big__Ostate_Itf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_It__Big__Ostate_Itf__a_J_Mtf__a_J_J,type,
    member2649312330005299536te_a_a: produc6994328786524016167te_a_a > set_Pr4734257497606733575te_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_Mt__Big__Ostate_Itf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_Mt__Big__Ostate_Itf__a_J_J_J,type,
    member6670390056965382224tate_a: produc4899590562927714343tate_a > set_Pr3534671552553933319tate_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_J_J,type,
    member1643692453717642000tate_a: produc5610530989376482535tate_a > set_Pr1690608023759948999tate_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_Mtf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_Mtf__a_J_J,type,
    member7027635665102103568te_a_a: produc8722129860559393511te_a_a > set_Pr5759549409764223431te_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_J,type,
    member1595334363549574352tate_a: produc5940350820068290983tate_a > set_Pr7963165934123475591tate_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_J_J,type,
    member8315558564207931920tate_a: produc786680722810446055tate_a > set_Pr5181609303329667015tate_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    member6330455413206600464od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J,type,
    member3175992478928454403tate_a: produc6972303929186420058tate_a > set_Pr4275752383657305402tate_a > $o ).

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

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J_J,type,
    member4824975370526228195tate_a: set_Pr4275752383657305402tate_a > set_se5193884918812770586tate_a > $o ).

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

thf(sy_v_big____,type,
    big: state_a ).

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

% Relevant facts (1276)
thf(fact_0_prod_Oinject,axiom,
    ! [X1: state_a,X2: a,Y1: state_a,Y2: a] :
      ( ( ( produc7328099264922328356te_a_a @ X1 @ X2 )
        = ( produc7328099264922328356te_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_1_prod_Oinject,axiom,
    ! [X1: state_a,X2: state_a,Y1: state_a,Y2: state_a] :
      ( ( ( produc6131900399494507223tate_a @ X1 @ X2 )
        = ( produc6131900399494507223tate_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_2_prod_Oinject,axiom,
    ! [X1: produc6972303929186420058tate_a,X2: produc6972303929186420058tate_a,Y1: produc6972303929186420058tate_a,Y2: produc6972303929186420058tate_a] :
      ( ( ( produc2143731062552654487tate_a @ X1 @ X2 )
        = ( produc2143731062552654487tate_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_3_prod_Oinject,axiom,
    ! [X1: a,X2: a,Y1: a,Y2: a] :
      ( ( ( product_Pair_a_a @ X1 @ X2 )
        = ( product_Pair_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_4_prod_Oinject,axiom,
    ! [X1: a,X2: state_a,Y1: a,Y2: state_a] :
      ( ( ( produc8641956578966763338tate_a @ X1 @ X2 )
        = ( produc8641956578966763338tate_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_5_old_Oprod_Oinject,axiom,
    ! [A: state_a,B: a,A2: state_a,B2: a] :
      ( ( ( produc7328099264922328356te_a_a @ A @ B )
        = ( produc7328099264922328356te_a_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_6_old_Oprod_Oinject,axiom,
    ! [A: state_a,B: state_a,A2: state_a,B2: state_a] :
      ( ( ( produc6131900399494507223tate_a @ A @ B )
        = ( produc6131900399494507223tate_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_7_old_Oprod_Oinject,axiom,
    ! [A: produc6972303929186420058tate_a,B: produc6972303929186420058tate_a,A2: produc6972303929186420058tate_a,B2: produc6972303929186420058tate_a] :
      ( ( ( produc2143731062552654487tate_a @ A @ B )
        = ( produc2143731062552654487tate_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_8_old_Oprod_Oinject,axiom,
    ! [A: a,B: a,A2: a,B2: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_9_old_Oprod_Oinject,axiom,
    ! [A: a,B: state_a,A2: a,B2: state_a] :
      ( ( ( produc8641956578966763338tate_a @ A @ B )
        = ( produc8641956578966763338tate_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_10_old_Oprod_Oexhaust,axiom,
    ! [Y: produc5126262593522865844te_a_a] :
      ~ ! [A3: state_a,B3: a] :
          ( Y
         != ( produc7328099264922328356te_a_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_11_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4774439622479004775tate_a] :
      ~ ! [A3: state_a,B3: state_a] :
          ( Y
         != ( produc6131900399494507223tate_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_12_old_Oprod_Oexhaust,axiom,
    ! [Y: produc5940350820068290983tate_a] :
      ~ ! [A3: produc6972303929186420058tate_a,B3: produc6972303929186420058tate_a] :
          ( Y
         != ( produc2143731062552654487tate_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_13_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_a] :
      ~ ! [A3: a,B3: a] :
          ( Y
         != ( product_Pair_a_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_14_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6972303929186420058tate_a] :
      ~ ! [A3: a,B3: state_a] :
          ( Y
         != ( produc8641956578966763338tate_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_15_surj__pair,axiom,
    ! [P: produc5126262593522865844te_a_a] :
    ? [X: state_a,Y3: a] :
      ( P
      = ( produc7328099264922328356te_a_a @ X @ Y3 ) ) ).

% surj_pair
thf(fact_16_surj__pair,axiom,
    ! [P: produc4774439622479004775tate_a] :
    ? [X: state_a,Y3: state_a] :
      ( P
      = ( produc6131900399494507223tate_a @ X @ Y3 ) ) ).

% surj_pair
thf(fact_17_surj__pair,axiom,
    ! [P: produc5940350820068290983tate_a] :
    ? [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a] :
      ( P
      = ( produc2143731062552654487tate_a @ X @ Y3 ) ) ).

% surj_pair
thf(fact_18_surj__pair,axiom,
    ! [P: product_prod_a_a] :
    ? [X: a,Y3: a] :
      ( P
      = ( product_Pair_a_a @ X @ Y3 ) ) ).

% surj_pair
thf(fact_19_surj__pair,axiom,
    ! [P: produc6972303929186420058tate_a] :
    ? [X: a,Y3: state_a] :
      ( P
      = ( produc8641956578966763338tate_a @ X @ Y3 ) ) ).

% surj_pair
thf(fact_20_prod__cases,axiom,
    ! [P2: produc5126262593522865844te_a_a > $o,P: produc5126262593522865844te_a_a] :
      ( ! [A3: state_a,B3: a] : ( P2 @ ( produc7328099264922328356te_a_a @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_21_prod__cases,axiom,
    ! [P2: produc4774439622479004775tate_a > $o,P: produc4774439622479004775tate_a] :
      ( ! [A3: state_a,B3: state_a] : ( P2 @ ( produc6131900399494507223tate_a @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_22_prod__cases,axiom,
    ! [P2: produc5940350820068290983tate_a > $o,P: produc5940350820068290983tate_a] :
      ( ! [A3: produc6972303929186420058tate_a,B3: produc6972303929186420058tate_a] : ( P2 @ ( produc2143731062552654487tate_a @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_23_prod__cases,axiom,
    ! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
      ( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_24_prod__cases,axiom,
    ! [P2: produc6972303929186420058tate_a > $o,P: produc6972303929186420058tate_a] :
      ( ! [A3: a,B3: state_a] : ( P2 @ ( produc8641956578966763338tate_a @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_25_Pair__inject,axiom,
    ! [A: state_a,B: a,A2: state_a,B2: a] :
      ( ( ( produc7328099264922328356te_a_a @ A @ B )
        = ( produc7328099264922328356te_a_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_26_Pair__inject,axiom,
    ! [A: state_a,B: state_a,A2: state_a,B2: state_a] :
      ( ( ( produc6131900399494507223tate_a @ A @ B )
        = ( produc6131900399494507223tate_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_27_Pair__inject,axiom,
    ! [A: produc6972303929186420058tate_a,B: produc6972303929186420058tate_a,A2: produc6972303929186420058tate_a,B2: produc6972303929186420058tate_a] :
      ( ( ( produc2143731062552654487tate_a @ A @ B )
        = ( produc2143731062552654487tate_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_28_Pair__inject,axiom,
    ! [A: a,B: a,A2: a,B2: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_29_Pair__inject,axiom,
    ! [A: a,B: state_a,A2: a,B2: state_a] :
      ( ( ( produc8641956578966763338tate_a @ A @ B )
        = ( produc8641956578966763338tate_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_30_prod__cases3,axiom,
    ! [Y: produc5940350820068290983tate_a] :
      ~ ! [A3: produc6972303929186420058tate_a,B3: a,C: state_a] :
          ( Y
         != ( produc2143731062552654487tate_a @ A3 @ ( produc8641956578966763338tate_a @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_31_prod__induct3,axiom,
    ! [P2: produc5940350820068290983tate_a > $o,X3: produc5940350820068290983tate_a] :
      ( ! [A3: produc6972303929186420058tate_a,B3: a,C: state_a] : ( P2 @ ( produc2143731062552654487tate_a @ A3 @ ( produc8641956578966763338tate_a @ B3 @ C ) ) )
     => ( P2 @ X3 ) ) ).

% prod_induct3
thf(fact_32_ssubst__Pair__rhs,axiom,
    ! [R: state_a,S: a,R2: set_Pr2749968112124914068te_a_a,S2: a] :
      ( ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_33_ssubst__Pair__rhs,axiom,
    ! [R: state_a,S: state_a,R2: set_Pr1753757909640466247tate_a,S2: state_a] :
      ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_34_ssubst__Pair__rhs,axiom,
    ! [R: produc6972303929186420058tate_a,S: produc6972303929186420058tate_a,R2: set_Pr7963165934123475591tate_a,S2: produc6972303929186420058tate_a] :
      ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_35_ssubst__Pair__rhs,axiom,
    ! [R: a,S: a,R2: set_Product_prod_a_a,S2: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_36_ssubst__Pair__rhs,axiom,
    ! [R: a,S: state_a,R2: set_Pr4275752383657305402tate_a,S2: state_a] :
      ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_37_curry__conv,axiom,
    ( produc6512344587036289050te_a_o
    = ( ^ [F: produc6972303929186420058tate_a > $o,A4: a,B4: state_a] : ( F @ ( produc8641956578966763338tate_a @ A4 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_38_curryI,axiom,
    ! [F2: produc5126262593522865844te_a_a > $o,A: state_a,B: a] :
      ( ( F2 @ ( produc7328099264922328356te_a_a @ A @ B ) )
     => ( produc3284770843760615360_a_a_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_39_curryI,axiom,
    ! [F2: produc4774439622479004775tate_a > $o,A: state_a,B: state_a] :
      ( ( F2 @ ( produc6131900399494507223tate_a @ A @ B ) )
     => ( produc7364888685945774285te_a_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_40_curryI,axiom,
    ! [F2: produc5940350820068290983tate_a > $o,A: produc6972303929186420058tate_a,B: produc6972303929186420058tate_a] :
      ( ( F2 @ ( produc2143731062552654487tate_a @ A @ B ) )
     => ( produc4157963810276359949te_a_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_41_curryI,axiom,
    ! [F2: product_prod_a_a > $o,A: a,B: a] :
      ( ( F2 @ ( product_Pair_a_a @ A @ B ) )
     => ( product_curry_a_a_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_42_curryI,axiom,
    ! [F2: produc6972303929186420058tate_a > $o,A: a,B: state_a] :
      ( ( F2 @ ( produc8641956578966763338tate_a @ A @ B ) )
     => ( produc6512344587036289050te_a_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_43_swap__simp,axiom,
    ! [X3: state_a,Y: state_a] :
      ( ( produc1839749624818714743tate_a @ ( produc6131900399494507223tate_a @ X3 @ Y ) )
      = ( produc6131900399494507223tate_a @ Y @ X3 ) ) ).

% swap_simp
thf(fact_44_swap__simp,axiom,
    ! [X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a] :
      ( ( produc1520385140544950327tate_a @ ( produc2143731062552654487tate_a @ X3 @ Y ) )
      = ( produc2143731062552654487tate_a @ Y @ X3 ) ) ).

% swap_simp
thf(fact_45_swap__simp,axiom,
    ! [X3: a,Y: a] :
      ( ( product_swap_a_a @ ( product_Pair_a_a @ X3 @ Y ) )
      = ( product_Pair_a_a @ Y @ X3 ) ) ).

% swap_simp
thf(fact_46_swap__simp,axiom,
    ! [X3: state_a,Y: a] :
      ( ( produc2589720147407833796te_a_a @ ( produc7328099264922328356te_a_a @ X3 @ Y ) )
      = ( produc8641956578966763338tate_a @ Y @ X3 ) ) ).

% swap_simp
thf(fact_47_swap__simp,axiom,
    ! [X3: a,Y: state_a] :
      ( ( produc3903577461452268778tate_a @ ( produc8641956578966763338tate_a @ X3 @ Y ) )
      = ( produc7328099264922328356te_a_a @ Y @ X3 ) ) ).

% swap_simp
thf(fact_48_apsnd__conv,axiom,
    ! [F2: state_a > a,X3: a,Y: state_a] :
      ( ( produc1259323963335463987_a_a_a @ F2 @ ( produc8641956578966763338tate_a @ X3 @ Y ) )
      = ( product_Pair_a_a @ X3 @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_49_apsnd__conv,axiom,
    ! [F2: a > a,X3: state_a,Y: a] :
      ( ( produc9038538116159941043tate_a @ F2 @ ( produc7328099264922328356te_a_a @ X3 @ Y ) )
      = ( produc7328099264922328356te_a_a @ X3 @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_50_apsnd__conv,axiom,
    ! [F2: a > state_a,X3: state_a,Y: a] :
      ( ( produc8426166416538473664tate_a @ F2 @ ( produc7328099264922328356te_a_a @ X3 @ Y ) )
      = ( produc6131900399494507223tate_a @ X3 @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_51_apsnd__conv,axiom,
    ! [F2: state_a > a,X3: state_a,Y: state_a] :
      ( ( produc5844437624964255334tate_a @ F2 @ ( produc6131900399494507223tate_a @ X3 @ Y ) )
      = ( produc7328099264922328356te_a_a @ X3 @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_52_apsnd__conv,axiom,
    ! [F2: state_a > state_a,X3: state_a,Y: state_a] :
      ( ( produc8319746412473484275tate_a @ F2 @ ( produc6131900399494507223tate_a @ X3 @ Y ) )
      = ( produc6131900399494507223tate_a @ X3 @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_53_apsnd__conv,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a] :
      ( ( produc6906016655019192284tate_a @ F2 @ ( produc2143731062552654487tate_a @ X3 @ Y ) )
      = ( produc2143731062552654487tate_a @ X3 @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_54_apsnd__conv,axiom,
    ! [F2: a > state_a,X3: a,Y: a] :
      ( ( produc7724680802115506061te_a_a @ F2 @ ( product_Pair_a_a @ X3 @ Y ) )
      = ( produc8641956578966763338tate_a @ X3 @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_55_apsnd__conv,axiom,
    ! [F2: a > a,X3: a,Y: a] :
      ( ( product_apsnd_a_a_a @ F2 @ ( product_Pair_a_a @ X3 @ Y ) )
      = ( product_Pair_a_a @ X3 @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_56_apsnd__conv,axiom,
    ! [F2: state_a > state_a,X3: a,Y: state_a] :
      ( ( produc4530580310919820352te_a_a @ F2 @ ( produc8641956578966763338tate_a @ X3 @ Y ) )
      = ( produc8641956578966763338tate_a @ X3 @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_57_apfst__conv,axiom,
    ! [F2: a > state_a,X3: a,Y: state_a] :
      ( ( produc8655383802206219650tate_a @ F2 @ ( produc8641956578966763338tate_a @ X3 @ Y ) )
      = ( produc6131900399494507223tate_a @ ( F2 @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_58_apfst__conv,axiom,
    ! [F2: state_a > state_a,X3: state_a,Y: a] :
      ( ( produc4759797696587566338te_a_a @ F2 @ ( produc7328099264922328356te_a_a @ X3 @ Y ) )
      = ( produc7328099264922328356te_a_a @ ( F2 @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_59_apfst__conv,axiom,
    ! [F2: state_a > a,X3: state_a,Y: a] :
      ( ( produc5432375924127299061_a_a_a @ F2 @ ( produc7328099264922328356te_a_a @ X3 @ Y ) )
      = ( product_Pair_a_a @ ( F2 @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_60_apfst__conv,axiom,
    ! [F2: state_a > a,X3: state_a,Y: state_a] :
      ( ( produc6073655010632001320tate_a @ F2 @ ( produc6131900399494507223tate_a @ X3 @ Y ) )
      = ( produc8641956578966763338tate_a @ ( F2 @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_61_apfst__conv,axiom,
    ! [F2: state_a > state_a,X3: state_a,Y: state_a] :
      ( ( produc6780963299072512949tate_a @ F2 @ ( produc6131900399494507223tate_a @ X3 @ Y ) )
      = ( produc6131900399494507223tate_a @ ( F2 @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_62_apfst__conv,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a] :
      ( ( produc3739228049781033374tate_a @ F2 @ ( produc2143731062552654487tate_a @ X3 @ Y ) )
      = ( produc2143731062552654487tate_a @ ( F2 @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_63_apfst__conv,axiom,
    ! [F2: a > state_a,X3: a,Y: a] :
      ( ( produc2674360726052565327te_a_a @ F2 @ ( product_Pair_a_a @ X3 @ Y ) )
      = ( produc7328099264922328356te_a_a @ ( F2 @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_64_apfst__conv,axiom,
    ! [F2: a > a,X3: a,Y: a] :
      ( ( product_apfst_a_a_a @ F2 @ ( product_Pair_a_a @ X3 @ Y ) )
      = ( product_Pair_a_a @ ( F2 @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_65_apfst__conv,axiom,
    ! [F2: a > a,X3: a,Y: state_a] :
      ( ( produc3988218040097000309tate_a @ F2 @ ( produc8641956578966763338tate_a @ X3 @ Y ) )
      = ( produc8641956578966763338tate_a @ ( F2 @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_66_curryE,axiom,
    ! [F2: produc5126262593522865844te_a_a > $o,A: state_a,B: a] :
      ( ( produc3284770843760615360_a_a_o @ F2 @ A @ B )
     => ( F2 @ ( produc7328099264922328356te_a_a @ A @ B ) ) ) ).

% curryE
thf(fact_67_curryE,axiom,
    ! [F2: produc4774439622479004775tate_a > $o,A: state_a,B: state_a] :
      ( ( produc7364888685945774285te_a_o @ F2 @ A @ B )
     => ( F2 @ ( produc6131900399494507223tate_a @ A @ B ) ) ) ).

% curryE
thf(fact_68_curryE,axiom,
    ! [F2: produc5940350820068290983tate_a > $o,A: produc6972303929186420058tate_a,B: produc6972303929186420058tate_a] :
      ( ( produc4157963810276359949te_a_o @ F2 @ A @ B )
     => ( F2 @ ( produc2143731062552654487tate_a @ A @ B ) ) ) ).

% curryE
thf(fact_69_curryE,axiom,
    ! [F2: product_prod_a_a > $o,A: a,B: a] :
      ( ( product_curry_a_a_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_a_a @ A @ B ) ) ) ).

% curryE
thf(fact_70_curryE,axiom,
    ! [F2: produc6972303929186420058tate_a > $o,A: a,B: state_a] :
      ( ( produc6512344587036289050te_a_o @ F2 @ A @ B )
     => ( F2 @ ( produc8641956578966763338tate_a @ A @ B ) ) ) ).

% curryE
thf(fact_71_swap__swap,axiom,
    ! [P: produc5126262593522865844te_a_a] :
      ( ( produc3903577461452268778tate_a @ ( produc2589720147407833796te_a_a @ P ) )
      = P ) ).

% swap_swap
thf(fact_72_swap__swap,axiom,
    ! [P: produc6972303929186420058tate_a] :
      ( ( produc2589720147407833796te_a_a @ ( produc3903577461452268778tate_a @ P ) )
      = P ) ).

% swap_swap
thf(fact_73_apsnd__apfst__commute,axiom,
    ! [F2: state_a > state_a,G: a > a,P: produc6972303929186420058tate_a] :
      ( ( produc4530580310919820352te_a_a @ F2 @ ( produc3988218040097000309tate_a @ G @ P ) )
      = ( produc3988218040097000309tate_a @ G @ ( produc4530580310919820352te_a_a @ F2 @ P ) ) ) ).

% apsnd_apfst_commute
thf(fact_74_curryD,axiom,
    ! [F2: produc5126262593522865844te_a_a > $o,A: state_a,B: a] :
      ( ( produc3284770843760615360_a_a_o @ F2 @ A @ B )
     => ( F2 @ ( produc7328099264922328356te_a_a @ A @ B ) ) ) ).

% curryD
thf(fact_75_curryD,axiom,
    ! [F2: produc4774439622479004775tate_a > $o,A: state_a,B: state_a] :
      ( ( produc7364888685945774285te_a_o @ F2 @ A @ B )
     => ( F2 @ ( produc6131900399494507223tate_a @ A @ B ) ) ) ).

% curryD
thf(fact_76_curryD,axiom,
    ! [F2: produc5940350820068290983tate_a > $o,A: produc6972303929186420058tate_a,B: produc6972303929186420058tate_a] :
      ( ( produc4157963810276359949te_a_o @ F2 @ A @ B )
     => ( F2 @ ( produc2143731062552654487tate_a @ A @ B ) ) ) ).

% curryD
thf(fact_77_curryD,axiom,
    ! [F2: product_prod_a_a > $o,A: a,B: a] :
      ( ( product_curry_a_a_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_a_a @ A @ B ) ) ) ).

% curryD
thf(fact_78_curryD,axiom,
    ! [F2: produc6972303929186420058tate_a > $o,A: a,B: state_a] :
      ( ( produc6512344587036289050te_a_o @ F2 @ A @ B )
     => ( F2 @ ( produc8641956578966763338tate_a @ A @ B ) ) ) ).

% curryD
thf(fact_79_pair__in__swap__image,axiom,
    ! [Y: state_a,X3: state_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y @ X3 ) @ ( image_6345842304067597253tate_a @ produc1839749624818714743tate_a @ A5 ) )
      = ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X3 @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_80_pair__in__swap__image,axiom,
    ! [Y: produc6972303929186420058tate_a,X3: produc6972303929186420058tate_a,A5: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y @ X3 ) @ ( image_1816496497719671749tate_a @ produc1520385140544950327tate_a @ A5 ) )
      = ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X3 @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_81_pair__in__swap__image,axiom,
    ! [Y: a,X3: a,A5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X3 ) @ ( image_4636654165204879301od_a_a @ product_swap_a_a @ A5 ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_82_pair__in__swap__image,axiom,
    ! [Y: state_a,X3: a,A5: set_Pr4275752383657305402tate_a] :
      ( ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ Y @ X3 ) @ ( image_71777787125004895te_a_a @ produc3903577461452268778tate_a @ A5 ) )
      = ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X3 @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_83_pair__in__swap__image,axiom,
    ! [Y: a,X3: state_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ Y @ X3 ) @ ( image_4497581360776675627tate_a @ produc2589720147407833796te_a_a @ A5 ) )
      = ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ X3 @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_84_internal__case__prod__def,axiom,
    produc8516608348040061219te_a_o = produc3717778009801768057te_a_o ).

% internal_case_prod_def
thf(fact_85_internal__case__prod__def,axiom,
    produc6455792976634893451tate_a = produc7343868943834020833tate_a ).

% internal_case_prod_def
thf(fact_86_internal__case__prod__def,axiom,
    produc4553788791886541029tate_a = produc5441864759085668411tate_a ).

% internal_case_prod_def
thf(fact_87_case__prod__curry,axiom,
    ! [F2: produc6972303929186420058tate_a > $o] :
      ( ( produc3717778009801768057te_a_o @ ( produc6512344587036289050te_a_o @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_88_case__prod__curry,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( produc7343868943834020833tate_a @ ( produc3060199777641404802tate_a @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_89_case__prod__curry,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( produc5441864759085668411tate_a @ ( produc1158195592893052380tate_a @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_90_curry__case__prod,axiom,
    ! [F2: a > state_a > $o] :
      ( ( produc6512344587036289050te_a_o @ ( produc3717778009801768057te_a_o @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_91_curry__case__prod,axiom,
    ! [F2: state_a > a > produc6972303929186420058tate_a] :
      ( ( produc3060199777641404802tate_a @ ( produc7343868943834020833tate_a @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_92_curry__case__prod,axiom,
    ! [F2: a > state_a > produc6972303929186420058tate_a] :
      ( ( produc1158195592893052380tate_a @ ( produc5441864759085668411tate_a @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_93_same__fstI,axiom,
    ! [P2: state_a > $o,X3: state_a,Y4: state_a,Y: state_a,R2: state_a > set_Pr1753757909640466247tate_a] :
      ( ( P2 @ X3 )
     => ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y4 @ Y ) @ ( R2 @ X3 ) )
       => ( member7923945571256933136tate_a @ ( produc1588899729123023703tate_a @ ( produc6131900399494507223tate_a @ X3 @ Y4 ) @ ( produc6131900399494507223tate_a @ X3 @ Y ) ) @ ( same_f6663734315287600300tate_a @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_94_same__fstI,axiom,
    ! [P2: produc6972303929186420058tate_a > $o,X3: produc6972303929186420058tate_a,Y4: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a,R2: produc6972303929186420058tate_a > set_Pr7963165934123475591tate_a] :
      ( ( P2 @ X3 )
     => ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y4 @ Y ) @ ( R2 @ X3 ) )
       => ( member1643692453717642000tate_a @ ( produc7335381768849070935tate_a @ ( produc2143731062552654487tate_a @ X3 @ Y4 ) @ ( produc2143731062552654487tate_a @ X3 @ Y ) ) @ ( same_f368192502419155564tate_a @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_95_same__fstI,axiom,
    ! [P2: state_a > $o,X3: state_a,Y4: a,Y: a,R2: state_a > set_Product_prod_a_a] :
      ( ( P2 @ X3 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Y ) @ ( R2 @ X3 ) )
       => ( member2649312330005299536te_a_a @ ( produc2877451964877216791te_a_a @ ( produc7328099264922328356te_a_a @ X3 @ Y4 ) @ ( produc7328099264922328356te_a_a @ X3 @ Y ) ) @ ( same_fst_state_a_a @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_96_same__fstI,axiom,
    ! [P2: a > $o,X3: a,Y4: a,Y: a,R2: a > set_Product_prod_a_a] :
      ( ( P2 @ X3 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Y ) @ ( R2 @ X3 ) )
       => ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( product_Pair_a_a @ X3 @ Y ) ) @ ( same_fst_a_a @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_97_same__fstI,axiom,
    ! [P2: a > $o,X3: a,Y4: state_a,Y: state_a,R2: a > set_Pr1753757909640466247tate_a] :
      ( ( P2 @ X3 )
     => ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y4 @ Y ) @ ( R2 @ X3 ) )
       => ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ ( produc8641956578966763338tate_a @ X3 @ Y4 ) @ ( produc8641956578966763338tate_a @ X3 @ Y ) ) @ ( same_fst_a_state_a @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_98_apfst__apsnd,axiom,
    ! [F2: a > state_a,G: state_a > a,X3: produc6972303929186420058tate_a] :
      ( ( produc2674360726052565327te_a_a @ F2 @ ( produc1259323963335463987_a_a_a @ G @ X3 ) )
      = ( produc7328099264922328356te_a_a @ ( F2 @ ( produc736293372669613878tate_a @ X3 ) ) @ ( G @ ( produc7615498795807706488tate_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_99_apfst__apsnd,axiom,
    ! [F2: a > produc6972303929186420058tate_a,G: state_a > produc6972303929186420058tate_a,X3: produc6972303929186420058tate_a] :
      ( ( produc6305817760558673730tate_a @ F2 @ ( produc8490830198457430743te_a_a @ G @ X3 ) )
      = ( produc2143731062552654487tate_a @ ( F2 @ ( produc736293372669613878tate_a @ X3 ) ) @ ( G @ ( produc7615498795807706488tate_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_100_apfst__apsnd,axiom,
    ! [F2: a > a,G: state_a > a,X3: produc6972303929186420058tate_a] :
      ( ( product_apfst_a_a_a @ F2 @ ( produc1259323963335463987_a_a_a @ G @ X3 ) )
      = ( product_Pair_a_a @ ( F2 @ ( produc736293372669613878tate_a @ X3 ) ) @ ( G @ ( produc7615498795807706488tate_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_101_apfst__apsnd,axiom,
    ! [F2: a > state_a,G: state_a > state_a,X3: produc6972303929186420058tate_a] :
      ( ( produc8655383802206219650tate_a @ F2 @ ( produc4530580310919820352te_a_a @ G @ X3 ) )
      = ( produc6131900399494507223tate_a @ ( F2 @ ( produc736293372669613878tate_a @ X3 ) ) @ ( G @ ( produc7615498795807706488tate_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_102_apfst__apsnd,axiom,
    ! [F2: a > a,G: state_a > state_a,X3: produc6972303929186420058tate_a] :
      ( ( produc3988218040097000309tate_a @ F2 @ ( produc4530580310919820352te_a_a @ G @ X3 ) )
      = ( produc8641956578966763338tate_a @ ( F2 @ ( produc736293372669613878tate_a @ X3 ) ) @ ( G @ ( produc7615498795807706488tate_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_103_apfst__apsnd,axiom,
    ! [F2: state_a > a,G: a > state_a,X3: produc5126262593522865844te_a_a] :
      ( ( produc6073655010632001320tate_a @ F2 @ ( produc8426166416538473664tate_a @ G @ X3 ) )
      = ( produc8641956578966763338tate_a @ ( F2 @ ( produc8645808095479954704te_a_a @ X3 ) ) @ ( G @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_104_apfst__apsnd,axiom,
    ! [F2: state_a > state_a,G: a > a,X3: produc5126262593522865844te_a_a] :
      ( ( produc4759797696587566338te_a_a @ F2 @ ( produc9038538116159941043tate_a @ G @ X3 ) )
      = ( produc7328099264922328356te_a_a @ ( F2 @ ( produc8645808095479954704te_a_a @ X3 ) ) @ ( G @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_105_apfst__apsnd,axiom,
    ! [F2: state_a > state_a,G: a > state_a,X3: produc5126262593522865844te_a_a] :
      ( ( produc6780963299072512949tate_a @ F2 @ ( produc8426166416538473664tate_a @ G @ X3 ) )
      = ( produc6131900399494507223tate_a @ ( F2 @ ( produc8645808095479954704te_a_a @ X3 ) ) @ ( G @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_106_apfst__apsnd,axiom,
    ! [F2: state_a > produc6972303929186420058tate_a,G: a > produc6972303929186420058tate_a,X3: produc5126262593522865844te_a_a] :
      ( ( produc3320078648867831669tate_a @ F2 @ ( produc2638113725217294039tate_a @ G @ X3 ) )
      = ( produc2143731062552654487tate_a @ ( F2 @ ( produc8645808095479954704te_a_a @ X3 ) ) @ ( G @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_107_apfst__apsnd,axiom,
    ! [F2: state_a > a,G: a > a,X3: produc5126262593522865844te_a_a] :
      ( ( produc5432375924127299061_a_a_a @ F2 @ ( produc9038538116159941043tate_a @ G @ X3 ) )
      = ( product_Pair_a_a @ ( F2 @ ( produc8645808095479954704te_a_a @ X3 ) ) @ ( G @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ) ).

% apfst_apsnd
thf(fact_108_apsnd__apfst,axiom,
    ! [F2: state_a > a,G: a > state_a,X3: produc6972303929186420058tate_a] :
      ( ( produc5844437624964255334tate_a @ F2 @ ( produc8655383802206219650tate_a @ G @ X3 ) )
      = ( produc7328099264922328356te_a_a @ ( G @ ( produc736293372669613878tate_a @ X3 ) ) @ ( F2 @ ( produc7615498795807706488tate_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_109_apsnd__apfst,axiom,
    ! [F2: state_a > state_a,G: a > state_a,X3: produc6972303929186420058tate_a] :
      ( ( produc8319746412473484275tate_a @ F2 @ ( produc8655383802206219650tate_a @ G @ X3 ) )
      = ( produc6131900399494507223tate_a @ ( G @ ( produc736293372669613878tate_a @ X3 ) ) @ ( F2 @ ( produc7615498795807706488tate_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_110_apsnd__apfst,axiom,
    ! [F2: state_a > produc6972303929186420058tate_a,G: a > produc6972303929186420058tate_a,X3: produc6972303929186420058tate_a] :
      ( ( produc2878013041908044723tate_a @ F2 @ ( produc606892987180027289tate_a @ G @ X3 ) )
      = ( produc2143731062552654487tate_a @ ( G @ ( produc736293372669613878tate_a @ X3 ) ) @ ( F2 @ ( produc7615498795807706488tate_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_111_apsnd__apfst,axiom,
    ! [F2: state_a > a,G: a > a,X3: produc6972303929186420058tate_a] :
      ( ( produc1259323963335463987_a_a_a @ F2 @ ( produc3988218040097000309tate_a @ G @ X3 ) )
      = ( product_Pair_a_a @ ( G @ ( produc736293372669613878tate_a @ X3 ) ) @ ( F2 @ ( produc7615498795807706488tate_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_112_apsnd__apfst,axiom,
    ! [F2: state_a > state_a,G: a > a,X3: produc6972303929186420058tate_a] :
      ( ( produc4530580310919820352te_a_a @ F2 @ ( produc3988218040097000309tate_a @ G @ X3 ) )
      = ( produc8641956578966763338tate_a @ ( G @ ( produc736293372669613878tate_a @ X3 ) ) @ ( F2 @ ( produc7615498795807706488tate_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_113_apsnd__apfst,axiom,
    ! [F2: a > state_a,G: state_a > a,X3: produc5126262593522865844te_a_a] :
      ( ( produc7724680802115506061te_a_a @ F2 @ ( produc5432375924127299061_a_a_a @ G @ X3 ) )
      = ( produc8641956578966763338tate_a @ ( G @ ( produc8645808095479954704te_a_a @ X3 ) ) @ ( F2 @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_114_apsnd__apfst,axiom,
    ! [F2: a > a,G: state_a > state_a,X3: produc5126262593522865844te_a_a] :
      ( ( produc9038538116159941043tate_a @ F2 @ ( produc4759797696587566338te_a_a @ G @ X3 ) )
      = ( produc7328099264922328356te_a_a @ ( G @ ( produc8645808095479954704te_a_a @ X3 ) ) @ ( F2 @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_115_apsnd__apfst,axiom,
    ! [F2: a > state_a,G: state_a > state_a,X3: produc5126262593522865844te_a_a] :
      ( ( produc8426166416538473664tate_a @ F2 @ ( produc4759797696587566338te_a_a @ G @ X3 ) )
      = ( produc6131900399494507223tate_a @ ( G @ ( produc8645808095479954704te_a_a @ X3 ) ) @ ( F2 @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_116_apsnd__apfst,axiom,
    ! [F2: a > produc6972303929186420058tate_a,G: state_a > produc6972303929186420058tate_a,X3: produc5126262593522865844te_a_a] :
      ( ( produc7040518746907032192tate_a @ F2 @ ( produc6459609460420163993te_a_a @ G @ X3 ) )
      = ( produc2143731062552654487tate_a @ ( G @ ( produc8645808095479954704te_a_a @ X3 ) ) @ ( F2 @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_117_apsnd__apfst,axiom,
    ! [F2: a > a,G: state_a > a,X3: produc5126262593522865844te_a_a] :
      ( ( product_apsnd_a_a_a @ F2 @ ( produc5432375924127299061_a_a_a @ G @ X3 ) )
      = ( product_Pair_a_a @ ( G @ ( produc8645808095479954704te_a_a @ X3 ) ) @ ( F2 @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ) ).

% apsnd_apfst
thf(fact_118_fst__apsnd,axiom,
    ! [F2: state_a > state_a,X3: produc6972303929186420058tate_a] :
      ( ( produc736293372669613878tate_a @ ( produc4530580310919820352te_a_a @ F2 @ X3 ) )
      = ( produc736293372669613878tate_a @ X3 ) ) ).

% fst_apsnd
thf(fact_119_fst__apsnd,axiom,
    ! [F2: a > a,X3: produc5126262593522865844te_a_a] :
      ( ( produc8645808095479954704te_a_a @ ( produc9038538116159941043tate_a @ F2 @ X3 ) )
      = ( produc8645808095479954704te_a_a @ X3 ) ) ).

% fst_apsnd
thf(fact_120_fst__apfst,axiom,
    ! [F2: a > a,X3: produc6972303929186420058tate_a] :
      ( ( produc736293372669613878tate_a @ ( produc3988218040097000309tate_a @ F2 @ X3 ) )
      = ( F2 @ ( produc736293372669613878tate_a @ X3 ) ) ) ).

% fst_apfst
thf(fact_121_fst__apfst,axiom,
    ! [F2: state_a > state_a,X3: produc5126262593522865844te_a_a] :
      ( ( produc8645808095479954704te_a_a @ ( produc4759797696587566338te_a_a @ F2 @ X3 ) )
      = ( F2 @ ( produc8645808095479954704te_a_a @ X3 ) ) ) ).

% fst_apfst
thf(fact_122_apfst__eq__conv,axiom,
    ! [F2: a > a,X3: produc6972303929186420058tate_a,G: a > a] :
      ( ( ( produc3988218040097000309tate_a @ F2 @ X3 )
        = ( produc3988218040097000309tate_a @ G @ X3 ) )
      = ( ( F2 @ ( produc736293372669613878tate_a @ X3 ) )
        = ( G @ ( produc736293372669613878tate_a @ X3 ) ) ) ) ).

% apfst_eq_conv
thf(fact_123_snd__apfst,axiom,
    ! [F2: a > a,X3: produc6972303929186420058tate_a] :
      ( ( produc7615498795807706488tate_a @ ( produc3988218040097000309tate_a @ F2 @ X3 ) )
      = ( produc7615498795807706488tate_a @ X3 ) ) ).

% snd_apfst
thf(fact_124_snd__apfst,axiom,
    ! [F2: state_a > state_a,X3: produc5126262593522865844te_a_a] :
      ( ( produc6301641481763271506te_a_a @ ( produc4759797696587566338te_a_a @ F2 @ X3 ) )
      = ( produc6301641481763271506te_a_a @ X3 ) ) ).

% snd_apfst
thf(fact_125_mem__Collect__eq,axiom,
    ! [A: produc6972303929186420058tate_a,P2: produc6972303929186420058tate_a > $o] :
      ( ( member3175992478928454403tate_a @ A @ ( collec8349941756605538373tate_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
    ! [A: produc5126262593522865844te_a_a,P2: produc5126262593522865844te_a_a > $o] :
      ( ( member1329951143264900189te_a_a @ A @ ( collec6503900420941984159te_a_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
    ! [A: produc4774439622479004775tate_a,P2: produc4774439622479004775tate_a > $o] :
      ( ( member3801464090328701328tate_a @ A @ ( collec5776123031894519250tate_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
    ! [A: produc5940350820068290983tate_a,P2: produc5940350820068290983tate_a > $o] :
      ( ( member1595334363549574352tate_a @ A @ ( collec4717139122855073042tate_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_129_mem__Collect__eq,axiom,
    ! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
      ( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_130_Collect__mem__eq,axiom,
    ! [A5: set_Pr4275752383657305402tate_a] :
      ( ( collec8349941756605538373tate_a
        @ ^ [X4: produc6972303929186420058tate_a] : ( member3175992478928454403tate_a @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a] :
      ( ( collec6503900420941984159te_a_a
        @ ^ [X4: produc5126262593522865844te_a_a] : ( member1329951143264900189te_a_a @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_132_Collect__mem__eq,axiom,
    ! [A5: set_Pr1753757909640466247tate_a] :
      ( ( collec5776123031894519250tate_a
        @ ^ [X4: produc4774439622479004775tate_a] : ( member3801464090328701328tate_a @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_133_Collect__mem__eq,axiom,
    ! [A5: set_Pr7963165934123475591tate_a] :
      ( ( collec4717139122855073042tate_a
        @ ^ [X4: produc5940350820068290983tate_a] : ( member1595334363549574352tate_a @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
    ! [A5: set_Product_prod_a_a] :
      ( ( collec3336397797384452498od_a_a
        @ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_135_apsnd__eq__conv,axiom,
    ! [F2: state_a > state_a,X3: produc6972303929186420058tate_a,G: state_a > state_a] :
      ( ( ( produc4530580310919820352te_a_a @ F2 @ X3 )
        = ( produc4530580310919820352te_a_a @ G @ X3 ) )
      = ( ( F2 @ ( produc7615498795807706488tate_a @ X3 ) )
        = ( G @ ( produc7615498795807706488tate_a @ X3 ) ) ) ) ).

% apsnd_eq_conv
thf(fact_136_snd__apsnd,axiom,
    ! [F2: state_a > state_a,X3: produc6972303929186420058tate_a] :
      ( ( produc7615498795807706488tate_a @ ( produc4530580310919820352te_a_a @ F2 @ X3 ) )
      = ( F2 @ ( produc7615498795807706488tate_a @ X3 ) ) ) ).

% snd_apsnd
thf(fact_137_snd__apsnd,axiom,
    ! [F2: a > a,X3: produc5126262593522865844te_a_a] :
      ( ( produc6301641481763271506te_a_a @ ( produc9038538116159941043tate_a @ F2 @ X3 ) )
      = ( F2 @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ).

% snd_apsnd
thf(fact_138_prod_Ocollapse,axiom,
    ! [Prod: produc4774439622479004775tate_a] :
      ( ( produc6131900399494507223tate_a @ ( produc2992983560240211139tate_a @ Prod ) @ ( produc7912480809654211077tate_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_139_prod_Ocollapse,axiom,
    ! [Prod: produc5940350820068290983tate_a] :
      ( ( produc2143731062552654487tate_a @ ( produc874779950727550083tate_a @ Prod ) @ ( produc7323807087048646085tate_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_140_prod_Ocollapse,axiom,
    ! [Prod: product_prod_a_a] :
      ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_141_prod_Ocollapse,axiom,
    ! [Prod: produc5126262593522865844te_a_a] :
      ( ( produc7328099264922328356te_a_a @ ( produc8645808095479954704te_a_a @ Prod ) @ ( produc6301641481763271506te_a_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_142_prod_Ocollapse,axiom,
    ! [Prod: produc6972303929186420058tate_a] :
      ( ( produc8641956578966763338tate_a @ ( produc736293372669613878tate_a @ Prod ) @ ( produc7615498795807706488tate_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_143_snd__swap,axiom,
    ! [X3: produc6972303929186420058tate_a] :
      ( ( produc6301641481763271506te_a_a @ ( produc3903577461452268778tate_a @ X3 ) )
      = ( produc736293372669613878tate_a @ X3 ) ) ).

% snd_swap
thf(fact_144_snd__swap,axiom,
    ! [X3: produc5126262593522865844te_a_a] :
      ( ( produc7615498795807706488tate_a @ ( produc2589720147407833796te_a_a @ X3 ) )
      = ( produc8645808095479954704te_a_a @ X3 ) ) ).

% snd_swap
thf(fact_145_fst__swap,axiom,
    ! [X3: produc6972303929186420058tate_a] :
      ( ( produc8645808095479954704te_a_a @ ( produc3903577461452268778tate_a @ X3 ) )
      = ( produc7615498795807706488tate_a @ X3 ) ) ).

% fst_swap
thf(fact_146_fst__swap,axiom,
    ! [X3: produc5126262593522865844te_a_a] :
      ( ( produc736293372669613878tate_a @ ( produc2589720147407833796te_a_a @ X3 ) )
      = ( produc6301641481763271506te_a_a @ X3 ) ) ).

% fst_swap
thf(fact_147_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc4774439622479004775tate_a,A5: state_a > state_a > $o] :
      ( ( member3801464090328701328tate_a @ X3 @ ( collec5776123031894519250tate_a @ ( produc6586738294279371180te_a_o @ A5 ) ) )
     => ( A5 @ ( produc2992983560240211139tate_a @ X3 ) @ ( produc7912480809654211077tate_a @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_148_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc5940350820068290983tate_a,A5: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o] :
      ( ( member1595334363549574352tate_a @ X3 @ ( collec4717139122855073042tate_a @ ( produc3822078179198691820te_a_o @ A5 ) ) )
     => ( A5 @ ( produc874779950727550083tate_a @ X3 ) @ ( produc7323807087048646085tate_a @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_149_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: product_prod_a_a,A5: a > a > $o] :
      ( ( member1426531477525435216od_a_a @ X3 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ A5 ) ) )
     => ( A5 @ ( product_fst_a_a @ X3 ) @ ( product_snd_a_a @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_150_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: state_a > a > $o] :
      ( ( member1329951143264900189te_a_a @ X3 @ ( collec6503900420941984159te_a_a @ ( produc490204266526094367_a_a_o @ A5 ) ) )
     => ( A5 @ ( produc8645808095479954704te_a_a @ X3 ) @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_151_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: a > state_a > $o] :
      ( ( member3175992478928454403tate_a @ X3 @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ A5 ) ) )
     => ( A5 @ ( produc736293372669613878tate_a @ X3 ) @ ( produc7615498795807706488tate_a @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_152_case__prod__beta,axiom,
    ( produc3717778009801768057te_a_o
    = ( ^ [F: a > state_a > $o,P3: produc6972303929186420058tate_a] : ( F @ ( produc736293372669613878tate_a @ P3 ) @ ( produc7615498795807706488tate_a @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_153_case__prod__beta,axiom,
    ( produc7343868943834020833tate_a
    = ( ^ [F: state_a > a > produc6972303929186420058tate_a,P3: produc5126262593522865844te_a_a] : ( F @ ( produc8645808095479954704te_a_a @ P3 ) @ ( produc6301641481763271506te_a_a @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_154_case__prod__beta,axiom,
    ( produc5441864759085668411tate_a
    = ( ^ [F: a > state_a > produc6972303929186420058tate_a,P3: produc6972303929186420058tate_a] : ( F @ ( produc736293372669613878tate_a @ P3 ) @ ( produc7615498795807706488tate_a @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_155_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc6972303929186420058tate_a,Z: produc6972303929186420058tate_a] : ( Y5 = Z ) )
    = ( ^ [S3: produc6972303929186420058tate_a,T: produc6972303929186420058tate_a] :
          ( ( ( produc736293372669613878tate_a @ S3 )
            = ( produc736293372669613878tate_a @ T ) )
          & ( ( produc7615498795807706488tate_a @ S3 )
            = ( produc7615498795807706488tate_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_156_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc5126262593522865844te_a_a,Z: produc5126262593522865844te_a_a] : ( Y5 = Z ) )
    = ( ^ [S3: produc5126262593522865844te_a_a,T: produc5126262593522865844te_a_a] :
          ( ( ( produc8645808095479954704te_a_a @ S3 )
            = ( produc8645808095479954704te_a_a @ T ) )
          & ( ( produc6301641481763271506te_a_a @ S3 )
            = ( produc6301641481763271506te_a_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_157_split__beta,axiom,
    ( produc3717778009801768057te_a_o
    = ( ^ [F: a > state_a > $o,Prod2: produc6972303929186420058tate_a] : ( F @ ( produc736293372669613878tate_a @ Prod2 ) @ ( produc7615498795807706488tate_a @ Prod2 ) ) ) ) ).

% split_beta
thf(fact_158_split__beta,axiom,
    ( produc7343868943834020833tate_a
    = ( ^ [F: state_a > a > produc6972303929186420058tate_a,Prod2: produc5126262593522865844te_a_a] : ( F @ ( produc8645808095479954704te_a_a @ Prod2 ) @ ( produc6301641481763271506te_a_a @ Prod2 ) ) ) ) ).

% split_beta
thf(fact_159_split__beta,axiom,
    ( produc5441864759085668411tate_a
    = ( ^ [F: a > state_a > produc6972303929186420058tate_a,Prod2: produc6972303929186420058tate_a] : ( F @ ( produc736293372669613878tate_a @ Prod2 ) @ ( produc7615498795807706488tate_a @ Prod2 ) ) ) ) ).

% split_beta
thf(fact_160_prod__eqI,axiom,
    ! [P: produc6972303929186420058tate_a,Q: produc6972303929186420058tate_a] :
      ( ( ( produc736293372669613878tate_a @ P )
        = ( produc736293372669613878tate_a @ Q ) )
     => ( ( ( produc7615498795807706488tate_a @ P )
          = ( produc7615498795807706488tate_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_161_prod__eqI,axiom,
    ! [P: produc5126262593522865844te_a_a,Q: produc5126262593522865844te_a_a] :
      ( ( ( produc8645808095479954704te_a_a @ P )
        = ( produc8645808095479954704te_a_a @ Q ) )
     => ( ( ( produc6301641481763271506te_a_a @ P )
          = ( produc6301641481763271506te_a_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_162_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F2: a > state_a > $o,Prod: produc6972303929186420058tate_a] :
      ( ( P2 @ ( produc3717778009801768057te_a_o @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc8641956578966763338tate_a @ ( produc736293372669613878tate_a @ Prod ) @ ( produc7615498795807706488tate_a @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( produc736293372669613878tate_a @ Prod ) @ ( produc7615498795807706488tate_a @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_163_prod_Osplit__sel__asm,axiom,
    ! [P2: produc6972303929186420058tate_a > $o,F2: state_a > a > produc6972303929186420058tate_a,Prod: produc5126262593522865844te_a_a] :
      ( ( P2 @ ( produc7343868943834020833tate_a @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc7328099264922328356te_a_a @ ( produc8645808095479954704te_a_a @ Prod ) @ ( produc6301641481763271506te_a_a @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( produc8645808095479954704te_a_a @ Prod ) @ ( produc6301641481763271506te_a_a @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_164_prod_Osplit__sel__asm,axiom,
    ! [P2: produc6972303929186420058tate_a > $o,F2: a > state_a > produc6972303929186420058tate_a,Prod: produc6972303929186420058tate_a] :
      ( ( P2 @ ( produc5441864759085668411tate_a @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc8641956578966763338tate_a @ ( produc736293372669613878tate_a @ Prod ) @ ( produc7615498795807706488tate_a @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( produc736293372669613878tate_a @ Prod ) @ ( produc7615498795807706488tate_a @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_165_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F2: a > state_a > $o,Prod: produc6972303929186420058tate_a] :
      ( ( P2 @ ( produc3717778009801768057te_a_o @ F2 @ Prod ) )
      = ( ( Prod
          = ( produc8641956578966763338tate_a @ ( produc736293372669613878tate_a @ Prod ) @ ( produc7615498795807706488tate_a @ Prod ) ) )
       => ( P2 @ ( F2 @ ( produc736293372669613878tate_a @ Prod ) @ ( produc7615498795807706488tate_a @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_166_prod_Osplit__sel,axiom,
    ! [P2: produc6972303929186420058tate_a > $o,F2: state_a > a > produc6972303929186420058tate_a,Prod: produc5126262593522865844te_a_a] :
      ( ( P2 @ ( produc7343868943834020833tate_a @ F2 @ Prod ) )
      = ( ( Prod
          = ( produc7328099264922328356te_a_a @ ( produc8645808095479954704te_a_a @ Prod ) @ ( produc6301641481763271506te_a_a @ Prod ) ) )
       => ( P2 @ ( F2 @ ( produc8645808095479954704te_a_a @ Prod ) @ ( produc6301641481763271506te_a_a @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_167_prod_Osplit__sel,axiom,
    ! [P2: produc6972303929186420058tate_a > $o,F2: a > state_a > produc6972303929186420058tate_a,Prod: produc6972303929186420058tate_a] :
      ( ( P2 @ ( produc5441864759085668411tate_a @ F2 @ Prod ) )
      = ( ( Prod
          = ( produc8641956578966763338tate_a @ ( produc736293372669613878tate_a @ Prod ) @ ( produc7615498795807706488tate_a @ Prod ) ) )
       => ( P2 @ ( F2 @ ( produc736293372669613878tate_a @ Prod ) @ ( produc7615498795807706488tate_a @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_168_prod_Oexpand,axiom,
    ! [Prod: produc6972303929186420058tate_a,Prod3: produc6972303929186420058tate_a] :
      ( ( ( ( produc736293372669613878tate_a @ Prod )
          = ( produc736293372669613878tate_a @ Prod3 ) )
        & ( ( produc7615498795807706488tate_a @ Prod )
          = ( produc7615498795807706488tate_a @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) ).

% prod.expand
thf(fact_169_prod_Oexpand,axiom,
    ! [Prod: produc5126262593522865844te_a_a,Prod3: produc5126262593522865844te_a_a] :
      ( ( ( ( produc8645808095479954704te_a_a @ Prod )
          = ( produc8645808095479954704te_a_a @ Prod3 ) )
        & ( ( produc6301641481763271506te_a_a @ Prod )
          = ( produc6301641481763271506te_a_a @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) ).

% prod.expand
thf(fact_170_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4774439622479004775tate_a] :
      ( Prod
      = ( produc6131900399494507223tate_a @ ( produc2992983560240211139tate_a @ Prod ) @ ( produc7912480809654211077tate_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_171_prod_Oexhaust__sel,axiom,
    ! [Prod: produc5940350820068290983tate_a] :
      ( Prod
      = ( produc2143731062552654487tate_a @ ( produc874779950727550083tate_a @ Prod ) @ ( produc7323807087048646085tate_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_172_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_a_a] :
      ( Prod
      = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_173_prod_Oexhaust__sel,axiom,
    ! [Prod: produc5126262593522865844te_a_a] :
      ( Prod
      = ( produc7328099264922328356te_a_a @ ( produc8645808095479954704te_a_a @ Prod ) @ ( produc6301641481763271506te_a_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_174_prod_Oexhaust__sel,axiom,
    ! [Prod: produc6972303929186420058tate_a] :
      ( Prod
      = ( produc8641956578966763338tate_a @ ( produc736293372669613878tate_a @ Prod ) @ ( produc7615498795807706488tate_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_175_surjective__pairing,axiom,
    ! [T2: produc4774439622479004775tate_a] :
      ( T2
      = ( produc6131900399494507223tate_a @ ( produc2992983560240211139tate_a @ T2 ) @ ( produc7912480809654211077tate_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_176_surjective__pairing,axiom,
    ! [T2: produc5940350820068290983tate_a] :
      ( T2
      = ( produc2143731062552654487tate_a @ ( produc874779950727550083tate_a @ T2 ) @ ( produc7323807087048646085tate_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_177_surjective__pairing,axiom,
    ! [T2: product_prod_a_a] :
      ( T2
      = ( product_Pair_a_a @ ( product_fst_a_a @ T2 ) @ ( product_snd_a_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_178_surjective__pairing,axiom,
    ! [T2: produc5126262593522865844te_a_a] :
      ( T2
      = ( produc7328099264922328356te_a_a @ ( produc8645808095479954704te_a_a @ T2 ) @ ( produc6301641481763271506te_a_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_179_surjective__pairing,axiom,
    ! [T2: produc6972303929186420058tate_a] :
      ( T2
      = ( produc8641956578966763338tate_a @ ( produc736293372669613878tate_a @ T2 ) @ ( produc7615498795807706488tate_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_180_prod_Oswap__def,axiom,
    ( produc1839749624818714743tate_a
    = ( ^ [P3: produc4774439622479004775tate_a] : ( produc6131900399494507223tate_a @ ( produc7912480809654211077tate_a @ P3 ) @ ( produc2992983560240211139tate_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_181_prod_Oswap__def,axiom,
    ( produc1520385140544950327tate_a
    = ( ^ [P3: produc5940350820068290983tate_a] : ( produc2143731062552654487tate_a @ ( produc7323807087048646085tate_a @ P3 ) @ ( produc874779950727550083tate_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_182_prod_Oswap__def,axiom,
    ( product_swap_a_a
    = ( ^ [P3: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P3 ) @ ( product_fst_a_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_183_prod_Oswap__def,axiom,
    ( produc3903577461452268778tate_a
    = ( ^ [P3: produc6972303929186420058tate_a] : ( produc7328099264922328356te_a_a @ ( produc7615498795807706488tate_a @ P3 ) @ ( produc736293372669613878tate_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_184_prod_Oswap__def,axiom,
    ( produc2589720147407833796te_a_a
    = ( ^ [P3: produc5126262593522865844te_a_a] : ( produc8641956578966763338tate_a @ ( produc6301641481763271506te_a_a @ P3 ) @ ( produc8645808095479954704te_a_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_185_snd__conv,axiom,
    ! [X1: state_a,X2: state_a] :
      ( ( produc7912480809654211077tate_a @ ( produc6131900399494507223tate_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_186_snd__conv,axiom,
    ! [X1: produc6972303929186420058tate_a,X2: produc6972303929186420058tate_a] :
      ( ( produc7323807087048646085tate_a @ ( produc2143731062552654487tate_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_187_snd__conv,axiom,
    ! [X1: a,X2: a] :
      ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_188_snd__conv,axiom,
    ! [X1: state_a,X2: a] :
      ( ( produc6301641481763271506te_a_a @ ( produc7328099264922328356te_a_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_189_snd__conv,axiom,
    ! [X1: a,X2: state_a] :
      ( ( produc7615498795807706488tate_a @ ( produc8641956578966763338tate_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_190_snd__eqD,axiom,
    ! [X3: state_a,Y: state_a,A: state_a] :
      ( ( ( produc7912480809654211077tate_a @ ( produc6131900399494507223tate_a @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_191_snd__eqD,axiom,
    ! [X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a,A: produc6972303929186420058tate_a] :
      ( ( ( produc7323807087048646085tate_a @ ( produc2143731062552654487tate_a @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_192_snd__eqD,axiom,
    ! [X3: a,Y: a,A: a] :
      ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_193_snd__eqD,axiom,
    ! [X3: state_a,Y: a,A: a] :
      ( ( ( produc6301641481763271506te_a_a @ ( produc7328099264922328356te_a_a @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_194_snd__eqD,axiom,
    ! [X3: a,Y: state_a,A: state_a] :
      ( ( ( produc7615498795807706488tate_a @ ( produc8641956578966763338tate_a @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_195_fst__conv,axiom,
    ! [X1: state_a,X2: state_a] :
      ( ( produc2992983560240211139tate_a @ ( produc6131900399494507223tate_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_196_fst__conv,axiom,
    ! [X1: produc6972303929186420058tate_a,X2: produc6972303929186420058tate_a] :
      ( ( produc874779950727550083tate_a @ ( produc2143731062552654487tate_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_197_fst__conv,axiom,
    ! [X1: a,X2: a] :
      ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_198_fst__conv,axiom,
    ! [X1: state_a,X2: a] :
      ( ( produc8645808095479954704te_a_a @ ( produc7328099264922328356te_a_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_199_fst__conv,axiom,
    ! [X1: a,X2: state_a] :
      ( ( produc736293372669613878tate_a @ ( produc8641956578966763338tate_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_200_fst__eqD,axiom,
    ! [X3: state_a,Y: state_a,A: state_a] :
      ( ( ( produc2992983560240211139tate_a @ ( produc6131900399494507223tate_a @ X3 @ Y ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_201_fst__eqD,axiom,
    ! [X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a,A: produc6972303929186420058tate_a] :
      ( ( ( produc874779950727550083tate_a @ ( produc2143731062552654487tate_a @ X3 @ Y ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_202_fst__eqD,axiom,
    ! [X3: a,Y: a,A: a] :
      ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X3 @ Y ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_203_fst__eqD,axiom,
    ! [X3: state_a,Y: a,A: state_a] :
      ( ( ( produc8645808095479954704te_a_a @ ( produc7328099264922328356te_a_a @ X3 @ Y ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_204_fst__eqD,axiom,
    ! [X3: a,Y: state_a,A: a] :
      ( ( ( produc736293372669613878tate_a @ ( produc8641956578966763338tate_a @ X3 @ Y ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_205_mem__case__prodE,axiom,
    ! [Z2: product_prod_a_a,C2: a > a > set_Product_prod_a_a,P: product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ Z2 @ ( produc5766521956407364827od_a_a @ C2 @ P ) )
     => ~ ! [X: a,Y3: a] :
            ( ( P
              = ( product_Pair_a_a @ X @ Y3 ) )
           => ~ ( member1426531477525435216od_a_a @ Z2 @ ( C2 @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_206_mem__case__prodE,axiom,
    ! [Z2: product_prod_a_a,C2: a > state_a > set_Product_prod_a_a,P: produc6972303929186420058tate_a] :
      ( ( member1426531477525435216od_a_a @ Z2 @ ( produc5733692637325658216od_a_a @ C2 @ P ) )
     => ~ ! [X: a,Y3: state_a] :
            ( ( P
              = ( produc8641956578966763338tate_a @ X @ Y3 ) )
           => ~ ( member1426531477525435216od_a_a @ Z2 @ ( C2 @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_207_mem__case__prodE,axiom,
    ! [Z2: product_prod_a_a,C2: state_a > a > set_Product_prod_a_a,P: produc5126262593522865844te_a_a] :
      ( ( member1426531477525435216od_a_a @ Z2 @ ( produc5706043332411121422od_a_a @ C2 @ P ) )
     => ~ ! [X: state_a,Y3: a] :
            ( ( P
              = ( produc7328099264922328356te_a_a @ X @ Y3 ) )
           => ~ ( member1426531477525435216od_a_a @ Z2 @ ( C2 @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_208_mem__case__prodE,axiom,
    ! [Z2: produc6972303929186420058tate_a,C2: a > a > set_Pr4275752383657305402tate_a,P: product_prod_a_a] :
      ( ( member3175992478928454403tate_a @ Z2 @ ( produc4072428802361025294tate_a @ C2 @ P ) )
     => ~ ! [X: a,Y3: a] :
            ( ( P
              = ( product_Pair_a_a @ X @ Y3 ) )
           => ~ ( member3175992478928454403tate_a @ Z2 @ ( C2 @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_209_mem__case__prodE,axiom,
    ! [Z2: produc5126262593522865844te_a_a,C2: a > a > set_Pr2749968112124914068te_a_a,P: product_prod_a_a] :
      ( ( member1329951143264900189te_a_a @ Z2 @ ( produc2546644530828633960te_a_a @ C2 @ P ) )
     => ~ ! [X: a,Y3: a] :
            ( ( P
              = ( product_Pair_a_a @ X @ Y3 ) )
           => ~ ( member1329951143264900189te_a_a @ Z2 @ ( C2 @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_210_mem__case__prodE,axiom,
    ! [Z2: produc6972303929186420058tate_a,C2: a > state_a > set_Pr4275752383657305402tate_a,P: produc6972303929186420058tate_a] :
      ( ( member3175992478928454403tate_a @ Z2 @ ( produc8858995663098442779tate_a @ C2 @ P ) )
     => ~ ! [X: a,Y3: state_a] :
            ( ( P
              = ( produc8641956578966763338tate_a @ X @ Y3 ) )
           => ~ ( member3175992478928454403tate_a @ Z2 @ ( C2 @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_211_mem__case__prodE,axiom,
    ! [Z2: produc5126262593522865844te_a_a,C2: a > state_a > set_Pr2749968112124914068te_a_a,P: produc6972303929186420058tate_a] :
      ( ( member1329951143264900189te_a_a @ Z2 @ ( produc7333211391566051445te_a_a @ C2 @ P ) )
     => ~ ! [X: a,Y3: state_a] :
            ( ( P
              = ( produc8641956578966763338tate_a @ X @ Y3 ) )
           => ~ ( member1329951143264900189te_a_a @ Z2 @ ( C2 @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_212_mem__case__prodE,axiom,
    ! [Z2: produc6972303929186420058tate_a,C2: state_a > a > set_Pr4275752383657305402tate_a,P: produc5126262593522865844te_a_a] :
      ( ( member3175992478928454403tate_a @ Z2 @ ( produc4004399222129377217tate_a @ C2 @ P ) )
     => ~ ! [X: state_a,Y3: a] :
            ( ( P
              = ( produc7328099264922328356te_a_a @ X @ Y3 ) )
           => ~ ( member3175992478928454403tate_a @ Z2 @ ( C2 @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_213_mem__case__prodE,axiom,
    ! [Z2: produc5126262593522865844te_a_a,C2: state_a > a > set_Pr2749968112124914068te_a_a,P: produc5126262593522865844te_a_a] :
      ( ( member1329951143264900189te_a_a @ Z2 @ ( produc2478614950596985883te_a_a @ C2 @ P ) )
     => ~ ! [X: state_a,Y3: a] :
            ( ( P
              = ( produc7328099264922328356te_a_a @ X @ Y3 ) )
           => ~ ( member1329951143264900189te_a_a @ Z2 @ ( C2 @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_214_mem__case__prodE,axiom,
    ! [Z2: product_prod_a_a,C2: state_a > state_a > set_Product_prod_a_a,P: produc4774439622479004775tate_a] :
      ( ( member1426531477525435216od_a_a @ Z2 @ ( produc5665663057094010139od_a_a @ C2 @ P ) )
     => ~ ! [X: state_a,Y3: state_a] :
            ( ( P
              = ( produc6131900399494507223tate_a @ X @ Y3 ) )
           => ~ ( member1426531477525435216od_a_a @ Z2 @ ( C2 @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_215_old_Oprod_Ocase,axiom,
    ! [F2: a > state_a > $o,X1: a,X2: state_a] :
      ( ( produc3717778009801768057te_a_o @ F2 @ ( produc8641956578966763338tate_a @ X1 @ X2 ) )
      = ( F2 @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_216_old_Oprod_Ocase,axiom,
    ! [F2: state_a > a > produc6972303929186420058tate_a,X1: state_a,X2: a] :
      ( ( produc7343868943834020833tate_a @ F2 @ ( produc7328099264922328356te_a_a @ X1 @ X2 ) )
      = ( F2 @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_217_old_Oprod_Ocase,axiom,
    ! [F2: a > state_a > produc6972303929186420058tate_a,X1: a,X2: state_a] :
      ( ( produc5441864759085668411tate_a @ F2 @ ( produc8641956578966763338tate_a @ X1 @ X2 ) )
      = ( F2 @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_218_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: state_a > state_a > $o,X3: state_a,Y: state_a,A: produc4774439622479004775tate_a] :
      ( ( P2 @ X3 @ Y )
     => ( ( A
          = ( produc6131900399494507223tate_a @ X3 @ Y ) )
       => ( P2 @ ( produc2992983560240211139tate_a @ A ) @ ( produc7912480809654211077tate_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_219_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o,X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a,A: produc5940350820068290983tate_a] :
      ( ( P2 @ X3 @ Y )
     => ( ( A
          = ( produc2143731062552654487tate_a @ X3 @ Y ) )
       => ( P2 @ ( produc874779950727550083tate_a @ A ) @ ( produc7323807087048646085tate_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_220_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: a > a > $o,X3: a,Y: a,A: product_prod_a_a] :
      ( ( P2 @ X3 @ Y )
     => ( ( A
          = ( product_Pair_a_a @ X3 @ Y ) )
       => ( P2 @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_221_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: state_a > a > $o,X3: state_a,Y: a,A: produc5126262593522865844te_a_a] :
      ( ( P2 @ X3 @ Y )
     => ( ( A
          = ( produc7328099264922328356te_a_a @ X3 @ Y ) )
       => ( P2 @ ( produc8645808095479954704te_a_a @ A ) @ ( produc6301641481763271506te_a_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_222_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: a > state_a > $o,X3: a,Y: state_a,A: produc6972303929186420058tate_a] :
      ( ( P2 @ X3 @ Y )
     => ( ( A
          = ( produc8641956578966763338tate_a @ X3 @ Y ) )
       => ( P2 @ ( produc736293372669613878tate_a @ A ) @ ( produc7615498795807706488tate_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_223_conjI__realizer,axiom,
    ! [P2: state_a > $o,P: state_a,Q2: state_a > $o,Q: state_a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc2992983560240211139tate_a @ ( produc6131900399494507223tate_a @ P @ Q ) ) )
          & ( Q2 @ ( produc7912480809654211077tate_a @ ( produc6131900399494507223tate_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_224_conjI__realizer,axiom,
    ! [P2: produc6972303929186420058tate_a > $o,P: produc6972303929186420058tate_a,Q2: produc6972303929186420058tate_a > $o,Q: produc6972303929186420058tate_a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc874779950727550083tate_a @ ( produc2143731062552654487tate_a @ P @ Q ) ) )
          & ( Q2 @ ( produc7323807087048646085tate_a @ ( produc2143731062552654487tate_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_225_conjI__realizer,axiom,
    ! [P2: a > $o,P: a,Q2: a > $o,Q: a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_a_a @ ( product_Pair_a_a @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_a_a @ ( product_Pair_a_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_226_conjI__realizer,axiom,
    ! [P2: state_a > $o,P: state_a,Q2: a > $o,Q: a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc8645808095479954704te_a_a @ ( produc7328099264922328356te_a_a @ P @ Q ) ) )
          & ( Q2 @ ( produc6301641481763271506te_a_a @ ( produc7328099264922328356te_a_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_227_conjI__realizer,axiom,
    ! [P2: a > $o,P: a,Q2: state_a > $o,Q: state_a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc736293372669613878tate_a @ ( produc8641956578966763338tate_a @ P @ Q ) ) )
          & ( Q2 @ ( produc7615498795807706488tate_a @ ( produc8641956578966763338tate_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_228_exI__realizer,axiom,
    ! [P2: state_a > state_a > $o,Y: state_a,X3: state_a] :
      ( ( P2 @ Y @ X3 )
     => ( P2 @ ( produc7912480809654211077tate_a @ ( produc6131900399494507223tate_a @ X3 @ Y ) ) @ ( produc2992983560240211139tate_a @ ( produc6131900399494507223tate_a @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_229_exI__realizer,axiom,
    ! [P2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o,Y: produc6972303929186420058tate_a,X3: produc6972303929186420058tate_a] :
      ( ( P2 @ Y @ X3 )
     => ( P2 @ ( produc7323807087048646085tate_a @ ( produc2143731062552654487tate_a @ X3 @ Y ) ) @ ( produc874779950727550083tate_a @ ( produc2143731062552654487tate_a @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_230_exI__realizer,axiom,
    ! [P2: a > a > $o,Y: a,X3: a] :
      ( ( P2 @ Y @ X3 )
     => ( P2 @ ( product_snd_a_a @ ( product_Pair_a_a @ X3 @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_231_exI__realizer,axiom,
    ! [P2: a > state_a > $o,Y: a,X3: state_a] :
      ( ( P2 @ Y @ X3 )
     => ( P2 @ ( produc6301641481763271506te_a_a @ ( produc7328099264922328356te_a_a @ X3 @ Y ) ) @ ( produc8645808095479954704te_a_a @ ( produc7328099264922328356te_a_a @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_232_exI__realizer,axiom,
    ! [P2: state_a > a > $o,Y: state_a,X3: a] :
      ( ( P2 @ Y @ X3 )
     => ( P2 @ ( produc7615498795807706488tate_a @ ( produc8641956578966763338tate_a @ X3 @ Y ) ) @ ( produc736293372669613878tate_a @ ( produc8641956578966763338tate_a @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_233_image__eqI,axiom,
    ! [B: product_prod_a_a,F2: product_prod_a_a > product_prod_a_a,X3: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member1426531477525435216od_a_a @ X3 @ A5 )
       => ( member1426531477525435216od_a_a @ B @ ( image_4636654165204879301od_a_a @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_234_image__eqI,axiom,
    ! [B: product_prod_a_a,F2: produc6972303929186420058tate_a > product_prod_a_a,X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member3175992478928454403tate_a @ X3 @ A5 )
       => ( member1426531477525435216od_a_a @ B @ ( image_8179770026730005074od_a_a @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_235_image__eqI,axiom,
    ! [B: product_prod_a_a,F2: produc5126262593522865844te_a_a > product_prod_a_a,X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
       => ( member1426531477525435216od_a_a @ B @ ( image_615948358740051320od_a_a @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_236_image__eqI,axiom,
    ! [B: produc6972303929186420058tate_a,F2: product_prod_a_a > produc6972303929186420058tate_a,X3: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member1426531477525435216od_a_a @ X3 @ A5 )
       => ( member3175992478928454403tate_a @ B @ ( image_1563038779069872376tate_a @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_237_image__eqI,axiom,
    ! [B: produc5126262593522865844te_a_a,F2: product_prod_a_a > produc5126262593522865844te_a_a,X3: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member1426531477525435216od_a_a @ X3 @ A5 )
       => ( member1329951143264900189te_a_a @ B @ ( image_8940369480261093970te_a_a @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_238_image__eqI,axiom,
    ! [B: produc6972303929186420058tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member3175992478928454403tate_a @ X3 @ A5 )
       => ( member3175992478928454403tate_a @ B @ ( image_1917819122788559109tate_a @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_239_image__eqI,axiom,
    ! [B: produc5126262593522865844te_a_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member3175992478928454403tate_a @ X3 @ A5 )
       => ( member1329951143264900189te_a_a @ B @ ( image_71777787125004895te_a_a @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_240_image__eqI,axiom,
    ! [B: produc6972303929186420058tate_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
       => ( member3175992478928454403tate_a @ B @ ( image_4497581360776675627tate_a @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_241_image__eqI,axiom,
    ! [B: produc5126262593522865844te_a_a,F2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a,X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
       => ( member1329951143264900189te_a_a @ B @ ( image_2651540025113121413te_a_a @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_242_image__eqI,axiom,
    ! [B: product_prod_a_a,F2: produc4774439622479004775tate_a > product_prod_a_a,X3: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member3801464090328701328tate_a @ X3 @ A5 )
       => ( member1426531477525435216od_a_a @ B @ ( image_1890940571582032517od_a_a @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_243_exE__realizer_H,axiom,
    ! [P2: state_a > a > $o,P: produc6972303929186420058tate_a] :
      ( ( P2 @ ( produc7615498795807706488tate_a @ P ) @ ( produc736293372669613878tate_a @ P ) )
     => ~ ! [X: a,Y3: state_a] :
            ~ ( P2 @ Y3 @ X ) ) ).

% exE_realizer'
thf(fact_244_exE__realizer_H,axiom,
    ! [P2: a > state_a > $o,P: produc5126262593522865844te_a_a] :
      ( ( P2 @ ( produc6301641481763271506te_a_a @ P ) @ ( produc8645808095479954704te_a_a @ P ) )
     => ~ ! [X: state_a,Y3: a] :
            ~ ( P2 @ Y3 @ X ) ) ).

% exE_realizer'
thf(fact_245_eq__snd__iff,axiom,
    ! [B: state_a,P: produc4774439622479004775tate_a] :
      ( ( B
        = ( produc7912480809654211077tate_a @ P ) )
      = ( ? [A4: state_a] :
            ( P
            = ( produc6131900399494507223tate_a @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_246_eq__snd__iff,axiom,
    ! [B: produc6972303929186420058tate_a,P: produc5940350820068290983tate_a] :
      ( ( B
        = ( produc7323807087048646085tate_a @ P ) )
      = ( ? [A4: produc6972303929186420058tate_a] :
            ( P
            = ( produc2143731062552654487tate_a @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_247_eq__snd__iff,axiom,
    ! [B: a,P: product_prod_a_a] :
      ( ( B
        = ( product_snd_a_a @ P ) )
      = ( ? [A4: a] :
            ( P
            = ( product_Pair_a_a @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_248_eq__snd__iff,axiom,
    ! [B: a,P: produc5126262593522865844te_a_a] :
      ( ( B
        = ( produc6301641481763271506te_a_a @ P ) )
      = ( ? [A4: state_a] :
            ( P
            = ( produc7328099264922328356te_a_a @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_249_eq__snd__iff,axiom,
    ! [B: state_a,P: produc6972303929186420058tate_a] :
      ( ( B
        = ( produc7615498795807706488tate_a @ P ) )
      = ( ? [A4: a] :
            ( P
            = ( produc8641956578966763338tate_a @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_250_sndI,axiom,
    ! [X3: produc4774439622479004775tate_a,Y: state_a,Z2: state_a] :
      ( ( X3
        = ( produc6131900399494507223tate_a @ Y @ Z2 ) )
     => ( ( produc7912480809654211077tate_a @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_251_sndI,axiom,
    ! [X3: produc5940350820068290983tate_a,Y: produc6972303929186420058tate_a,Z2: produc6972303929186420058tate_a] :
      ( ( X3
        = ( produc2143731062552654487tate_a @ Y @ Z2 ) )
     => ( ( produc7323807087048646085tate_a @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_252_sndI,axiom,
    ! [X3: product_prod_a_a,Y: a,Z2: a] :
      ( ( X3
        = ( product_Pair_a_a @ Y @ Z2 ) )
     => ( ( product_snd_a_a @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_253_sndI,axiom,
    ! [X3: produc5126262593522865844te_a_a,Y: state_a,Z2: a] :
      ( ( X3
        = ( produc7328099264922328356te_a_a @ Y @ Z2 ) )
     => ( ( produc6301641481763271506te_a_a @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_254_sndI,axiom,
    ! [X3: produc6972303929186420058tate_a,Y: a,Z2: state_a] :
      ( ( X3
        = ( produc8641956578966763338tate_a @ Y @ Z2 ) )
     => ( ( produc7615498795807706488tate_a @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_255_eq__fst__iff,axiom,
    ! [A: state_a,P: produc4774439622479004775tate_a] :
      ( ( A
        = ( produc2992983560240211139tate_a @ P ) )
      = ( ? [B4: state_a] :
            ( P
            = ( produc6131900399494507223tate_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_256_eq__fst__iff,axiom,
    ! [A: produc6972303929186420058tate_a,P: produc5940350820068290983tate_a] :
      ( ( A
        = ( produc874779950727550083tate_a @ P ) )
      = ( ? [B4: produc6972303929186420058tate_a] :
            ( P
            = ( produc2143731062552654487tate_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_257_eq__fst__iff,axiom,
    ! [A: a,P: product_prod_a_a] :
      ( ( A
        = ( product_fst_a_a @ P ) )
      = ( ? [B4: a] :
            ( P
            = ( product_Pair_a_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_258_eq__fst__iff,axiom,
    ! [A: state_a,P: produc5126262593522865844te_a_a] :
      ( ( A
        = ( produc8645808095479954704te_a_a @ P ) )
      = ( ? [B4: a] :
            ( P
            = ( produc7328099264922328356te_a_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_259_eq__fst__iff,axiom,
    ! [A: a,P: produc6972303929186420058tate_a] :
      ( ( A
        = ( produc736293372669613878tate_a @ P ) )
      = ( ? [B4: state_a] :
            ( P
            = ( produc8641956578966763338tate_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_260_fstI,axiom,
    ! [X3: produc4774439622479004775tate_a,Y: state_a,Z2: state_a] :
      ( ( X3
        = ( produc6131900399494507223tate_a @ Y @ Z2 ) )
     => ( ( produc2992983560240211139tate_a @ X3 )
        = Y ) ) ).

% fstI
thf(fact_261_fstI,axiom,
    ! [X3: produc5940350820068290983tate_a,Y: produc6972303929186420058tate_a,Z2: produc6972303929186420058tate_a] :
      ( ( X3
        = ( produc2143731062552654487tate_a @ Y @ Z2 ) )
     => ( ( produc874779950727550083tate_a @ X3 )
        = Y ) ) ).

% fstI
thf(fact_262_fstI,axiom,
    ! [X3: product_prod_a_a,Y: a,Z2: a] :
      ( ( X3
        = ( product_Pair_a_a @ Y @ Z2 ) )
     => ( ( product_fst_a_a @ X3 )
        = Y ) ) ).

% fstI
thf(fact_263_fstI,axiom,
    ! [X3: produc5126262593522865844te_a_a,Y: state_a,Z2: a] :
      ( ( X3
        = ( produc7328099264922328356te_a_a @ Y @ Z2 ) )
     => ( ( produc8645808095479954704te_a_a @ X3 )
        = Y ) ) ).

% fstI
thf(fact_264_fstI,axiom,
    ! [X3: produc6972303929186420058tate_a,Y: a,Z2: state_a] :
      ( ( X3
        = ( produc8641956578966763338tate_a @ Y @ Z2 ) )
     => ( ( produc736293372669613878tate_a @ X3 )
        = Y ) ) ).

% fstI
thf(fact_265_split__cong,axiom,
    ! [Q: produc6972303929186420058tate_a,F2: a > state_a > $o,G: a > state_a > $o,P: produc6972303929186420058tate_a] :
      ( ! [X: a,Y3: state_a] :
          ( ( ( produc8641956578966763338tate_a @ X @ Y3 )
            = Q )
         => ( ( F2 @ X @ Y3 )
            = ( G @ X @ Y3 ) ) )
     => ( ( P = Q )
       => ( ( produc3717778009801768057te_a_o @ F2 @ P )
          = ( produc3717778009801768057te_a_o @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_266_split__cong,axiom,
    ! [Q: produc5126262593522865844te_a_a,F2: state_a > a > produc6972303929186420058tate_a,G: state_a > a > produc6972303929186420058tate_a,P: produc5126262593522865844te_a_a] :
      ( ! [X: state_a,Y3: a] :
          ( ( ( produc7328099264922328356te_a_a @ X @ Y3 )
            = Q )
         => ( ( F2 @ X @ Y3 )
            = ( G @ X @ Y3 ) ) )
     => ( ( P = Q )
       => ( ( produc7343868943834020833tate_a @ F2 @ P )
          = ( produc7343868943834020833tate_a @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_267_split__cong,axiom,
    ! [Q: produc6972303929186420058tate_a,F2: a > state_a > produc6972303929186420058tate_a,G: a > state_a > produc6972303929186420058tate_a,P: produc6972303929186420058tate_a] :
      ( ! [X: a,Y3: state_a] :
          ( ( ( produc8641956578966763338tate_a @ X @ Y3 )
            = Q )
         => ( ( F2 @ X @ Y3 )
            = ( G @ X @ Y3 ) ) )
     => ( ( P = Q )
       => ( ( produc5441864759085668411tate_a @ F2 @ P )
          = ( produc5441864759085668411tate_a @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_268_surj__swap,axiom,
    ( ( image_4497581360776675627tate_a @ produc2589720147407833796te_a_a @ top_to6931181444738162404te_a_a )
    = top_to8456965716270553738tate_a ) ).

% surj_swap
thf(fact_269_surj__swap,axiom,
    ( ( image_71777787125004895te_a_a @ produc3903577461452268778tate_a @ top_to8456965716270553738tate_a )
    = top_to6931181444738162404te_a_a ) ).

% surj_swap
thf(fact_270_UNIV__I,axiom,
    ! [X3: produc5126262593522865844te_a_a] : ( member1329951143264900189te_a_a @ X3 @ top_to6931181444738162404te_a_a ) ).

% UNIV_I
thf(fact_271_UNIV__I,axiom,
    ! [X3: produc4774439622479004775tate_a] : ( member3801464090328701328tate_a @ X3 @ top_to496764914387262615tate_a ) ).

% UNIV_I
thf(fact_272_UNIV__I,axiom,
    ! [X3: produc5940350820068290983tate_a] : ( member1595334363549574352tate_a @ X3 @ top_to93540037818290135tate_a ) ).

% UNIV_I
thf(fact_273_UNIV__I,axiom,
    ! [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ top_to8063371432257647191od_a_a ) ).

% UNIV_I
thf(fact_274_UNIV__I,axiom,
    ! [X3: produc6972303929186420058tate_a] : ( member3175992478928454403tate_a @ X3 @ top_to8456965716270553738tate_a ) ).

% UNIV_I
thf(fact_275_UNIV__witness,axiom,
    ? [X: produc5126262593522865844te_a_a] : ( member1329951143264900189te_a_a @ X @ top_to6931181444738162404te_a_a ) ).

% UNIV_witness
thf(fact_276_UNIV__witness,axiom,
    ? [X: produc4774439622479004775tate_a] : ( member3801464090328701328tate_a @ X @ top_to496764914387262615tate_a ) ).

% UNIV_witness
thf(fact_277_UNIV__witness,axiom,
    ? [X: produc5940350820068290983tate_a] : ( member1595334363549574352tate_a @ X @ top_to93540037818290135tate_a ) ).

% UNIV_witness
thf(fact_278_UNIV__witness,axiom,
    ? [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ top_to8063371432257647191od_a_a ) ).

% UNIV_witness
thf(fact_279_UNIV__witness,axiom,
    ? [X: produc6972303929186420058tate_a] : ( member3175992478928454403tate_a @ X @ top_to8456965716270553738tate_a ) ).

% UNIV_witness
thf(fact_280_UNIV__eq__I,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a] :
      ( ! [X: produc5126262593522865844te_a_a] : ( member1329951143264900189te_a_a @ X @ A5 )
     => ( top_to6931181444738162404te_a_a = A5 ) ) ).

% UNIV_eq_I
thf(fact_281_UNIV__eq__I,axiom,
    ! [A5: set_Pr1753757909640466247tate_a] :
      ( ! [X: produc4774439622479004775tate_a] : ( member3801464090328701328tate_a @ X @ A5 )
     => ( top_to496764914387262615tate_a = A5 ) ) ).

% UNIV_eq_I
thf(fact_282_UNIV__eq__I,axiom,
    ! [A5: set_Pr7963165934123475591tate_a] :
      ( ! [X: produc5940350820068290983tate_a] : ( member1595334363549574352tate_a @ X @ A5 )
     => ( top_to93540037818290135tate_a = A5 ) ) ).

% UNIV_eq_I
thf(fact_283_UNIV__eq__I,axiom,
    ! [A5: set_Product_prod_a_a] :
      ( ! [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ A5 )
     => ( top_to8063371432257647191od_a_a = A5 ) ) ).

% UNIV_eq_I
thf(fact_284_UNIV__eq__I,axiom,
    ! [A5: set_Pr4275752383657305402tate_a] :
      ( ! [X: produc6972303929186420058tate_a] : ( member3175992478928454403tate_a @ X @ A5 )
     => ( top_to8456965716270553738tate_a = A5 ) ) ).

% UNIV_eq_I
thf(fact_285_range__eqI,axiom,
    ! [B: produc6972303929186420058tate_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,X3: produc5126262593522865844te_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( member3175992478928454403tate_a @ B @ ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a ) ) ) ).

% range_eqI
thf(fact_286_range__eqI,axiom,
    ! [B: produc6972303929186420058tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,X3: produc6972303929186420058tate_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( member3175992478928454403tate_a @ B @ ( image_1917819122788559109tate_a @ F2 @ top_to8456965716270553738tate_a ) ) ) ).

% range_eqI
thf(fact_287_range__eqI,axiom,
    ! [B: produc5126262593522865844te_a_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,X3: produc6972303929186420058tate_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( member1329951143264900189te_a_a @ B @ ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a ) ) ) ).

% range_eqI
thf(fact_288_range__eqI,axiom,
    ! [B: produc4774439622479004775tate_a,F2: produc6972303929186420058tate_a > produc4774439622479004775tate_a,X3: produc6972303929186420058tate_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( member3801464090328701328tate_a @ B @ ( image_5672933159226955410tate_a @ F2 @ top_to8456965716270553738tate_a ) ) ) ).

% range_eqI
thf(fact_289_range__eqI,axiom,
    ! [B: produc5940350820068290983tate_a,F2: produc6972303929186420058tate_a > produc5940350820068290983tate_a,X3: produc6972303929186420058tate_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( member1595334363549574352tate_a @ B @ ( image_8048040396972735954tate_a @ F2 @ top_to8456965716270553738tate_a ) ) ) ).

% range_eqI
thf(fact_290_range__eqI,axiom,
    ! [B: product_prod_a_a,F2: produc6972303929186420058tate_a > product_prod_a_a,X3: produc6972303929186420058tate_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( member1426531477525435216od_a_a @ B @ ( image_8179770026730005074od_a_a @ F2 @ top_to8456965716270553738tate_a ) ) ) ).

% range_eqI
thf(fact_291_rangeI,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,X3: produc5126262593522865844te_a_a] : ( member3175992478928454403tate_a @ ( F2 @ X3 ) @ ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a ) ) ).

% rangeI
thf(fact_292_rangeI,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,X3: produc6972303929186420058tate_a] : ( member3175992478928454403tate_a @ ( F2 @ X3 ) @ ( image_1917819122788559109tate_a @ F2 @ top_to8456965716270553738tate_a ) ) ).

% rangeI
thf(fact_293_rangeI,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,X3: produc6972303929186420058tate_a] : ( member1329951143264900189te_a_a @ ( F2 @ X3 ) @ ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a ) ) ).

% rangeI
thf(fact_294_rangeI,axiom,
    ! [F2: produc6972303929186420058tate_a > produc4774439622479004775tate_a,X3: produc6972303929186420058tate_a] : ( member3801464090328701328tate_a @ ( F2 @ X3 ) @ ( image_5672933159226955410tate_a @ F2 @ top_to8456965716270553738tate_a ) ) ).

% rangeI
thf(fact_295_rangeI,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5940350820068290983tate_a,X3: produc6972303929186420058tate_a] : ( member1595334363549574352tate_a @ ( F2 @ X3 ) @ ( image_8048040396972735954tate_a @ F2 @ top_to8456965716270553738tate_a ) ) ).

% rangeI
thf(fact_296_rangeI,axiom,
    ! [F2: produc6972303929186420058tate_a > product_prod_a_a,X3: produc6972303929186420058tate_a] : ( member1426531477525435216od_a_a @ ( F2 @ X3 ) @ ( image_8179770026730005074od_a_a @ F2 @ top_to8456965716270553738tate_a ) ) ).

% rangeI
thf(fact_297_rev__image__eqI,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a,B: product_prod_a_a,F2: product_prod_a_a > product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member1426531477525435216od_a_a @ B @ ( image_4636654165204879301od_a_a @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_298_rev__image__eqI,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B: product_prod_a_a,F2: produc6972303929186420058tate_a > product_prod_a_a] :
      ( ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member1426531477525435216od_a_a @ B @ ( image_8179770026730005074od_a_a @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_299_rev__image__eqI,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,B: product_prod_a_a,F2: produc5126262593522865844te_a_a > product_prod_a_a] :
      ( ( member1329951143264900189te_a_a @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member1426531477525435216od_a_a @ B @ ( image_615948358740051320od_a_a @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_300_rev__image__eqI,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a,B: produc6972303929186420058tate_a,F2: product_prod_a_a > produc6972303929186420058tate_a] :
      ( ( member1426531477525435216od_a_a @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member3175992478928454403tate_a @ B @ ( image_1563038779069872376tate_a @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_301_rev__image__eqI,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a,B: produc5126262593522865844te_a_a,F2: product_prod_a_a > produc5126262593522865844te_a_a] :
      ( ( member1426531477525435216od_a_a @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member1329951143264900189te_a_a @ B @ ( image_8940369480261093970te_a_a @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_302_rev__image__eqI,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B: produc6972303929186420058tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member3175992478928454403tate_a @ B @ ( image_1917819122788559109tate_a @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_303_rev__image__eqI,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B: produc5126262593522865844te_a_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member1329951143264900189te_a_a @ B @ ( image_71777787125004895te_a_a @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_304_rev__image__eqI,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,B: produc6972303929186420058tate_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( member1329951143264900189te_a_a @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member3175992478928454403tate_a @ B @ ( image_4497581360776675627tate_a @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_305_rev__image__eqI,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,B: produc5126262593522865844te_a_a,F2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a] :
      ( ( member1329951143264900189te_a_a @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member1329951143264900189te_a_a @ B @ ( image_2651540025113121413te_a_a @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_306_rev__image__eqI,axiom,
    ! [X3: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a,B: product_prod_a_a,F2: produc4774439622479004775tate_a > product_prod_a_a] :
      ( ( member3801464090328701328tate_a @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member1426531477525435216od_a_a @ B @ ( image_1890940571582032517od_a_a @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_307_ball__imageD,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,P2: produc6972303929186420058tate_a > $o] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ ( image_4497581360776675627tate_a @ F2 @ A5 ) )
         => ( P2 @ X ) )
     => ! [X5: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X5 @ A5 )
         => ( P2 @ ( F2 @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_308_ball__imageD,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,P2: produc5126262593522865844te_a_a > $o] :
      ( ! [X: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ ( image_71777787125004895te_a_a @ F2 @ A5 ) )
         => ( P2 @ X ) )
     => ! [X5: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X5 @ A5 )
         => ( P2 @ ( F2 @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_309_ball__imageD,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,P2: produc6972303929186420058tate_a > $o] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ ( image_1917819122788559109tate_a @ F2 @ A5 ) )
         => ( P2 @ X ) )
     => ! [X5: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X5 @ A5 )
         => ( P2 @ ( F2 @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_310_image__cong,axiom,
    ! [M: set_Pr4275752383657305402tate_a,N: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,G: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ( M = N )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X @ N )
           => ( ( F2 @ X )
              = ( G @ X ) ) )
       => ( ( image_71777787125004895te_a_a @ F2 @ M )
          = ( image_71777787125004895te_a_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_311_image__cong,axiom,
    ! [M: set_Pr4275752383657305402tate_a,N: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,G: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( M = N )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X @ N )
           => ( ( F2 @ X )
              = ( G @ X ) ) )
       => ( ( image_1917819122788559109tate_a @ F2 @ M )
          = ( image_1917819122788559109tate_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_312_image__cong,axiom,
    ! [M: set_Pr2749968112124914068te_a_a,N: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,G: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( M = N )
     => ( ! [X: produc5126262593522865844te_a_a] :
            ( ( member1329951143264900189te_a_a @ X @ N )
           => ( ( F2 @ X )
              = ( G @ X ) ) )
       => ( ( image_4497581360776675627tate_a @ F2 @ M )
          = ( image_4497581360776675627tate_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_313_bex__imageD,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,P2: produc6972303929186420058tate_a > $o] :
      ( ? [X5: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X5 @ ( image_4497581360776675627tate_a @ F2 @ A5 ) )
          & ( P2 @ X5 ) )
     => ? [X: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ A5 )
          & ( P2 @ ( F2 @ X ) ) ) ) ).

% bex_imageD
thf(fact_314_bex__imageD,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,P2: produc5126262593522865844te_a_a > $o] :
      ( ? [X5: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X5 @ ( image_71777787125004895te_a_a @ F2 @ A5 ) )
          & ( P2 @ X5 ) )
     => ? [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ A5 )
          & ( P2 @ ( F2 @ X ) ) ) ) ).

% bex_imageD
thf(fact_315_bex__imageD,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,P2: produc6972303929186420058tate_a > $o] :
      ( ? [X5: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X5 @ ( image_1917819122788559109tate_a @ F2 @ A5 ) )
          & ( P2 @ X5 ) )
     => ? [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ A5 )
          & ( P2 @ ( F2 @ X ) ) ) ) ).

% bex_imageD
thf(fact_316_image__iff,axiom,
    ! [Z2: produc6972303929186420058tate_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( member3175992478928454403tate_a @ Z2 @ ( image_4497581360776675627tate_a @ F2 @ A5 ) )
      = ( ? [X4: produc5126262593522865844te_a_a] :
            ( ( member1329951143264900189te_a_a @ X4 @ A5 )
            & ( Z2
              = ( F2 @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_317_image__iff,axiom,
    ! [Z2: produc6972303929186420058tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ Z2 @ ( image_1917819122788559109tate_a @ F2 @ A5 ) )
      = ( ? [X4: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X4 @ A5 )
            & ( Z2
              = ( F2 @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_318_image__iff,axiom,
    ! [Z2: produc5126262593522865844te_a_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( member1329951143264900189te_a_a @ Z2 @ ( image_71777787125004895te_a_a @ F2 @ A5 ) )
      = ( ? [X4: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X4 @ A5 )
            & ( Z2
              = ( F2 @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_319_imageI,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a,F2: product_prod_a_a > product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ X3 @ A5 )
     => ( member1426531477525435216od_a_a @ ( F2 @ X3 ) @ ( image_4636654165204879301od_a_a @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_320_imageI,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > product_prod_a_a] :
      ( ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( member1426531477525435216od_a_a @ ( F2 @ X3 ) @ ( image_8179770026730005074od_a_a @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_321_imageI,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > product_prod_a_a] :
      ( ( member1329951143264900189te_a_a @ X3 @ A5 )
     => ( member1426531477525435216od_a_a @ ( F2 @ X3 ) @ ( image_615948358740051320od_a_a @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_322_imageI,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a,F2: product_prod_a_a > produc6972303929186420058tate_a] :
      ( ( member1426531477525435216od_a_a @ X3 @ A5 )
     => ( member3175992478928454403tate_a @ ( F2 @ X3 ) @ ( image_1563038779069872376tate_a @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_323_imageI,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a,F2: product_prod_a_a > produc5126262593522865844te_a_a] :
      ( ( member1426531477525435216od_a_a @ X3 @ A5 )
     => ( member1329951143264900189te_a_a @ ( F2 @ X3 ) @ ( image_8940369480261093970te_a_a @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_324_imageI,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( member3175992478928454403tate_a @ ( F2 @ X3 ) @ ( image_1917819122788559109tate_a @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_325_imageI,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( member1329951143264900189te_a_a @ ( F2 @ X3 ) @ ( image_71777787125004895te_a_a @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_326_imageI,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( member1329951143264900189te_a_a @ X3 @ A5 )
     => ( member3175992478928454403tate_a @ ( F2 @ X3 ) @ ( image_4497581360776675627tate_a @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_327_imageI,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a] :
      ( ( member1329951143264900189te_a_a @ X3 @ A5 )
     => ( member1329951143264900189te_a_a @ ( F2 @ X3 ) @ ( image_2651540025113121413te_a_a @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_328_imageI,axiom,
    ! [X3: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a,F2: produc4774439622479004775tate_a > product_prod_a_a] :
      ( ( member3801464090328701328tate_a @ X3 @ A5 )
     => ( member1426531477525435216od_a_a @ ( F2 @ X3 ) @ ( image_1890940571582032517od_a_a @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_329_range__snd,axiom,
    ( ( image_3052321768533554255_a_a_a @ produc6301641481763271506te_a_a @ top_to6931181444738162404te_a_a )
    = top_top_set_a ) ).

% range_snd
thf(fact_330_range__snd,axiom,
    ( ( image_2364236133221778012tate_a @ produc7615498795807706488tate_a @ top_to8456965716270553738tate_a )
    = top_top_set_state_a ) ).

% range_snd
thf(fact_331_range__fst,axiom,
    ( ( image_6150641562437185666tate_a @ produc8645808095479954704te_a_a @ top_to6931181444738162404te_a_a )
    = top_top_set_state_a ) ).

% range_fst
thf(fact_332_range__fst,axiom,
    ( ( image_2134324207772891945te_a_a @ produc736293372669613878tate_a @ top_to8456965716270553738tate_a )
    = top_top_set_a ) ).

% range_fst
thf(fact_333_iso__tuple__UNIV__I,axiom,
    ! [X3: produc5126262593522865844te_a_a] : ( member1329951143264900189te_a_a @ X3 @ top_to6931181444738162404te_a_a ) ).

% iso_tuple_UNIV_I
thf(fact_334_iso__tuple__UNIV__I,axiom,
    ! [X3: produc4774439622479004775tate_a] : ( member3801464090328701328tate_a @ X3 @ top_to496764914387262615tate_a ) ).

% iso_tuple_UNIV_I
thf(fact_335_iso__tuple__UNIV__I,axiom,
    ! [X3: produc5940350820068290983tate_a] : ( member1595334363549574352tate_a @ X3 @ top_to93540037818290135tate_a ) ).

% iso_tuple_UNIV_I
thf(fact_336_iso__tuple__UNIV__I,axiom,
    ! [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ top_to8063371432257647191od_a_a ) ).

% iso_tuple_UNIV_I
thf(fact_337_iso__tuple__UNIV__I,axiom,
    ! [X3: produc6972303929186420058tate_a] : ( member3175992478928454403tate_a @ X3 @ top_to8456965716270553738tate_a ) ).

% iso_tuple_UNIV_I
thf(fact_338_top__apply,axiom,
    ( top_top_a_state_a_o
    = ( ^ [X4: a] : top_top_state_a_o ) ) ).

% top_apply
thf(fact_339_surj__def,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a )
        = top_to8456965716270553738tate_a )
      = ( ! [Y6: produc6972303929186420058tate_a] :
          ? [X4: produc5126262593522865844te_a_a] :
            ( Y6
            = ( F2 @ X4 ) ) ) ) ).

% surj_def
thf(fact_340_surj__def,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ( ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to6931181444738162404te_a_a )
      = ( ! [Y6: produc5126262593522865844te_a_a] :
          ? [X4: produc6972303929186420058tate_a] :
            ( Y6
            = ( F2 @ X4 ) ) ) ) ).

% surj_def
thf(fact_341_surj__def,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( ( image_1917819122788559109tate_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to8456965716270553738tate_a )
      = ( ! [Y6: produc6972303929186420058tate_a] :
          ? [X4: produc6972303929186420058tate_a] :
            ( Y6
            = ( F2 @ X4 ) ) ) ) ).

% surj_def
thf(fact_342_surjI,axiom,
    ! [G: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( G @ ( F2 @ X ) )
          = X )
     => ( ( image_4497581360776675627tate_a @ G @ top_to6931181444738162404te_a_a )
        = top_to8456965716270553738tate_a ) ) ).

% surjI
thf(fact_343_surjI,axiom,
    ! [G: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ! [X: produc5126262593522865844te_a_a] :
          ( ( G @ ( F2 @ X ) )
          = X )
     => ( ( image_71777787125004895te_a_a @ G @ top_to8456965716270553738tate_a )
        = top_to6931181444738162404te_a_a ) ) ).

% surjI
thf(fact_344_surjI,axiom,
    ! [G: produc6972303929186420058tate_a > produc6972303929186420058tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( G @ ( F2 @ X ) )
          = X )
     => ( ( image_1917819122788559109tate_a @ G @ top_to8456965716270553738tate_a )
        = top_to8456965716270553738tate_a ) ) ).

% surjI
thf(fact_345_surjE,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a] :
      ( ( ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a )
        = top_to8456965716270553738tate_a )
     => ~ ! [X: produc5126262593522865844te_a_a] :
            ( Y
           != ( F2 @ X ) ) ) ).

% surjE
thf(fact_346_surjE,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,Y: produc5126262593522865844te_a_a] :
      ( ( ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to6931181444738162404te_a_a )
     => ~ ! [X: produc6972303929186420058tate_a] :
            ( Y
           != ( F2 @ X ) ) ) ).

% surjE
thf(fact_347_surjE,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a] :
      ( ( ( image_1917819122788559109tate_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to8456965716270553738tate_a )
     => ~ ! [X: produc6972303929186420058tate_a] :
            ( Y
           != ( F2 @ X ) ) ) ).

% surjE
thf(fact_348_surjD,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a] :
      ( ( ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a )
        = top_to8456965716270553738tate_a )
     => ? [X: produc5126262593522865844te_a_a] :
          ( Y
          = ( F2 @ X ) ) ) ).

% surjD
thf(fact_349_surjD,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,Y: produc5126262593522865844te_a_a] :
      ( ( ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to6931181444738162404te_a_a )
     => ? [X: produc6972303929186420058tate_a] :
          ( Y
          = ( F2 @ X ) ) ) ).

% surjD
thf(fact_350_surjD,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a] :
      ( ( ( image_1917819122788559109tate_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to8456965716270553738tate_a )
     => ? [X: produc6972303929186420058tate_a] :
          ( Y
          = ( F2 @ X ) ) ) ).

% surjD
thf(fact_351_Collect__split__mono__strong,axiom,
    ! [X6: set_state_a,A5: set_Pr2749968112124914068te_a_a,Y7: set_a,P2: state_a > a > $o,Q2: state_a > a > $o] :
      ( ( X6
        = ( image_6150641562437185666tate_a @ produc8645808095479954704te_a_a @ A5 ) )
     => ( ( Y7
          = ( image_3052321768533554255_a_a_a @ produc6301641481763271506te_a_a @ A5 ) )
       => ( ! [X: state_a] :
              ( ( member_state_a @ X @ X6 )
             => ! [Xa: a] :
                  ( ( member_a @ Xa @ Y7 )
                 => ( ( P2 @ X @ Xa )
                   => ( Q2 @ X @ Xa ) ) ) )
         => ( ( ord_le5819720210775101748te_a_a @ A5 @ ( collec6503900420941984159te_a_a @ ( produc490204266526094367_a_a_o @ P2 ) ) )
           => ( ord_le5819720210775101748te_a_a @ A5 @ ( collec6503900420941984159te_a_a @ ( produc490204266526094367_a_a_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_352_Collect__split__mono__strong,axiom,
    ! [X6: set_a,A5: set_Pr4275752383657305402tate_a,Y7: set_state_a,P2: a > state_a > $o,Q2: a > state_a > $o] :
      ( ( X6
        = ( image_2134324207772891945te_a_a @ produc736293372669613878tate_a @ A5 ) )
     => ( ( Y7
          = ( image_2364236133221778012tate_a @ produc7615498795807706488tate_a @ A5 ) )
       => ( ! [X: a] :
              ( ( member_a @ X @ X6 )
             => ! [Xa: state_a] :
                  ( ( member_state_a @ Xa @ Y7 )
                 => ( ( P2 @ X @ Xa )
                   => ( Q2 @ X @ Xa ) ) ) )
         => ( ( ord_le7345504482307493082tate_a @ A5 @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ P2 ) ) )
           => ( ord_le7345504482307493082tate_a @ A5 @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_353_map__prod__surj,axiom,
    ! [F2: state_a > a,G: a > state_a] :
      ( ( ( image_state_a_a @ F2 @ top_top_set_state_a )
        = top_top_set_a )
     => ( ( ( image_a_state_a @ G @ top_top_set_a )
          = top_top_set_state_a )
       => ( ( image_4497581360776675627tate_a @ ( produc5926476928146254390tate_a @ F2 @ G ) @ top_to6931181444738162404te_a_a )
          = top_to8456965716270553738tate_a ) ) ) ).

% map_prod_surj
thf(fact_354_map__prod__surj,axiom,
    ! [F2: a > state_a,G: state_a > a] :
      ( ( ( image_a_state_a @ F2 @ top_top_set_a )
        = top_top_set_state_a )
     => ( ( ( image_state_a_a @ G @ top_top_set_state_a )
          = top_top_set_a )
       => ( ( image_71777787125004895te_a_a @ ( produc2865014304882136118te_a_a @ F2 @ G ) @ top_to8456965716270553738tate_a )
          = top_to6931181444738162404te_a_a ) ) ) ).

% map_prod_surj
thf(fact_355_map__prod__surj,axiom,
    ! [F2: a > a,G: state_a > state_a] :
      ( ( ( image_a_a @ F2 @ top_top_set_a )
        = top_top_set_a )
     => ( ( ( image_3877308803806453061tate_a @ G @ top_top_set_state_a )
          = top_top_set_state_a )
       => ( ( image_1917819122788559109tate_a @ ( produc6760600410500789430tate_a @ F2 @ G ) @ top_to8456965716270553738tate_a )
          = top_to8456965716270553738tate_a ) ) ) ).

% map_prod_surj
thf(fact_356_map__prod__surj,axiom,
    ! [F2: a > produc6972303929186420058tate_a,G: state_a > produc6972303929186420058tate_a] :
      ( ( ( image_5502445033238569825tate_a @ F2 @ top_top_set_a )
        = top_to8456965716270553738tate_a )
     => ( ( ( image_459697210324464366tate_a @ G @ top_top_set_state_a )
          = top_to8456965716270553738tate_a )
       => ( ( image_8048040396972735954tate_a @ ( produc3648729650937203131tate_a @ F2 @ G ) @ top_to8456965716270553738tate_a )
          = top_to93540037818290135tate_a ) ) ) ).

% map_prod_surj
thf(fact_357_map__prod__surj,axiom,
    ! [F2: produc6972303929186420058tate_a > a,G: produc6972303929186420058tate_a > state_a] :
      ( ( ( image_2134324207772891945te_a_a @ F2 @ top_to8456965716270553738tate_a )
        = top_top_set_a )
     => ( ( ( image_2364236133221778012tate_a @ G @ top_to8456965716270553738tate_a )
          = top_top_set_state_a )
       => ( ( image_3711197217765773944tate_a @ ( produc8720770522196761841tate_a @ F2 @ G ) @ top_to93540037818290135tate_a )
          = top_to8456965716270553738tate_a ) ) ) ).

% map_prod_surj
thf(fact_358_map__prod__surj,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,G: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a )
        = top_to8456965716270553738tate_a )
     => ( ( ( image_4497581360776675627tate_a @ G @ top_to6931181444738162404te_a_a )
          = top_to8456965716270553738tate_a )
       => ( ( image_8117903017743089477tate_a @ ( produc1508824635387789378tate_a @ F2 @ G ) @ top_to6088003638156323927te_a_a )
          = top_to93540037818290135tate_a ) ) ) ).

% map_prod_surj
thf(fact_359_map__prod__surj,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,G: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ( ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a )
        = top_to8456965716270553738tate_a )
     => ( ( ( image_71777787125004895te_a_a @ G @ top_to8456965716270553738tate_a )
          = top_to6931181444738162404te_a_a )
       => ( ( image_1454221569763251577te_a_a @ ( produc6306393098590894454te_a_a @ F2 @ G ) @ top_to2613737104116118781tate_a )
          = top_to3567806571858495281te_a_a ) ) ) ).

% map_prod_surj
thf(fact_360_map__prod__surj,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,G: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a )
        = top_to8456965716270553738tate_a )
     => ( ( ( image_1917819122788559109tate_a @ G @ top_to8456965716270553738tate_a )
          = top_to8456965716270553738tate_a )
       => ( ( image_2980005841295642911tate_a @ ( produc8152434434254448668tate_a @ F2 @ G ) @ top_to2613737104116118781tate_a )
          = top_to93540037818290135tate_a ) ) ) ).

% map_prod_surj
thf(fact_361_map__prod__surj,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,G: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to6931181444738162404te_a_a )
     => ( ( ( image_4497581360776675627tate_a @ G @ top_to6931181444738162404te_a_a )
          = top_to8456965716270553738tate_a )
       => ( ( image_310783875300459025tate_a @ ( produc4842458045871297398tate_a @ F2 @ G ) @ top_to3567806571858495281te_a_a )
          = top_to2613737104116118781tate_a ) ) ) ).

% map_prod_surj
thf(fact_362_map__prod__surj,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,G: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ( ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to6931181444738162404te_a_a )
     => ( ( ( image_71777787125004895te_a_a @ G @ top_to8456965716270553738tate_a )
          = top_to6931181444738162404te_a_a )
       => ( ( image_2870474464175396933te_a_a @ ( produc416654472219626666te_a_a @ F2 @ G ) @ top_to93540037818290135tate_a )
          = top_to6088003638156323927te_a_a ) ) ) ).

% map_prod_surj
thf(fact_363_order__refl,axiom,
    ! [X3: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ X3 @ X3 ) ).

% order_refl
thf(fact_364_dual__order_Orefl,axiom,
    ! [A: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ A @ A ) ).

% dual_order.refl
thf(fact_365_subset__antisym,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
     => ( ( ord_le7345504482307493082tate_a @ B5 @ A5 )
       => ( A5 = B5 ) ) ) ).

% subset_antisym
thf(fact_366_subsetI,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ! [X: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ A5 )
         => ( member1329951143264900189te_a_a @ X @ B5 ) )
     => ( ord_le5819720210775101748te_a_a @ A5 @ B5 ) ) ).

% subsetI
thf(fact_367_subsetI,axiom,
    ! [A5: set_Pr1753757909640466247tate_a,B5: set_Pr1753757909640466247tate_a] :
      ( ! [X: produc4774439622479004775tate_a] :
          ( ( member3801464090328701328tate_a @ X @ A5 )
         => ( member3801464090328701328tate_a @ X @ B5 ) )
     => ( ord_le5561929874677720807tate_a @ A5 @ B5 ) ) ).

% subsetI
thf(fact_368_subsetI,axiom,
    ! [A5: set_Pr7963165934123475591tate_a,B5: set_Pr7963165934123475591tate_a] :
      ( ! [X: produc5940350820068290983tate_a] :
          ( ( member1595334363549574352tate_a @ X @ A5 )
         => ( member1595334363549574352tate_a @ X @ B5 ) )
     => ( ord_le4349072690221431335tate_a @ A5 @ B5 ) ) ).

% subsetI
thf(fact_369_subsetI,axiom,
    ! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ! [X: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ A5 )
         => ( member1426531477525435216od_a_a @ X @ B5 ) )
     => ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ).

% subsetI
thf(fact_370_subsetI,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ A5 )
         => ( member3175992478928454403tate_a @ X @ B5 ) )
     => ( ord_le7345504482307493082tate_a @ A5 @ B5 ) ) ).

% subsetI
thf(fact_371_map__prod__simp,axiom,
    ! [F2: a > a,G: state_a > state_a,A: a,B: state_a] :
      ( ( produc6760600410500789430tate_a @ F2 @ G @ ( produc8641956578966763338tate_a @ A @ B ) )
      = ( produc8641956578966763338tate_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_372_map__prod__simp,axiom,
    ! [F2: a > a,G: a > a,A: a,B: a] :
      ( ( produc4018698449564827638_a_a_a @ F2 @ G @ ( product_Pair_a_a @ A @ B ) )
      = ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_373_map__prod__simp,axiom,
    ! [F2: a > a,G: state_a > a,A: a,B: state_a] :
      ( ( produc7512501727144780419te_a_a @ F2 @ G @ ( produc8641956578966763338tate_a @ A @ B ) )
      = ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_374_map__prod__simp,axiom,
    ! [F2: state_a > a,G: a > a,A: state_a,B: a] :
      ( ( produc7010603597014767619_a_a_a @ F2 @ G @ ( produc7328099264922328356te_a_a @ A @ B ) )
      = ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_375_map__prod__simp,axiom,
    ! [F2: a > a,G: a > state_a,A: a,B: a] :
      ( ( produc8826359041189215401tate_a @ F2 @ G @ ( product_Pair_a_a @ A @ B ) )
      = ( produc8641956578966763338tate_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_376_map__prod__simp,axiom,
    ! [F2: a > state_a,G: a > a,A: a,B: a] :
      ( ( produc1047144888364738345_a_a_a @ F2 @ G @ ( product_Pair_a_a @ A @ B ) )
      = ( produc7328099264922328356te_a_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_377_map__prod__simp,axiom,
    ! [F2: a > state_a,G: state_a > a,A: a,B: state_a] :
      ( ( produc2865014304882136118te_a_a @ F2 @ G @ ( produc8641956578966763338tate_a @ A @ B ) )
      = ( produc7328099264922328356te_a_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_378_map__prod__simp,axiom,
    ! [F2: state_a > a,G: a > state_a,A: state_a,B: a] :
      ( ( produc5926476928146254390tate_a @ F2 @ G @ ( produc7328099264922328356te_a_a @ A @ B ) )
      = ( produc8641956578966763338tate_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_379_map__prod__simp,axiom,
    ! [F2: state_a > state_a,G: a > a,A: state_a,B: a] :
      ( ( produc7370634812176553142_a_a_a @ F2 @ G @ ( produc7328099264922328356te_a_a @ A @ B ) )
      = ( produc7328099264922328356te_a_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_380_map__prod__simp,axiom,
    ! [F2: state_a > a,G: state_a > a,A: state_a,B: state_a] :
      ( ( produc4612619614101819408te_a_a @ F2 @ G @ ( produc6131900399494507223tate_a @ A @ B ) )
      = ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_381_fst__map__prod,axiom,
    ! [F2: a > a,G: state_a > state_a,X3: produc6972303929186420058tate_a] :
      ( ( produc736293372669613878tate_a @ ( produc6760600410500789430tate_a @ F2 @ G @ X3 ) )
      = ( F2 @ ( produc736293372669613878tate_a @ X3 ) ) ) ).

% fst_map_prod
thf(fact_382_fst__map__prod,axiom,
    ! [F2: state_a > a,G: a > state_a,X3: produc5126262593522865844te_a_a] :
      ( ( produc736293372669613878tate_a @ ( produc5926476928146254390tate_a @ F2 @ G @ X3 ) )
      = ( F2 @ ( produc8645808095479954704te_a_a @ X3 ) ) ) ).

% fst_map_prod
thf(fact_383_fst__map__prod,axiom,
    ! [F2: a > state_a,G: state_a > a,X3: produc6972303929186420058tate_a] :
      ( ( produc8645808095479954704te_a_a @ ( produc2865014304882136118te_a_a @ F2 @ G @ X3 ) )
      = ( F2 @ ( produc736293372669613878tate_a @ X3 ) ) ) ).

% fst_map_prod
thf(fact_384_fst__map__prod,axiom,
    ! [F2: state_a > state_a,G: a > a,X3: produc5126262593522865844te_a_a] :
      ( ( produc8645808095479954704te_a_a @ ( produc7370634812176553142_a_a_a @ F2 @ G @ X3 ) )
      = ( F2 @ ( produc8645808095479954704te_a_a @ X3 ) ) ) ).

% fst_map_prod
thf(fact_385_snd__map__prod,axiom,
    ! [F2: a > a,G: state_a > state_a,X3: produc6972303929186420058tate_a] :
      ( ( produc7615498795807706488tate_a @ ( produc6760600410500789430tate_a @ F2 @ G @ X3 ) )
      = ( G @ ( produc7615498795807706488tate_a @ X3 ) ) ) ).

% snd_map_prod
thf(fact_386_snd__map__prod,axiom,
    ! [F2: state_a > a,G: a > state_a,X3: produc5126262593522865844te_a_a] :
      ( ( produc7615498795807706488tate_a @ ( produc5926476928146254390tate_a @ F2 @ G @ X3 ) )
      = ( G @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ).

% snd_map_prod
thf(fact_387_snd__map__prod,axiom,
    ! [F2: a > state_a,G: state_a > a,X3: produc6972303929186420058tate_a] :
      ( ( produc6301641481763271506te_a_a @ ( produc2865014304882136118te_a_a @ F2 @ G @ X3 ) )
      = ( G @ ( produc7615498795807706488tate_a @ X3 ) ) ) ).

% snd_map_prod
thf(fact_388_snd__map__prod,axiom,
    ! [F2: state_a > state_a,G: a > a,X3: produc5126262593522865844te_a_a] :
      ( ( produc6301641481763271506te_a_a @ ( produc7370634812176553142_a_a_a @ F2 @ G @ X3 ) )
      = ( G @ ( produc6301641481763271506te_a_a @ X3 ) ) ) ).

% snd_map_prod
thf(fact_389_map__prod__imageI,axiom,
    ! [A: a,B: state_a,R2: set_Pr4275752383657305402tate_a,F2: a > a,G: state_a > state_a] :
      ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A @ B ) @ R2 )
     => ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_1917819122788559109tate_a @ ( produc6760600410500789430tate_a @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_390_map__prod__imageI,axiom,
    ! [A: a,B: a,R2: set_Product_prod_a_a,F2: a > a,G: a > a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_4636654165204879301od_a_a @ ( produc4018698449564827638_a_a_a @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_391_map__prod__imageI,axiom,
    ! [A: a,B: state_a,R2: set_Pr4275752383657305402tate_a,F2: a > a,G: state_a > a] :
      ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A @ B ) @ R2 )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_8179770026730005074od_a_a @ ( produc7512501727144780419te_a_a @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_392_map__prod__imageI,axiom,
    ! [A: state_a,B: a,R2: set_Pr2749968112124914068te_a_a,F2: state_a > a,G: a > a] :
      ( ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ A @ B ) @ R2 )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_615948358740051320od_a_a @ ( produc7010603597014767619_a_a_a @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_393_map__prod__imageI,axiom,
    ! [A: a,B: a,R2: set_Product_prod_a_a,F2: a > a,G: a > state_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
     => ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_1563038779069872376tate_a @ ( produc8826359041189215401tate_a @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_394_map__prod__imageI,axiom,
    ! [A: a,B: a,R2: set_Product_prod_a_a,F2: a > state_a,G: a > a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
     => ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_8940369480261093970te_a_a @ ( produc1047144888364738345_a_a_a @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_395_map__prod__imageI,axiom,
    ! [A: a,B: state_a,R2: set_Pr4275752383657305402tate_a,F2: a > state_a,G: state_a > a] :
      ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A @ B ) @ R2 )
     => ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_71777787125004895te_a_a @ ( produc2865014304882136118te_a_a @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_396_map__prod__imageI,axiom,
    ! [A: state_a,B: a,R2: set_Pr2749968112124914068te_a_a,F2: state_a > a,G: a > state_a] :
      ( ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ A @ B ) @ R2 )
     => ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_4497581360776675627tate_a @ ( produc5926476928146254390tate_a @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_397_map__prod__imageI,axiom,
    ! [A: state_a,B: a,R2: set_Pr2749968112124914068te_a_a,F2: state_a > state_a,G: a > a] :
      ( ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ A @ B ) @ R2 )
     => ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_2651540025113121413te_a_a @ ( produc7370634812176553142_a_a_a @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_398_map__prod__imageI,axiom,
    ! [A: state_a,B: state_a,R2: set_Pr1753757909640466247tate_a,F2: state_a > a,G: state_a > a] :
      ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ A @ B ) @ R2 )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_1890940571582032517od_a_a @ ( produc4612619614101819408te_a_a @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_399_Collect__mono__iff,axiom,
    ! [P2: produc6972303929186420058tate_a > $o,Q2: produc6972303929186420058tate_a > $o] :
      ( ( ord_le7345504482307493082tate_a @ ( collec8349941756605538373tate_a @ P2 ) @ ( collec8349941756605538373tate_a @ Q2 ) )
      = ( ! [X4: produc6972303929186420058tate_a] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_400_set__eq__subset,axiom,
    ( ( ^ [Y5: set_Pr4275752383657305402tate_a,Z: set_Pr4275752383657305402tate_a] : ( Y5 = Z ) )
    = ( ^ [A6: set_Pr4275752383657305402tate_a,B6: set_Pr4275752383657305402tate_a] :
          ( ( ord_le7345504482307493082tate_a @ A6 @ B6 )
          & ( ord_le7345504482307493082tate_a @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_401_subset__trans,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,C3: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
     => ( ( ord_le7345504482307493082tate_a @ B5 @ C3 )
       => ( ord_le7345504482307493082tate_a @ A5 @ C3 ) ) ) ).

% subset_trans
thf(fact_402_Collect__mono,axiom,
    ! [P2: produc6972303929186420058tate_a > $o,Q2: produc6972303929186420058tate_a > $o] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( P2 @ X )
         => ( Q2 @ X ) )
     => ( ord_le7345504482307493082tate_a @ ( collec8349941756605538373tate_a @ P2 ) @ ( collec8349941756605538373tate_a @ Q2 ) ) ) ).

% Collect_mono
thf(fact_403_subset__refl,axiom,
    ! [A5: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ A5 @ A5 ) ).

% subset_refl
thf(fact_404_subset__iff,axiom,
    ( ord_le5819720210775101748te_a_a
    = ( ^ [A6: set_Pr2749968112124914068te_a_a,B6: set_Pr2749968112124914068te_a_a] :
        ! [T: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ T @ A6 )
         => ( member1329951143264900189te_a_a @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_405_subset__iff,axiom,
    ( ord_le5561929874677720807tate_a
    = ( ^ [A6: set_Pr1753757909640466247tate_a,B6: set_Pr1753757909640466247tate_a] :
        ! [T: produc4774439622479004775tate_a] :
          ( ( member3801464090328701328tate_a @ T @ A6 )
         => ( member3801464090328701328tate_a @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_406_subset__iff,axiom,
    ( ord_le4349072690221431335tate_a
    = ( ^ [A6: set_Pr7963165934123475591tate_a,B6: set_Pr7963165934123475591tate_a] :
        ! [T: produc5940350820068290983tate_a] :
          ( ( member1595334363549574352tate_a @ T @ A6 )
         => ( member1595334363549574352tate_a @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_407_subset__iff,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
        ! [T: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ T @ A6 )
         => ( member1426531477525435216od_a_a @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_408_subset__iff,axiom,
    ( ord_le7345504482307493082tate_a
    = ( ^ [A6: set_Pr4275752383657305402tate_a,B6: set_Pr4275752383657305402tate_a] :
        ! [T: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ T @ A6 )
         => ( member3175992478928454403tate_a @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_409_equalityD2,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( A5 = B5 )
     => ( ord_le7345504482307493082tate_a @ B5 @ A5 ) ) ).

% equalityD2
thf(fact_410_equalityD1,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( A5 = B5 )
     => ( ord_le7345504482307493082tate_a @ A5 @ B5 ) ) ).

% equalityD1
thf(fact_411_subset__eq,axiom,
    ( ord_le5819720210775101748te_a_a
    = ( ^ [A6: set_Pr2749968112124914068te_a_a,B6: set_Pr2749968112124914068te_a_a] :
        ! [X4: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X4 @ A6 )
         => ( member1329951143264900189te_a_a @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_412_subset__eq,axiom,
    ( ord_le5561929874677720807tate_a
    = ( ^ [A6: set_Pr1753757909640466247tate_a,B6: set_Pr1753757909640466247tate_a] :
        ! [X4: produc4774439622479004775tate_a] :
          ( ( member3801464090328701328tate_a @ X4 @ A6 )
         => ( member3801464090328701328tate_a @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_413_subset__eq,axiom,
    ( ord_le4349072690221431335tate_a
    = ( ^ [A6: set_Pr7963165934123475591tate_a,B6: set_Pr7963165934123475591tate_a] :
        ! [X4: produc5940350820068290983tate_a] :
          ( ( member1595334363549574352tate_a @ X4 @ A6 )
         => ( member1595334363549574352tate_a @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_414_subset__eq,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
        ! [X4: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ A6 )
         => ( member1426531477525435216od_a_a @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_415_subset__eq,axiom,
    ( ord_le7345504482307493082tate_a
    = ( ^ [A6: set_Pr4275752383657305402tate_a,B6: set_Pr4275752383657305402tate_a] :
        ! [X4: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X4 @ A6 )
         => ( member3175992478928454403tate_a @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_416_equalityE,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( A5 = B5 )
     => ~ ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
         => ~ ( ord_le7345504482307493082tate_a @ B5 @ A5 ) ) ) ).

% equalityE
thf(fact_417_subsetD,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a,C2: produc5126262593522865844te_a_a] :
      ( ( ord_le5819720210775101748te_a_a @ A5 @ B5 )
     => ( ( member1329951143264900189te_a_a @ C2 @ A5 )
       => ( member1329951143264900189te_a_a @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_418_subsetD,axiom,
    ! [A5: set_Pr1753757909640466247tate_a,B5: set_Pr1753757909640466247tate_a,C2: produc4774439622479004775tate_a] :
      ( ( ord_le5561929874677720807tate_a @ A5 @ B5 )
     => ( ( member3801464090328701328tate_a @ C2 @ A5 )
       => ( member3801464090328701328tate_a @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_419_subsetD,axiom,
    ! [A5: set_Pr7963165934123475591tate_a,B5: set_Pr7963165934123475591tate_a,C2: produc5940350820068290983tate_a] :
      ( ( ord_le4349072690221431335tate_a @ A5 @ B5 )
     => ( ( member1595334363549574352tate_a @ C2 @ A5 )
       => ( member1595334363549574352tate_a @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_420_subsetD,axiom,
    ! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,C2: product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
     => ( ( member1426531477525435216od_a_a @ C2 @ A5 )
       => ( member1426531477525435216od_a_a @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_421_subsetD,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,C2: produc6972303929186420058tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
     => ( ( member3175992478928454403tate_a @ C2 @ A5 )
       => ( member3175992478928454403tate_a @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_422_in__mono,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a,X3: produc5126262593522865844te_a_a] :
      ( ( ord_le5819720210775101748te_a_a @ A5 @ B5 )
     => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
       => ( member1329951143264900189te_a_a @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_423_in__mono,axiom,
    ! [A5: set_Pr1753757909640466247tate_a,B5: set_Pr1753757909640466247tate_a,X3: produc4774439622479004775tate_a] :
      ( ( ord_le5561929874677720807tate_a @ A5 @ B5 )
     => ( ( member3801464090328701328tate_a @ X3 @ A5 )
       => ( member3801464090328701328tate_a @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_424_in__mono,axiom,
    ! [A5: set_Pr7963165934123475591tate_a,B5: set_Pr7963165934123475591tate_a,X3: produc5940350820068290983tate_a] :
      ( ( ord_le4349072690221431335tate_a @ A5 @ B5 )
     => ( ( member1595334363549574352tate_a @ X3 @ A5 )
       => ( member1595334363549574352tate_a @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_425_in__mono,axiom,
    ! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,X3: product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
     => ( ( member1426531477525435216od_a_a @ X3 @ A5 )
       => ( member1426531477525435216od_a_a @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_426_in__mono,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,X3: produc6972303929186420058tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
     => ( ( member3175992478928454403tate_a @ X3 @ A5 )
       => ( member3175992478928454403tate_a @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_427_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_Pr4275752383657305402tate_a,Z: set_Pr4275752383657305402tate_a] : ( Y5 = Z ) )
    = ( ^ [X4: set_Pr4275752383657305402tate_a,Y6: set_Pr4275752383657305402tate_a] :
          ( ( ord_le7345504482307493082tate_a @ X4 @ Y6 )
          & ( ord_le7345504482307493082tate_a @ Y6 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_428_ord__eq__le__trans,axiom,
    ! [A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a,C2: set_Pr4275752383657305402tate_a] :
      ( ( A = B )
     => ( ( ord_le7345504482307493082tate_a @ B @ C2 )
       => ( ord_le7345504482307493082tate_a @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_429_ord__le__eq__trans,axiom,
    ! [A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a,C2: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A @ B )
     => ( ( B = C2 )
       => ( ord_le7345504482307493082tate_a @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_430_order__antisym,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ X3 @ Y )
     => ( ( ord_le7345504482307493082tate_a @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_431_order_Otrans,axiom,
    ! [A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a,C2: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A @ B )
     => ( ( ord_le7345504482307493082tate_a @ B @ C2 )
       => ( ord_le7345504482307493082tate_a @ A @ C2 ) ) ) ).

% order.trans
thf(fact_432_order__trans,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a,Z2: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ X3 @ Y )
     => ( ( ord_le7345504482307493082tate_a @ Y @ Z2 )
       => ( ord_le7345504482307493082tate_a @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_433_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_Pr4275752383657305402tate_a,Z: set_Pr4275752383657305402tate_a] : ( Y5 = Z ) )
    = ( ^ [A4: set_Pr4275752383657305402tate_a,B4: set_Pr4275752383657305402tate_a] :
          ( ( ord_le7345504482307493082tate_a @ B4 @ A4 )
          & ( ord_le7345504482307493082tate_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_434_dual__order_Oantisym,axiom,
    ! [B: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ B @ A )
     => ( ( ord_le7345504482307493082tate_a @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_435_dual__order_Otrans,axiom,
    ! [B: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a,C2: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ B @ A )
     => ( ( ord_le7345504482307493082tate_a @ C2 @ B )
       => ( ord_le7345504482307493082tate_a @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_436_antisym,axiom,
    ! [A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A @ B )
     => ( ( ord_le7345504482307493082tate_a @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_437_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_Pr4275752383657305402tate_a,Z: set_Pr4275752383657305402tate_a] : ( Y5 = Z ) )
    = ( ^ [A4: set_Pr4275752383657305402tate_a,B4: set_Pr4275752383657305402tate_a] :
          ( ( ord_le7345504482307493082tate_a @ A4 @ B4 )
          & ( ord_le7345504482307493082tate_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_438_order__subst1,axiom,
    ! [A: set_Pr4275752383657305402tate_a,F2: set_Pr4275752383657305402tate_a > set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a,C2: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A @ ( F2 @ B ) )
     => ( ( ord_le7345504482307493082tate_a @ B @ C2 )
       => ( ! [X: set_Pr4275752383657305402tate_a,Y3: set_Pr4275752383657305402tate_a] :
              ( ( ord_le7345504482307493082tate_a @ X @ Y3 )
             => ( ord_le7345504482307493082tate_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
         => ( ord_le7345504482307493082tate_a @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_439_order__subst2,axiom,
    ! [A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a,F2: set_Pr4275752383657305402tate_a > set_Pr4275752383657305402tate_a,C2: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A @ B )
     => ( ( ord_le7345504482307493082tate_a @ ( F2 @ B ) @ C2 )
       => ( ! [X: set_Pr4275752383657305402tate_a,Y3: set_Pr4275752383657305402tate_a] :
              ( ( ord_le7345504482307493082tate_a @ X @ Y3 )
             => ( ord_le7345504482307493082tate_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
         => ( ord_le7345504482307493082tate_a @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_440_order__eq__refl,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a] :
      ( ( X3 = Y )
     => ( ord_le7345504482307493082tate_a @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_441_ord__eq__le__subst,axiom,
    ! [A: set_Pr4275752383657305402tate_a,F2: set_Pr4275752383657305402tate_a > set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a,C2: set_Pr4275752383657305402tate_a] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_le7345504482307493082tate_a @ B @ C2 )
       => ( ! [X: set_Pr4275752383657305402tate_a,Y3: set_Pr4275752383657305402tate_a] :
              ( ( ord_le7345504482307493082tate_a @ X @ Y3 )
             => ( ord_le7345504482307493082tate_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
         => ( ord_le7345504482307493082tate_a @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_442_ord__le__eq__subst,axiom,
    ! [A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a,F2: set_Pr4275752383657305402tate_a > set_Pr4275752383657305402tate_a,C2: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X: set_Pr4275752383657305402tate_a,Y3: set_Pr4275752383657305402tate_a] :
              ( ( ord_le7345504482307493082tate_a @ X @ Y3 )
             => ( ord_le7345504482307493082tate_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
         => ( ord_le7345504482307493082tate_a @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_443_order__antisym__conv,axiom,
    ! [Y: set_Pr4275752383657305402tate_a,X3: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ Y @ X3 )
     => ( ( ord_le7345504482307493082tate_a @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_444_top__empty__eq,axiom,
    ( top_to8831379791598551801_a_a_o
    = ( ^ [X4: produc5126262593522865844te_a_a] : ( member1329951143264900189te_a_a @ X4 @ top_to6931181444738162404te_a_a ) ) ) ).

% top_empty_eq
thf(fact_445_top__empty__eq,axiom,
    ( top_to4864679581356359430te_a_o
    = ( ^ [X4: produc4774439622479004775tate_a] : ( member3801464090328701328tate_a @ X4 @ top_to496764914387262615tate_a ) ) ) ).

% top_empty_eq
thf(fact_446_top__empty__eq,axiom,
    ( top_to8009086793514025926te_a_o
    = ( ^ [X4: produc5940350820068290983tate_a] : ( member1595334363549574352tate_a @ X4 @ top_to93540037818290135tate_a ) ) ) ).

% top_empty_eq
thf(fact_447_top__empty__eq,axiom,
    ( top_to8687885267596698950_a_a_o
    = ( ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ top_to8063371432257647191od_a_a ) ) ) ).

% top_empty_eq
thf(fact_448_top__empty__eq,axiom,
    ( top_to4950562248668698323te_a_o
    = ( ^ [X4: produc6972303929186420058tate_a] : ( member3175992478928454403tate_a @ X4 @ top_to8456965716270553738tate_a ) ) ) ).

% top_empty_eq
thf(fact_449_subrelI,axiom,
    ! [R: set_Pr2749968112124914068te_a_a,S: set_Pr2749968112124914068te_a_a] :
      ( ! [X: state_a,Y3: a] :
          ( ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ X @ Y3 ) @ R )
         => ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ X @ Y3 ) @ S ) )
     => ( ord_le5819720210775101748te_a_a @ R @ S ) ) ).

% subrelI
thf(fact_450_subrelI,axiom,
    ! [R: set_Pr1753757909640466247tate_a,S: set_Pr1753757909640466247tate_a] :
      ( ! [X: state_a,Y3: state_a] :
          ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X @ Y3 ) @ R )
         => ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X @ Y3 ) @ S ) )
     => ( ord_le5561929874677720807tate_a @ R @ S ) ) ).

% subrelI
thf(fact_451_subrelI,axiom,
    ! [R: set_Pr7963165934123475591tate_a,S: set_Pr7963165934123475591tate_a] :
      ( ! [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a] :
          ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X @ Y3 ) @ R )
         => ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X @ Y3 ) @ S ) )
     => ( ord_le4349072690221431335tate_a @ R @ S ) ) ).

% subrelI
thf(fact_452_subrelI,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ! [X: a,Y3: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ S ) )
     => ( ord_le746702958409616551od_a_a @ R @ S ) ) ).

% subrelI
thf(fact_453_subrelI,axiom,
    ! [R: set_Pr4275752383657305402tate_a,S: set_Pr4275752383657305402tate_a] :
      ( ! [X: a,Y3: state_a] :
          ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X @ Y3 ) @ R )
         => ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X @ Y3 ) @ S ) )
     => ( ord_le7345504482307493082tate_a @ R @ S ) ) ).

% subrelI
thf(fact_454_top__greatest,axiom,
    ! [A: a > state_a > $o] : ( ord_le5391092903712282779te_a_o @ A @ top_top_a_state_a_o ) ).

% top_greatest
thf(fact_455_top__greatest,axiom,
    ! [A: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ A @ top_to8456965716270553738tate_a ) ).

% top_greatest
thf(fact_456_top_Oextremum__unique,axiom,
    ! [A: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ top_top_a_state_a_o @ A )
      = ( A = top_top_a_state_a_o ) ) ).

% top.extremum_unique
thf(fact_457_top_Oextremum__unique,axiom,
    ! [A: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ top_to8456965716270553738tate_a @ A )
      = ( A = top_to8456965716270553738tate_a ) ) ).

% top.extremum_unique
thf(fact_458_top_Oextremum__uniqueI,axiom,
    ! [A: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ top_top_a_state_a_o @ A )
     => ( A = top_top_a_state_a_o ) ) ).

% top.extremum_uniqueI
thf(fact_459_top_Oextremum__uniqueI,axiom,
    ! [A: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ top_to8456965716270553738tate_a @ A )
     => ( A = top_to8456965716270553738tate_a ) ) ).

% top.extremum_uniqueI
thf(fact_460_top__set__def,axiom,
    ( top_to8456965716270553738tate_a
    = ( collec8349941756605538373tate_a @ top_to4950562248668698323te_a_o ) ) ).

% top_set_def
thf(fact_461_image__mono,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( ord_le5819720210775101748te_a_a @ A5 @ B5 )
     => ( ord_le7345504482307493082tate_a @ ( image_4497581360776675627tate_a @ F2 @ A5 ) @ ( image_4497581360776675627tate_a @ F2 @ B5 ) ) ) ).

% image_mono
thf(fact_462_image__mono,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
     => ( ord_le5819720210775101748te_a_a @ ( image_71777787125004895te_a_a @ F2 @ A5 ) @ ( image_71777787125004895te_a_a @ F2 @ B5 ) ) ) ).

% image_mono
thf(fact_463_image__mono,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
     => ( ord_le7345504482307493082tate_a @ ( image_1917819122788559109tate_a @ F2 @ A5 ) @ ( image_1917819122788559109tate_a @ F2 @ B5 ) ) ) ).

% image_mono
thf(fact_464_image__subsetI,axiom,
    ! [A5: set_Product_prod_a_a,F2: product_prod_a_a > product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ! [X: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ A5 )
         => ( member1426531477525435216od_a_a @ ( F2 @ X ) @ B5 ) )
     => ( ord_le746702958409616551od_a_a @ ( image_4636654165204879301od_a_a @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_465_image__subsetI,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ A5 )
         => ( member1426531477525435216od_a_a @ ( F2 @ X ) @ B5 ) )
     => ( ord_le746702958409616551od_a_a @ ( image_8179770026730005074od_a_a @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_466_image__subsetI,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ! [X: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ A5 )
         => ( member1426531477525435216od_a_a @ ( F2 @ X ) @ B5 ) )
     => ( ord_le746702958409616551od_a_a @ ( image_615948358740051320od_a_a @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_467_image__subsetI,axiom,
    ! [A5: set_Product_prod_a_a,F2: product_prod_a_a > produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ! [X: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ A5 )
         => ( member1329951143264900189te_a_a @ ( F2 @ X ) @ B5 ) )
     => ( ord_le5819720210775101748te_a_a @ ( image_8940369480261093970te_a_a @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_468_image__subsetI,axiom,
    ! [A5: set_Product_prod_a_a,F2: product_prod_a_a > produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ! [X: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ A5 )
         => ( member3175992478928454403tate_a @ ( F2 @ X ) @ B5 ) )
     => ( ord_le7345504482307493082tate_a @ ( image_1563038779069872376tate_a @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_469_image__subsetI,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ A5 )
         => ( member1329951143264900189te_a_a @ ( F2 @ X ) @ B5 ) )
     => ( ord_le5819720210775101748te_a_a @ ( image_71777787125004895te_a_a @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_470_image__subsetI,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ! [X: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ A5 )
         => ( member1329951143264900189te_a_a @ ( F2 @ X ) @ B5 ) )
     => ( ord_le5819720210775101748te_a_a @ ( image_2651540025113121413te_a_a @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_471_image__subsetI,axiom,
    ! [A5: set_Pr1753757909640466247tate_a,F2: produc4774439622479004775tate_a > product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ! [X: produc4774439622479004775tate_a] :
          ( ( member3801464090328701328tate_a @ X @ A5 )
         => ( member1426531477525435216od_a_a @ ( F2 @ X ) @ B5 ) )
     => ( ord_le746702958409616551od_a_a @ ( image_1890940571582032517od_a_a @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_472_image__subsetI,axiom,
    ! [A5: set_Product_prod_a_a,F2: product_prod_a_a > produc4774439622479004775tate_a,B5: set_Pr1753757909640466247tate_a] :
      ( ! [X: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ A5 )
         => ( member3801464090328701328tate_a @ ( F2 @ X ) @ B5 ) )
     => ( ord_le5561929874677720807tate_a @ ( image_1794547935558701317tate_a @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_473_image__subsetI,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ A5 )
         => ( member3175992478928454403tate_a @ ( F2 @ X ) @ B5 ) )
     => ( ord_le7345504482307493082tate_a @ ( image_1917819122788559109tate_a @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_474_subset__imageE,axiom,
    ! [B5: set_Pr2749968112124914068te_a_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le5819720210775101748te_a_a @ B5 @ ( image_71777787125004895te_a_a @ F2 @ A5 ) )
     => ~ ! [C4: set_Pr4275752383657305402tate_a] :
            ( ( ord_le7345504482307493082tate_a @ C4 @ A5 )
           => ( B5
             != ( image_71777787125004895te_a_a @ F2 @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_475_subset__imageE,axiom,
    ! [B5: set_Pr4275752383657305402tate_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( ord_le7345504482307493082tate_a @ B5 @ ( image_4497581360776675627tate_a @ F2 @ A5 ) )
     => ~ ! [C4: set_Pr2749968112124914068te_a_a] :
            ( ( ord_le5819720210775101748te_a_a @ C4 @ A5 )
           => ( B5
             != ( image_4497581360776675627tate_a @ F2 @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_476_subset__imageE,axiom,
    ! [B5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ B5 @ ( image_1917819122788559109tate_a @ F2 @ A5 ) )
     => ~ ! [C4: set_Pr4275752383657305402tate_a] :
            ( ( ord_le7345504482307493082tate_a @ C4 @ A5 )
           => ( B5
             != ( image_1917819122788559109tate_a @ F2 @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_477_image__subset__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( ord_le5819720210775101748te_a_a @ ( image_71777787125004895te_a_a @ F2 @ A5 ) @ B5 )
      = ( ! [X4: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X4 @ A5 )
           => ( member1329951143264900189te_a_a @ ( F2 @ X4 ) @ B5 ) ) ) ) ).

% image_subset_iff
thf(fact_478_image__subset__iff,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( image_4497581360776675627tate_a @ F2 @ A5 ) @ B5 )
      = ( ! [X4: produc5126262593522865844te_a_a] :
            ( ( member1329951143264900189te_a_a @ X4 @ A5 )
           => ( member3175992478928454403tate_a @ ( F2 @ X4 ) @ B5 ) ) ) ) ).

% image_subset_iff
thf(fact_479_image__subset__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( image_1917819122788559109tate_a @ F2 @ A5 ) @ B5 )
      = ( ! [X4: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X4 @ A5 )
           => ( member3175992478928454403tate_a @ ( F2 @ X4 ) @ B5 ) ) ) ) ).

% image_subset_iff
thf(fact_480_subset__image__iff,axiom,
    ! [B5: set_Pr2749968112124914068te_a_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le5819720210775101748te_a_a @ B5 @ ( image_71777787125004895te_a_a @ F2 @ A5 ) )
      = ( ? [AA: set_Pr4275752383657305402tate_a] :
            ( ( ord_le7345504482307493082tate_a @ AA @ A5 )
            & ( B5
              = ( image_71777787125004895te_a_a @ F2 @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_481_subset__image__iff,axiom,
    ! [B5: set_Pr4275752383657305402tate_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( ord_le7345504482307493082tate_a @ B5 @ ( image_4497581360776675627tate_a @ F2 @ A5 ) )
      = ( ? [AA: set_Pr2749968112124914068te_a_a] :
            ( ( ord_le5819720210775101748te_a_a @ AA @ A5 )
            & ( B5
              = ( image_4497581360776675627tate_a @ F2 @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_482_subset__image__iff,axiom,
    ! [B5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ B5 @ ( image_1917819122788559109tate_a @ F2 @ A5 ) )
      = ( ? [AA: set_Pr4275752383657305402tate_a] :
            ( ( ord_le7345504482307493082tate_a @ AA @ A5 )
            & ( B5
              = ( image_1917819122788559109tate_a @ F2 @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_483_subset__UNIV,axiom,
    ! [A5: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ A5 @ top_to8456965716270553738tate_a ) ).

% subset_UNIV
thf(fact_484_range__subsetD,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a,I: produc6972303929186420058tate_a] :
      ( ( ord_le5819720210775101748te_a_a @ ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a ) @ B5 )
     => ( member1329951143264900189te_a_a @ ( F2 @ I ) @ B5 ) ) ).

% range_subsetD
thf(fact_485_range__subsetD,axiom,
    ! [F2: produc6972303929186420058tate_a > produc4774439622479004775tate_a,B5: set_Pr1753757909640466247tate_a,I: produc6972303929186420058tate_a] :
      ( ( ord_le5561929874677720807tate_a @ ( image_5672933159226955410tate_a @ F2 @ top_to8456965716270553738tate_a ) @ B5 )
     => ( member3801464090328701328tate_a @ ( F2 @ I ) @ B5 ) ) ).

% range_subsetD
thf(fact_486_range__subsetD,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5940350820068290983tate_a,B5: set_Pr7963165934123475591tate_a,I: produc6972303929186420058tate_a] :
      ( ( ord_le4349072690221431335tate_a @ ( image_8048040396972735954tate_a @ F2 @ top_to8456965716270553738tate_a ) @ B5 )
     => ( member1595334363549574352tate_a @ ( F2 @ I ) @ B5 ) ) ).

% range_subsetD
thf(fact_487_range__subsetD,axiom,
    ! [F2: produc6972303929186420058tate_a > product_prod_a_a,B5: set_Product_prod_a_a,I: produc6972303929186420058tate_a] :
      ( ( ord_le746702958409616551od_a_a @ ( image_8179770026730005074od_a_a @ F2 @ top_to8456965716270553738tate_a ) @ B5 )
     => ( member1426531477525435216od_a_a @ ( F2 @ I ) @ B5 ) ) ).

% range_subsetD
thf(fact_488_range__subsetD,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a,I: produc5126262593522865844te_a_a] :
      ( ( ord_le7345504482307493082tate_a @ ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a ) @ B5 )
     => ( member3175992478928454403tate_a @ ( F2 @ I ) @ B5 ) ) ).

% range_subsetD
thf(fact_489_range__subsetD,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a,I: produc6972303929186420058tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( image_1917819122788559109tate_a @ F2 @ top_to8456965716270553738tate_a ) @ B5 )
     => ( member3175992478928454403tate_a @ ( F2 @ I ) @ B5 ) ) ).

% range_subsetD
thf(fact_490_prod__fun__imageE,axiom,
    ! [C2: produc6972303929186420058tate_a,F2: a > a,G: state_a > state_a,R2: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ C2 @ ( image_1917819122788559109tate_a @ ( produc6760600410500789430tate_a @ F2 @ G ) @ R2 ) )
     => ~ ! [X: a,Y3: state_a] :
            ( ( C2
              = ( produc8641956578966763338tate_a @ ( F2 @ X ) @ ( G @ Y3 ) ) )
           => ~ ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_491_prod__fun__imageE,axiom,
    ! [C2: product_prod_a_a,F2: a > a,G: a > a,R2: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C2 @ ( image_4636654165204879301od_a_a @ ( produc4018698449564827638_a_a_a @ F2 @ G ) @ R2 ) )
     => ~ ! [X: a,Y3: a] :
            ( ( C2
              = ( product_Pair_a_a @ ( F2 @ X ) @ ( G @ Y3 ) ) )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_492_prod__fun__imageE,axiom,
    ! [C2: produc6972303929186420058tate_a,F2: a > a,G: a > state_a,R2: set_Product_prod_a_a] :
      ( ( member3175992478928454403tate_a @ C2 @ ( image_1563038779069872376tate_a @ ( produc8826359041189215401tate_a @ F2 @ G ) @ R2 ) )
     => ~ ! [X: a,Y3: a] :
            ( ( C2
              = ( produc8641956578966763338tate_a @ ( F2 @ X ) @ ( G @ Y3 ) ) )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_493_prod__fun__imageE,axiom,
    ! [C2: produc5126262593522865844te_a_a,F2: a > state_a,G: a > a,R2: set_Product_prod_a_a] :
      ( ( member1329951143264900189te_a_a @ C2 @ ( image_8940369480261093970te_a_a @ ( produc1047144888364738345_a_a_a @ F2 @ G ) @ R2 ) )
     => ~ ! [X: a,Y3: a] :
            ( ( C2
              = ( produc7328099264922328356te_a_a @ ( F2 @ X ) @ ( G @ Y3 ) ) )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_494_prod__fun__imageE,axiom,
    ! [C2: product_prod_a_a,F2: a > a,G: state_a > a,R2: set_Pr4275752383657305402tate_a] :
      ( ( member1426531477525435216od_a_a @ C2 @ ( image_8179770026730005074od_a_a @ ( produc7512501727144780419te_a_a @ F2 @ G ) @ R2 ) )
     => ~ ! [X: a,Y3: state_a] :
            ( ( C2
              = ( product_Pair_a_a @ ( F2 @ X ) @ ( G @ Y3 ) ) )
           => ~ ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_495_prod__fun__imageE,axiom,
    ! [C2: product_prod_a_a,F2: state_a > a,G: a > a,R2: set_Pr2749968112124914068te_a_a] :
      ( ( member1426531477525435216od_a_a @ C2 @ ( image_615948358740051320od_a_a @ ( produc7010603597014767619_a_a_a @ F2 @ G ) @ R2 ) )
     => ~ ! [X: state_a,Y3: a] :
            ( ( C2
              = ( product_Pair_a_a @ ( F2 @ X ) @ ( G @ Y3 ) ) )
           => ~ ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ X @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_496_prod__fun__imageE,axiom,
    ! [C2: produc6972303929186420058tate_a,F2: state_a > a,G: a > state_a,R2: set_Pr2749968112124914068te_a_a] :
      ( ( member3175992478928454403tate_a @ C2 @ ( image_4497581360776675627tate_a @ ( produc5926476928146254390tate_a @ F2 @ G ) @ R2 ) )
     => ~ ! [X: state_a,Y3: a] :
            ( ( C2
              = ( produc8641956578966763338tate_a @ ( F2 @ X ) @ ( G @ Y3 ) ) )
           => ~ ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ X @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_497_prod__fun__imageE,axiom,
    ! [C2: produc5126262593522865844te_a_a,F2: a > state_a,G: state_a > a,R2: set_Pr4275752383657305402tate_a] :
      ( ( member1329951143264900189te_a_a @ C2 @ ( image_71777787125004895te_a_a @ ( produc2865014304882136118te_a_a @ F2 @ G ) @ R2 ) )
     => ~ ! [X: a,Y3: state_a] :
            ( ( C2
              = ( produc7328099264922328356te_a_a @ ( F2 @ X ) @ ( G @ Y3 ) ) )
           => ~ ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_498_prod__fun__imageE,axiom,
    ! [C2: produc5126262593522865844te_a_a,F2: state_a > state_a,G: a > a,R2: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ C2 @ ( image_2651540025113121413te_a_a @ ( produc7370634812176553142_a_a_a @ F2 @ G ) @ R2 ) )
     => ~ ! [X: state_a,Y3: a] :
            ( ( C2
              = ( produc7328099264922328356te_a_a @ ( F2 @ X ) @ ( G @ Y3 ) ) )
           => ~ ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ X @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_499_prod__fun__imageE,axiom,
    ! [C2: produc4774439622479004775tate_a,F2: a > state_a,G: a > state_a,R2: set_Product_prod_a_a] :
      ( ( member3801464090328701328tate_a @ C2 @ ( image_1794547935558701317tate_a @ ( produc4178871618926571100tate_a @ F2 @ G ) @ R2 ) )
     => ~ ! [X: a,Y3: a] :
            ( ( C2
              = ( produc6131900399494507223tate_a @ ( F2 @ X ) @ ( G @ Y3 ) ) )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_500_all__subset__image,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,P2: set_Pr2749968112124914068te_a_a > $o] :
      ( ( ! [B6: set_Pr2749968112124914068te_a_a] :
            ( ( ord_le5819720210775101748te_a_a @ B6 @ ( image_71777787125004895te_a_a @ F2 @ A5 ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_Pr4275752383657305402tate_a] :
            ( ( ord_le7345504482307493082tate_a @ B6 @ A5 )
           => ( P2 @ ( image_71777787125004895te_a_a @ F2 @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_501_all__subset__image,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,P2: set_Pr4275752383657305402tate_a > $o] :
      ( ( ! [B6: set_Pr4275752383657305402tate_a] :
            ( ( ord_le7345504482307493082tate_a @ B6 @ ( image_4497581360776675627tate_a @ F2 @ A5 ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_Pr2749968112124914068te_a_a] :
            ( ( ord_le5819720210775101748te_a_a @ B6 @ A5 )
           => ( P2 @ ( image_4497581360776675627tate_a @ F2 @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_502_all__subset__image,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,P2: set_Pr4275752383657305402tate_a > $o] :
      ( ( ! [B6: set_Pr4275752383657305402tate_a] :
            ( ( ord_le7345504482307493082tate_a @ B6 @ ( image_1917819122788559109tate_a @ F2 @ A5 ) )
           => ( P2 @ B6 ) ) )
      = ( ! [B6: set_Pr4275752383657305402tate_a] :
            ( ( ord_le7345504482307493082tate_a @ B6 @ A5 )
           => ( P2 @ ( image_1917819122788559109tate_a @ F2 @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_503_relChain__def,axiom,
    ( bNF_Ca1648762892614795885tate_a
    = ( ^ [R3: set_Pr1753757909640466247tate_a,As: state_a > set_Pr4275752383657305402tate_a] :
        ! [I2: state_a,J: state_a] :
          ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ I2 @ J ) @ R3 )
         => ( ord_le7345504482307493082tate_a @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_504_relChain__def,axiom,
    ( bNF_Ca2826069026209883524tate_a
    = ( ^ [R3: set_Pr7963165934123475591tate_a,As: produc6972303929186420058tate_a > set_Pr4275752383657305402tate_a] :
        ! [I2: produc6972303929186420058tate_a,J: produc6972303929186420058tate_a] :
          ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ I2 @ J ) @ R3 )
         => ( ord_le7345504482307493082tate_a @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_505_relChain__def,axiom,
    ( bNF_Ca1942141597700211296tate_a
    = ( ^ [R3: set_Product_prod_a_a,As: a > set_Pr4275752383657305402tate_a] :
        ! [I2: a,J: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I2 @ J ) @ R3 )
         => ( ord_le7345504482307493082tate_a @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_506_Collect__case__prod__in__rel__leI,axiom,
    ! [X6: set_Pr4275752383657305402tate_a,Y7: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ X6 @ Y7 )
     => ( ord_le7345504482307493082tate_a @ X6 @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ ( fun_in_rel_a_state_a @ Y7 ) ) ) ) ) ).

% Collect_case_prod_in_rel_leI
thf(fact_507_Collect__case__prod__in__rel__leE,axiom,
    ! [X6: set_Pr4275752383657305402tate_a,Y7: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ X6 @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ ( fun_in_rel_a_state_a @ Y7 ) ) ) )
     => ( ord_le7345504482307493082tate_a @ X6 @ Y7 ) ) ).

% Collect_case_prod_in_rel_leE
thf(fact_508_sndOp__def,axiom,
    ( bNF_sn3184738459333431869tate_a
    = ( ^ [P4: a > a > $o,Q3: a > state_a > $o,Ac: produc6972303929186420058tate_a] : ( produc8641956578966763338tate_a @ ( bNF_pi4221486096819041613tate_a @ P4 @ Q3 @ ( produc736293372669613878tate_a @ Ac ) @ ( produc7615498795807706488tate_a @ Ac ) ) @ ( produc7615498795807706488tate_a @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_509_sndOp__def,axiom,
    ( bNF_sn36956615848800330tate_a
    = ( ^ [P4: a > state_a > $o,Q3: state_a > state_a > $o,Ac: produc6972303929186420058tate_a] : ( produc6131900399494507223tate_a @ ( bNF_pi2511082644353977178tate_a @ P4 @ Q3 @ ( produc736293372669613878tate_a @ Ac ) @ ( produc7615498795807706488tate_a @ Ac ) ) @ ( produc7615498795807706488tate_a @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_510_sndOp__def,axiom,
    ( bNF_sn5364742547084922826te_a_a
    = ( ^ [P4: state_a > state_a > $o,Q3: state_a > a > $o,Ac: produc5126262593522865844te_a_a] : ( produc7328099264922328356te_a_a @ ( bNF_pi7838868575590099674te_a_a @ P4 @ Q3 @ ( produc8645808095479954704te_a_a @ Ac ) @ ( produc6301641481763271506te_a_a @ Ac ) ) @ ( produc6301641481763271506te_a_a @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_511_sndOp__def,axiom,
    ( bNF_sn4628896343363730621_a_a_a
    = ( ^ [P4: state_a > a > $o,Q3: a > a > $o,Ac: produc5126262593522865844te_a_a] : ( product_Pair_a_a @ ( bNF_pi5665643980849340365_a_a_a @ P4 @ Q3 @ ( produc8645808095479954704te_a_a @ Ac ) @ ( produc6301641481763271506te_a_a @ Ac ) ) @ ( produc6301641481763271506te_a_a @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_512_fstOp__def,axiom,
    ( bNF_fs1355894354835818508tate_a
    = ( ^ [P4: a > state_a > $o,Q3: state_a > state_a > $o,Ac: produc6972303929186420058tate_a] : ( produc8641956578966763338tate_a @ ( produc736293372669613878tate_a @ Ac ) @ ( bNF_pi2511082644353977178tate_a @ P4 @ Q3 @ ( produc736293372669613878tate_a @ Ac ) @ ( produc7615498795807706488tate_a @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_513_fstOp__def,axiom,
    ( bNF_fs1569747998200427263tate_a
    = ( ^ [P4: a > a > $o,Q3: a > state_a > $o,Ac: produc6972303929186420058tate_a] : ( product_Pair_a_a @ ( produc736293372669613878tate_a @ Ac ) @ ( bNF_pi4221486096819041613tate_a @ P4 @ Q3 @ ( produc736293372669613878tate_a @ Ac ) @ ( produc7615498795807706488tate_a @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_514_fstOp__def,axiom,
    ( bNF_fs3013905882230726015_a_a_a
    = ( ^ [P4: state_a > a > $o,Q3: a > a > $o,Ac: produc5126262593522865844te_a_a] : ( produc7328099264922328356te_a_a @ ( produc8645808095479954704te_a_a @ Ac ) @ ( bNF_pi5665643980849340365_a_a_a @ P4 @ Q3 @ ( produc8645808095479954704te_a_a @ Ac ) @ ( produc6301641481763271506te_a_a @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_515_fstOp__def,axiom,
    ( bNF_fs6683680286071941004te_a_a
    = ( ^ [P4: state_a > state_a > $o,Q3: state_a > a > $o,Ac: produc5126262593522865844te_a_a] : ( produc6131900399494507223tate_a @ ( produc8645808095479954704te_a_a @ Ac ) @ ( bNF_pi7838868575590099674te_a_a @ P4 @ Q3 @ ( produc8645808095479954704te_a_a @ Ac ) @ ( produc6301641481763271506te_a_a @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_516_image2__eqI,axiom,
    ! [B: a,F2: product_prod_a_a > a,X3: product_prod_a_a,C2: a,G: product_prod_a_a > a,A5: set_Product_prod_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member1426531477525435216od_a_a @ X3 @ A5 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C2 ) @ ( bNF_Gr1149069696037075021_a_a_a @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_517_image2__eqI,axiom,
    ! [B: a,F2: product_prod_a_a > a,X3: product_prod_a_a,C2: state_a,G: product_prod_a_a > state_a,A5: set_Product_prod_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member1426531477525435216od_a_a @ X3 @ A5 )
         => ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ B @ C2 ) @ ( bNF_Gr5127297355883978112tate_a @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_518_image2__eqI,axiom,
    ! [B: state_a,F2: product_prod_a_a > state_a,X3: product_prod_a_a,C2: a,G: product_prod_a_a > a,A5: set_Product_prod_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member1426531477525435216od_a_a @ X3 @ A5 )
         => ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ B @ C2 ) @ ( bNF_Gr3813440041839543130te_a_a @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_519_image2__eqI,axiom,
    ! [B: a,F2: produc6972303929186420058tate_a > a,X3: produc6972303929186420058tate_a,C2: a,G: produc6972303929186420058tate_a > a,A5: set_Pr4275752383657305402tate_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member3175992478928454403tate_a @ X3 @ A5 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C2 ) @ ( bNF_Gr2278677502387744768_a_a_a @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_520_image2__eqI,axiom,
    ! [B: a,F2: produc5126262593522865844te_a_a > a,X3: produc5126262593522865844te_a_a,C2: a,G: produc5126262593522865844te_a_a > a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C2 ) @ ( bNF_Gr6620124013041519706_a_a_a @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_521_image2__eqI,axiom,
    ! [B: a,F2: produc6972303929186420058tate_a > a,X3: produc6972303929186420058tate_a,C2: state_a,G: produc6972303929186420058tate_a > state_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member3175992478928454403tate_a @ X3 @ A5 )
         => ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ B @ C2 ) @ ( bNF_Gr5089472653393977267tate_a @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_522_image2__eqI,axiom,
    ! [B: a,F2: produc5126262593522865844te_a_a > a,X3: produc5126262593522865844te_a_a,C2: state_a,G: produc5126262593522865844te_a_a > state_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
         => ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ B @ C2 ) @ ( bNF_Gr6519342181441242893tate_a @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_523_image2__eqI,axiom,
    ! [B: state_a,F2: produc6972303929186420058tate_a > state_a,X3: produc6972303929186420058tate_a,C2: a,G: produc6972303929186420058tate_a > a,A5: set_Pr4275752383657305402tate_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member3175992478928454403tate_a @ X3 @ A5 )
         => ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ B @ C2 ) @ ( bNF_Gr3775615339349542285te_a_a @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_524_image2__eqI,axiom,
    ! [B: state_a,F2: produc5126262593522865844te_a_a > state_a,X3: produc5126262593522865844te_a_a,C2: a,G: produc5126262593522865844te_a_a > a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
         => ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ B @ C2 ) @ ( bNF_Gr5205484867396807911te_a_a @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_525_image2__eqI,axiom,
    ! [B: state_a,F2: product_prod_a_a > state_a,X3: product_prod_a_a,C2: state_a,G: product_prod_a_a > state_a,A5: set_Product_prod_a_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C2
          = ( G @ X3 ) )
       => ( ( member1426531477525435216od_a_a @ X3 @ A5 )
         => ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ B @ C2 ) @ ( bNF_Gr8527404693569682957tate_a @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_526_GreatestI2__order,axiom,
    ! [P2: set_Pr4275752383657305402tate_a > $o,X3: set_Pr4275752383657305402tate_a,Q2: set_Pr4275752383657305402tate_a > $o] :
      ( ( P2 @ X3 )
     => ( ! [Y3: set_Pr4275752383657305402tate_a] :
            ( ( P2 @ Y3 )
           => ( ord_le7345504482307493082tate_a @ Y3 @ X3 ) )
       => ( ! [X: set_Pr4275752383657305402tate_a] :
              ( ( P2 @ X )
             => ( ! [Y8: set_Pr4275752383657305402tate_a] :
                    ( ( P2 @ Y8 )
                   => ( ord_le7345504482307493082tate_a @ Y8 @ X ) )
               => ( Q2 @ X ) ) )
         => ( Q2 @ ( order_2208051748804461971tate_a @ P2 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_527_Greatest__equality,axiom,
    ! [P2: set_Pr4275752383657305402tate_a > $o,X3: set_Pr4275752383657305402tate_a] :
      ( ( P2 @ X3 )
     => ( ! [Y3: set_Pr4275752383657305402tate_a] :
            ( ( P2 @ Y3 )
           => ( ord_le7345504482307493082tate_a @ Y3 @ X3 ) )
       => ( ( order_2208051748804461971tate_a @ P2 )
          = X3 ) ) ) ).

% Greatest_equality
thf(fact_528_asymI,axiom,
    ! [R: set_Pr1753757909640466247tate_a] :
      ( ! [X: state_a,Y3: state_a] :
          ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X @ Y3 ) @ R )
         => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y3 @ X ) @ R ) )
     => ( asym_on_state_a @ top_top_set_state_a @ R ) ) ).

% asymI
thf(fact_529_asymI,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ! [X: a,Y3: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
         => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R ) )
     => ( asym_on_a @ top_top_set_a @ R ) ) ).

% asymI
thf(fact_530_asymI,axiom,
    ! [R: set_Pr7963165934123475591tate_a] :
      ( ! [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a] :
          ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X @ Y3 ) @ R )
         => ~ ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y3 @ X ) @ R ) )
     => ( asym_o3617238991104122252tate_a @ top_to8456965716270553738tate_a @ R ) ) ).

% asymI
thf(fact_531_asym__onI,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,R: set_Pr4734257497606733575te_a_a] :
      ( ! [X: produc5126262593522865844te_a_a,Y3: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ A5 )
         => ( ( member1329951143264900189te_a_a @ Y3 @ A5 )
           => ( ( member2649312330005299536te_a_a @ ( produc2877451964877216791te_a_a @ X @ Y3 ) @ R )
             => ~ ( member2649312330005299536te_a_a @ ( produc2877451964877216791te_a_a @ Y3 @ X ) @ R ) ) ) )
     => ( asym_o1771197655440568038te_a_a @ A5 @ R ) ) ).

% asym_onI
thf(fact_532_asym__onI,axiom,
    ! [A5: set_Pr1753757909640466247tate_a,R: set_Pr2656942177312230599tate_a] :
      ( ! [X: produc4774439622479004775tate_a,Y3: produc4774439622479004775tate_a] :
          ( ( member3801464090328701328tate_a @ X @ A5 )
         => ( ( member3801464090328701328tate_a @ Y3 @ A5 )
           => ( ( member7923945571256933136tate_a @ ( produc1588899729123023703tate_a @ X @ Y3 ) @ R )
             => ~ ( member7923945571256933136tate_a @ ( produc1588899729123023703tate_a @ Y3 @ X ) @ R ) ) ) )
     => ( asym_o5616783792318651289tate_a @ A5 @ R ) ) ).

% asym_onI
thf(fact_533_asym__onI,axiom,
    ! [A5: set_Pr7963165934123475591tate_a,R: set_Pr1690608023759948999tate_a] :
      ( ! [X: produc5940350820068290983tate_a,Y3: produc5940350820068290983tate_a] :
          ( ( member1595334363549574352tate_a @ X @ A5 )
         => ( ( member1595334363549574352tate_a @ Y3 @ A5 )
           => ( ( member1643692453717642000tate_a @ ( produc7335381768849070935tate_a @ X @ Y3 ) @ R )
             => ~ ( member1643692453717642000tate_a @ ( produc7335381768849070935tate_a @ Y3 @ X ) @ R ) ) ) )
     => ( asym_o7314948133871076569tate_a @ A5 @ R ) ) ).

% asym_onI
thf(fact_534_asym__onI,axiom,
    ! [A5: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
      ( ! [X: product_prod_a_a,Y3: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ A5 )
         => ( ( member1426531477525435216od_a_a @ Y3 @ A5 )
           => ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ R )
             => ~ ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ X ) @ R ) ) ) )
     => ( asym_o3869890535549240921od_a_a @ A5 @ R ) ) ).

% asym_onI
thf(fact_535_asym__onI,axiom,
    ! [A5: set_state_a,R: set_Pr1753757909640466247tate_a] :
      ( ! [X: state_a,Y3: state_a] :
          ( ( member_state_a @ X @ A5 )
         => ( ( member_state_a @ Y3 @ A5 )
           => ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X @ Y3 ) @ R )
             => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y3 @ X ) @ R ) ) ) )
     => ( asym_on_state_a @ A5 @ R ) ) ).

% asym_onI
thf(fact_536_asym__onI,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,R: set_Pr7963165934123475591tate_a] :
      ( ! [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ A5 )
         => ( ( member3175992478928454403tate_a @ Y3 @ A5 )
           => ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X @ Y3 ) @ R )
             => ~ ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y3 @ X ) @ R ) ) ) )
     => ( asym_o3617238991104122252tate_a @ A5 @ R ) ) ).

% asym_onI
thf(fact_537_asym__onI,axiom,
    ! [A5: set_a,R: set_Product_prod_a_a] :
      ( ! [X: a,Y3: a] :
          ( ( member_a @ X @ A5 )
         => ( ( member_a @ Y3 @ A5 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
             => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R ) ) ) )
     => ( asym_on_a @ A5 @ R ) ) ).

% asym_onI
thf(fact_538_asym__onD,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,R: set_Pr4734257497606733575te_a_a,X3: produc5126262593522865844te_a_a,Y: produc5126262593522865844te_a_a] :
      ( ( asym_o1771197655440568038te_a_a @ A5 @ R )
     => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
       => ( ( member1329951143264900189te_a_a @ Y @ A5 )
         => ( ( member2649312330005299536te_a_a @ ( produc2877451964877216791te_a_a @ X3 @ Y ) @ R )
           => ~ ( member2649312330005299536te_a_a @ ( produc2877451964877216791te_a_a @ Y @ X3 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_539_asym__onD,axiom,
    ! [A5: set_Pr1753757909640466247tate_a,R: set_Pr2656942177312230599tate_a,X3: produc4774439622479004775tate_a,Y: produc4774439622479004775tate_a] :
      ( ( asym_o5616783792318651289tate_a @ A5 @ R )
     => ( ( member3801464090328701328tate_a @ X3 @ A5 )
       => ( ( member3801464090328701328tate_a @ Y @ A5 )
         => ( ( member7923945571256933136tate_a @ ( produc1588899729123023703tate_a @ X3 @ Y ) @ R )
           => ~ ( member7923945571256933136tate_a @ ( produc1588899729123023703tate_a @ Y @ X3 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_540_asym__onD,axiom,
    ! [A5: set_Pr7963165934123475591tate_a,R: set_Pr1690608023759948999tate_a,X3: produc5940350820068290983tate_a,Y: produc5940350820068290983tate_a] :
      ( ( asym_o7314948133871076569tate_a @ A5 @ R )
     => ( ( member1595334363549574352tate_a @ X3 @ A5 )
       => ( ( member1595334363549574352tate_a @ Y @ A5 )
         => ( ( member1643692453717642000tate_a @ ( produc7335381768849070935tate_a @ X3 @ Y ) @ R )
           => ~ ( member1643692453717642000tate_a @ ( produc7335381768849070935tate_a @ Y @ X3 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_541_asym__onD,axiom,
    ! [A5: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a,X3: product_prod_a_a,Y: product_prod_a_a] :
      ( ( asym_o3869890535549240921od_a_a @ A5 @ R )
     => ( ( member1426531477525435216od_a_a @ X3 @ A5 )
       => ( ( member1426531477525435216od_a_a @ Y @ A5 )
         => ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y ) @ R )
           => ~ ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y @ X3 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_542_asym__onD,axiom,
    ! [A5: set_state_a,R: set_Pr1753757909640466247tate_a,X3: state_a,Y: state_a] :
      ( ( asym_on_state_a @ A5 @ R )
     => ( ( member_state_a @ X3 @ A5 )
       => ( ( member_state_a @ Y @ A5 )
         => ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X3 @ Y ) @ R )
           => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y @ X3 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_543_asym__onD,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,R: set_Pr7963165934123475591tate_a,X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a] :
      ( ( asym_o3617238991104122252tate_a @ A5 @ R )
     => ( ( member3175992478928454403tate_a @ X3 @ A5 )
       => ( ( member3175992478928454403tate_a @ Y @ A5 )
         => ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X3 @ Y ) @ R )
           => ~ ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y @ X3 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_544_asym__onD,axiom,
    ! [A5: set_a,R: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( asym_on_a @ A5 @ R )
     => ( ( member_a @ X3 @ A5 )
       => ( ( member_a @ Y @ A5 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X3 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_545_asym__on__def,axiom,
    ( asym_on_state_a
    = ( ^ [A6: set_state_a,R3: set_Pr1753757909640466247tate_a] :
        ! [X4: state_a] :
          ( ( member_state_a @ X4 @ A6 )
         => ! [Y6: state_a] :
              ( ( member_state_a @ Y6 @ A6 )
             => ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X4 @ Y6 ) @ R3 )
               => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y6 @ X4 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_546_asym__on__def,axiom,
    ( asym_o3617238991104122252tate_a
    = ( ^ [A6: set_Pr4275752383657305402tate_a,R3: set_Pr7963165934123475591tate_a] :
        ! [X4: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X4 @ A6 )
         => ! [Y6: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ Y6 @ A6 )
             => ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X4 @ Y6 ) @ R3 )
               => ~ ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y6 @ X4 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_547_asym__on__def,axiom,
    ( asym_on_a
    = ( ^ [A6: set_a,R3: set_Product_prod_a_a] :
        ! [X4: a] :
          ( ( member_a @ X4 @ A6 )
         => ! [Y6: a] :
              ( ( member_a @ Y6 @ A6 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ R3 )
               => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y6 @ X4 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_548_asym__on__subset,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,R: set_Pr7963165934123475591tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( asym_o3617238991104122252tate_a @ A5 @ R )
     => ( ( ord_le7345504482307493082tate_a @ B5 @ A5 )
       => ( asym_o3617238991104122252tate_a @ B5 @ R ) ) ) ).

% asym_on_subset
thf(fact_549_in__rel__def,axiom,
    ( fun_in_rel_state_a_a
    = ( ^ [R4: set_Pr2749968112124914068te_a_a,X4: state_a,Y6: a] : ( member1329951143264900189te_a_a @ ( produc7328099264922328356te_a_a @ X4 @ Y6 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_550_in__rel__def,axiom,
    ( fun_in6186945512769221122tate_a
    = ( ^ [R4: set_Pr1753757909640466247tate_a,X4: state_a,Y6: state_a] : ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X4 @ Y6 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_551_in__rel__def,axiom,
    ( fun_in2474912675648533442tate_a
    = ( ^ [R4: set_Pr7963165934123475591tate_a,X4: produc6972303929186420058tate_a,Y6: produc6972303929186420058tate_a] : ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X4 @ Y6 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_552_in__rel__def,axiom,
    ( fun_in_rel_a_a
    = ( ^ [R4: set_Product_prod_a_a,X4: a,Y6: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_553_in__rel__def,axiom,
    ( fun_in_rel_a_state_a
    = ( ^ [R4: set_Pr4275752383657305402tate_a,X4: a,Y6: state_a] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y6 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_554_in__rel__Collect__case__prod__eq,axiom,
    ! [X6: a > state_a > $o] :
      ( ( fun_in_rel_a_state_a @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ X6 ) ) )
      = X6 ) ).

% in_rel_Collect_case_prod_eq
thf(fact_555_Collect__case__prod__mono,axiom,
    ! [A5: a > state_a > $o,B5: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ A5 @ B5 )
     => ( ord_le7345504482307493082tate_a @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ A5 ) ) @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_556_asym__iff,axiom,
    ! [R: set_Pr1753757909640466247tate_a] :
      ( ( asym_on_state_a @ top_top_set_state_a @ R )
      = ( ! [X4: state_a,Y6: state_a] :
            ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X4 @ Y6 ) @ R )
           => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y6 @ X4 ) @ R ) ) ) ) ).

% asym_iff
thf(fact_557_asym__iff,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ( asym_on_a @ top_top_set_a @ R )
      = ( ! [X4: a,Y6: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ R )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y6 @ X4 ) @ R ) ) ) ) ).

% asym_iff
thf(fact_558_asym__iff,axiom,
    ! [R: set_Pr7963165934123475591tate_a] :
      ( ( asym_o3617238991104122252tate_a @ top_to8456965716270553738tate_a @ R )
      = ( ! [X4: produc6972303929186420058tate_a,Y6: produc6972303929186420058tate_a] :
            ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X4 @ Y6 ) @ R )
           => ~ ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y6 @ X4 ) @ R ) ) ) ) ).

% asym_iff
thf(fact_559_asymD,axiom,
    ! [R: set_Pr1753757909640466247tate_a,X3: state_a,Y: state_a] :
      ( ( asym_on_state_a @ top_top_set_state_a @ R )
     => ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X3 @ Y ) @ R )
       => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y @ X3 ) @ R ) ) ) ).

% asymD
thf(fact_560_asymD,axiom,
    ! [R: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( asym_on_a @ top_top_set_a @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R )
       => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X3 ) @ R ) ) ) ).

% asymD
thf(fact_561_asymD,axiom,
    ! [R: set_Pr7963165934123475591tate_a,X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a] :
      ( ( asym_o3617238991104122252tate_a @ top_to8456965716270553738tate_a @ R )
     => ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X3 @ Y ) @ R )
       => ~ ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y @ X3 ) @ R ) ) ) ).

% asymD
thf(fact_562_csquare__fstOp__sndOp,axiom,
    ! [F2: ( a > a > $o ) > product_prod_a_a > $o,P2: a > state_a > $o,Q2: state_a > a > $o] : ( bNF_cs992974942529369663te_a_a @ ( collec3336397797384452498od_a_a @ ( F2 @ ( relcompp_a_state_a_a @ P2 @ Q2 ) ) ) @ produc7615498795807706488tate_a @ produc8645808095479954704te_a_a @ ( bNF_fs255890684155992281te_a_a @ P2 @ Q2 ) @ ( bNF_sn1870881145288996887te_a_a @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_563_csquare__fstOp__sndOp,axiom,
    ! [F2: ( state_a > state_a > $o ) > produc4774439622479004775tate_a > $o,P2: state_a > a > $o,Q2: a > state_a > $o] : ( bNF_cs4436417150143650034tate_a @ ( collec5776123031894519250tate_a @ ( F2 @ ( relcom7673254116290642291tate_a @ P2 @ Q2 ) ) ) @ produc6301641481763271506te_a_a @ produc736293372669613878tate_a @ ( bNF_fs7997537600116375986tate_a @ P2 @ Q2 ) @ ( bNF_sn6678599861129357808tate_a @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_564_asym__inv__image,axiom,
    ! [R2: set_Pr7963165934123475591tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( asym_o3617238991104122252tate_a @ top_to8456965716270553738tate_a @ R2 )
     => ( asym_o3617238991104122252tate_a @ top_to8456965716270553738tate_a @ ( inv_im4402535653060257874tate_a @ R2 @ F2 ) ) ) ).

% asym_inv_image
thf(fact_565_image__Fpow__mono,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( ord_le5819720210775101748te_a_a @ ( image_71777787125004895te_a_a @ F2 @ A5 ) @ B5 )
     => ( ord_le2176902880835264532te_a_a @ ( image_5360107183579672927te_a_a @ ( image_71777787125004895te_a_a @ F2 ) @ ( finite8761379122080342720tate_a @ A5 ) ) @ ( finite6915337786416788506te_a_a @ B5 ) ) ) ).

% image_Fpow_mono
thf(fact_566_image__Fpow__mono,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( image_4497581360776675627tate_a @ F2 @ A5 ) @ B5 )
     => ( ord_le7926008383649835194tate_a @ ( image_1561333526596188715tate_a @ ( image_4497581360776675627tate_a @ F2 ) @ ( finite6915337786416788506te_a_a @ A5 ) ) @ ( finite8761379122080342720tate_a @ B5 ) ) ) ).

% image_Fpow_mono
thf(fact_567_image__Fpow__mono,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( image_1917819122788559109tate_a @ F2 @ A5 ) @ B5 )
     => ( ord_le7926008383649835194tate_a @ ( image_6885891455112064261tate_a @ ( image_1917819122788559109tate_a @ F2 ) @ ( finite8761379122080342720tate_a @ A5 ) ) @ ( finite8761379122080342720tate_a @ B5 ) ) ) ).

% image_Fpow_mono
thf(fact_568_asym__lex__prod,axiom,
    ! [R_A: set_Product_prod_a_a,R_B: set_Pr1753757909640466247tate_a] :
      ( ( asym_on_a @ top_top_set_a @ R_A )
     => ( ( asym_on_state_a @ top_top_set_state_a @ R_B )
       => ( asym_o3617238991104122252tate_a @ top_to8456965716270553738tate_a @ ( lex_prod_a_state_a @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_569_asym__lex__prod,axiom,
    ! [R_A: set_Pr7963165934123475591tate_a,R_B: set_Pr7963165934123475591tate_a] :
      ( ( asym_o3617238991104122252tate_a @ top_to8456965716270553738tate_a @ R_A )
     => ( ( asym_o3617238991104122252tate_a @ top_to8456965716270553738tate_a @ R_B )
       => ( asym_o7314948133871076569tate_a @ top_to93540037818290135tate_a @ ( lex_pr5802797673463856815tate_a @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_570_Sup_OSUP__cong,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,C3: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,D: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,Sup: set_Pr2749968112124914068te_a_a > produc5126262593522865844te_a_a] :
      ( ( A5 = B5 )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X @ B5 )
           => ( ( C3 @ X )
              = ( D @ X ) ) )
       => ( ( Sup @ ( image_71777787125004895te_a_a @ C3 @ A5 ) )
          = ( Sup @ ( image_71777787125004895te_a_a @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_571_Sup_OSUP__cong,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,C3: produc6972303929186420058tate_a > produc6972303929186420058tate_a,D: produc6972303929186420058tate_a > produc6972303929186420058tate_a,Sup: set_Pr4275752383657305402tate_a > produc6972303929186420058tate_a] :
      ( ( A5 = B5 )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X @ B5 )
           => ( ( C3 @ X )
              = ( D @ X ) ) )
       => ( ( Sup @ ( image_1917819122788559109tate_a @ C3 @ A5 ) )
          = ( Sup @ ( image_1917819122788559109tate_a @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_572_Sup_OSUP__cong,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a,C3: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,D: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,Sup: set_Pr4275752383657305402tate_a > produc6972303929186420058tate_a] :
      ( ( A5 = B5 )
     => ( ! [X: produc5126262593522865844te_a_a] :
            ( ( member1329951143264900189te_a_a @ X @ B5 )
           => ( ( C3 @ X )
              = ( D @ X ) ) )
       => ( ( Sup @ ( image_4497581360776675627tate_a @ C3 @ A5 ) )
          = ( Sup @ ( image_4497581360776675627tate_a @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_573_Inf_OINF__cong,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,C3: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,D: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,Inf: set_Pr2749968112124914068te_a_a > produc5126262593522865844te_a_a] :
      ( ( A5 = B5 )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X @ B5 )
           => ( ( C3 @ X )
              = ( D @ X ) ) )
       => ( ( Inf @ ( image_71777787125004895te_a_a @ C3 @ A5 ) )
          = ( Inf @ ( image_71777787125004895te_a_a @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_574_Inf_OINF__cong,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,C3: produc6972303929186420058tate_a > produc6972303929186420058tate_a,D: produc6972303929186420058tate_a > produc6972303929186420058tate_a,Inf: set_Pr4275752383657305402tate_a > produc6972303929186420058tate_a] :
      ( ( A5 = B5 )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X @ B5 )
           => ( ( C3 @ X )
              = ( D @ X ) ) )
       => ( ( Inf @ ( image_1917819122788559109tate_a @ C3 @ A5 ) )
          = ( Inf @ ( image_1917819122788559109tate_a @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_575_Inf_OINF__cong,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a,C3: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,D: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,Inf: set_Pr4275752383657305402tate_a > produc6972303929186420058tate_a] :
      ( ( A5 = B5 )
     => ( ! [X: produc5126262593522865844te_a_a] :
            ( ( member1329951143264900189te_a_a @ X @ B5 )
           => ( ( C3 @ X )
              = ( D @ X ) ) )
       => ( ( Inf @ ( image_4497581360776675627tate_a @ C3 @ A5 ) )
          = ( Inf @ ( image_4497581360776675627tate_a @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_576_in__inv__image,axiom,
    ! [X3: state_a,Y: state_a,R: set_Pr1753757909640466247tate_a,F2: state_a > state_a] :
      ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X3 @ Y ) @ ( inv_im4751046366479140498tate_a @ R @ F2 ) )
      = ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_577_in__inv__image,axiom,
    ! [X3: state_a,Y: state_a,R: set_Pr7963165934123475591tate_a,F2: state_a > produc6972303929186420058tate_a] :
      ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X3 @ Y ) @ ( inv_im6450695643621578409tate_a @ R @ F2 ) )
      = ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_578_in__inv__image,axiom,
    ! [X3: state_a,Y: state_a,R: set_Product_prod_a_a,F2: state_a > a] :
      ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X3 @ Y ) @ ( inv_image_a_state_a @ R @ F2 ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_579_in__inv__image,axiom,
    ! [X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a,R: set_Pr1753757909640466247tate_a,F2: produc6972303929186420058tate_a > state_a] :
      ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X3 @ Y ) @ ( inv_im4546156720724264763tate_a @ R @ F2 ) )
      = ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_580_in__inv__image,axiom,
    ! [X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a,R: set_Pr7963165934123475591tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X3 @ Y ) @ ( inv_im4402535653060257874tate_a @ R @ F2 ) )
      = ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_581_in__inv__image,axiom,
    ! [X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a,R: set_Product_prod_a_a,F2: produc6972303929186420058tate_a > a] :
      ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X3 @ Y ) @ ( inv_im4128934058405003822tate_a @ R @ F2 ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_582_in__inv__image,axiom,
    ! [X3: a,Y: a,R: set_Pr1753757909640466247tate_a,F2: a > state_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( inv_image_state_a_a @ R @ F2 ) )
      = ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_583_in__inv__image,axiom,
    ! [X3: a,Y: a,R: set_Pr7963165934123475591tate_a,F2: a > produc6972303929186420058tate_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( inv_im760813232939325942te_a_a @ R @ F2 ) )
      = ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_584_in__inv__image,axiom,
    ! [X3: a,Y: a,R: set_Product_prod_a_a,F2: a > a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( inv_image_a_a @ R @ F2 ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_585_in__lex__prod,axiom,
    ! [A: state_a,B: produc6972303929186420058tate_a,A2: state_a,B2: produc6972303929186420058tate_a,R: set_Pr1753757909640466247tate_a,S: set_Pr7963165934123475591tate_a] :
      ( ( member761508830101964240tate_a @ ( produc5385980883598410391tate_a @ ( produc3709553252826914432tate_a @ A @ B ) @ ( produc3709553252826914432tate_a @ A2 @ B2 ) ) @ ( lex_pr8341538044225208984tate_a @ R @ S ) )
      = ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_586_in__lex__prod,axiom,
    ! [A: produc6972303929186420058tate_a,B: state_a,A2: produc6972303929186420058tate_a,B2: state_a,R: set_Pr7963165934123475591tate_a,S: set_Pr1753757909640466247tate_a] :
      ( ( member6670390056965382224tate_a @ ( produc6525781772916077591tate_a @ ( produc5614092175724228078tate_a @ A @ B ) @ ( produc5614092175724228078tate_a @ A2 @ B2 ) ) @ ( lex_pr1022704930267746822tate_a @ R @ S ) )
      = ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_587_in__lex__prod,axiom,
    ! [A: produc6972303929186420058tate_a,B: a,A2: produc6972303929186420058tate_a,B2: a,R: set_Pr7963165934123475591tate_a,S: set_Product_prod_a_a] :
      ( ( member7027635665102103568te_a_a @ ( produc98301680644887127te_a_a @ ( produc8779472934055335867te_a_a @ A @ B ) @ ( produc8779472934055335867te_a_a @ A2 @ B2 ) ) @ ( lex_pr3849822815040038867te_a_a @ R @ S ) )
      = ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_588_in__lex__prod,axiom,
    ! [A: a,B: produc6972303929186420058tate_a,A2: a,B2: produc6972303929186420058tate_a,R: set_Product_prod_a_a,S: set_Pr7963165934123475591tate_a] :
      ( ( member8315558564207931920tate_a @ ( produc8717265729681587287tate_a @ ( produc2924221722666237939tate_a @ A @ B ) @ ( produc2924221722666237939tate_a @ A2 @ B2 ) ) @ ( lex_pr7217943640505716747tate_a @ R @ S ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_589_in__lex__prod,axiom,
    ! [A: state_a,B: a,A2: state_a,B2: a,R: set_Pr1753757909640466247tate_a,S: set_Product_prod_a_a] :
      ( ( member2649312330005299536te_a_a @ ( produc2877451964877216791te_a_a @ ( produc7328099264922328356te_a_a @ A @ B ) @ ( produc7328099264922328356te_a_a @ A2 @ B2 ) ) @ ( lex_prod_state_a_a @ R @ S ) )
      = ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_590_in__lex__prod,axiom,
    ! [A: state_a,B: state_a,A2: state_a,B2: state_a,R: set_Pr1753757909640466247tate_a,S: set_Pr1753757909640466247tate_a] :
      ( ( member7923945571256933136tate_a @ ( produc1588899729123023703tate_a @ ( produc6131900399494507223tate_a @ A @ B ) @ ( produc6131900399494507223tate_a @ A2 @ B2 ) ) @ ( lex_pr7692759003852220143tate_a @ R @ S ) )
      = ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_591_in__lex__prod,axiom,
    ! [A: produc6972303929186420058tate_a,B: produc6972303929186420058tate_a,A2: produc6972303929186420058tate_a,B2: produc6972303929186420058tate_a,R: set_Pr7963165934123475591tate_a,S: set_Pr7963165934123475591tate_a] :
      ( ( member1643692453717642000tate_a @ ( produc7335381768849070935tate_a @ ( produc2143731062552654487tate_a @ A @ B ) @ ( produc2143731062552654487tate_a @ A2 @ B2 ) ) @ ( lex_pr5802797673463856815tate_a @ R @ S ) )
      = ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_592_in__lex__prod,axiom,
    ! [A: a,B: a,A2: a,B2: a,R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Pair_a_a @ A2 @ B2 ) ) @ ( lex_prod_a_a @ R @ S ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_593_in__lex__prod,axiom,
    ! [A: a,B: state_a,A2: a,B2: state_a,R: set_Product_prod_a_a,S: set_Pr1753757909640466247tate_a] :
      ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ ( produc8641956578966763338tate_a @ A @ B ) @ ( produc8641956578966763338tate_a @ A2 @ B2 ) ) @ ( lex_prod_a_state_a @ R @ S ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_594_Fpow__mono,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
     => ( ord_le7926008383649835194tate_a @ ( finite8761379122080342720tate_a @ A5 ) @ ( finite8761379122080342720tate_a @ B5 ) ) ) ).

% Fpow_mono
thf(fact_595_sndOp__in,axiom,
    ! [Ac2: produc5126262593522865844te_a_a,P2: state_a > state_a > $o,Q2: state_a > a > $o] :
      ( ( member1329951143264900189te_a_a @ Ac2 @ ( collec6503900420941984159te_a_a @ ( produc490204266526094367_a_a_o @ ( relcom6359396802246207309te_a_a @ P2 @ Q2 ) ) ) )
     => ( member1329951143264900189te_a_a @ ( bNF_sn5364742547084922826te_a_a @ P2 @ Q2 @ Ac2 ) @ ( collec6503900420941984159te_a_a @ ( produc490204266526094367_a_a_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_596_sndOp__in,axiom,
    ! [Ac2: produc5126262593522865844te_a_a,P2: state_a > a > $o,Q2: a > a > $o] :
      ( ( member1329951143264900189te_a_a @ Ac2 @ ( collec6503900420941984159te_a_a @ ( produc490204266526094367_a_a_o @ ( relcompp_state_a_a_a @ P2 @ Q2 ) ) ) )
     => ( member1426531477525435216od_a_a @ ( bNF_sn4628896343363730621_a_a_a @ P2 @ Q2 @ Ac2 ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_597_sndOp__in,axiom,
    ! [Ac2: produc4774439622479004775tate_a,P2: state_a > state_a > $o,Q2: state_a > state_a > $o] :
      ( ( member3801464090328701328tate_a @ Ac2 @ ( collec5776123031894519250tate_a @ ( produc6586738294279371180te_a_o @ ( relcom6754413068187290752tate_a @ P2 @ Q2 ) ) ) )
     => ( member3801464090328701328tate_a @ ( bNF_sn1791734581026771581tate_a @ P2 @ Q2 @ Ac2 ) @ ( collec5776123031894519250tate_a @ ( produc6586738294279371180te_a_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_598_sndOp__in,axiom,
    ! [Ac2: produc5940350820068290983tate_a,P2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o,Q2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o] :
      ( ( member1595334363549574352tate_a @ Ac2 @ ( collec4717139122855073042tate_a @ ( produc3822078179198691820te_a_o @ ( relcom4255716151527258473tate_a @ P2 @ Q2 ) ) ) )
     => ( member1595334363549574352tate_a @ ( bNF_sn1595019482075934822tate_a @ P2 @ Q2 @ Ac2 ) @ ( collec4717139122855073042tate_a @ ( produc3822078179198691820te_a_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_599_sndOp__in,axiom,
    ! [Ac2: product_prod_a_a,P2: a > state_a > $o,Q2: state_a > a > $o] :
      ( ( member1426531477525435216od_a_a @ Ac2 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( relcompp_a_state_a_a @ P2 @ Q2 ) ) ) )
     => ( member1329951143264900189te_a_a @ ( bNF_sn1870881145288996887te_a_a @ P2 @ Q2 @ Ac2 ) @ ( collec6503900420941984159te_a_a @ ( produc490204266526094367_a_a_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_600_sndOp__in,axiom,
    ! [Ac2: product_prod_a_a,P2: a > a > $o,Q2: a > a > $o] :
      ( ( member1426531477525435216od_a_a @ Ac2 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( relcompp_a_a_a @ P2 @ Q2 ) ) ) )
     => ( member1426531477525435216od_a_a @ ( bNF_sndOp_a_a_a @ P2 @ Q2 @ Ac2 ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_601_sndOp__in,axiom,
    ! [Ac2: produc4774439622479004775tate_a,P2: state_a > a > $o,Q2: a > state_a > $o] :
      ( ( member3801464090328701328tate_a @ Ac2 @ ( collec5776123031894519250tate_a @ ( produc6586738294279371180te_a_o @ ( relcom7673254116290642291tate_a @ P2 @ Q2 ) ) ) )
     => ( member3175992478928454403tate_a @ ( bNF_sn6678599861129357808tate_a @ P2 @ Q2 @ Ac2 ) @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_602_sndOp__in,axiom,
    ! [Ac2: produc6972303929186420058tate_a,P2: a > state_a > $o,Q2: state_a > state_a > $o] :
      ( ( member3175992478928454403tate_a @ Ac2 @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ ( relcom1031610871010084813tate_a @ P2 @ Q2 ) ) ) )
     => ( member3801464090328701328tate_a @ ( bNF_sn36956615848800330tate_a @ P2 @ Q2 @ Ac2 ) @ ( collec5776123031894519250tate_a @ ( produc6586738294279371180te_a_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_603_sndOp__in,axiom,
    ! [Ac2: produc6972303929186420058tate_a,P2: a > a > $o,Q2: a > state_a > $o] :
      ( ( member3175992478928454403tate_a @ Ac2 @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ ( relcompp_a_a_state_a @ P2 @ Q2 ) ) ) )
     => ( member3175992478928454403tate_a @ ( bNF_sn3184738459333431869tate_a @ P2 @ Q2 @ Ac2 ) @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_604_fstOp__in,axiom,
    ! [Ac2: produc5126262593522865844te_a_a,P2: state_a > a > $o,Q2: a > a > $o] :
      ( ( member1329951143264900189te_a_a @ Ac2 @ ( collec6503900420941984159te_a_a @ ( produc490204266526094367_a_a_o @ ( relcompp_state_a_a_a @ P2 @ Q2 ) ) ) )
     => ( member1329951143264900189te_a_a @ ( bNF_fs3013905882230726015_a_a_a @ P2 @ Q2 @ Ac2 ) @ ( collec6503900420941984159te_a_a @ ( produc490204266526094367_a_a_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_605_fstOp__in,axiom,
    ! [Ac2: produc5126262593522865844te_a_a,P2: state_a > state_a > $o,Q2: state_a > a > $o] :
      ( ( member1329951143264900189te_a_a @ Ac2 @ ( collec6503900420941984159te_a_a @ ( produc490204266526094367_a_a_o @ ( relcom6359396802246207309te_a_a @ P2 @ Q2 ) ) ) )
     => ( member3801464090328701328tate_a @ ( bNF_fs6683680286071941004te_a_a @ P2 @ Q2 @ Ac2 ) @ ( collec5776123031894519250tate_a @ ( produc6586738294279371180te_a_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_606_fstOp__in,axiom,
    ! [Ac2: produc4774439622479004775tate_a,P2: state_a > a > $o,Q2: a > state_a > $o] :
      ( ( member3801464090328701328tate_a @ Ac2 @ ( collec5776123031894519250tate_a @ ( produc6586738294279371180te_a_o @ ( relcom7673254116290642291tate_a @ P2 @ Q2 ) ) ) )
     => ( member1329951143264900189te_a_a @ ( bNF_fs7997537600116375986tate_a @ P2 @ Q2 @ Ac2 ) @ ( collec6503900420941984159te_a_a @ ( produc490204266526094367_a_a_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_607_fstOp__in,axiom,
    ! [Ac2: produc4774439622479004775tate_a,P2: state_a > state_a > $o,Q2: state_a > state_a > $o] :
      ( ( member3801464090328701328tate_a @ Ac2 @ ( collec5776123031894519250tate_a @ ( produc6586738294279371180te_a_o @ ( relcom6754413068187290752tate_a @ P2 @ Q2 ) ) ) )
     => ( member3801464090328701328tate_a @ ( bNF_fs8172768232217780543tate_a @ P2 @ Q2 @ Ac2 ) @ ( collec5776123031894519250tate_a @ ( produc6586738294279371180te_a_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_608_fstOp__in,axiom,
    ! [Ac2: produc5940350820068290983tate_a,P2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o,Q2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o] :
      ( ( member1595334363549574352tate_a @ Ac2 @ ( collec4717139122855073042tate_a @ ( produc3822078179198691820te_a_o @ ( relcom4255716151527258473tate_a @ P2 @ Q2 ) ) ) )
     => ( member1595334363549574352tate_a @ ( bNF_fs1927333335074883880tate_a @ P2 @ Q2 @ Ac2 ) @ ( collec4717139122855073042tate_a @ ( produc3822078179198691820te_a_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_609_fstOp__in,axiom,
    ! [Ac2: product_prod_a_a,P2: a > a > $o,Q2: a > a > $o] :
      ( ( member1426531477525435216od_a_a @ Ac2 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( relcompp_a_a_a @ P2 @ Q2 ) ) ) )
     => ( member1426531477525435216od_a_a @ ( bNF_fstOp_a_a_a @ P2 @ Q2 @ Ac2 ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_610_fstOp__in,axiom,
    ! [Ac2: product_prod_a_a,P2: a > state_a > $o,Q2: state_a > a > $o] :
      ( ( member1426531477525435216od_a_a @ Ac2 @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ ( relcompp_a_state_a_a @ P2 @ Q2 ) ) ) )
     => ( member3175992478928454403tate_a @ ( bNF_fs255890684155992281te_a_a @ P2 @ Q2 @ Ac2 ) @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_611_fstOp__in,axiom,
    ! [Ac2: produc6972303929186420058tate_a,P2: a > a > $o,Q2: a > state_a > $o] :
      ( ( member3175992478928454403tate_a @ Ac2 @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ ( relcompp_a_a_state_a @ P2 @ Q2 ) ) ) )
     => ( member1426531477525435216od_a_a @ ( bNF_fs1569747998200427263tate_a @ P2 @ Q2 @ Ac2 ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_612_fstOp__in,axiom,
    ! [Ac2: produc6972303929186420058tate_a,P2: a > state_a > $o,Q2: state_a > state_a > $o] :
      ( ( member3175992478928454403tate_a @ Ac2 @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ ( relcom1031610871010084813tate_a @ P2 @ Q2 ) ) ) )
     => ( member3175992478928454403tate_a @ ( bNF_fs1355894354835818508tate_a @ P2 @ Q2 @ Ac2 ) @ ( collec8349941756605538373tate_a @ ( produc3717778009801768057te_a_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_613_image__Pow__mono,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( ord_le5819720210775101748te_a_a @ ( image_71777787125004895te_a_a @ F2 @ A5 ) @ B5 )
     => ( ord_le2176902880835264532te_a_a @ ( image_5360107183579672927te_a_a @ ( image_71777787125004895te_a_a @ F2 ) @ ( pow_Pr2998936717439131767tate_a @ A5 ) ) @ ( pow_Pr1152895381775577553te_a_a @ B5 ) ) ) ).

% image_Pow_mono
thf(fact_614_image__Pow__mono,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( image_4497581360776675627tate_a @ F2 @ A5 ) @ B5 )
     => ( ord_le7926008383649835194tate_a @ ( image_1561333526596188715tate_a @ ( image_4497581360776675627tate_a @ F2 ) @ ( pow_Pr1152895381775577553te_a_a @ A5 ) ) @ ( pow_Pr2998936717439131767tate_a @ B5 ) ) ) ).

% image_Pow_mono
thf(fact_615_image__Pow__mono,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( image_1917819122788559109tate_a @ F2 @ A5 ) @ B5 )
     => ( ord_le7926008383649835194tate_a @ ( image_6885891455112064261tate_a @ ( image_1917819122788559109tate_a @ F2 ) @ ( pow_Pr2998936717439131767tate_a @ A5 ) ) @ ( pow_Pr2998936717439131767tate_a @ B5 ) ) ) ).

% image_Pow_mono
thf(fact_616_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_617_le__rel__bool__arg__iff,axiom,
    ( ord_le1503166539455550823tate_a
    = ( ^ [X7: $o > set_Pr4275752383657305402tate_a,Y9: $o > set_Pr4275752383657305402tate_a] :
          ( ( ord_le7345504482307493082tate_a @ ( X7 @ $false ) @ ( Y9 @ $false ) )
          & ( ord_le7345504482307493082tate_a @ ( X7 @ $true ) @ ( Y9 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_618_transp__relcompp__less__eq,axiom,
    ! [R: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o] :
      ( ( transp2263347561435144970tate_a @ top_to8456965716270553738tate_a @ R )
     => ( ord_le1957189445062774030te_a_o @ ( relcom4255716151527258473tate_a @ R @ R ) @ R ) ) ).

% transp_relcompp_less_eq
thf(fact_619_transp__relcompp,axiom,
    ! [R: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o] :
      ( ( transp2263347561435144970tate_a @ top_to8456965716270553738tate_a @ R )
      = ( ord_le1957189445062774030te_a_o @ ( relcom4255716151527258473tate_a @ R @ R ) @ R ) ) ).

% transp_relcompp
thf(fact_620_surj__Compl__image__subset,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to6931181444738162404te_a_a )
     => ( ord_le5819720210775101748te_a_a @ ( uminus2465391768719355147te_a_a @ ( image_71777787125004895te_a_a @ F2 @ A5 ) ) @ ( image_71777787125004895te_a_a @ F2 @ ( uminus3991176040251746481tate_a @ A5 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_621_surj__Compl__image__subset,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a )
        = top_to8456965716270553738tate_a )
     => ( ord_le7345504482307493082tate_a @ ( uminus3991176040251746481tate_a @ ( image_4497581360776675627tate_a @ F2 @ A5 ) ) @ ( image_4497581360776675627tate_a @ F2 @ ( uminus2465391768719355147te_a_a @ A5 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_622_surj__Compl__image__subset,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( ( image_1917819122788559109tate_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to8456965716270553738tate_a )
     => ( ord_le7345504482307493082tate_a @ ( uminus3991176040251746481tate_a @ ( image_1917819122788559109tate_a @ F2 @ A5 ) ) @ ( image_1917819122788559109tate_a @ F2 @ ( uminus3991176040251746481tate_a @ A5 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_623_ComplI,axiom,
    ! [C2: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ~ ( member3175992478928454403tate_a @ C2 @ A5 )
     => ( member3175992478928454403tate_a @ C2 @ ( uminus3991176040251746481tate_a @ A5 ) ) ) ).

% ComplI
thf(fact_624_ComplI,axiom,
    ! [C2: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ~ ( member1329951143264900189te_a_a @ C2 @ A5 )
     => ( member1329951143264900189te_a_a @ C2 @ ( uminus2465391768719355147te_a_a @ A5 ) ) ) ).

% ComplI
thf(fact_625_ComplI,axiom,
    ! [C2: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ~ ( member3801464090328701328tate_a @ C2 @ A5 )
     => ( member3801464090328701328tate_a @ C2 @ ( uminus4943716833861577534tate_a @ A5 ) ) ) ).

% ComplI
thf(fact_626_ComplI,axiom,
    ! [C2: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a] :
      ( ~ ( member1595334363549574352tate_a @ C2 @ A5 )
     => ( member1595334363549574352tate_a @ C2 @ ( uminus6615162123411005566tate_a @ A5 ) ) ) ).

% ComplI
thf(fact_627_ComplI,axiom,
    ! [C2: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ C2 @ A5 )
     => ( member1426531477525435216od_a_a @ C2 @ ( uminus5530930396987473918od_a_a @ A5 ) ) ) ).

% ComplI
thf(fact_628_Compl__iff,axiom,
    ! [C2: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ C2 @ ( uminus3991176040251746481tate_a @ A5 ) )
      = ( ~ ( member3175992478928454403tate_a @ C2 @ A5 ) ) ) ).

% Compl_iff
thf(fact_629_Compl__iff,axiom,
    ! [C2: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ C2 @ ( uminus2465391768719355147te_a_a @ A5 ) )
      = ( ~ ( member1329951143264900189te_a_a @ C2 @ A5 ) ) ) ).

% Compl_iff
thf(fact_630_Compl__iff,axiom,
    ! [C2: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ C2 @ ( uminus4943716833861577534tate_a @ A5 ) )
      = ( ~ ( member3801464090328701328tate_a @ C2 @ A5 ) ) ) ).

% Compl_iff
thf(fact_631_Compl__iff,axiom,
    ! [C2: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ C2 @ ( uminus6615162123411005566tate_a @ A5 ) )
      = ( ~ ( member1595334363549574352tate_a @ C2 @ A5 ) ) ) ).

% Compl_iff
thf(fact_632_Compl__iff,axiom,
    ! [C2: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C2 @ ( uminus5530930396987473918od_a_a @ A5 ) )
      = ( ~ ( member1426531477525435216od_a_a @ C2 @ A5 ) ) ) ).

% Compl_iff
thf(fact_633_Compl__anti__mono,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
     => ( ord_le7345504482307493082tate_a @ ( uminus3991176040251746481tate_a @ B5 ) @ ( uminus3991176040251746481tate_a @ A5 ) ) ) ).

% Compl_anti_mono
thf(fact_634_Compl__subset__Compl__iff,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( uminus3991176040251746481tate_a @ A5 ) @ ( uminus3991176040251746481tate_a @ B5 ) )
      = ( ord_le7345504482307493082tate_a @ B5 @ A5 ) ) ).

% Compl_subset_Compl_iff
thf(fact_635_Pow__UNIV,axiom,
    ( ( pow_Pr2998936717439131767tate_a @ top_to8456965716270553738tate_a )
    = top_to4898740227593795178tate_a ) ).

% Pow_UNIV
thf(fact_636_PowI,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
     => ( member4824975370526228195tate_a @ A5 @ ( pow_Pr2998936717439131767tate_a @ B5 ) ) ) ).

% PowI
thf(fact_637_Pow__iff,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( member4824975370526228195tate_a @ A5 @ ( pow_Pr2998936717439131767tate_a @ B5 ) )
      = ( ord_le7345504482307493082tate_a @ A5 @ B5 ) ) ).

% Pow_iff
thf(fact_638_transp__onD,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,R2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o,X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a,Z2: produc6972303929186420058tate_a] :
      ( ( transp2263347561435144970tate_a @ A5 @ R2 )
     => ( ( member3175992478928454403tate_a @ X3 @ A5 )
       => ( ( member3175992478928454403tate_a @ Y @ A5 )
         => ( ( member3175992478928454403tate_a @ Z2 @ A5 )
           => ( ( R2 @ X3 @ Y )
             => ( ( R2 @ Y @ Z2 )
               => ( R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).

% transp_onD
thf(fact_639_transp__onD,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,R2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a > $o,X3: produc5126262593522865844te_a_a,Y: produc5126262593522865844te_a_a,Z2: produc5126262593522865844te_a_a] :
      ( ( transp417306225771590756te_a_a @ A5 @ R2 )
     => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
       => ( ( member1329951143264900189te_a_a @ Y @ A5 )
         => ( ( member1329951143264900189te_a_a @ Z2 @ A5 )
           => ( ( R2 @ X3 @ Y )
             => ( ( R2 @ Y @ Z2 )
               => ( R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).

% transp_onD
thf(fact_640_transp__onD,axiom,
    ! [A5: set_Pr1753757909640466247tate_a,R2: produc4774439622479004775tate_a > produc4774439622479004775tate_a > $o,X3: produc4774439622479004775tate_a,Y: produc4774439622479004775tate_a,Z2: produc4774439622479004775tate_a] :
      ( ( transp895483406694337047tate_a @ A5 @ R2 )
     => ( ( member3801464090328701328tate_a @ X3 @ A5 )
       => ( ( member3801464090328701328tate_a @ Y @ A5 )
         => ( ( member3801464090328701328tate_a @ Z2 @ A5 )
           => ( ( R2 @ X3 @ Y )
             => ( ( R2 @ Y @ Z2 )
               => ( R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).

% transp_onD
thf(fact_641_transp__onD,axiom,
    ! [A5: set_Pr7963165934123475591tate_a,R2: produc5940350820068290983tate_a > produc5940350820068290983tate_a > $o,X3: produc5940350820068290983tate_a,Y: produc5940350820068290983tate_a,Z2: produc5940350820068290983tate_a] :
      ( ( transp2232037258560083799tate_a @ A5 @ R2 )
     => ( ( member1595334363549574352tate_a @ X3 @ A5 )
       => ( ( member1595334363549574352tate_a @ Y @ A5 )
         => ( ( member1595334363549574352tate_a @ Z2 @ A5 )
           => ( ( R2 @ X3 @ Y )
             => ( ( R2 @ Y @ Z2 )
               => ( R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).

% transp_onD
thf(fact_642_transp__onD,axiom,
    ! [A5: set_Product_prod_a_a,R2: product_prod_a_a > product_prod_a_a > $o,X3: product_prod_a_a,Y: product_prod_a_a,Z2: product_prod_a_a] :
      ( ( transp3378240297350259415od_a_a @ A5 @ R2 )
     => ( ( member1426531477525435216od_a_a @ X3 @ A5 )
       => ( ( member1426531477525435216od_a_a @ Y @ A5 )
         => ( ( member1426531477525435216od_a_a @ Z2 @ A5 )
           => ( ( R2 @ X3 @ Y )
             => ( ( R2 @ Y @ Z2 )
               => ( R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).

% transp_onD
thf(fact_643_transp__onI,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,R2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o] :
      ( ! [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a,Z3: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ A5 )
         => ( ( member3175992478928454403tate_a @ Y3 @ A5 )
           => ( ( member3175992478928454403tate_a @ Z3 @ A5 )
             => ( ( R2 @ X @ Y3 )
               => ( ( R2 @ Y3 @ Z3 )
                 => ( R2 @ X @ Z3 ) ) ) ) ) )
     => ( transp2263347561435144970tate_a @ A5 @ R2 ) ) ).

% transp_onI
thf(fact_644_transp__onI,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,R2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a > $o] :
      ( ! [X: produc5126262593522865844te_a_a,Y3: produc5126262593522865844te_a_a,Z3: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ A5 )
         => ( ( member1329951143264900189te_a_a @ Y3 @ A5 )
           => ( ( member1329951143264900189te_a_a @ Z3 @ A5 )
             => ( ( R2 @ X @ Y3 )
               => ( ( R2 @ Y3 @ Z3 )
                 => ( R2 @ X @ Z3 ) ) ) ) ) )
     => ( transp417306225771590756te_a_a @ A5 @ R2 ) ) ).

% transp_onI
thf(fact_645_transp__onI,axiom,
    ! [A5: set_Pr1753757909640466247tate_a,R2: produc4774439622479004775tate_a > produc4774439622479004775tate_a > $o] :
      ( ! [X: produc4774439622479004775tate_a,Y3: produc4774439622479004775tate_a,Z3: produc4774439622479004775tate_a] :
          ( ( member3801464090328701328tate_a @ X @ A5 )
         => ( ( member3801464090328701328tate_a @ Y3 @ A5 )
           => ( ( member3801464090328701328tate_a @ Z3 @ A5 )
             => ( ( R2 @ X @ Y3 )
               => ( ( R2 @ Y3 @ Z3 )
                 => ( R2 @ X @ Z3 ) ) ) ) ) )
     => ( transp895483406694337047tate_a @ A5 @ R2 ) ) ).

% transp_onI
thf(fact_646_transp__onI,axiom,
    ! [A5: set_Pr7963165934123475591tate_a,R2: produc5940350820068290983tate_a > produc5940350820068290983tate_a > $o] :
      ( ! [X: produc5940350820068290983tate_a,Y3: produc5940350820068290983tate_a,Z3: produc5940350820068290983tate_a] :
          ( ( member1595334363549574352tate_a @ X @ A5 )
         => ( ( member1595334363549574352tate_a @ Y3 @ A5 )
           => ( ( member1595334363549574352tate_a @ Z3 @ A5 )
             => ( ( R2 @ X @ Y3 )
               => ( ( R2 @ Y3 @ Z3 )
                 => ( R2 @ X @ Z3 ) ) ) ) ) )
     => ( transp2232037258560083799tate_a @ A5 @ R2 ) ) ).

% transp_onI
thf(fact_647_transp__onI,axiom,
    ! [A5: set_Product_prod_a_a,R2: product_prod_a_a > product_prod_a_a > $o] :
      ( ! [X: product_prod_a_a,Y3: product_prod_a_a,Z3: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ A5 )
         => ( ( member1426531477525435216od_a_a @ Y3 @ A5 )
           => ( ( member1426531477525435216od_a_a @ Z3 @ A5 )
             => ( ( R2 @ X @ Y3 )
               => ( ( R2 @ Y3 @ Z3 )
                 => ( R2 @ X @ Z3 ) ) ) ) ) )
     => ( transp3378240297350259415od_a_a @ A5 @ R2 ) ) ).

% transp_onI
thf(fact_648_ComplD,axiom,
    ! [C2: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ C2 @ ( uminus3991176040251746481tate_a @ A5 ) )
     => ~ ( member3175992478928454403tate_a @ C2 @ A5 ) ) ).

% ComplD
thf(fact_649_ComplD,axiom,
    ! [C2: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ C2 @ ( uminus2465391768719355147te_a_a @ A5 ) )
     => ~ ( member1329951143264900189te_a_a @ C2 @ A5 ) ) ).

% ComplD
thf(fact_650_ComplD,axiom,
    ! [C2: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ C2 @ ( uminus4943716833861577534tate_a @ A5 ) )
     => ~ ( member3801464090328701328tate_a @ C2 @ A5 ) ) ).

% ComplD
thf(fact_651_ComplD,axiom,
    ! [C2: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ C2 @ ( uminus6615162123411005566tate_a @ A5 ) )
     => ~ ( member1595334363549574352tate_a @ C2 @ A5 ) ) ).

% ComplD
thf(fact_652_ComplD,axiom,
    ! [C2: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C2 @ ( uminus5530930396987473918od_a_a @ A5 ) )
     => ~ ( member1426531477525435216od_a_a @ C2 @ A5 ) ) ).

% ComplD
thf(fact_653_transp__on__le,axiom,
    ! [A5: set_se5193884918812770586tate_a] : ( transp6220366192284844778tate_a @ A5 @ ord_le7345504482307493082tate_a ) ).

% transp_on_le
thf(fact_654_transp__equality,axiom,
    ( transp2263347561435144970tate_a @ top_to8456965716270553738tate_a
    @ ^ [Y5: produc6972303929186420058tate_a,Z: produc6972303929186420058tate_a] : ( Y5 = Z ) ) ).

% transp_equality
thf(fact_655_transp__def,axiom,
    ! [R2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o] :
      ( ( transp2263347561435144970tate_a @ top_to8456965716270553738tate_a @ R2 )
      = ( ! [X4: produc6972303929186420058tate_a,Y6: produc6972303929186420058tate_a,Z4: produc6972303929186420058tate_a] :
            ( ( R2 @ X4 @ Y6 )
           => ( ( R2 @ Y6 @ Z4 )
             => ( R2 @ X4 @ Z4 ) ) ) ) ) ).

% transp_def
thf(fact_656_transpI,axiom,
    ! [R2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o] :
      ( ! [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a,Z3: produc6972303929186420058tate_a] :
          ( ( R2 @ X @ Y3 )
         => ( ( R2 @ Y3 @ Z3 )
           => ( R2 @ X @ Z3 ) ) )
     => ( transp2263347561435144970tate_a @ top_to8456965716270553738tate_a @ R2 ) ) ).

% transpI
thf(fact_657_transpE,axiom,
    ! [R: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o,X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a,Z2: produc6972303929186420058tate_a] :
      ( ( transp2263347561435144970tate_a @ top_to8456965716270553738tate_a @ R )
     => ( ( R @ X3 @ Y )
       => ( ( R @ Y @ Z2 )
         => ( R @ X3 @ Z2 ) ) ) ) ).

% transpE
thf(fact_658_transpD,axiom,
    ! [R2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o,X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a,Z2: produc6972303929186420058tate_a] :
      ( ( transp2263347561435144970tate_a @ top_to8456965716270553738tate_a @ R2 )
     => ( ( R2 @ X3 @ Y )
       => ( ( R2 @ Y @ Z2 )
         => ( R2 @ X3 @ Z2 ) ) ) ) ).

% transpD
thf(fact_659_transp__on__subset,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,R2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o,B5: set_Pr4275752383657305402tate_a] :
      ( ( transp2263347561435144970tate_a @ A5 @ R2 )
     => ( ( ord_le7345504482307493082tate_a @ B5 @ A5 )
       => ( transp2263347561435144970tate_a @ B5 @ R2 ) ) ) ).

% transp_on_subset
thf(fact_660_PowD,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( member4824975370526228195tate_a @ A5 @ ( pow_Pr2998936717439131767tate_a @ B5 ) )
     => ( ord_le7345504482307493082tate_a @ A5 @ B5 ) ) ).

% PowD
thf(fact_661_Pow__mono,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
     => ( ord_le7926008383649835194tate_a @ ( pow_Pr2998936717439131767tate_a @ A5 ) @ ( pow_Pr2998936717439131767tate_a @ B5 ) ) ) ).

% Pow_mono
thf(fact_662_image__Pow__surj,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ( image_4497581360776675627tate_a @ F2 @ A5 )
        = B5 )
     => ( ( image_1561333526596188715tate_a @ ( image_4497581360776675627tate_a @ F2 ) @ ( pow_Pr1152895381775577553te_a_a @ A5 ) )
        = ( pow_Pr2998936717439131767tate_a @ B5 ) ) ) ).

% image_Pow_surj
thf(fact_663_image__Pow__surj,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( ( image_71777787125004895te_a_a @ F2 @ A5 )
        = B5 )
     => ( ( image_5360107183579672927te_a_a @ ( image_71777787125004895te_a_a @ F2 ) @ ( pow_Pr2998936717439131767tate_a @ A5 ) )
        = ( pow_Pr1152895381775577553te_a_a @ B5 ) ) ) ).

% image_Pow_surj
thf(fact_664_image__Pow__surj,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ( image_1917819122788559109tate_a @ F2 @ A5 )
        = B5 )
     => ( ( image_6885891455112064261tate_a @ ( image_1917819122788559109tate_a @ F2 ) @ ( pow_Pr2998936717439131767tate_a @ A5 ) )
        = ( pow_Pr2998936717439131767tate_a @ B5 ) ) ) ).

% image_Pow_surj
thf(fact_665_compl__le__compl__iff,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( uminus3991176040251746481tate_a @ X3 ) @ ( uminus3991176040251746481tate_a @ Y ) )
      = ( ord_le7345504482307493082tate_a @ Y @ X3 ) ) ).

% compl_le_compl_iff
thf(fact_666_compl__le__swap2,axiom,
    ! [Y: set_Pr4275752383657305402tate_a,X3: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( uminus3991176040251746481tate_a @ Y ) @ X3 )
     => ( ord_le7345504482307493082tate_a @ ( uminus3991176040251746481tate_a @ X3 ) @ Y ) ) ).

% compl_le_swap2
thf(fact_667_compl__le__swap1,axiom,
    ! [Y: set_Pr4275752383657305402tate_a,X3: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ Y @ ( uminus3991176040251746481tate_a @ X3 ) )
     => ( ord_le7345504482307493082tate_a @ X3 @ ( uminus3991176040251746481tate_a @ Y ) ) ) ).

% compl_le_swap1
thf(fact_668_compl__mono,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ X3 @ Y )
     => ( ord_le7345504482307493082tate_a @ ( uminus3991176040251746481tate_a @ Y ) @ ( uminus3991176040251746481tate_a @ X3 ) ) ) ).

% compl_mono
thf(fact_669_inj__image__Compl__subset,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ord_le5819720210775101748te_a_a @ ( image_71777787125004895te_a_a @ F2 @ ( uminus3991176040251746481tate_a @ A5 ) ) @ ( uminus2465391768719355147te_a_a @ ( image_71777787125004895te_a_a @ F2 @ A5 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_670_inj__image__Compl__subset,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
     => ( ord_le7345504482307493082tate_a @ ( image_5502445033238569825tate_a @ F2 @ ( uminus_uminus_set_a @ A5 ) ) @ ( uminus3991176040251746481tate_a @ ( image_5502445033238569825tate_a @ F2 @ A5 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_671_inj__image__Compl__subset,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
     => ( ord_le7345504482307493082tate_a @ ( image_4497581360776675627tate_a @ F2 @ ( uminus2465391768719355147te_a_a @ A5 ) ) @ ( uminus3991176040251746481tate_a @ ( image_4497581360776675627tate_a @ F2 @ A5 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_672_inj__image__Compl__subset,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ord_le7345504482307493082tate_a @ ( image_1917819122788559109tate_a @ F2 @ ( uminus3991176040251746481tate_a @ A5 ) ) @ ( uminus3991176040251746481tate_a @ ( image_1917819122788559109tate_a @ F2 @ A5 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_673_sup__shunt,axiom,
    ! [X3: a > state_a > $o,Y: a > state_a > $o] :
      ( ( ( sup_sup_a_state_a_o @ X3 @ Y )
        = top_top_a_state_a_o )
      = ( ord_le5391092903712282779te_a_o @ ( uminus9152238405698698866te_a_o @ X3 ) @ Y ) ) ).

% sup_shunt
thf(fact_674_sup__shunt,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a] :
      ( ( ( sup_su1662690461080552718tate_a @ X3 @ Y )
        = top_to8456965716270553738tate_a )
      = ( ord_le7345504482307493082tate_a @ ( uminus3991176040251746481tate_a @ X3 ) @ Y ) ) ).

% sup_shunt
thf(fact_675_antisymp__less__eq,axiom,
    ! [R: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o,S: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o] :
      ( ( ord_le1957189445062774030te_a_o @ R @ S )
     => ( ( antisy2379887893289268487tate_a @ top_to8456965716270553738tate_a @ S )
       => ( antisy2379887893289268487tate_a @ top_to8456965716270553738tate_a @ R ) ) ) ).

% antisymp_less_eq
thf(fact_676_bij__image__Compl__eq,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( bij_be957954900305576860tate_a @ F2 @ top_to6931181444738162404te_a_a @ top_to8456965716270553738tate_a )
     => ( ( image_4497581360776675627tate_a @ F2 @ ( uminus2465391768719355147te_a_a @ A5 ) )
        = ( uminus3991176040251746481tate_a @ ( image_4497581360776675627tate_a @ F2 @ A5 ) ) ) ) ).

% bij_image_Compl_eq
thf(fact_677_bij__image__Compl__eq,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( bij_be5755523363508681936te_a_a @ F2 @ top_to8456965716270553738tate_a @ top_to6931181444738162404te_a_a )
     => ( ( image_71777787125004895te_a_a @ F2 @ ( uminus3991176040251746481tate_a @ A5 ) )
        = ( uminus2465391768719355147te_a_a @ ( image_71777787125004895te_a_a @ F2 @ A5 ) ) ) ) ).

% bij_image_Compl_eq
thf(fact_678_bij__image__Compl__eq,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( bij_be7601564699172236150tate_a @ F2 @ top_to8456965716270553738tate_a @ top_to8456965716270553738tate_a )
     => ( ( image_1917819122788559109tate_a @ F2 @ ( uminus3991176040251746481tate_a @ A5 ) )
        = ( uminus3991176040251746481tate_a @ ( image_1917819122788559109tate_a @ F2 @ A5 ) ) ) ) ).

% bij_image_Compl_eq
thf(fact_679_inj__image__subset__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( ord_le5819720210775101748te_a_a @ ( image_71777787125004895te_a_a @ F2 @ A5 ) @ ( image_71777787125004895te_a_a @ F2 @ B5 ) )
        = ( ord_le7345504482307493082tate_a @ A5 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_680_inj__image__subset__iff,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,B5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
     => ( ( ord_le7345504482307493082tate_a @ ( image_5502445033238569825tate_a @ F2 @ A5 ) @ ( image_5502445033238569825tate_a @ F2 @ B5 ) )
        = ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_681_inj__image__subset__iff,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
     => ( ( ord_le7345504482307493082tate_a @ ( image_4497581360776675627tate_a @ F2 @ A5 ) @ ( image_4497581360776675627tate_a @ F2 @ B5 ) )
        = ( ord_le5819720210775101748te_a_a @ A5 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_682_inj__image__subset__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( ord_le7345504482307493082tate_a @ ( image_1917819122788559109tate_a @ F2 @ A5 ) @ ( image_1917819122788559109tate_a @ F2 @ B5 ) )
        = ( ord_le7345504482307493082tate_a @ A5 @ B5 ) ) ) ).

% inj_image_subset_iff
thf(fact_683_Un__subset__iff,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,C3: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) @ C3 )
      = ( ( ord_le7345504482307493082tate_a @ A5 @ C3 )
        & ( ord_le7345504482307493082tate_a @ B5 @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_684_relcompp__distrib,axiom,
    ! [R2: a > a > $o,S4: a > state_a > $o,T3: a > state_a > $o] :
      ( ( relcompp_a_a_state_a @ R2 @ ( sup_sup_a_state_a_o @ S4 @ T3 ) )
      = ( sup_sup_a_state_a_o @ ( relcompp_a_a_state_a @ R2 @ S4 ) @ ( relcompp_a_a_state_a @ R2 @ T3 ) ) ) ).

% relcompp_distrib
thf(fact_685_relcompp__distrib2,axiom,
    ! [S4: a > state_a > $o,T3: a > state_a > $o,R2: state_a > state_a > $o] :
      ( ( relcom1031610871010084813tate_a @ ( sup_sup_a_state_a_o @ S4 @ T3 ) @ R2 )
      = ( sup_sup_a_state_a_o @ ( relcom1031610871010084813tate_a @ S4 @ R2 ) @ ( relcom1031610871010084813tate_a @ T3 @ R2 ) ) ) ).

% relcompp_distrib2
thf(fact_686_boolean__algebra_Odisj__one__left,axiom,
    ! [X3: a > state_a > $o] :
      ( ( sup_sup_a_state_a_o @ top_top_a_state_a_o @ X3 )
      = top_top_a_state_a_o ) ).

% boolean_algebra.disj_one_left
thf(fact_687_boolean__algebra_Odisj__one__left,axiom,
    ! [X3: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ top_to8456965716270553738tate_a @ X3 )
      = top_to8456965716270553738tate_a ) ).

% boolean_algebra.disj_one_left
thf(fact_688_boolean__algebra_Odisj__one__right,axiom,
    ! [X3: a > state_a > $o] :
      ( ( sup_sup_a_state_a_o @ X3 @ top_top_a_state_a_o )
      = top_top_a_state_a_o ) ).

% boolean_algebra.disj_one_right
thf(fact_689_boolean__algebra_Odisj__one__right,axiom,
    ! [X3: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ X3 @ top_to8456965716270553738tate_a )
      = top_to8456965716270553738tate_a ) ).

% boolean_algebra.disj_one_right
thf(fact_690_sup__compl__top__left1,axiom,
    ! [X3: a > state_a > $o,Y: a > state_a > $o] :
      ( ( sup_sup_a_state_a_o @ ( uminus9152238405698698866te_a_o @ X3 ) @ ( sup_sup_a_state_a_o @ X3 @ Y ) )
      = top_top_a_state_a_o ) ).

% sup_compl_top_left1
thf(fact_691_sup__compl__top__left1,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ ( uminus3991176040251746481tate_a @ X3 ) @ ( sup_su1662690461080552718tate_a @ X3 @ Y ) )
      = top_to8456965716270553738tate_a ) ).

% sup_compl_top_left1
thf(fact_692_sup__compl__top__left2,axiom,
    ! [X3: a > state_a > $o,Y: a > state_a > $o] :
      ( ( sup_sup_a_state_a_o @ X3 @ ( sup_sup_a_state_a_o @ ( uminus9152238405698698866te_a_o @ X3 ) @ Y ) )
      = top_top_a_state_a_o ) ).

% sup_compl_top_left2
thf(fact_693_sup__compl__top__left2,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ X3 @ ( sup_su1662690461080552718tate_a @ ( uminus3991176040251746481tate_a @ X3 ) @ Y ) )
      = top_to8456965716270553738tate_a ) ).

% sup_compl_top_left2
thf(fact_694_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X3: a > state_a > $o] :
      ( ( sup_sup_a_state_a_o @ ( uminus9152238405698698866te_a_o @ X3 ) @ X3 )
      = top_top_a_state_a_o ) ).

% boolean_algebra.disj_cancel_left
thf(fact_695_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X3: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ ( uminus3991176040251746481tate_a @ X3 ) @ X3 )
      = top_to8456965716270553738tate_a ) ).

% boolean_algebra.disj_cancel_left
thf(fact_696_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X3: a > state_a > $o] :
      ( ( sup_sup_a_state_a_o @ X3 @ ( uminus9152238405698698866te_a_o @ X3 ) )
      = top_top_a_state_a_o ) ).

% boolean_algebra.disj_cancel_right
thf(fact_697_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X3: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ X3 @ ( uminus3991176040251746481tate_a @ X3 ) )
      = top_to8456965716270553738tate_a ) ).

% boolean_algebra.disj_cancel_right
thf(fact_698_inj__apfst,axiom,
    ! [F2: a > produc6972303929186420058tate_a] :
      ( ( inj_on206654034347362069tate_a @ ( produc606892987180027289tate_a @ F2 ) @ top_to8456965716270553738tate_a )
      = ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a ) ) ).

% inj_apfst
thf(fact_699_inj__apfst,axiom,
    ! [F2: a > a] :
      ( ( inj_on4126798524766655601tate_a @ ( produc3988218040097000309tate_a @ F2 ) @ top_to8456965716270553738tate_a )
      = ( inj_on_a_a @ F2 @ top_top_set_a ) ) ).

% inj_apfst
thf(fact_700_inj__apsnd,axiom,
    ! [F2: state_a > state_a] :
      ( ( inj_on4126798524766655601tate_a @ ( produc4530580310919820352te_a_a @ F2 ) @ top_to8456965716270553738tate_a )
      = ( inj_on5418343104458039473tate_a @ F2 @ top_top_set_state_a ) ) ).

% inj_apsnd
thf(fact_701_bij__is__inj,axiom,
    ! [F2: a > produc6972303929186420058tate_a] :
      ( ( bij_be5628941944454835026tate_a @ F2 @ top_top_set_a @ top_to8456965716270553738tate_a )
     => ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a ) ) ).

% bij_is_inj
thf(fact_702_bij__is__inj,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( bij_be957954900305576860tate_a @ F2 @ top_to6931181444738162404te_a_a @ top_to8456965716270553738tate_a )
     => ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a ) ) ).

% bij_is_inj
thf(fact_703_bij__is__inj,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( bij_be7601564699172236150tate_a @ F2 @ top_to8456965716270553738tate_a @ top_to8456965716270553738tate_a )
     => ( inj_on4126798524766655601tate_a @ F2 @ top_to8456965716270553738tate_a ) ) ).

% bij_is_inj
thf(fact_704_bij__betw__def,axiom,
    ( bij_be5755523363508681936te_a_a
    = ( ^ [F: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A6: set_Pr4275752383657305402tate_a,B6: set_Pr2749968112124914068te_a_a] :
          ( ( inj_on2280757189103101387te_a_a @ F @ A6 )
          & ( ( image_71777787125004895te_a_a @ F @ A6 )
            = B6 ) ) ) ) ).

% bij_betw_def
thf(fact_705_bij__betw__def,axiom,
    ( bij_be7601564699172236150tate_a
    = ( ^ [F: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A6: set_Pr4275752383657305402tate_a,B6: set_Pr4275752383657305402tate_a] :
          ( ( inj_on4126798524766655601tate_a @ F @ A6 )
          & ( ( image_1917819122788559109tate_a @ F @ A6 )
            = B6 ) ) ) ) ).

% bij_betw_def
thf(fact_706_bij__betw__def,axiom,
    ( bij_be5628941944454835026tate_a
    = ( ^ [F: a > produc6972303929186420058tate_a,A6: set_a,B6: set_Pr4275752383657305402tate_a] :
          ( ( inj_on3780297825070694605tate_a @ F @ A6 )
          & ( ( image_5502445033238569825tate_a @ F @ A6 )
            = B6 ) ) ) ) ).

% bij_betw_def
thf(fact_707_bij__betw__def,axiom,
    ( bij_be957954900305576860tate_a
    = ( ^ [F: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A6: set_Pr2749968112124914068te_a_a,B6: set_Pr4275752383657305402tate_a] :
          ( ( inj_on6706560762754772119tate_a @ F @ A6 )
          & ( ( image_4497581360776675627tate_a @ F @ A6 )
            = B6 ) ) ) ) ).

% bij_betw_def
thf(fact_708_bij__betw__imageI,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ A5 )
     => ( ( ( image_71777787125004895te_a_a @ F2 @ A5 )
          = B5 )
       => ( bij_be5755523363508681936te_a_a @ F2 @ A5 @ B5 ) ) ) ).

% bij_betw_imageI
thf(fact_709_bij__betw__imageI,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ A5 )
     => ( ( ( image_1917819122788559109tate_a @ F2 @ A5 )
          = B5 )
       => ( bij_be7601564699172236150tate_a @ F2 @ A5 @ B5 ) ) ) ).

% bij_betw_imageI
thf(fact_710_bij__betw__imageI,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( ( ( image_5502445033238569825tate_a @ F2 @ A5 )
          = B5 )
       => ( bij_be5628941944454835026tate_a @ F2 @ A5 @ B5 ) ) ) ).

% bij_betw_imageI
thf(fact_711_bij__betw__imageI,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( ( ( image_4497581360776675627tate_a @ F2 @ A5 )
          = B5 )
       => ( bij_be957954900305576860tate_a @ F2 @ A5 @ B5 ) ) ) ).

% bij_betw_imageI
thf(fact_712_inj__on__imp__bij__betw,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ A5 )
     => ( bij_be5755523363508681936te_a_a @ F2 @ A5 @ ( image_71777787125004895te_a_a @ F2 @ A5 ) ) ) ).

% inj_on_imp_bij_betw
thf(fact_713_inj__on__imp__bij__betw,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ A5 )
     => ( bij_be7601564699172236150tate_a @ F2 @ A5 @ ( image_1917819122788559109tate_a @ F2 @ A5 ) ) ) ).

% inj_on_imp_bij_betw
thf(fact_714_inj__on__imp__bij__betw,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( bij_be5628941944454835026tate_a @ F2 @ A5 @ ( image_5502445033238569825tate_a @ F2 @ A5 ) ) ) ).

% inj_on_imp_bij_betw
thf(fact_715_inj__on__imp__bij__betw,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( bij_be957954900305576860tate_a @ F2 @ A5 @ ( image_4497581360776675627tate_a @ F2 @ A5 ) ) ) ).

% inj_on_imp_bij_betw
thf(fact_716_inj__on__Un__image__eq__iff,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,B5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ ( sup_sup_set_a @ A5 @ B5 ) )
     => ( ( ( image_5502445033238569825tate_a @ F2 @ A5 )
          = ( image_5502445033238569825tate_a @ F2 @ B5 ) )
        = ( A5 = B5 ) ) ) ).

% inj_on_Un_image_eq_iff
thf(fact_717_inj__on__Un__image__eq__iff,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ ( sup_su136906189548161384te_a_a @ A5 @ B5 ) )
     => ( ( ( image_4497581360776675627tate_a @ F2 @ A5 )
          = ( image_4497581360776675627tate_a @ F2 @ B5 ) )
        = ( A5 = B5 ) ) ) ).

% inj_on_Un_image_eq_iff
thf(fact_718_inj__on__Un__image__eq__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) )
     => ( ( ( image_71777787125004895te_a_a @ F2 @ A5 )
          = ( image_71777787125004895te_a_a @ F2 @ B5 ) )
        = ( A5 = B5 ) ) ) ).

% inj_on_Un_image_eq_iff
thf(fact_719_inj__on__Un__image__eq__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) )
     => ( ( ( image_1917819122788559109tate_a @ F2 @ A5 )
          = ( image_1917819122788559109tate_a @ F2 @ B5 ) )
        = ( A5 = B5 ) ) ) ).

% inj_on_Un_image_eq_iff
thf(fact_720_inj__onD,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,X3: a,Y: a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y ) )
       => ( ( member_a @ X3 @ A5 )
         => ( ( member_a @ Y @ A5 )
           => ( X3 = Y ) ) ) ) ) ).

% inj_onD
thf(fact_721_inj__onD,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,X3: produc5126262593522865844te_a_a,Y: produc5126262593522865844te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y ) )
       => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
         => ( ( member1329951143264900189te_a_a @ Y @ A5 )
           => ( X3 = Y ) ) ) ) ) ).

% inj_onD
thf(fact_722_inj__onI,axiom,
    ! [A5: set_a,F2: a > produc6972303929186420058tate_a] :
      ( ! [X: a,Y3: a] :
          ( ( member_a @ X @ A5 )
         => ( ( member_a @ Y3 @ A5 )
           => ( ( ( F2 @ X )
                = ( F2 @ Y3 ) )
             => ( X = Y3 ) ) ) )
     => ( inj_on3780297825070694605tate_a @ F2 @ A5 ) ) ).

% inj_onI
thf(fact_723_inj__onI,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ! [X: produc5126262593522865844te_a_a,Y3: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ A5 )
         => ( ( member1329951143264900189te_a_a @ Y3 @ A5 )
           => ( ( ( F2 @ X )
                = ( F2 @ Y3 ) )
             => ( X = Y3 ) ) ) )
     => ( inj_on6706560762754772119tate_a @ F2 @ A5 ) ) ).

% inj_onI
thf(fact_724_inj__on__def,axiom,
    ( inj_on3780297825070694605tate_a
    = ( ^ [F: a > produc6972303929186420058tate_a,A6: set_a] :
        ! [X4: a] :
          ( ( member_a @ X4 @ A6 )
         => ! [Y6: a] :
              ( ( member_a @ Y6 @ A6 )
             => ( ( ( F @ X4 )
                  = ( F @ Y6 ) )
               => ( X4 = Y6 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_725_inj__on__def,axiom,
    ( inj_on6706560762754772119tate_a
    = ( ^ [F: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A6: set_Pr2749968112124914068te_a_a] :
        ! [X4: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X4 @ A6 )
         => ! [Y6: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ Y6 @ A6 )
             => ( ( ( F @ X4 )
                  = ( F @ Y6 ) )
               => ( X4 = Y6 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_726_inj__on__cong,axiom,
    ! [A5: set_a,F2: a > produc6972303929186420058tate_a,G: a > produc6972303929186420058tate_a] :
      ( ! [A3: a] :
          ( ( member_a @ A3 @ A5 )
         => ( ( F2 @ A3 )
            = ( G @ A3 ) ) )
     => ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
        = ( inj_on3780297825070694605tate_a @ G @ A5 ) ) ) ).

% inj_on_cong
thf(fact_727_inj__on__cong,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,G: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ! [A3: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ A3 @ A5 )
         => ( ( F2 @ A3 )
            = ( G @ A3 ) ) )
     => ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
        = ( inj_on6706560762754772119tate_a @ G @ A5 ) ) ) ).

% inj_on_cong
thf(fact_728_inj__on__eq__iff,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,X3: a,Y: a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( ( member_a @ X3 @ A5 )
       => ( ( member_a @ Y @ A5 )
         => ( ( ( F2 @ X3 )
              = ( F2 @ Y ) )
            = ( X3 = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_729_inj__on__eq__iff,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,X3: produc5126262593522865844te_a_a,Y: produc5126262593522865844te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
       => ( ( member1329951143264900189te_a_a @ Y @ A5 )
         => ( ( ( F2 @ X3 )
              = ( F2 @ Y ) )
            = ( X3 = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_730_bij__betw__apply,axiom,
    ! [F2: product_prod_a_a > product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,A: product_prod_a_a] :
      ( ( bij_be2600099163980227382od_a_a @ F2 @ A5 @ B5 )
     => ( ( member1426531477525435216od_a_a @ A @ A5 )
       => ( member1426531477525435216od_a_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_731_bij__betw__apply,axiom,
    ! [F2: produc6972303929186420058tate_a > product_prod_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Product_prod_a_a,A: produc6972303929186420058tate_a] :
      ( ( bij_be7000454981477323075od_a_a @ F2 @ A5 @ B5 )
     => ( ( member3175992478928454403tate_a @ A @ A5 )
       => ( member1426531477525435216od_a_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_732_bij__betw__apply,axiom,
    ! [F2: produc5126262593522865844te_a_a > product_prod_a_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Product_prod_a_a,A: produc5126262593522865844te_a_a] :
      ( ( bij_be8660005350342145129od_a_a @ F2 @ A5 @ B5 )
     => ( ( member1329951143264900189te_a_a @ A @ A5 )
       => ( member1426531477525435216od_a_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_733_bij__betw__apply,axiom,
    ! [F2: product_prod_a_a > produc6972303929186420058tate_a,A5: set_Product_prod_a_a,B5: set_Pr4275752383657305402tate_a,A: product_prod_a_a] :
      ( ( bij_be383723733817190377tate_a @ F2 @ A5 @ B5 )
     => ( ( member1426531477525435216od_a_a @ A @ A5 )
       => ( member3175992478928454403tate_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_734_bij__betw__apply,axiom,
    ! [F2: product_prod_a_a > produc5126262593522865844te_a_a,A5: set_Product_prod_a_a,B5: set_Pr2749968112124914068te_a_a,A: product_prod_a_a] :
      ( ( bij_be7761054435008411971te_a_a @ F2 @ A5 @ B5 )
     => ( ( member1426531477525435216od_a_a @ A @ A5 )
       => ( member1329951143264900189te_a_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_735_bij__betw__apply,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,A: produc6972303929186420058tate_a] :
      ( ( bij_be7601564699172236150tate_a @ F2 @ A5 @ B5 )
     => ( ( member3175992478928454403tate_a @ A @ A5 )
       => ( member3175992478928454403tate_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_736_bij__betw__apply,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr2749968112124914068te_a_a,A: produc6972303929186420058tate_a] :
      ( ( bij_be5755523363508681936te_a_a @ F2 @ A5 @ B5 )
     => ( ( member3175992478928454403tate_a @ A @ A5 )
       => ( member1329951143264900189te_a_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_737_bij__betw__apply,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr4275752383657305402tate_a,A: produc5126262593522865844te_a_a] :
      ( ( bij_be957954900305576860tate_a @ F2 @ A5 @ B5 )
     => ( ( member1329951143264900189te_a_a @ A @ A5 )
       => ( member3175992478928454403tate_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_738_bij__betw__apply,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a,A: produc5126262593522865844te_a_a] :
      ( ( bij_be8335285601496798454te_a_a @ F2 @ A5 @ B5 )
     => ( ( member1329951143264900189te_a_a @ A @ A5 )
       => ( member1329951143264900189te_a_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_739_bij__betw__apply,axiom,
    ! [F2: produc4774439622479004775tate_a > product_prod_a_a,A5: set_Pr1753757909640466247tate_a,B5: set_Product_prod_a_a,A: produc4774439622479004775tate_a] :
      ( ( bij_be7574686147965709558od_a_a @ F2 @ A5 @ B5 )
     => ( ( member3801464090328701328tate_a @ A @ A5 )
       => ( member1426531477525435216od_a_a @ ( F2 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_740_inj__on__contraD,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,X3: a,Y: a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( ( X3 != Y )
       => ( ( member_a @ X3 @ A5 )
         => ( ( member_a @ Y @ A5 )
           => ( ( F2 @ X3 )
             != ( F2 @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_741_inj__on__contraD,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,X3: produc5126262593522865844te_a_a,Y: produc5126262593522865844te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( ( X3 != Y )
       => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
         => ( ( member1329951143264900189te_a_a @ Y @ A5 )
           => ( ( F2 @ X3 )
             != ( F2 @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_742_inj__on__inverseI,axiom,
    ! [A5: set_a,G: produc6972303929186420058tate_a > a,F2: a > produc6972303929186420058tate_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A5 )
         => ( ( G @ ( F2 @ X ) )
            = X ) )
     => ( inj_on3780297825070694605tate_a @ F2 @ A5 ) ) ).

% inj_on_inverseI
thf(fact_743_inj__on__inverseI,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,G: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ! [X: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ A5 )
         => ( ( G @ ( F2 @ X ) )
            = X ) )
     => ( inj_on6706560762754772119tate_a @ F2 @ A5 ) ) ).

% inj_on_inverseI
thf(fact_744_antisymp__onD,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,R2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o,X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a] :
      ( ( antisy2379887893289268487tate_a @ A5 @ R2 )
     => ( ( member3175992478928454403tate_a @ X3 @ A5 )
       => ( ( member3175992478928454403tate_a @ Y @ A5 )
         => ( ( R2 @ X3 @ Y )
           => ( ( R2 @ Y @ X3 )
             => ( X3 = Y ) ) ) ) ) ) ).

% antisymp_onD
thf(fact_745_antisymp__onD,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,R2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a > $o,X3: produc5126262593522865844te_a_a,Y: produc5126262593522865844te_a_a] :
      ( ( antisy533846557625714273te_a_a @ A5 @ R2 )
     => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
       => ( ( member1329951143264900189te_a_a @ Y @ A5 )
         => ( ( R2 @ X3 @ Y )
           => ( ( R2 @ Y @ X3 )
             => ( X3 = Y ) ) ) ) ) ) ).

% antisymp_onD
thf(fact_746_antisymp__onD,axiom,
    ! [A5: set_Pr1753757909640466247tate_a,R2: produc4774439622479004775tate_a > produc4774439622479004775tate_a > $o,X3: produc4774439622479004775tate_a,Y: produc4774439622479004775tate_a] :
      ( ( antisy1085239330457909652tate_a @ A5 @ R2 )
     => ( ( member3801464090328701328tate_a @ X3 @ A5 )
       => ( ( member3801464090328701328tate_a @ Y @ A5 )
         => ( ( R2 @ X3 @ Y )
           => ( ( R2 @ Y @ X3 )
             => ( X3 = Y ) ) ) ) ) ) ).

% antisymp_onD
thf(fact_747_antisymp__onD,axiom,
    ! [A5: set_Pr7963165934123475591tate_a,R2: produc5940350820068290983tate_a > produc5940350820068290983tate_a > $o,X3: produc5940350820068290983tate_a,Y: produc5940350820068290983tate_a] :
      ( ( antisy2485538517233384660tate_a @ A5 @ R2 )
     => ( ( member1595334363549574352tate_a @ X3 @ A5 )
       => ( ( member1595334363549574352tate_a @ Y @ A5 )
         => ( ( R2 @ X3 @ Y )
           => ( ( R2 @ Y @ X3 )
             => ( X3 = Y ) ) ) ) ) ) ).

% antisymp_onD
thf(fact_748_antisymp__onD,axiom,
    ! [A5: set_Product_prod_a_a,R2: product_prod_a_a > product_prod_a_a > $o,X3: product_prod_a_a,Y: product_prod_a_a] :
      ( ( antisy8373617738294204244od_a_a @ A5 @ R2 )
     => ( ( member1426531477525435216od_a_a @ X3 @ A5 )
       => ( ( member1426531477525435216od_a_a @ Y @ A5 )
         => ( ( R2 @ X3 @ Y )
           => ( ( R2 @ Y @ X3 )
             => ( X3 = Y ) ) ) ) ) ) ).

% antisymp_onD
thf(fact_749_antisymp__onI,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,R2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o] :
      ( ! [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ A5 )
         => ( ( member3175992478928454403tate_a @ Y3 @ A5 )
           => ( ( R2 @ X @ Y3 )
             => ( ( R2 @ Y3 @ X )
               => ( X = Y3 ) ) ) ) )
     => ( antisy2379887893289268487tate_a @ A5 @ R2 ) ) ).

% antisymp_onI
thf(fact_750_antisymp__onI,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,R2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a > $o] :
      ( ! [X: produc5126262593522865844te_a_a,Y3: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ A5 )
         => ( ( member1329951143264900189te_a_a @ Y3 @ A5 )
           => ( ( R2 @ X @ Y3 )
             => ( ( R2 @ Y3 @ X )
               => ( X = Y3 ) ) ) ) )
     => ( antisy533846557625714273te_a_a @ A5 @ R2 ) ) ).

% antisymp_onI
thf(fact_751_antisymp__onI,axiom,
    ! [A5: set_Pr1753757909640466247tate_a,R2: produc4774439622479004775tate_a > produc4774439622479004775tate_a > $o] :
      ( ! [X: produc4774439622479004775tate_a,Y3: produc4774439622479004775tate_a] :
          ( ( member3801464090328701328tate_a @ X @ A5 )
         => ( ( member3801464090328701328tate_a @ Y3 @ A5 )
           => ( ( R2 @ X @ Y3 )
             => ( ( R2 @ Y3 @ X )
               => ( X = Y3 ) ) ) ) )
     => ( antisy1085239330457909652tate_a @ A5 @ R2 ) ) ).

% antisymp_onI
thf(fact_752_antisymp__onI,axiom,
    ! [A5: set_Pr7963165934123475591tate_a,R2: produc5940350820068290983tate_a > produc5940350820068290983tate_a > $o] :
      ( ! [X: produc5940350820068290983tate_a,Y3: produc5940350820068290983tate_a] :
          ( ( member1595334363549574352tate_a @ X @ A5 )
         => ( ( member1595334363549574352tate_a @ Y3 @ A5 )
           => ( ( R2 @ X @ Y3 )
             => ( ( R2 @ Y3 @ X )
               => ( X = Y3 ) ) ) ) )
     => ( antisy2485538517233384660tate_a @ A5 @ R2 ) ) ).

% antisymp_onI
thf(fact_753_antisymp__onI,axiom,
    ! [A5: set_Product_prod_a_a,R2: product_prod_a_a > product_prod_a_a > $o] :
      ( ! [X: product_prod_a_a,Y3: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ A5 )
         => ( ( member1426531477525435216od_a_a @ Y3 @ A5 )
           => ( ( R2 @ X @ Y3 )
             => ( ( R2 @ Y3 @ X )
               => ( X = Y3 ) ) ) ) )
     => ( antisy8373617738294204244od_a_a @ A5 @ R2 ) ) ).

% antisymp_onI
thf(fact_754_bij__betw__imp__inj__on,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( bij_be5628941944454835026tate_a @ F2 @ A5 @ B5 )
     => ( inj_on3780297825070694605tate_a @ F2 @ A5 ) ) ).

% bij_betw_imp_inj_on
thf(fact_755_bij__betw__imp__inj__on,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( bij_be957954900305576860tate_a @ F2 @ A5 @ B5 )
     => ( inj_on6706560762754772119tate_a @ F2 @ A5 ) ) ).

% bij_betw_imp_inj_on
thf(fact_756_bij__betw__iff__bijections,axiom,
    ( bij_be2600099163980227382od_a_a
    = ( ^ [F: product_prod_a_a > product_prod_a_a,A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
        ? [G2: product_prod_a_a > product_prod_a_a] :
          ( ! [X4: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X4 @ A6 )
             => ( ( member1426531477525435216od_a_a @ ( F @ X4 ) @ B6 )
                & ( ( G2 @ ( F @ X4 ) )
                  = X4 ) ) )
          & ! [X4: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X4 @ B6 )
             => ( ( member1426531477525435216od_a_a @ ( G2 @ X4 ) @ A6 )
                & ( ( F @ ( G2 @ X4 ) )
                  = X4 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_757_bij__betw__iff__bijections,axiom,
    ( bij_be383723733817190377tate_a
    = ( ^ [F: product_prod_a_a > produc6972303929186420058tate_a,A6: set_Product_prod_a_a,B6: set_Pr4275752383657305402tate_a] :
        ? [G2: produc6972303929186420058tate_a > product_prod_a_a] :
          ( ! [X4: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X4 @ A6 )
             => ( ( member3175992478928454403tate_a @ ( F @ X4 ) @ B6 )
                & ( ( G2 @ ( F @ X4 ) )
                  = X4 ) ) )
          & ! [X4: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ X4 @ B6 )
             => ( ( member1426531477525435216od_a_a @ ( G2 @ X4 ) @ A6 )
                & ( ( F @ ( G2 @ X4 ) )
                  = X4 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_758_bij__betw__iff__bijections,axiom,
    ( bij_be7761054435008411971te_a_a
    = ( ^ [F: product_prod_a_a > produc5126262593522865844te_a_a,A6: set_Product_prod_a_a,B6: set_Pr2749968112124914068te_a_a] :
        ? [G2: produc5126262593522865844te_a_a > product_prod_a_a] :
          ( ! [X4: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X4 @ A6 )
             => ( ( member1329951143264900189te_a_a @ ( F @ X4 ) @ B6 )
                & ( ( G2 @ ( F @ X4 ) )
                  = X4 ) ) )
          & ! [X4: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ X4 @ B6 )
             => ( ( member1426531477525435216od_a_a @ ( G2 @ X4 ) @ A6 )
                & ( ( F @ ( G2 @ X4 ) )
                  = X4 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_759_bij__betw__iff__bijections,axiom,
    ( bij_be7000454981477323075od_a_a
    = ( ^ [F: produc6972303929186420058tate_a > product_prod_a_a,A6: set_Pr4275752383657305402tate_a,B6: set_Product_prod_a_a] :
        ? [G2: product_prod_a_a > produc6972303929186420058tate_a] :
          ( ! [X4: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ X4 @ A6 )
             => ( ( member1426531477525435216od_a_a @ ( F @ X4 ) @ B6 )
                & ( ( G2 @ ( F @ X4 ) )
                  = X4 ) ) )
          & ! [X4: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X4 @ B6 )
             => ( ( member3175992478928454403tate_a @ ( G2 @ X4 ) @ A6 )
                & ( ( F @ ( G2 @ X4 ) )
                  = X4 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_760_bij__betw__iff__bijections,axiom,
    ( bij_be8660005350342145129od_a_a
    = ( ^ [F: produc5126262593522865844te_a_a > product_prod_a_a,A6: set_Pr2749968112124914068te_a_a,B6: set_Product_prod_a_a] :
        ? [G2: product_prod_a_a > produc5126262593522865844te_a_a] :
          ( ! [X4: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ X4 @ A6 )
             => ( ( member1426531477525435216od_a_a @ ( F @ X4 ) @ B6 )
                & ( ( G2 @ ( F @ X4 ) )
                  = X4 ) ) )
          & ! [X4: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X4 @ B6 )
             => ( ( member1329951143264900189te_a_a @ ( G2 @ X4 ) @ A6 )
                & ( ( F @ ( G2 @ X4 ) )
                  = X4 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_761_bij__betw__iff__bijections,axiom,
    ( bij_be7601564699172236150tate_a
    = ( ^ [F: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A6: set_Pr4275752383657305402tate_a,B6: set_Pr4275752383657305402tate_a] :
        ? [G2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
          ( ! [X4: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ X4 @ A6 )
             => ( ( member3175992478928454403tate_a @ ( F @ X4 ) @ B6 )
                & ( ( G2 @ ( F @ X4 ) )
                  = X4 ) ) )
          & ! [X4: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ X4 @ B6 )
             => ( ( member3175992478928454403tate_a @ ( G2 @ X4 ) @ A6 )
                & ( ( F @ ( G2 @ X4 ) )
                  = X4 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_762_bij__betw__iff__bijections,axiom,
    ( bij_be957954900305576860tate_a
    = ( ^ [F: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A6: set_Pr2749968112124914068te_a_a,B6: set_Pr4275752383657305402tate_a] :
        ? [G2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
          ( ! [X4: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ X4 @ A6 )
             => ( ( member3175992478928454403tate_a @ ( F @ X4 ) @ B6 )
                & ( ( G2 @ ( F @ X4 ) )
                  = X4 ) ) )
          & ! [X4: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ X4 @ B6 )
             => ( ( member1329951143264900189te_a_a @ ( G2 @ X4 ) @ A6 )
                & ( ( F @ ( G2 @ X4 ) )
                  = X4 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_763_bij__betw__iff__bijections,axiom,
    ( bij_be5755523363508681936te_a_a
    = ( ^ [F: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A6: set_Pr4275752383657305402tate_a,B6: set_Pr2749968112124914068te_a_a] :
        ? [G2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
          ( ! [X4: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ X4 @ A6 )
             => ( ( member1329951143264900189te_a_a @ ( F @ X4 ) @ B6 )
                & ( ( G2 @ ( F @ X4 ) )
                  = X4 ) ) )
          & ! [X4: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ X4 @ B6 )
             => ( ( member3175992478928454403tate_a @ ( G2 @ X4 ) @ A6 )
                & ( ( F @ ( G2 @ X4 ) )
                  = X4 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_764_bij__betw__iff__bijections,axiom,
    ( bij_be8335285601496798454te_a_a
    = ( ^ [F: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a,A6: set_Pr2749968112124914068te_a_a,B6: set_Pr2749968112124914068te_a_a] :
        ? [G2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a] :
          ( ! [X4: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ X4 @ A6 )
             => ( ( member1329951143264900189te_a_a @ ( F @ X4 ) @ B6 )
                & ( ( G2 @ ( F @ X4 ) )
                  = X4 ) ) )
          & ! [X4: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ X4 @ B6 )
             => ( ( member1329951143264900189te_a_a @ ( G2 @ X4 ) @ A6 )
                & ( ( F @ ( G2 @ X4 ) )
                  = X4 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_765_bij__betw__iff__bijections,axiom,
    ( bij_be7478293511942378358tate_a
    = ( ^ [F: product_prod_a_a > produc4774439622479004775tate_a,A6: set_Product_prod_a_a,B6: set_Pr1753757909640466247tate_a] :
        ? [G2: produc4774439622479004775tate_a > product_prod_a_a] :
          ( ! [X4: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X4 @ A6 )
             => ( ( member3801464090328701328tate_a @ ( F @ X4 ) @ B6 )
                & ( ( G2 @ ( F @ X4 ) )
                  = X4 ) ) )
          & ! [X4: produc4774439622479004775tate_a] :
              ( ( member3801464090328701328tate_a @ X4 @ B6 )
             => ( ( member1426531477525435216od_a_a @ ( G2 @ X4 ) @ A6 )
                & ( ( F @ ( G2 @ X4 ) )
                  = X4 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_766_bij__betwI_H,axiom,
    ! [X6: set_Product_prod_a_a,F2: product_prod_a_a > product_prod_a_a,Y7: set_Product_prod_a_a] :
      ( ! [X: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ X6 )
         => ! [Y3: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ Y3 @ X6 )
             => ( ( ( F2 @ X )
                  = ( F2 @ Y3 ) )
                = ( X = Y3 ) ) ) )
     => ( ! [X: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X @ X6 )
           => ( member1426531477525435216od_a_a @ ( F2 @ X ) @ Y7 ) )
       => ( ! [Y3: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ Y3 @ Y7 )
             => ? [X5: product_prod_a_a] :
                  ( ( member1426531477525435216od_a_a @ X5 @ X6 )
                  & ( Y3
                    = ( F2 @ X5 ) ) ) )
         => ( bij_be2600099163980227382od_a_a @ F2 @ X6 @ Y7 ) ) ) ) ).

% bij_betwI'
thf(fact_767_bij__betwI_H,axiom,
    ! [X6: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > product_prod_a_a,Y7: set_Product_prod_a_a] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ X6 )
         => ! [Y3: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ Y3 @ X6 )
             => ( ( ( F2 @ X )
                  = ( F2 @ Y3 ) )
                = ( X = Y3 ) ) ) )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X @ X6 )
           => ( member1426531477525435216od_a_a @ ( F2 @ X ) @ Y7 ) )
       => ( ! [Y3: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ Y3 @ Y7 )
             => ? [X5: produc6972303929186420058tate_a] :
                  ( ( member3175992478928454403tate_a @ X5 @ X6 )
                  & ( Y3
                    = ( F2 @ X5 ) ) ) )
         => ( bij_be7000454981477323075od_a_a @ F2 @ X6 @ Y7 ) ) ) ) ).

% bij_betwI'
thf(fact_768_bij__betwI_H,axiom,
    ! [X6: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > product_prod_a_a,Y7: set_Product_prod_a_a] :
      ( ! [X: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ X6 )
         => ! [Y3: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ Y3 @ X6 )
             => ( ( ( F2 @ X )
                  = ( F2 @ Y3 ) )
                = ( X = Y3 ) ) ) )
     => ( ! [X: produc5126262593522865844te_a_a] :
            ( ( member1329951143264900189te_a_a @ X @ X6 )
           => ( member1426531477525435216od_a_a @ ( F2 @ X ) @ Y7 ) )
       => ( ! [Y3: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ Y3 @ Y7 )
             => ? [X5: produc5126262593522865844te_a_a] :
                  ( ( member1329951143264900189te_a_a @ X5 @ X6 )
                  & ( Y3
                    = ( F2 @ X5 ) ) ) )
         => ( bij_be8660005350342145129od_a_a @ F2 @ X6 @ Y7 ) ) ) ) ).

% bij_betwI'
thf(fact_769_bij__betwI_H,axiom,
    ! [X6: set_Product_prod_a_a,F2: product_prod_a_a > produc6972303929186420058tate_a,Y7: set_Pr4275752383657305402tate_a] :
      ( ! [X: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ X6 )
         => ! [Y3: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ Y3 @ X6 )
             => ( ( ( F2 @ X )
                  = ( F2 @ Y3 ) )
                = ( X = Y3 ) ) ) )
     => ( ! [X: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X @ X6 )
           => ( member3175992478928454403tate_a @ ( F2 @ X ) @ Y7 ) )
       => ( ! [Y3: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ Y3 @ Y7 )
             => ? [X5: product_prod_a_a] :
                  ( ( member1426531477525435216od_a_a @ X5 @ X6 )
                  & ( Y3
                    = ( F2 @ X5 ) ) ) )
         => ( bij_be383723733817190377tate_a @ F2 @ X6 @ Y7 ) ) ) ) ).

% bij_betwI'
thf(fact_770_bij__betwI_H,axiom,
    ! [X6: set_Product_prod_a_a,F2: product_prod_a_a > produc5126262593522865844te_a_a,Y7: set_Pr2749968112124914068te_a_a] :
      ( ! [X: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ X6 )
         => ! [Y3: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ Y3 @ X6 )
             => ( ( ( F2 @ X )
                  = ( F2 @ Y3 ) )
                = ( X = Y3 ) ) ) )
     => ( ! [X: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X @ X6 )
           => ( member1329951143264900189te_a_a @ ( F2 @ X ) @ Y7 ) )
       => ( ! [Y3: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ Y3 @ Y7 )
             => ? [X5: product_prod_a_a] :
                  ( ( member1426531477525435216od_a_a @ X5 @ X6 )
                  & ( Y3
                    = ( F2 @ X5 ) ) ) )
         => ( bij_be7761054435008411971te_a_a @ F2 @ X6 @ Y7 ) ) ) ) ).

% bij_betwI'
thf(fact_771_bij__betwI_H,axiom,
    ! [X6: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,Y7: set_Pr4275752383657305402tate_a] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ X6 )
         => ! [Y3: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ Y3 @ X6 )
             => ( ( ( F2 @ X )
                  = ( F2 @ Y3 ) )
                = ( X = Y3 ) ) ) )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X @ X6 )
           => ( member3175992478928454403tate_a @ ( F2 @ X ) @ Y7 ) )
       => ( ! [Y3: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ Y3 @ Y7 )
             => ? [X5: produc6972303929186420058tate_a] :
                  ( ( member3175992478928454403tate_a @ X5 @ X6 )
                  & ( Y3
                    = ( F2 @ X5 ) ) ) )
         => ( bij_be7601564699172236150tate_a @ F2 @ X6 @ Y7 ) ) ) ) ).

% bij_betwI'
thf(fact_772_bij__betwI_H,axiom,
    ! [X6: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,Y7: set_Pr2749968112124914068te_a_a] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ X6 )
         => ! [Y3: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ Y3 @ X6 )
             => ( ( ( F2 @ X )
                  = ( F2 @ Y3 ) )
                = ( X = Y3 ) ) ) )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X @ X6 )
           => ( member1329951143264900189te_a_a @ ( F2 @ X ) @ Y7 ) )
       => ( ! [Y3: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ Y3 @ Y7 )
             => ? [X5: produc6972303929186420058tate_a] :
                  ( ( member3175992478928454403tate_a @ X5 @ X6 )
                  & ( Y3
                    = ( F2 @ X5 ) ) ) )
         => ( bij_be5755523363508681936te_a_a @ F2 @ X6 @ Y7 ) ) ) ) ).

% bij_betwI'
thf(fact_773_bij__betwI_H,axiom,
    ! [X6: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,Y7: set_Pr4275752383657305402tate_a] :
      ( ! [X: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ X6 )
         => ! [Y3: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ Y3 @ X6 )
             => ( ( ( F2 @ X )
                  = ( F2 @ Y3 ) )
                = ( X = Y3 ) ) ) )
     => ( ! [X: produc5126262593522865844te_a_a] :
            ( ( member1329951143264900189te_a_a @ X @ X6 )
           => ( member3175992478928454403tate_a @ ( F2 @ X ) @ Y7 ) )
       => ( ! [Y3: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ Y3 @ Y7 )
             => ? [X5: produc5126262593522865844te_a_a] :
                  ( ( member1329951143264900189te_a_a @ X5 @ X6 )
                  & ( Y3
                    = ( F2 @ X5 ) ) ) )
         => ( bij_be957954900305576860tate_a @ F2 @ X6 @ Y7 ) ) ) ) ).

% bij_betwI'
thf(fact_774_bij__betwI_H,axiom,
    ! [X6: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a,Y7: set_Pr2749968112124914068te_a_a] :
      ( ! [X: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ X6 )
         => ! [Y3: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ Y3 @ X6 )
             => ( ( ( F2 @ X )
                  = ( F2 @ Y3 ) )
                = ( X = Y3 ) ) ) )
     => ( ! [X: produc5126262593522865844te_a_a] :
            ( ( member1329951143264900189te_a_a @ X @ X6 )
           => ( member1329951143264900189te_a_a @ ( F2 @ X ) @ Y7 ) )
       => ( ! [Y3: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ Y3 @ Y7 )
             => ? [X5: produc5126262593522865844te_a_a] :
                  ( ( member1329951143264900189te_a_a @ X5 @ X6 )
                  & ( Y3
                    = ( F2 @ X5 ) ) ) )
         => ( bij_be8335285601496798454te_a_a @ F2 @ X6 @ Y7 ) ) ) ) ).

% bij_betwI'
thf(fact_775_bij__betwI_H,axiom,
    ! [X6: set_Pr1753757909640466247tate_a,F2: produc4774439622479004775tate_a > product_prod_a_a,Y7: set_Product_prod_a_a] :
      ( ! [X: produc4774439622479004775tate_a] :
          ( ( member3801464090328701328tate_a @ X @ X6 )
         => ! [Y3: produc4774439622479004775tate_a] :
              ( ( member3801464090328701328tate_a @ Y3 @ X6 )
             => ( ( ( F2 @ X )
                  = ( F2 @ Y3 ) )
                = ( X = Y3 ) ) ) )
     => ( ! [X: produc4774439622479004775tate_a] :
            ( ( member3801464090328701328tate_a @ X @ X6 )
           => ( member1426531477525435216od_a_a @ ( F2 @ X ) @ Y7 ) )
       => ( ! [Y3: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ Y3 @ Y7 )
             => ? [X5: produc4774439622479004775tate_a] :
                  ( ( member3801464090328701328tate_a @ X5 @ X6 )
                  & ( Y3
                    = ( F2 @ X5 ) ) ) )
         => ( bij_be7574686147965709558od_a_a @ F2 @ X6 @ Y7 ) ) ) ) ).

% bij_betwI'
thf(fact_776_bijI,axiom,
    ! [F2: a > produc6972303929186420058tate_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
     => ( ( ( image_5502445033238569825tate_a @ F2 @ top_top_set_a )
          = top_to8456965716270553738tate_a )
       => ( bij_be5628941944454835026tate_a @ F2 @ top_top_set_a @ top_to8456965716270553738tate_a ) ) ) ).

% bijI
thf(fact_777_bijI,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
     => ( ( ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a )
          = top_to8456965716270553738tate_a )
       => ( bij_be957954900305576860tate_a @ F2 @ top_to6931181444738162404te_a_a @ top_to8456965716270553738tate_a ) ) ) ).

% bijI
thf(fact_778_bijI,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a )
          = top_to6931181444738162404te_a_a )
       => ( bij_be5755523363508681936te_a_a @ F2 @ top_to8456965716270553738tate_a @ top_to6931181444738162404te_a_a ) ) ) ).

% bijI
thf(fact_779_bijI,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( ( image_1917819122788559109tate_a @ F2 @ top_to8456965716270553738tate_a )
          = top_to8456965716270553738tate_a )
       => ( bij_be7601564699172236150tate_a @ F2 @ top_to8456965716270553738tate_a @ top_to8456965716270553738tate_a ) ) ) ).

% bijI
thf(fact_780_bij__def,axiom,
    ! [F2: a > produc6972303929186420058tate_a] :
      ( ( bij_be5628941944454835026tate_a @ F2 @ top_top_set_a @ top_to8456965716270553738tate_a )
      = ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
        & ( ( image_5502445033238569825tate_a @ F2 @ top_top_set_a )
          = top_to8456965716270553738tate_a ) ) ) ).

% bij_def
thf(fact_781_bij__def,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( bij_be957954900305576860tate_a @ F2 @ top_to6931181444738162404te_a_a @ top_to8456965716270553738tate_a )
      = ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
        & ( ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a )
          = top_to8456965716270553738tate_a ) ) ) ).

% bij_def
thf(fact_782_bij__def,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ( bij_be5755523363508681936te_a_a @ F2 @ top_to8456965716270553738tate_a @ top_to6931181444738162404te_a_a )
      = ( ( inj_on2280757189103101387te_a_a @ F2 @ top_to8456965716270553738tate_a )
        & ( ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a )
          = top_to6931181444738162404te_a_a ) ) ) ).

% bij_def
thf(fact_783_bij__def,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( bij_be7601564699172236150tate_a @ F2 @ top_to8456965716270553738tate_a @ top_to8456965716270553738tate_a )
      = ( ( inj_on4126798524766655601tate_a @ F2 @ top_to8456965716270553738tate_a )
        & ( ( image_1917819122788559109tate_a @ F2 @ top_to8456965716270553738tate_a )
          = top_to8456965716270553738tate_a ) ) ) ).

% bij_def
thf(fact_784_Schroeder__Bernstein,axiom,
    ! [F2: produc6972303929186420058tate_a > a,A5: set_Pr4275752383657305402tate_a,B5: set_a,G: a > produc6972303929186420058tate_a] :
      ( ( inj_on412176999605016725te_a_a @ F2 @ A5 )
     => ( ( ord_less_eq_set_a @ ( image_2134324207772891945te_a_a @ F2 @ A5 ) @ B5 )
       => ( ( inj_on3780297825070694605tate_a @ G @ B5 )
         => ( ( ord_le7345504482307493082tate_a @ ( image_5502445033238569825tate_a @ G @ B5 ) @ A5 )
           => ? [H: produc6972303929186420058tate_a > a] : ( bij_be2260821118989157146te_a_a @ H @ A5 @ B5 ) ) ) ) ) ).

% Schroeder_Bernstein
thf(fact_785_Schroeder__Bernstein,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr2749968112124914068te_a_a,G: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ A5 )
     => ( ( ord_le5819720210775101748te_a_a @ ( image_71777787125004895te_a_a @ F2 @ A5 ) @ B5 )
       => ( ( inj_on6706560762754772119tate_a @ G @ B5 )
         => ( ( ord_le7345504482307493082tate_a @ ( image_4497581360776675627tate_a @ G @ B5 ) @ A5 )
           => ? [H: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] : ( bij_be5755523363508681936te_a_a @ H @ A5 @ B5 ) ) ) ) ) ).

% Schroeder_Bernstein
thf(fact_786_Schroeder__Bernstein,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,B5: set_Pr4275752383657305402tate_a,G: produc6972303929186420058tate_a > a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( ( ord_le7345504482307493082tate_a @ ( image_5502445033238569825tate_a @ F2 @ A5 ) @ B5 )
       => ( ( inj_on412176999605016725te_a_a @ G @ B5 )
         => ( ( ord_less_eq_set_a @ ( image_2134324207772891945te_a_a @ G @ B5 ) @ A5 )
           => ? [H: a > produc6972303929186420058tate_a] : ( bij_be5628941944454835026tate_a @ H @ A5 @ B5 ) ) ) ) ) ).

% Schroeder_Bernstein
thf(fact_787_Schroeder__Bernstein,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr4275752383657305402tate_a,G: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( ( ord_le7345504482307493082tate_a @ ( image_4497581360776675627tate_a @ F2 @ A5 ) @ B5 )
       => ( ( inj_on2280757189103101387te_a_a @ G @ B5 )
         => ( ( ord_le5819720210775101748te_a_a @ ( image_71777787125004895te_a_a @ G @ B5 ) @ A5 )
           => ? [H: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] : ( bij_be957954900305576860tate_a @ H @ A5 @ B5 ) ) ) ) ) ).

% Schroeder_Bernstein
thf(fact_788_Schroeder__Bernstein,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,G: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ A5 )
     => ( ( ord_le7345504482307493082tate_a @ ( image_1917819122788559109tate_a @ F2 @ A5 ) @ B5 )
       => ( ( inj_on4126798524766655601tate_a @ G @ B5 )
         => ( ( ord_le7345504482307493082tate_a @ ( image_1917819122788559109tate_a @ G @ B5 ) @ A5 )
           => ? [H: produc6972303929186420058tate_a > produc6972303929186420058tate_a] : ( bij_be7601564699172236150tate_a @ H @ A5 @ B5 ) ) ) ) ) ).

% Schroeder_Bernstein
thf(fact_789_bij__betw__Pow,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( bij_be957954900305576860tate_a @ F2 @ A5 @ B5 )
     => ( bij_be5362636009243931804tate_a @ ( image_4497581360776675627tate_a @ F2 ) @ ( pow_Pr1152895381775577553te_a_a @ A5 ) @ ( pow_Pr2998936717439131767tate_a @ B5 ) ) ) ).

% bij_betw_Pow
thf(fact_790_bij__betw__Pow,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( bij_be5755523363508681936te_a_a @ F2 @ A5 @ B5 )
     => ( bij_be9161409666227416016te_a_a @ ( image_71777787125004895te_a_a @ F2 ) @ ( pow_Pr2998936717439131767tate_a @ A5 ) @ ( pow_Pr1152895381775577553te_a_a @ B5 ) ) ) ).

% bij_betw_Pow
thf(fact_791_bij__betw__Pow,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( bij_be7601564699172236150tate_a @ F2 @ A5 @ B5 )
     => ( bij_be1463821900905031542tate_a @ ( image_1917819122788559109tate_a @ F2 ) @ ( pow_Pr2998936717439131767tate_a @ A5 ) @ ( pow_Pr2998936717439131767tate_a @ B5 ) ) ) ).

% bij_betw_Pow
thf(fact_792_inj__on__image__Pow,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ A5 )
     => ( inj_on779827642740309195te_a_a @ ( image_71777787125004895te_a_a @ F2 ) @ ( pow_Pr2998936717439131767tate_a @ A5 ) ) ) ).

% inj_on_image_Pow
thf(fact_793_inj__on__image__Pow,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ A5 )
     => ( inj_on2305611914272700529tate_a @ ( image_1917819122788559109tate_a @ F2 ) @ ( pow_Pr2998936717439131767tate_a @ A5 ) ) ) ).

% inj_on_image_Pow
thf(fact_794_inj__on__image__Pow,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( inj_on8057499159837936461tate_a @ ( image_5502445033238569825tate_a @ F2 ) @ ( pow_a @ A5 ) ) ) ).

% inj_on_image_Pow
thf(fact_795_inj__on__image__Pow,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( inj_on6204426022611600791tate_a @ ( image_4497581360776675627tate_a @ F2 ) @ ( pow_Pr1152895381775577553te_a_a @ A5 ) ) ) ).

% inj_on_image_Pow
thf(fact_796_bij__betw__imp__surj__on,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( bij_be957954900305576860tate_a @ F2 @ A5 @ B5 )
     => ( ( image_4497581360776675627tate_a @ F2 @ A5 )
        = B5 ) ) ).

% bij_betw_imp_surj_on
thf(fact_797_bij__betw__imp__surj__on,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( bij_be5755523363508681936te_a_a @ F2 @ A5 @ B5 )
     => ( ( image_71777787125004895te_a_a @ F2 @ A5 )
        = B5 ) ) ).

% bij_betw_imp_surj_on
thf(fact_798_bij__betw__imp__surj__on,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( bij_be7601564699172236150tate_a @ F2 @ A5 @ B5 )
     => ( ( image_1917819122788559109tate_a @ F2 @ A5 )
        = B5 ) ) ).

% bij_betw_imp_surj_on
thf(fact_799_bijI_H,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ! [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a] :
          ( ( ( F2 @ X )
            = ( F2 @ Y3 ) )
          = ( X = Y3 ) )
     => ( ! [Y3: produc6972303929186420058tate_a] :
          ? [X5: produc6972303929186420058tate_a] :
            ( Y3
            = ( F2 @ X5 ) )
       => ( bij_be7601564699172236150tate_a @ F2 @ top_to8456965716270553738tate_a @ top_to8456965716270553738tate_a ) ) ) ).

% bijI'
thf(fact_800_bij__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( bij_be7601564699172236150tate_a @ F2 @ top_to8456965716270553738tate_a @ top_to8456965716270553738tate_a )
      = ( ! [X4: produc6972303929186420058tate_a] :
          ? [Y6: produc6972303929186420058tate_a] :
            ( ( ( F2 @ Y6 )
              = X4 )
            & ! [Z4: produc6972303929186420058tate_a] :
                ( ( ( F2 @ Z4 )
                  = X4 )
               => ( Z4 = Y6 ) ) ) ) ) ).

% bij_iff
thf(fact_801_bij__pointE,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a] :
      ( ( bij_be7601564699172236150tate_a @ F2 @ top_to8456965716270553738tate_a @ top_to8456965716270553738tate_a )
     => ~ ! [X: produc6972303929186420058tate_a] :
            ( ( Y
              = ( F2 @ X ) )
           => ~ ! [X8: produc6972303929186420058tate_a] :
                  ( ( Y
                    = ( F2 @ X8 ) )
                 => ( X8 = X ) ) ) ) ).

% bij_pointE
thf(fact_802_involuntory__imp__bij,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( F2 @ ( F2 @ X ) )
          = X )
     => ( bij_be7601564699172236150tate_a @ F2 @ top_to8456965716270553738tate_a @ top_to8456965716270553738tate_a ) ) ).

% involuntory_imp_bij
thf(fact_803_inj__on__image__iff,axiom,
    ! [A5: set_a,G: a > produc6972303929186420058tate_a,F2: a > a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A5 )
         => ! [Xa: a] :
              ( ( member_a @ Xa @ A5 )
             => ( ( ( G @ ( F2 @ X ) )
                  = ( G @ ( F2 @ Xa ) ) )
                = ( ( G @ X )
                  = ( G @ Xa ) ) ) ) )
     => ( ( inj_on_a_a @ F2 @ A5 )
       => ( ( inj_on3780297825070694605tate_a @ G @ ( image_a_a @ F2 @ A5 ) )
          = ( inj_on3780297825070694605tate_a @ G @ A5 ) ) ) ) ).

% inj_on_image_iff
thf(fact_804_inj__on__image__iff,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,G: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,F2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a] :
      ( ! [X: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ A5 )
         => ! [Xa: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ Xa @ A5 )
             => ( ( ( G @ ( F2 @ X ) )
                  = ( G @ ( F2 @ Xa ) ) )
                = ( ( G @ X )
                  = ( G @ Xa ) ) ) ) )
     => ( ( inj_on4860519427091217905te_a_a @ F2 @ A5 )
       => ( ( inj_on6706560762754772119tate_a @ G @ ( image_2651540025113121413te_a_a @ F2 @ A5 ) )
          = ( inj_on6706560762754772119tate_a @ G @ A5 ) ) ) ) ).

% inj_on_image_iff
thf(fact_805_inj__def,axiom,
    ! [F2: a > produc6972303929186420058tate_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
      = ( ! [X4: a,Y6: a] :
            ( ( ( F2 @ X4 )
              = ( F2 @ Y6 ) )
           => ( X4 = Y6 ) ) ) ) ).

% inj_def
thf(fact_806_inj__def,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
      = ( ! [X4: produc5126262593522865844te_a_a,Y6: produc5126262593522865844te_a_a] :
            ( ( ( F2 @ X4 )
              = ( F2 @ Y6 ) )
           => ( X4 = Y6 ) ) ) ) ).

% inj_def
thf(fact_807_inj__eq,axiom,
    ! [F2: a > produc6972303929186420058tate_a,X3: a,Y: a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y ) )
        = ( X3 = Y ) ) ) ).

% inj_eq
thf(fact_808_inj__eq,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,X3: produc5126262593522865844te_a_a,Y: produc5126262593522865844te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y ) )
        = ( X3 = Y ) ) ) ).

% inj_eq
thf(fact_809_injI,axiom,
    ! [F2: a > produc6972303929186420058tate_a] :
      ( ! [X: a,Y3: a] :
          ( ( ( F2 @ X )
            = ( F2 @ Y3 ) )
         => ( X = Y3 ) )
     => ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a ) ) ).

% injI
thf(fact_810_injI,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ! [X: produc5126262593522865844te_a_a,Y3: produc5126262593522865844te_a_a] :
          ( ( ( F2 @ X )
            = ( F2 @ Y3 ) )
         => ( X = Y3 ) )
     => ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a ) ) ).

% injI
thf(fact_811_injD,axiom,
    ! [F2: a > produc6972303929186420058tate_a,X3: a,Y: a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y ) )
       => ( X3 = Y ) ) ) ).

% injD
thf(fact_812_injD,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,X3: produc5126262593522865844te_a_a,Y: produc5126262593522865844te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
     => ( ( ( F2 @ X3 )
          = ( F2 @ Y ) )
       => ( X3 = Y ) ) ) ).

% injD
thf(fact_813_subset__inj__on,axiom,
    ! [F2: a > produc6972303929186420058tate_a,B5: set_a,A5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ B5 )
     => ( ( ord_less_eq_set_a @ A5 @ B5 )
       => ( inj_on3780297825070694605tate_a @ F2 @ A5 ) ) ) ).

% subset_inj_on
thf(fact_814_subset__inj__on,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,B5: set_Pr2749968112124914068te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ B5 )
     => ( ( ord_le5819720210775101748te_a_a @ A5 @ B5 )
       => ( inj_on6706560762754772119tate_a @ F2 @ A5 ) ) ) ).

% subset_inj_on
thf(fact_815_inj__on__subset,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,B5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( ( ord_less_eq_set_a @ B5 @ A5 )
       => ( inj_on3780297825070694605tate_a @ F2 @ B5 ) ) ) ).

% inj_on_subset
thf(fact_816_inj__on__subset,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( ( ord_le5819720210775101748te_a_a @ B5 @ A5 )
       => ( inj_on6706560762754772119tate_a @ F2 @ B5 ) ) ) ).

% inj_on_subset
thf(fact_817_inj__on__image__Fpow,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ A5 )
     => ( inj_on779827642740309195te_a_a @ ( image_71777787125004895te_a_a @ F2 ) @ ( finite8761379122080342720tate_a @ A5 ) ) ) ).

% inj_on_image_Fpow
thf(fact_818_inj__on__image__Fpow,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ A5 )
     => ( inj_on2305611914272700529tate_a @ ( image_1917819122788559109tate_a @ F2 ) @ ( finite8761379122080342720tate_a @ A5 ) ) ) ).

% inj_on_image_Fpow
thf(fact_819_inj__on__image__Fpow,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( inj_on8057499159837936461tate_a @ ( image_5502445033238569825tate_a @ F2 ) @ ( finite_Fpow_a @ A5 ) ) ) ).

% inj_on_image_Fpow
thf(fact_820_inj__on__image__Fpow,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( inj_on6204426022611600791tate_a @ ( image_4497581360776675627tate_a @ F2 ) @ ( finite6915337786416788506te_a_a @ A5 ) ) ) ).

% inj_on_image_Fpow
thf(fact_821_Un__Pow__subset,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] : ( ord_le7926008383649835194tate_a @ ( sup_su530886023631425774tate_a @ ( pow_Pr2998936717439131767tate_a @ A5 ) @ ( pow_Pr2998936717439131767tate_a @ B5 ) ) @ ( pow_Pr2998936717439131767tate_a @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) ) ) ).

% Un_Pow_subset
thf(fact_822_image__Un,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( image_4497581360776675627tate_a @ F2 @ ( sup_su136906189548161384te_a_a @ A5 @ B5 ) )
      = ( sup_su1662690461080552718tate_a @ ( image_4497581360776675627tate_a @ F2 @ A5 ) @ ( image_4497581360776675627tate_a @ F2 @ B5 ) ) ) ).

% image_Un
thf(fact_823_image__Un,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( image_71777787125004895te_a_a @ F2 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) )
      = ( sup_su136906189548161384te_a_a @ ( image_71777787125004895te_a_a @ F2 @ A5 ) @ ( image_71777787125004895te_a_a @ F2 @ B5 ) ) ) ).

% image_Un
thf(fact_824_image__Un,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( image_1917819122788559109tate_a @ F2 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) )
      = ( sup_su1662690461080552718tate_a @ ( image_1917819122788559109tate_a @ F2 @ A5 ) @ ( image_1917819122788559109tate_a @ F2 @ B5 ) ) ) ).

% image_Un
thf(fact_825_Un__UNIV__right,axiom,
    ! [A5: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ A5 @ top_to8456965716270553738tate_a )
      = top_to8456965716270553738tate_a ) ).

% Un_UNIV_right
thf(fact_826_Un__UNIV__left,axiom,
    ! [B5: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ top_to8456965716270553738tate_a @ B5 )
      = top_to8456965716270553738tate_a ) ).

% Un_UNIV_left
thf(fact_827_Un__mono,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,C3: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,D: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ C3 )
     => ( ( ord_le7345504482307493082tate_a @ B5 @ D )
       => ( ord_le7345504482307493082tate_a @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) @ ( sup_su1662690461080552718tate_a @ C3 @ D ) ) ) ) ).

% Un_mono
thf(fact_828_Un__least,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,C3: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ C3 )
     => ( ( ord_le7345504482307493082tate_a @ B5 @ C3 )
       => ( ord_le7345504482307493082tate_a @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) @ C3 ) ) ) ).

% Un_least
thf(fact_829_Un__upper1,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ A5 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) ) ).

% Un_upper1
thf(fact_830_Un__upper2,axiom,
    ! [B5: set_Pr4275752383657305402tate_a,A5: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ B5 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) ) ).

% Un_upper2
thf(fact_831_Un__absorb1,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
     => ( ( sup_su1662690461080552718tate_a @ A5 @ B5 )
        = B5 ) ) ).

% Un_absorb1
thf(fact_832_Un__absorb2,axiom,
    ! [B5: set_Pr4275752383657305402tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ B5 @ A5 )
     => ( ( sup_su1662690461080552718tate_a @ A5 @ B5 )
        = A5 ) ) ).

% Un_absorb2
thf(fact_833_subset__UnE,axiom,
    ! [C3: set_Pr4275752383657305402tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ C3 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) )
     => ~ ! [A7: set_Pr4275752383657305402tate_a] :
            ( ( ord_le7345504482307493082tate_a @ A7 @ A5 )
           => ! [B7: set_Pr4275752383657305402tate_a] :
                ( ( ord_le7345504482307493082tate_a @ B7 @ B5 )
               => ( C3
                 != ( sup_su1662690461080552718tate_a @ A7 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_834_subset__Un__eq,axiom,
    ( ord_le7345504482307493082tate_a
    = ( ^ [A6: set_Pr4275752383657305402tate_a,B6: set_Pr4275752383657305402tate_a] :
          ( ( sup_su1662690461080552718tate_a @ A6 @ B6 )
          = B6 ) ) ) ).

% subset_Un_eq
thf(fact_835_inj__swap,axiom,
    ! [A5: set_Pr4275752383657305402tate_a] : ( inj_on2280757189103101387te_a_a @ produc3903577461452268778tate_a @ A5 ) ).

% inj_swap
thf(fact_836_inj__swap,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a] : ( inj_on6706560762754772119tate_a @ produc2589720147407833796te_a_a @ A5 ) ).

% inj_swap
thf(fact_837_antisymp__on__le,axiom,
    ! [A5: set_se5193884918812770586tate_a] : ( antisy4329420592516833511tate_a @ A5 @ ord_le7345504482307493082tate_a ) ).

% antisymp_on_le
thf(fact_838_antisymp__equality,axiom,
    ( antisy2379887893289268487tate_a @ top_to8456965716270553738tate_a
    @ ^ [Y5: produc6972303929186420058tate_a,Z: produc6972303929186420058tate_a] : ( Y5 = Z ) ) ).

% antisymp_equality
thf(fact_839_antisympI,axiom,
    ! [R2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o] :
      ( ! [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a] :
          ( ( R2 @ X @ Y3 )
         => ( ( R2 @ Y3 @ X )
           => ( X = Y3 ) ) )
     => ( antisy2379887893289268487tate_a @ top_to8456965716270553738tate_a @ R2 ) ) ).

% antisympI
thf(fact_840_antisympD,axiom,
    ! [R2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o,X3: produc6972303929186420058tate_a,Y: produc6972303929186420058tate_a] :
      ( ( antisy2379887893289268487tate_a @ top_to8456965716270553738tate_a @ R2 )
     => ( ( R2 @ X3 @ Y )
       => ( ( R2 @ Y @ X3 )
         => ( X3 = Y ) ) ) ) ).

% antisympD
thf(fact_841_antisymp__on__subset,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,R2: produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o,B5: set_Pr4275752383657305402tate_a] :
      ( ( antisy2379887893289268487tate_a @ A5 @ R2 )
     => ( ( ord_le7345504482307493082tate_a @ B5 @ A5 )
       => ( antisy2379887893289268487tate_a @ B5 @ R2 ) ) ) ).

% antisymp_on_subset
thf(fact_842_bij__is__surj,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( bij_be957954900305576860tate_a @ F2 @ top_to6931181444738162404te_a_a @ top_to8456965716270553738tate_a )
     => ( ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a )
        = top_to8456965716270553738tate_a ) ) ).

% bij_is_surj
thf(fact_843_bij__is__surj,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ( bij_be5755523363508681936te_a_a @ F2 @ top_to8456965716270553738tate_a @ top_to6931181444738162404te_a_a )
     => ( ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to6931181444738162404te_a_a ) ) ).

% bij_is_surj
thf(fact_844_bij__is__surj,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( bij_be7601564699172236150tate_a @ F2 @ top_to8456965716270553738tate_a @ top_to8456965716270553738tate_a )
     => ( ( image_1917819122788559109tate_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to8456965716270553738tate_a ) ) ).

% bij_is_surj
thf(fact_845_bij__betw__imp__surj,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( bij_be5755523363508681936te_a_a @ F2 @ A5 @ top_to6931181444738162404te_a_a )
     => ( ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to6931181444738162404te_a_a ) ) ).

% bij_betw_imp_surj
thf(fact_846_bij__betw__imp__surj,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( bij_be957954900305576860tate_a @ F2 @ A5 @ top_to8456965716270553738tate_a )
     => ( ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a )
        = top_to8456965716270553738tate_a ) ) ).

% bij_betw_imp_surj
thf(fact_847_bij__betw__imp__surj,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( bij_be7601564699172236150tate_a @ F2 @ A5 @ top_to8456965716270553738tate_a )
     => ( ( image_1917819122788559109tate_a @ F2 @ top_to8456965716270553738tate_a )
        = top_to8456965716270553738tate_a ) ) ).

% bij_betw_imp_surj
thf(fact_848_bij__betw__byWitness,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,F3: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A8: set_Pr2749968112124914068te_a_a] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ A5 )
         => ( ( F3 @ ( F2 @ X ) )
            = X ) )
     => ( ! [X: produc5126262593522865844te_a_a] :
            ( ( member1329951143264900189te_a_a @ X @ A8 )
           => ( ( F2 @ ( F3 @ X ) )
              = X ) )
       => ( ( ord_le5819720210775101748te_a_a @ ( image_71777787125004895te_a_a @ F2 @ A5 ) @ A8 )
         => ( ( ord_le7345504482307493082tate_a @ ( image_4497581360776675627tate_a @ F3 @ A8 ) @ A5 )
           => ( bij_be5755523363508681936te_a_a @ F2 @ A5 @ A8 ) ) ) ) ) ).

% bij_betw_byWitness
thf(fact_849_bij__betw__byWitness,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,F3: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A8: set_Pr4275752383657305402tate_a] :
      ( ! [X: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ A5 )
         => ( ( F3 @ ( F2 @ X ) )
            = X ) )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X @ A8 )
           => ( ( F2 @ ( F3 @ X ) )
              = X ) )
       => ( ( ord_le7345504482307493082tate_a @ ( image_4497581360776675627tate_a @ F2 @ A5 ) @ A8 )
         => ( ( ord_le5819720210775101748te_a_a @ ( image_71777787125004895te_a_a @ F3 @ A8 ) @ A5 )
           => ( bij_be957954900305576860tate_a @ F2 @ A5 @ A8 ) ) ) ) ) ).

% bij_betw_byWitness
thf(fact_850_bij__betw__byWitness,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,F3: produc6972303929186420058tate_a > produc6972303929186420058tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A8: set_Pr4275752383657305402tate_a] :
      ( ! [X: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ A5 )
         => ( ( F3 @ ( F2 @ X ) )
            = X ) )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X @ A8 )
           => ( ( F2 @ ( F3 @ X ) )
              = X ) )
       => ( ( ord_le7345504482307493082tate_a @ ( image_1917819122788559109tate_a @ F2 @ A5 ) @ A8 )
         => ( ( ord_le7345504482307493082tate_a @ ( image_1917819122788559109tate_a @ F3 @ A8 ) @ A5 )
           => ( bij_be7601564699172236150tate_a @ F2 @ A5 @ A8 ) ) ) ) ) ).

% bij_betw_byWitness
thf(fact_851_bij__betw__subset,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,A8: set_Pr4275752383657305402tate_a,B5: set_Pr2749968112124914068te_a_a,B8: set_Pr4275752383657305402tate_a] :
      ( ( bij_be957954900305576860tate_a @ F2 @ A5 @ A8 )
     => ( ( ord_le5819720210775101748te_a_a @ B5 @ A5 )
       => ( ( ( image_4497581360776675627tate_a @ F2 @ B5 )
            = B8 )
         => ( bij_be957954900305576860tate_a @ F2 @ B5 @ B8 ) ) ) ) ).

% bij_betw_subset
thf(fact_852_bij__betw__subset,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,A8: set_Pr2749968112124914068te_a_a,B5: set_Pr4275752383657305402tate_a,B8: set_Pr2749968112124914068te_a_a] :
      ( ( bij_be5755523363508681936te_a_a @ F2 @ A5 @ A8 )
     => ( ( ord_le7345504482307493082tate_a @ B5 @ A5 )
       => ( ( ( image_71777787125004895te_a_a @ F2 @ B5 )
            = B8 )
         => ( bij_be5755523363508681936te_a_a @ F2 @ B5 @ B8 ) ) ) ) ).

% bij_betw_subset
thf(fact_853_bij__betw__subset,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,A8: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,B8: set_Pr4275752383657305402tate_a] :
      ( ( bij_be7601564699172236150tate_a @ F2 @ A5 @ A8 )
     => ( ( ord_le7345504482307493082tate_a @ B5 @ A5 )
       => ( ( ( image_1917819122788559109tate_a @ F2 @ B5 )
            = B8 )
         => ( bij_be7601564699172236150tate_a @ F2 @ B5 @ B8 ) ) ) ) ).

% bij_betw_subset
thf(fact_854_range__ex1__eq,axiom,
    ! [F2: a > produc6972303929186420058tate_a,B: produc6972303929186420058tate_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
     => ( ( member3175992478928454403tate_a @ B @ ( image_5502445033238569825tate_a @ F2 @ top_top_set_a ) )
        = ( ? [X4: a] :
              ( ( B
                = ( F2 @ X4 ) )
              & ! [Y6: a] :
                  ( ( B
                    = ( F2 @ Y6 ) )
                 => ( Y6 = X4 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_855_range__ex1__eq,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,B: produc6972303929186420058tate_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
     => ( ( member3175992478928454403tate_a @ B @ ( image_4497581360776675627tate_a @ F2 @ top_to6931181444738162404te_a_a ) )
        = ( ? [X4: produc5126262593522865844te_a_a] :
              ( ( B
                = ( F2 @ X4 ) )
              & ! [Y6: produc5126262593522865844te_a_a] :
                  ( ( B
                    = ( F2 @ Y6 ) )
                 => ( Y6 = X4 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_856_range__ex1__eq,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,B: produc6972303929186420058tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( member3175992478928454403tate_a @ B @ ( image_1917819122788559109tate_a @ F2 @ top_to8456965716270553738tate_a ) )
        = ( ? [X4: produc6972303929186420058tate_a] :
              ( ( B
                = ( F2 @ X4 ) )
              & ! [Y6: produc6972303929186420058tate_a] :
                  ( ( B
                    = ( F2 @ Y6 ) )
                 => ( Y6 = X4 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_857_range__ex1__eq,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,B: produc5126262593522865844te_a_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( member1329951143264900189te_a_a @ B @ ( image_71777787125004895te_a_a @ F2 @ top_to8456965716270553738tate_a ) )
        = ( ? [X4: produc6972303929186420058tate_a] :
              ( ( B
                = ( F2 @ X4 ) )
              & ! [Y6: produc6972303929186420058tate_a] :
                  ( ( B
                    = ( F2 @ Y6 ) )
                 => ( Y6 = X4 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_858_range__ex1__eq,axiom,
    ! [F2: produc6972303929186420058tate_a > produc4774439622479004775tate_a,B: produc4774439622479004775tate_a] :
      ( ( inj_on6627312848323107838tate_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( member3801464090328701328tate_a @ B @ ( image_5672933159226955410tate_a @ F2 @ top_to8456965716270553738tate_a ) )
        = ( ? [X4: produc6972303929186420058tate_a] :
              ( ( B
                = ( F2 @ X4 ) )
              & ! [Y6: produc6972303929186420058tate_a] :
                  ( ( B
                    = ( F2 @ Y6 ) )
                 => ( Y6 = X4 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_859_range__ex1__eq,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5940350820068290983tate_a,B: produc5940350820068290983tate_a] :
      ( ( inj_on4734429731042995006tate_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( member1595334363549574352tate_a @ B @ ( image_8048040396972735954tate_a @ F2 @ top_to8456965716270553738tate_a ) )
        = ( ? [X4: produc6972303929186420058tate_a] :
              ( ( B
                = ( F2 @ X4 ) )
              & ! [Y6: produc6972303929186420058tate_a] :
                  ( ( B
                    = ( F2 @ Y6 ) )
                 => ( Y6 = X4 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_860_range__ex1__eq,axiom,
    ! [F2: produc6972303929186420058tate_a > product_prod_a_a,B: product_prod_a_a] :
      ( ( inj_on4637469111408784318od_a_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( member1426531477525435216od_a_a @ B @ ( image_8179770026730005074od_a_a @ F2 @ top_to8456965716270553738tate_a ) )
        = ( ? [X4: produc6972303929186420058tate_a] :
              ( ( B
                = ( F2 @ X4 ) )
              & ! [Y6: produc6972303929186420058tate_a] :
                  ( ( B
                    = ( F2 @ Y6 ) )
                 => ( Y6 = X4 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_861_inj__image__eq__iff,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,B5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
     => ( ( ( image_5502445033238569825tate_a @ F2 @ A5 )
          = ( image_5502445033238569825tate_a @ F2 @ B5 ) )
        = ( A5 = B5 ) ) ) ).

% inj_image_eq_iff
thf(fact_862_inj__image__eq__iff,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
     => ( ( ( image_4497581360776675627tate_a @ F2 @ A5 )
          = ( image_4497581360776675627tate_a @ F2 @ B5 ) )
        = ( A5 = B5 ) ) ) ).

% inj_image_eq_iff
thf(fact_863_inj__image__eq__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( ( image_71777787125004895te_a_a @ F2 @ A5 )
          = ( image_71777787125004895te_a_a @ F2 @ B5 ) )
        = ( A5 = B5 ) ) ) ).

% inj_image_eq_iff
thf(fact_864_inj__image__eq__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( ( image_1917819122788559109tate_a @ F2 @ A5 )
          = ( image_1917819122788559109tate_a @ F2 @ B5 ) )
        = ( A5 = B5 ) ) ) ).

% inj_image_eq_iff
thf(fact_865_inj__image__mem__iff,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A: a,A5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
     => ( ( member3175992478928454403tate_a @ ( F2 @ A ) @ ( image_5502445033238569825tate_a @ F2 @ A5 ) )
        = ( member_a @ A @ A5 ) ) ) ).

% inj_image_mem_iff
thf(fact_866_inj__image__mem__iff,axiom,
    ! [F2: product_prod_a_a > product_prod_a_a,A: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( inj_on2566144670800592689od_a_a @ F2 @ top_to8063371432257647191od_a_a )
     => ( ( member1426531477525435216od_a_a @ ( F2 @ A ) @ ( image_4636654165204879301od_a_a @ F2 @ A5 ) )
        = ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ).

% inj_image_mem_iff
thf(fact_867_inj__image__mem__iff,axiom,
    ! [F2: product_prod_a_a > produc6972303929186420058tate_a,A: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( inj_on7244109900603427428tate_a @ F2 @ top_to8063371432257647191od_a_a )
     => ( ( member3175992478928454403tate_a @ ( F2 @ A ) @ ( image_1563038779069872376tate_a @ F2 @ A5 ) )
        = ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ).

% inj_image_mem_iff
thf(fact_868_inj__image__mem__iff,axiom,
    ! [F2: product_prod_a_a > produc5126262593522865844te_a_a,A: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( inj_on5398068564939873214te_a_a @ F2 @ top_to8063371432257647191od_a_a )
     => ( ( member1329951143264900189te_a_a @ ( F2 @ A ) @ ( image_8940369480261093970te_a_a @ F2 @ A5 ) )
        = ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ).

% inj_image_mem_iff
thf(fact_869_inj__image__mem__iff,axiom,
    ! [F2: produc5126262593522865844te_a_a > product_prod_a_a,A: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6297019480273606372od_a_a @ F2 @ top_to6931181444738162404te_a_a )
     => ( ( member1426531477525435216od_a_a @ ( F2 @ A ) @ ( image_615948358740051320od_a_a @ F2 @ A5 ) )
        = ( member1329951143264900189te_a_a @ A @ A5 ) ) ) ).

% inj_image_mem_iff
thf(fact_870_inj__image__mem__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > product_prod_a_a,A: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4637469111408784318od_a_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( member1426531477525435216od_a_a @ ( F2 @ A ) @ ( image_8179770026730005074od_a_a @ F2 @ A5 ) )
        = ( member3175992478928454403tate_a @ A @ A5 ) ) ) ).

% inj_image_mem_iff
thf(fact_871_inj__image__mem__iff,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
     => ( ( member3175992478928454403tate_a @ ( F2 @ A ) @ ( image_4497581360776675627tate_a @ F2 @ A5 ) )
        = ( member1329951143264900189te_a_a @ A @ A5 ) ) ) ).

% inj_image_mem_iff
thf(fact_872_inj__image__mem__iff,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a,A: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on4860519427091217905te_a_a @ F2 @ top_to6931181444738162404te_a_a )
     => ( ( member1329951143264900189te_a_a @ ( F2 @ A ) @ ( image_2651540025113121413te_a_a @ F2 @ A5 ) )
        = ( member1329951143264900189te_a_a @ A @ A5 ) ) ) ).

% inj_image_mem_iff
thf(fact_873_inj__image__mem__iff,axiom,
    ! [F2: product_prod_a_a > produc4774439622479004775tate_a,A: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( inj_on4003527337536797809tate_a @ F2 @ top_to8063371432257647191od_a_a )
     => ( ( member3801464090328701328tate_a @ ( F2 @ A ) @ ( image_1794547935558701317tate_a @ F2 @ A5 ) )
        = ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ).

% inj_image_mem_iff
thf(fact_874_inj__image__mem__iff,axiom,
    ! [F2: produc4774439622479004775tate_a > product_prod_a_a,A: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( inj_on4099919973560129009od_a_a @ F2 @ top_to496764914387262615tate_a )
     => ( ( member1426531477525435216od_a_a @ ( F2 @ A ) @ ( image_1890940571582032517od_a_a @ F2 @ A5 ) )
        = ( member3801464090328701328tate_a @ A @ A5 ) ) ) ).

% inj_image_mem_iff
thf(fact_875_inj__on__image__mem__iff,axiom,
    ! [F2: a > produc6972303929186420058tate_a,B5: set_a,A: a,A5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ B5 )
     => ( ( member_a @ A @ B5 )
       => ( ( ord_less_eq_set_a @ A5 @ B5 )
         => ( ( member3175992478928454403tate_a @ ( F2 @ A ) @ ( image_5502445033238569825tate_a @ F2 @ A5 ) )
            = ( member_a @ A @ A5 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_876_inj__on__image__mem__iff,axiom,
    ! [F2: product_prod_a_a > product_prod_a_a,B5: set_Product_prod_a_a,A: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( inj_on2566144670800592689od_a_a @ F2 @ B5 )
     => ( ( member1426531477525435216od_a_a @ A @ B5 )
       => ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
         => ( ( member1426531477525435216od_a_a @ ( F2 @ A ) @ ( image_4636654165204879301od_a_a @ F2 @ A5 ) )
            = ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_877_inj__on__image__mem__iff,axiom,
    ! [F2: produc5126262593522865844te_a_a > product_prod_a_a,B5: set_Pr2749968112124914068te_a_a,A: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6297019480273606372od_a_a @ F2 @ B5 )
     => ( ( member1329951143264900189te_a_a @ A @ B5 )
       => ( ( ord_le5819720210775101748te_a_a @ A5 @ B5 )
         => ( ( member1426531477525435216od_a_a @ ( F2 @ A ) @ ( image_615948358740051320od_a_a @ F2 @ A5 ) )
            = ( member1329951143264900189te_a_a @ A @ A5 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_878_inj__on__image__mem__iff,axiom,
    ! [F2: product_prod_a_a > produc6972303929186420058tate_a,B5: set_Product_prod_a_a,A: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( inj_on7244109900603427428tate_a @ F2 @ B5 )
     => ( ( member1426531477525435216od_a_a @ A @ B5 )
       => ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
         => ( ( member3175992478928454403tate_a @ ( F2 @ A ) @ ( image_1563038779069872376tate_a @ F2 @ A5 ) )
            = ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_879_inj__on__image__mem__iff,axiom,
    ! [F2: product_prod_a_a > produc5126262593522865844te_a_a,B5: set_Product_prod_a_a,A: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( inj_on5398068564939873214te_a_a @ F2 @ B5 )
     => ( ( member1426531477525435216od_a_a @ A @ B5 )
       => ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
         => ( ( member1329951143264900189te_a_a @ ( F2 @ A ) @ ( image_8940369480261093970te_a_a @ F2 @ A5 ) )
            = ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_880_inj__on__image__mem__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > product_prod_a_a,B5: set_Pr4275752383657305402tate_a,A: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4637469111408784318od_a_a @ F2 @ B5 )
     => ( ( member3175992478928454403tate_a @ A @ B5 )
       => ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
         => ( ( member1426531477525435216od_a_a @ ( F2 @ A ) @ ( image_8179770026730005074od_a_a @ F2 @ A5 ) )
            = ( member3175992478928454403tate_a @ A @ A5 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_881_inj__on__image__mem__iff,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,B5: set_Pr2749968112124914068te_a_a,A: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ B5 )
     => ( ( member1329951143264900189te_a_a @ A @ B5 )
       => ( ( ord_le5819720210775101748te_a_a @ A5 @ B5 )
         => ( ( member3175992478928454403tate_a @ ( F2 @ A ) @ ( image_4497581360776675627tate_a @ F2 @ A5 ) )
            = ( member1329951143264900189te_a_a @ A @ A5 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_882_inj__on__image__mem__iff,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a,A: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on4860519427091217905te_a_a @ F2 @ B5 )
     => ( ( member1329951143264900189te_a_a @ A @ B5 )
       => ( ( ord_le5819720210775101748te_a_a @ A5 @ B5 )
         => ( ( member1329951143264900189te_a_a @ ( F2 @ A ) @ ( image_2651540025113121413te_a_a @ F2 @ A5 ) )
            = ( member1329951143264900189te_a_a @ A @ A5 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_883_inj__on__image__mem__iff,axiom,
    ! [F2: produc4774439622479004775tate_a > product_prod_a_a,B5: set_Pr1753757909640466247tate_a,A: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( inj_on4099919973560129009od_a_a @ F2 @ B5 )
     => ( ( member3801464090328701328tate_a @ A @ B5 )
       => ( ( ord_le5561929874677720807tate_a @ A5 @ B5 )
         => ( ( member1426531477525435216od_a_a @ ( F2 @ A ) @ ( image_1890940571582032517od_a_a @ F2 @ A5 ) )
            = ( member3801464090328701328tate_a @ A @ A5 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_884_inj__on__image__mem__iff,axiom,
    ! [F2: product_prod_a_a > produc4774439622479004775tate_a,B5: set_Product_prod_a_a,A: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( inj_on4003527337536797809tate_a @ F2 @ B5 )
     => ( ( member1426531477525435216od_a_a @ A @ B5 )
       => ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
         => ( ( member3801464090328701328tate_a @ ( F2 @ A ) @ ( image_1794547935558701317tate_a @ F2 @ A5 ) )
            = ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_885_inj__on__image__eq__iff,axiom,
    ! [F2: a > produc6972303929186420058tate_a,C3: set_a,A5: set_a,B5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ C3 )
     => ( ( ord_less_eq_set_a @ A5 @ C3 )
       => ( ( ord_less_eq_set_a @ B5 @ C3 )
         => ( ( ( image_5502445033238569825tate_a @ F2 @ A5 )
              = ( image_5502445033238569825tate_a @ F2 @ B5 ) )
            = ( A5 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_886_inj__on__image__eq__iff,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,C3: set_Pr2749968112124914068te_a_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ C3 )
     => ( ( ord_le5819720210775101748te_a_a @ A5 @ C3 )
       => ( ( ord_le5819720210775101748te_a_a @ B5 @ C3 )
         => ( ( ( image_4497581360776675627tate_a @ F2 @ A5 )
              = ( image_4497581360776675627tate_a @ F2 @ B5 ) )
            = ( A5 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_887_inj__on__image__eq__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,C3: set_Pr4275752383657305402tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ C3 )
     => ( ( ord_le7345504482307493082tate_a @ A5 @ C3 )
       => ( ( ord_le7345504482307493082tate_a @ B5 @ C3 )
         => ( ( ( image_71777787125004895te_a_a @ F2 @ A5 )
              = ( image_71777787125004895te_a_a @ F2 @ B5 ) )
            = ( A5 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_888_inj__on__image__eq__iff,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,C3: set_Pr4275752383657305402tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ C3 )
     => ( ( ord_le7345504482307493082tate_a @ A5 @ C3 )
       => ( ( ord_le7345504482307493082tate_a @ B5 @ C3 )
         => ( ( ( image_1917819122788559109tate_a @ F2 @ A5 )
              = ( image_1917819122788559109tate_a @ F2 @ B5 ) )
            = ( A5 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_889_sup__cancel__left1,axiom,
    ! [X3: a > state_a > $o,A: a > state_a > $o,B: a > state_a > $o] :
      ( ( sup_sup_a_state_a_o @ ( sup_sup_a_state_a_o @ X3 @ A ) @ ( sup_sup_a_state_a_o @ ( uminus9152238405698698866te_a_o @ X3 ) @ B ) )
      = top_top_a_state_a_o ) ).

% sup_cancel_left1
thf(fact_890_sup__cancel__left1,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ ( sup_su1662690461080552718tate_a @ X3 @ A ) @ ( sup_su1662690461080552718tate_a @ ( uminus3991176040251746481tate_a @ X3 ) @ B ) )
      = top_to8456965716270553738tate_a ) ).

% sup_cancel_left1
thf(fact_891_sup__cancel__left2,axiom,
    ! [X3: a > state_a > $o,A: a > state_a > $o,B: a > state_a > $o] :
      ( ( sup_sup_a_state_a_o @ ( sup_sup_a_state_a_o @ ( uminus9152238405698698866te_a_o @ X3 ) @ A ) @ ( sup_sup_a_state_a_o @ X3 @ B ) )
      = top_top_a_state_a_o ) ).

% sup_cancel_left2
thf(fact_892_sup__cancel__left2,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ ( sup_su1662690461080552718tate_a @ ( uminus3991176040251746481tate_a @ X3 ) @ A ) @ ( sup_su1662690461080552718tate_a @ X3 @ B ) )
      = top_to8456965716270553738tate_a ) ).

% sup_cancel_left2
thf(fact_893_Compl__partition,axiom,
    ! [A5: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ A5 @ ( uminus3991176040251746481tate_a @ A5 ) )
      = top_to8456965716270553738tate_a ) ).

% Compl_partition
thf(fact_894_Compl__partition2,axiom,
    ! [A5: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ ( uminus3991176040251746481tate_a @ A5 ) @ A5 )
      = top_to8456965716270553738tate_a ) ).

% Compl_partition2
thf(fact_895_bij__swap,axiom,
    bij_be957954900305576860tate_a @ produc2589720147407833796te_a_a @ top_to6931181444738162404te_a_a @ top_to8456965716270553738tate_a ).

% bij_swap
thf(fact_896_bij__swap,axiom,
    bij_be5755523363508681936te_a_a @ produc3903577461452268778tate_a @ top_to8456965716270553738tate_a @ top_to6931181444738162404te_a_a ).

% bij_swap
thf(fact_897_sup__top__right,axiom,
    ! [X3: a > state_a > $o] :
      ( ( sup_sup_a_state_a_o @ X3 @ top_top_a_state_a_o )
      = top_top_a_state_a_o ) ).

% sup_top_right
thf(fact_898_sup__top__right,axiom,
    ! [X3: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ X3 @ top_to8456965716270553738tate_a )
      = top_to8456965716270553738tate_a ) ).

% sup_top_right
thf(fact_899_sup__top__left,axiom,
    ! [X3: a > state_a > $o] :
      ( ( sup_sup_a_state_a_o @ top_top_a_state_a_o @ X3 )
      = top_top_a_state_a_o ) ).

% sup_top_left
thf(fact_900_sup__top__left,axiom,
    ! [X3: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ top_to8456965716270553738tate_a @ X3 )
      = top_to8456965716270553738tate_a ) ).

% sup_top_left
thf(fact_901_le__sup__iff,axiom,
    ! [X3: a > state_a > $o,Y: a > state_a > $o,Z2: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ ( sup_sup_a_state_a_o @ X3 @ Y ) @ Z2 )
      = ( ( ord_le5391092903712282779te_a_o @ X3 @ Z2 )
        & ( ord_le5391092903712282779te_a_o @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_902_le__sup__iff,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a,Z2: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( sup_su1662690461080552718tate_a @ X3 @ Y ) @ Z2 )
      = ( ( ord_le7345504482307493082tate_a @ X3 @ Z2 )
        & ( ord_le7345504482307493082tate_a @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_903_sup_Obounded__iff,axiom,
    ! [B: a > state_a > $o,C2: a > state_a > $o,A: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ ( sup_sup_a_state_a_o @ B @ C2 ) @ A )
      = ( ( ord_le5391092903712282779te_a_o @ B @ A )
        & ( ord_le5391092903712282779te_a_o @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_904_sup_Obounded__iff,axiom,
    ! [B: set_Pr4275752383657305402tate_a,C2: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( sup_su1662690461080552718tate_a @ B @ C2 ) @ A )
      = ( ( ord_le7345504482307493082tate_a @ B @ A )
        & ( ord_le7345504482307493082tate_a @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_905_prod_Oinj__map,axiom,
    ! [F1: state_a > a,F22: a > state_a] :
      ( ( inj_on_state_a_a @ F1 @ top_top_set_state_a )
     => ( ( inj_on_a_state_a @ F22 @ top_top_set_a )
       => ( inj_on6706560762754772119tate_a @ ( produc5926476928146254390tate_a @ F1 @ F22 ) @ top_to6931181444738162404te_a_a ) ) ) ).

% prod.inj_map
thf(fact_906_prod_Oinj__map,axiom,
    ! [F1: a > produc6972303929186420058tate_a,F22: a > produc6972303929186420058tate_a] :
      ( ( inj_on3780297825070694605tate_a @ F1 @ top_top_set_a )
     => ( ( inj_on3780297825070694605tate_a @ F22 @ top_top_set_a )
       => ( inj_on4812055642248623537tate_a @ ( produc3303943909295981742tate_a @ F1 @ F22 ) @ top_to8063371432257647191od_a_a ) ) ) ).

% prod.inj_map
thf(fact_907_prod_Oinj__map,axiom,
    ! [F1: a > produc6972303929186420058tate_a,F22: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( inj_on3780297825070694605tate_a @ F1 @ top_top_set_a )
     => ( ( inj_on6706560762754772119tate_a @ F22 @ top_to6931181444738162404te_a_a )
       => ( inj_on2314482362798833787tate_a @ ( produc3016041650167546616tate_a @ F1 @ F22 ) @ top_to4945334307497793421te_a_a ) ) ) ).

% prod.inj_map
thf(fact_908_prod_Oinj__map,axiom,
    ! [F1: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,F22: a > produc6972303929186420058tate_a] :
      ( ( inj_on6706560762754772119tate_a @ F1 @ top_to6931181444738162404te_a_a )
     => ( ( inj_on3780297825070694605tate_a @ F22 @ top_top_set_a )
       => ( inj_on1003780114035383527tate_a @ ( produc2006019012592617720tate_a @ F1 @ F22 ) @ top_to3053490249063943713_a_a_a ) ) ) ).

% prod.inj_map
thf(fact_909_prod_Oinj__map,axiom,
    ! [F1: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,F22: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( inj_on6706560762754772119tate_a @ F1 @ top_to6931181444738162404te_a_a )
     => ( ( inj_on6706560762754772119tate_a @ F22 @ top_to6931181444738162404te_a_a )
       => ( inj_on8497762983094103729tate_a @ ( produc1508824635387789378tate_a @ F1 @ F22 ) @ top_to6088003638156323927te_a_a ) ) ) ).

% prod.inj_map
thf(fact_910_prod_Oinj__map,axiom,
    ! [F1: a > a,F22: state_a > state_a] :
      ( ( inj_on_a_a @ F1 @ top_top_set_a )
     => ( ( inj_on5418343104458039473tate_a @ F22 @ top_top_set_state_a )
       => ( inj_on4126798524766655601tate_a @ ( produc6760600410500789430tate_a @ F1 @ F22 ) @ top_to8456965716270553738tate_a ) ) ) ).

% prod.inj_map
thf(fact_911_subset__image__inj,axiom,
    ! [S4: set_Pr2749968112124914068te_a_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,T3: set_Pr4275752383657305402tate_a] :
      ( ( ord_le5819720210775101748te_a_a @ S4 @ ( image_71777787125004895te_a_a @ F2 @ T3 ) )
      = ( ? [U: set_Pr4275752383657305402tate_a] :
            ( ( ord_le7345504482307493082tate_a @ U @ T3 )
            & ( inj_on2280757189103101387te_a_a @ F2 @ U )
            & ( S4
              = ( image_71777787125004895te_a_a @ F2 @ U ) ) ) ) ) ).

% subset_image_inj
thf(fact_912_subset__image__inj,axiom,
    ! [S4: set_Pr4275752383657305402tate_a,F2: a > produc6972303929186420058tate_a,T3: set_a] :
      ( ( ord_le7345504482307493082tate_a @ S4 @ ( image_5502445033238569825tate_a @ F2 @ T3 ) )
      = ( ? [U: set_a] :
            ( ( ord_less_eq_set_a @ U @ T3 )
            & ( inj_on3780297825070694605tate_a @ F2 @ U )
            & ( S4
              = ( image_5502445033238569825tate_a @ F2 @ U ) ) ) ) ) ).

% subset_image_inj
thf(fact_913_subset__image__inj,axiom,
    ! [S4: set_Pr4275752383657305402tate_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,T3: set_Pr2749968112124914068te_a_a] :
      ( ( ord_le7345504482307493082tate_a @ S4 @ ( image_4497581360776675627tate_a @ F2 @ T3 ) )
      = ( ? [U: set_Pr2749968112124914068te_a_a] :
            ( ( ord_le5819720210775101748te_a_a @ U @ T3 )
            & ( inj_on6706560762754772119tate_a @ F2 @ U )
            & ( S4
              = ( image_4497581360776675627tate_a @ F2 @ U ) ) ) ) ) ).

% subset_image_inj
thf(fact_914_subset__image__inj,axiom,
    ! [S4: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,T3: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ S4 @ ( image_1917819122788559109tate_a @ F2 @ T3 ) )
      = ( ? [U: set_Pr4275752383657305402tate_a] :
            ( ( ord_le7345504482307493082tate_a @ U @ T3 )
            & ( inj_on4126798524766655601tate_a @ F2 @ U )
            & ( S4
              = ( image_1917819122788559109tate_a @ F2 @ U ) ) ) ) ) ).

% subset_image_inj
thf(fact_915_sup_OcoboundedI2,axiom,
    ! [C2: a > state_a > $o,B: a > state_a > $o,A: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ C2 @ B )
     => ( ord_le5391092903712282779te_a_o @ C2 @ ( sup_sup_a_state_a_o @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_916_sup_OcoboundedI2,axiom,
    ! [C2: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ C2 @ B )
     => ( ord_le7345504482307493082tate_a @ C2 @ ( sup_su1662690461080552718tate_a @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_917_UnCI,axiom,
    ! [C2: produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( ~ ( member1329951143264900189te_a_a @ C2 @ B5 )
       => ( member1329951143264900189te_a_a @ C2 @ A5 ) )
     => ( member1329951143264900189te_a_a @ C2 @ ( sup_su136906189548161384te_a_a @ A5 @ B5 ) ) ) ).

% UnCI
thf(fact_918_UnCI,axiom,
    ! [C2: produc4774439622479004775tate_a,B5: set_Pr1753757909640466247tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( ~ ( member3801464090328701328tate_a @ C2 @ B5 )
       => ( member3801464090328701328tate_a @ C2 @ A5 ) )
     => ( member3801464090328701328tate_a @ C2 @ ( sup_su7146074582217225499tate_a @ A5 @ B5 ) ) ) ).

% UnCI
thf(fact_919_UnCI,axiom,
    ! [C2: produc5940350820068290983tate_a,B5: set_Pr7963165934123475591tate_a,A5: set_Pr7963165934123475591tate_a] :
      ( ( ~ ( member1595334363549574352tate_a @ C2 @ B5 )
       => ( member1595334363549574352tate_a @ C2 @ A5 ) )
     => ( member1595334363549574352tate_a @ C2 @ ( sup_su7932848497031159899tate_a @ A5 @ B5 ) ) ) ).

% UnCI
thf(fact_920_UnCI,axiom,
    ! [C2: product_prod_a_a,B5: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( ~ ( member1426531477525435216od_a_a @ C2 @ B5 )
       => ( member1426531477525435216od_a_a @ C2 @ A5 ) )
     => ( member1426531477525435216od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) ) ) ).

% UnCI
thf(fact_921_UnCI,axiom,
    ! [C2: produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( ~ ( member3175992478928454403tate_a @ C2 @ B5 )
       => ( member3175992478928454403tate_a @ C2 @ A5 ) )
     => ( member3175992478928454403tate_a @ C2 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) ) ) ).

% UnCI
thf(fact_922_Un__iff,axiom,
    ! [C2: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ C2 @ ( sup_su136906189548161384te_a_a @ A5 @ B5 ) )
      = ( ( member1329951143264900189te_a_a @ C2 @ A5 )
        | ( member1329951143264900189te_a_a @ C2 @ B5 ) ) ) ).

% Un_iff
thf(fact_923_Un__iff,axiom,
    ! [C2: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a,B5: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ C2 @ ( sup_su7146074582217225499tate_a @ A5 @ B5 ) )
      = ( ( member3801464090328701328tate_a @ C2 @ A5 )
        | ( member3801464090328701328tate_a @ C2 @ B5 ) ) ) ).

% Un_iff
thf(fact_924_Un__iff,axiom,
    ! [C2: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a,B5: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ C2 @ ( sup_su7932848497031159899tate_a @ A5 @ B5 ) )
      = ( ( member1595334363549574352tate_a @ C2 @ A5 )
        | ( member1595334363549574352tate_a @ C2 @ B5 ) ) ) ).

% Un_iff
thf(fact_925_Un__iff,axiom,
    ! [C2: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) )
      = ( ( member1426531477525435216od_a_a @ C2 @ A5 )
        | ( member1426531477525435216od_a_a @ C2 @ B5 ) ) ) ).

% Un_iff
thf(fact_926_Un__iff,axiom,
    ! [C2: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ C2 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) )
      = ( ( member3175992478928454403tate_a @ C2 @ A5 )
        | ( member3175992478928454403tate_a @ C2 @ B5 ) ) ) ).

% Un_iff
thf(fact_927_UnE,axiom,
    ! [C2: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ C2 @ ( sup_su136906189548161384te_a_a @ A5 @ B5 ) )
     => ( ~ ( member1329951143264900189te_a_a @ C2 @ A5 )
       => ( member1329951143264900189te_a_a @ C2 @ B5 ) ) ) ).

% UnE
thf(fact_928_UnE,axiom,
    ! [C2: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a,B5: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ C2 @ ( sup_su7146074582217225499tate_a @ A5 @ B5 ) )
     => ( ~ ( member3801464090328701328tate_a @ C2 @ A5 )
       => ( member3801464090328701328tate_a @ C2 @ B5 ) ) ) ).

% UnE
thf(fact_929_UnE,axiom,
    ! [C2: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a,B5: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ C2 @ ( sup_su7932848497031159899tate_a @ A5 @ B5 ) )
     => ( ~ ( member1595334363549574352tate_a @ C2 @ A5 )
       => ( member1595334363549574352tate_a @ C2 @ B5 ) ) ) ).

% UnE
thf(fact_930_UnE,axiom,
    ! [C2: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) )
     => ( ~ ( member1426531477525435216od_a_a @ C2 @ A5 )
       => ( member1426531477525435216od_a_a @ C2 @ B5 ) ) ) ).

% UnE
thf(fact_931_UnE,axiom,
    ! [C2: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ C2 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) )
     => ( ~ ( member3175992478928454403tate_a @ C2 @ A5 )
       => ( member3175992478928454403tate_a @ C2 @ B5 ) ) ) ).

% UnE
thf(fact_932_UnI1,axiom,
    ! [C2: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ C2 @ A5 )
     => ( member1329951143264900189te_a_a @ C2 @ ( sup_su136906189548161384te_a_a @ A5 @ B5 ) ) ) ).

% UnI1
thf(fact_933_UnI1,axiom,
    ! [C2: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a,B5: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ C2 @ A5 )
     => ( member3801464090328701328tate_a @ C2 @ ( sup_su7146074582217225499tate_a @ A5 @ B5 ) ) ) ).

% UnI1
thf(fact_934_UnI1,axiom,
    ! [C2: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a,B5: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ C2 @ A5 )
     => ( member1595334363549574352tate_a @ C2 @ ( sup_su7932848497031159899tate_a @ A5 @ B5 ) ) ) ).

% UnI1
thf(fact_935_UnI1,axiom,
    ! [C2: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C2 @ A5 )
     => ( member1426531477525435216od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) ) ) ).

% UnI1
thf(fact_936_UnI1,axiom,
    ! [C2: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ C2 @ A5 )
     => ( member3175992478928454403tate_a @ C2 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) ) ) ).

% UnI1
thf(fact_937_UnI2,axiom,
    ! [C2: produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ C2 @ B5 )
     => ( member1329951143264900189te_a_a @ C2 @ ( sup_su136906189548161384te_a_a @ A5 @ B5 ) ) ) ).

% UnI2
thf(fact_938_UnI2,axiom,
    ! [C2: produc4774439622479004775tate_a,B5: set_Pr1753757909640466247tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ C2 @ B5 )
     => ( member3801464090328701328tate_a @ C2 @ ( sup_su7146074582217225499tate_a @ A5 @ B5 ) ) ) ).

% UnI2
thf(fact_939_UnI2,axiom,
    ! [C2: produc5940350820068290983tate_a,B5: set_Pr7963165934123475591tate_a,A5: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ C2 @ B5 )
     => ( member1595334363549574352tate_a @ C2 @ ( sup_su7932848497031159899tate_a @ A5 @ B5 ) ) ) ).

% UnI2
thf(fact_940_UnI2,axiom,
    ! [C2: product_prod_a_a,B5: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C2 @ B5 )
     => ( member1426531477525435216od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) ) ) ).

% UnI2
thf(fact_941_UnI2,axiom,
    ! [C2: produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ C2 @ B5 )
     => ( member3175992478928454403tate_a @ C2 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) ) ) ).

% UnI2
thf(fact_942_bex__Un,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,P2: produc6972303929186420058tate_a > $o] :
      ( ( ? [X4: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X4 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [X4: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X4 @ A5 )
            & ( P2 @ X4 ) )
        | ? [X4: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X4 @ B5 )
            & ( P2 @ X4 ) ) ) ) ).

% bex_Un
thf(fact_943_ball__Un,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,P2: produc6972303929186420058tate_a > $o] :
      ( ( ! [X4: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X4 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) )
           => ( P2 @ X4 ) ) )
      = ( ! [X4: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X4 @ A5 )
           => ( P2 @ X4 ) )
        & ! [X4: produc6972303929186420058tate_a] :
            ( ( member3175992478928454403tate_a @ X4 @ B5 )
           => ( P2 @ X4 ) ) ) ) ).

% ball_Un
thf(fact_944_Un__assoc,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,C3: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) @ C3 )
      = ( sup_su1662690461080552718tate_a @ A5 @ ( sup_su1662690461080552718tate_a @ B5 @ C3 ) ) ) ).

% Un_assoc
thf(fact_945_Un__absorb,axiom,
    ! [A5: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ A5 @ A5 )
      = A5 ) ).

% Un_absorb
thf(fact_946_Un__commute,axiom,
    ( sup_su1662690461080552718tate_a
    = ( ^ [A6: set_Pr4275752383657305402tate_a,B6: set_Pr4275752383657305402tate_a] : ( sup_su1662690461080552718tate_a @ B6 @ A6 ) ) ) ).

% Un_commute
thf(fact_947_Un__left__absorb,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ A5 @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) )
      = ( sup_su1662690461080552718tate_a @ A5 @ B5 ) ) ).

% Un_left_absorb
thf(fact_948_Un__left__commute,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,C3: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ A5 @ ( sup_su1662690461080552718tate_a @ B5 @ C3 ) )
      = ( sup_su1662690461080552718tate_a @ B5 @ ( sup_su1662690461080552718tate_a @ A5 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_949_inf__sup__ord_I4_J,axiom,
    ! [Y: a > state_a > $o,X3: a > state_a > $o] : ( ord_le5391092903712282779te_a_o @ Y @ ( sup_sup_a_state_a_o @ X3 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_950_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr4275752383657305402tate_a,X3: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ Y @ ( sup_su1662690461080552718tate_a @ X3 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_951_inf__sup__ord_I3_J,axiom,
    ! [X3: a > state_a > $o,Y: a > state_a > $o] : ( ord_le5391092903712282779te_a_o @ X3 @ ( sup_sup_a_state_a_o @ X3 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_952_inf__sup__ord_I3_J,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ X3 @ ( sup_su1662690461080552718tate_a @ X3 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_953_le__supE,axiom,
    ! [A: a > state_a > $o,B: a > state_a > $o,X3: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ ( sup_sup_a_state_a_o @ A @ B ) @ X3 )
     => ~ ( ( ord_le5391092903712282779te_a_o @ A @ X3 )
         => ~ ( ord_le5391092903712282779te_a_o @ B @ X3 ) ) ) ).

% le_supE
thf(fact_954_le__supE,axiom,
    ! [A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a,X3: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( sup_su1662690461080552718tate_a @ A @ B ) @ X3 )
     => ~ ( ( ord_le7345504482307493082tate_a @ A @ X3 )
         => ~ ( ord_le7345504482307493082tate_a @ B @ X3 ) ) ) ).

% le_supE
thf(fact_955_le__supI,axiom,
    ! [A: a > state_a > $o,X3: a > state_a > $o,B: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ A @ X3 )
     => ( ( ord_le5391092903712282779te_a_o @ B @ X3 )
       => ( ord_le5391092903712282779te_a_o @ ( sup_sup_a_state_a_o @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_956_le__supI,axiom,
    ! [A: set_Pr4275752383657305402tate_a,X3: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A @ X3 )
     => ( ( ord_le7345504482307493082tate_a @ B @ X3 )
       => ( ord_le7345504482307493082tate_a @ ( sup_su1662690461080552718tate_a @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_957_sup__ge1,axiom,
    ! [X3: a > state_a > $o,Y: a > state_a > $o] : ( ord_le5391092903712282779te_a_o @ X3 @ ( sup_sup_a_state_a_o @ X3 @ Y ) ) ).

% sup_ge1
thf(fact_958_sup__ge1,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ X3 @ ( sup_su1662690461080552718tate_a @ X3 @ Y ) ) ).

% sup_ge1
thf(fact_959_sup__ge2,axiom,
    ! [Y: a > state_a > $o,X3: a > state_a > $o] : ( ord_le5391092903712282779te_a_o @ Y @ ( sup_sup_a_state_a_o @ X3 @ Y ) ) ).

% sup_ge2
thf(fact_960_sup__ge2,axiom,
    ! [Y: set_Pr4275752383657305402tate_a,X3: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ Y @ ( sup_su1662690461080552718tate_a @ X3 @ Y ) ) ).

% sup_ge2
thf(fact_961_le__supI1,axiom,
    ! [X3: a > state_a > $o,A: a > state_a > $o,B: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ X3 @ A )
     => ( ord_le5391092903712282779te_a_o @ X3 @ ( sup_sup_a_state_a_o @ A @ B ) ) ) ).

% le_supI1
thf(fact_962_le__supI1,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ X3 @ A )
     => ( ord_le7345504482307493082tate_a @ X3 @ ( sup_su1662690461080552718tate_a @ A @ B ) ) ) ).

% le_supI1
thf(fact_963_le__supI2,axiom,
    ! [X3: a > state_a > $o,B: a > state_a > $o,A: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ X3 @ B )
     => ( ord_le5391092903712282779te_a_o @ X3 @ ( sup_sup_a_state_a_o @ A @ B ) ) ) ).

% le_supI2
thf(fact_964_le__supI2,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ X3 @ B )
     => ( ord_le7345504482307493082tate_a @ X3 @ ( sup_su1662690461080552718tate_a @ A @ B ) ) ) ).

% le_supI2
thf(fact_965_sup_Omono,axiom,
    ! [C2: a > state_a > $o,A: a > state_a > $o,D2: a > state_a > $o,B: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ C2 @ A )
     => ( ( ord_le5391092903712282779te_a_o @ D2 @ B )
       => ( ord_le5391092903712282779te_a_o @ ( sup_sup_a_state_a_o @ C2 @ D2 ) @ ( sup_sup_a_state_a_o @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_966_sup_Omono,axiom,
    ! [C2: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a,D2: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ C2 @ A )
     => ( ( ord_le7345504482307493082tate_a @ D2 @ B )
       => ( ord_le7345504482307493082tate_a @ ( sup_su1662690461080552718tate_a @ C2 @ D2 ) @ ( sup_su1662690461080552718tate_a @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_967_sup__mono,axiom,
    ! [A: a > state_a > $o,C2: a > state_a > $o,B: a > state_a > $o,D2: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ A @ C2 )
     => ( ( ord_le5391092903712282779te_a_o @ B @ D2 )
       => ( ord_le5391092903712282779te_a_o @ ( sup_sup_a_state_a_o @ A @ B ) @ ( sup_sup_a_state_a_o @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_968_sup__mono,axiom,
    ! [A: set_Pr4275752383657305402tate_a,C2: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a,D2: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A @ C2 )
     => ( ( ord_le7345504482307493082tate_a @ B @ D2 )
       => ( ord_le7345504482307493082tate_a @ ( sup_su1662690461080552718tate_a @ A @ B ) @ ( sup_su1662690461080552718tate_a @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_969_sup__least,axiom,
    ! [Y: a > state_a > $o,X3: a > state_a > $o,Z2: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ Y @ X3 )
     => ( ( ord_le5391092903712282779te_a_o @ Z2 @ X3 )
       => ( ord_le5391092903712282779te_a_o @ ( sup_sup_a_state_a_o @ Y @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_970_sup__least,axiom,
    ! [Y: set_Pr4275752383657305402tate_a,X3: set_Pr4275752383657305402tate_a,Z2: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ Y @ X3 )
     => ( ( ord_le7345504482307493082tate_a @ Z2 @ X3 )
       => ( ord_le7345504482307493082tate_a @ ( sup_su1662690461080552718tate_a @ Y @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_971_le__iff__sup,axiom,
    ( ord_le5391092903712282779te_a_o
    = ( ^ [X4: a > state_a > $o,Y6: a > state_a > $o] :
          ( ( sup_sup_a_state_a_o @ X4 @ Y6 )
          = Y6 ) ) ) ).

% le_iff_sup
thf(fact_972_le__iff__sup,axiom,
    ( ord_le7345504482307493082tate_a
    = ( ^ [X4: set_Pr4275752383657305402tate_a,Y6: set_Pr4275752383657305402tate_a] :
          ( ( sup_su1662690461080552718tate_a @ X4 @ Y6 )
          = Y6 ) ) ) ).

% le_iff_sup
thf(fact_973_sup_OorderE,axiom,
    ! [B: a > state_a > $o,A: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ B @ A )
     => ( A
        = ( sup_sup_a_state_a_o @ A @ B ) ) ) ).

% sup.orderE
thf(fact_974_sup_OorderE,axiom,
    ! [B: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ B @ A )
     => ( A
        = ( sup_su1662690461080552718tate_a @ A @ B ) ) ) ).

% sup.orderE
thf(fact_975_sup_OorderI,axiom,
    ! [A: a > state_a > $o,B: a > state_a > $o] :
      ( ( A
        = ( sup_sup_a_state_a_o @ A @ B ) )
     => ( ord_le5391092903712282779te_a_o @ B @ A ) ) ).

% sup.orderI
thf(fact_976_sup_OorderI,axiom,
    ! [A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a] :
      ( ( A
        = ( sup_su1662690461080552718tate_a @ A @ B ) )
     => ( ord_le7345504482307493082tate_a @ B @ A ) ) ).

% sup.orderI
thf(fact_977_sup__unique,axiom,
    ! [F2: ( a > state_a > $o ) > ( a > state_a > $o ) > a > state_a > $o,X3: a > state_a > $o,Y: a > state_a > $o] :
      ( ! [X: a > state_a > $o,Y3: a > state_a > $o] : ( ord_le5391092903712282779te_a_o @ X @ ( F2 @ X @ Y3 ) )
     => ( ! [X: a > state_a > $o,Y3: a > state_a > $o] : ( ord_le5391092903712282779te_a_o @ Y3 @ ( F2 @ X @ Y3 ) )
       => ( ! [X: a > state_a > $o,Y3: a > state_a > $o,Z3: a > state_a > $o] :
              ( ( ord_le5391092903712282779te_a_o @ Y3 @ X )
             => ( ( ord_le5391092903712282779te_a_o @ Z3 @ X )
               => ( ord_le5391092903712282779te_a_o @ ( F2 @ Y3 @ Z3 ) @ X ) ) )
         => ( ( sup_sup_a_state_a_o @ X3 @ Y )
            = ( F2 @ X3 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_978_sup__unique,axiom,
    ! [F2: set_Pr4275752383657305402tate_a > set_Pr4275752383657305402tate_a > set_Pr4275752383657305402tate_a,X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a] :
      ( ! [X: set_Pr4275752383657305402tate_a,Y3: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ X @ ( F2 @ X @ Y3 ) )
     => ( ! [X: set_Pr4275752383657305402tate_a,Y3: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ Y3 @ ( F2 @ X @ Y3 ) )
       => ( ! [X: set_Pr4275752383657305402tate_a,Y3: set_Pr4275752383657305402tate_a,Z3: set_Pr4275752383657305402tate_a] :
              ( ( ord_le7345504482307493082tate_a @ Y3 @ X )
             => ( ( ord_le7345504482307493082tate_a @ Z3 @ X )
               => ( ord_le7345504482307493082tate_a @ ( F2 @ Y3 @ Z3 ) @ X ) ) )
         => ( ( sup_su1662690461080552718tate_a @ X3 @ Y )
            = ( F2 @ X3 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_979_sup_Oabsorb1,axiom,
    ! [B: a > state_a > $o,A: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ B @ A )
     => ( ( sup_sup_a_state_a_o @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_980_sup_Oabsorb1,axiom,
    ! [B: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ B @ A )
     => ( ( sup_su1662690461080552718tate_a @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_981_sup_Oabsorb2,axiom,
    ! [A: a > state_a > $o,B: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ A @ B )
     => ( ( sup_sup_a_state_a_o @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_982_sup_Oabsorb2,axiom,
    ! [A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A @ B )
     => ( ( sup_su1662690461080552718tate_a @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_983_sup__absorb1,axiom,
    ! [Y: a > state_a > $o,X3: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ Y @ X3 )
     => ( ( sup_sup_a_state_a_o @ X3 @ Y )
        = X3 ) ) ).

% sup_absorb1
thf(fact_984_sup__absorb1,axiom,
    ! [Y: set_Pr4275752383657305402tate_a,X3: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ Y @ X3 )
     => ( ( sup_su1662690461080552718tate_a @ X3 @ Y )
        = X3 ) ) ).

% sup_absorb1
thf(fact_985_sup__absorb2,axiom,
    ! [X3: a > state_a > $o,Y: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ X3 @ Y )
     => ( ( sup_sup_a_state_a_o @ X3 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_986_sup__absorb2,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,Y: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ X3 @ Y )
     => ( ( sup_su1662690461080552718tate_a @ X3 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_987_sup_OboundedE,axiom,
    ! [B: a > state_a > $o,C2: a > state_a > $o,A: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ ( sup_sup_a_state_a_o @ B @ C2 ) @ A )
     => ~ ( ( ord_le5391092903712282779te_a_o @ B @ A )
         => ~ ( ord_le5391092903712282779te_a_o @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_988_sup_OboundedE,axiom,
    ! [B: set_Pr4275752383657305402tate_a,C2: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( sup_su1662690461080552718tate_a @ B @ C2 ) @ A )
     => ~ ( ( ord_le7345504482307493082tate_a @ B @ A )
         => ~ ( ord_le7345504482307493082tate_a @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_989_sup_OboundedI,axiom,
    ! [B: a > state_a > $o,A: a > state_a > $o,C2: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ B @ A )
     => ( ( ord_le5391092903712282779te_a_o @ C2 @ A )
       => ( ord_le5391092903712282779te_a_o @ ( sup_sup_a_state_a_o @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_990_sup_OboundedI,axiom,
    ! [B: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a,C2: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ B @ A )
     => ( ( ord_le7345504482307493082tate_a @ C2 @ A )
       => ( ord_le7345504482307493082tate_a @ ( sup_su1662690461080552718tate_a @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_991_sup_Oorder__iff,axiom,
    ( ord_le5391092903712282779te_a_o
    = ( ^ [B4: a > state_a > $o,A4: a > state_a > $o] :
          ( A4
          = ( sup_sup_a_state_a_o @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_992_sup_Oorder__iff,axiom,
    ( ord_le7345504482307493082tate_a
    = ( ^ [B4: set_Pr4275752383657305402tate_a,A4: set_Pr4275752383657305402tate_a] :
          ( A4
          = ( sup_su1662690461080552718tate_a @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_993_sup_Ocobounded1,axiom,
    ! [A: a > state_a > $o,B: a > state_a > $o] : ( ord_le5391092903712282779te_a_o @ A @ ( sup_sup_a_state_a_o @ A @ B ) ) ).

% sup.cobounded1
thf(fact_994_sup_Ocobounded1,axiom,
    ! [A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ A @ ( sup_su1662690461080552718tate_a @ A @ B ) ) ).

% sup.cobounded1
thf(fact_995_sup_Ocobounded2,axiom,
    ! [B: a > state_a > $o,A: a > state_a > $o] : ( ord_le5391092903712282779te_a_o @ B @ ( sup_sup_a_state_a_o @ A @ B ) ) ).

% sup.cobounded2
thf(fact_996_sup_Ocobounded2,axiom,
    ! [B: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a] : ( ord_le7345504482307493082tate_a @ B @ ( sup_su1662690461080552718tate_a @ A @ B ) ) ).

% sup.cobounded2
thf(fact_997_sup_Oabsorb__iff1,axiom,
    ( ord_le5391092903712282779te_a_o
    = ( ^ [B4: a > state_a > $o,A4: a > state_a > $o] :
          ( ( sup_sup_a_state_a_o @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_998_sup_Oabsorb__iff1,axiom,
    ( ord_le7345504482307493082tate_a
    = ( ^ [B4: set_Pr4275752383657305402tate_a,A4: set_Pr4275752383657305402tate_a] :
          ( ( sup_su1662690461080552718tate_a @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_999_sup_Oabsorb__iff2,axiom,
    ( ord_le5391092903712282779te_a_o
    = ( ^ [A4: a > state_a > $o,B4: a > state_a > $o] :
          ( ( sup_sup_a_state_a_o @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_1000_sup_Oabsorb__iff2,axiom,
    ( ord_le7345504482307493082tate_a
    = ( ^ [A4: set_Pr4275752383657305402tate_a,B4: set_Pr4275752383657305402tate_a] :
          ( ( sup_su1662690461080552718tate_a @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_1001_sup_OcoboundedI1,axiom,
    ! [C2: a > state_a > $o,A: a > state_a > $o,B: a > state_a > $o] :
      ( ( ord_le5391092903712282779te_a_o @ C2 @ A )
     => ( ord_le5391092903712282779te_a_o @ C2 @ ( sup_sup_a_state_a_o @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1002_sup_OcoboundedI1,axiom,
    ! [C2: set_Pr4275752383657305402tate_a,A: set_Pr4275752383657305402tate_a,B: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ C2 @ A )
     => ( ord_le7345504482307493082tate_a @ C2 @ ( sup_su1662690461080552718tate_a @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1003_the__inv__into__into,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,X3: produc6972303929186420058tate_a,B5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( ( member3175992478928454403tate_a @ X3 @ ( image_5502445033238569825tate_a @ F2 @ A5 ) )
       => ( ( ord_less_eq_set_a @ A5 @ B5 )
         => ( member_a @ ( the_in5147833725387914767tate_a @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1004_the__inv__into__into,axiom,
    ! [F2: product_prod_a_a > product_prod_a_a,A5: set_Product_prod_a_a,X3: product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ( inj_on2566144670800592689od_a_a @ F2 @ A5 )
     => ( ( member1426531477525435216od_a_a @ X3 @ ( image_4636654165204879301od_a_a @ F2 @ A5 ) )
       => ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
         => ( member1426531477525435216od_a_a @ ( the_in6395559570202473331od_a_a @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1005_the__inv__into__into,axiom,
    ! [F2: product_prod_a_a > produc6972303929186420058tate_a,A5: set_Product_prod_a_a,X3: produc6972303929186420058tate_a,B5: set_Product_prod_a_a] :
      ( ( inj_on7244109900603427428tate_a @ F2 @ A5 )
     => ( ( member3175992478928454403tate_a @ X3 @ ( image_1563038779069872376tate_a @ F2 @ A5 ) )
       => ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
         => ( member1426531477525435216od_a_a @ ( the_in3243614416948917670tate_a @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1006_the__inv__into__into,axiom,
    ! [F2: product_prod_a_a > produc5126262593522865844te_a_a,A5: set_Product_prod_a_a,X3: produc5126262593522865844te_a_a,B5: set_Product_prod_a_a] :
      ( ( inj_on5398068564939873214te_a_a @ F2 @ A5 )
     => ( ( member1329951143264900189te_a_a @ X3 @ ( image_8940369480261093970te_a_a @ F2 @ A5 ) )
       => ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
         => ( member1426531477525435216od_a_a @ ( the_in1397573081285363456te_a_a @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1007_the__inv__into__into,axiom,
    ! [F2: produc5126262593522865844te_a_a > product_prod_a_a,A5: set_Pr2749968112124914068te_a_a,X3: product_prod_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6297019480273606372od_a_a @ F2 @ A5 )
     => ( ( member1426531477525435216od_a_a @ X3 @ ( image_615948358740051320od_a_a @ F2 @ A5 ) )
       => ( ( ord_le5819720210775101748te_a_a @ A5 @ B5 )
         => ( member1329951143264900189te_a_a @ ( the_in2296523996619096614od_a_a @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1008_the__inv__into__into,axiom,
    ! [F2: produc6972303929186420058tate_a > product_prod_a_a,A5: set_Pr4275752383657305402tate_a,X3: product_prod_a_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4637469111408784318od_a_a @ F2 @ A5 )
     => ( ( member1426531477525435216od_a_a @ X3 @ ( image_8179770026730005074od_a_a @ F2 @ A5 ) )
       => ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
         => ( member3175992478928454403tate_a @ ( the_in636973627754274560od_a_a @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1009_the__inv__into__into,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,X3: produc6972303929186420058tate_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( ( member3175992478928454403tate_a @ X3 @ ( image_4497581360776675627tate_a @ F2 @ A5 ) )
       => ( ( ord_le5819720210775101748te_a_a @ A5 @ B5 )
         => ( member1329951143264900189te_a_a @ ( the_in3438972003123105497tate_a @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1010_the__inv__into__into,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,X3: produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on4860519427091217905te_a_a @ F2 @ A5 )
     => ( ( member1329951143264900189te_a_a @ X3 @ ( image_2651540025113121413te_a_a @ F2 @ A5 ) )
       => ( ( ord_le5819720210775101748te_a_a @ A5 @ B5 )
         => ( member1329951143264900189te_a_a @ ( the_in1592930667459551283te_a_a @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1011_the__inv__into__into,axiom,
    ! [F2: product_prod_a_a > produc4774439622479004775tate_a,A5: set_Product_prod_a_a,X3: produc4774439622479004775tate_a,B5: set_Product_prod_a_a] :
      ( ( inj_on4003527337536797809tate_a @ F2 @ A5 )
     => ( ( member3801464090328701328tate_a @ X3 @ ( image_1794547935558701317tate_a @ F2 @ A5 ) )
       => ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
         => ( member1426531477525435216od_a_a @ ( the_in735938577905131187tate_a @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1012_the__inv__into__into,axiom,
    ! [F2: produc4774439622479004775tate_a > product_prod_a_a,A5: set_Pr1753757909640466247tate_a,X3: product_prod_a_a,B5: set_Pr1753757909640466247tate_a] :
      ( ( inj_on4099919973560129009od_a_a @ F2 @ A5 )
     => ( ( member1426531477525435216od_a_a @ X3 @ ( image_1890940571582032517od_a_a @ F2 @ A5 ) )
       => ( ( ord_le5561929874677720807tate_a @ A5 @ B5 )
         => ( member3801464090328701328tate_a @ ( the_in832331213928462387od_a_a @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_1013_wf__map__prod__image,axiom,
    ! [R: set_Product_prod_a_a,F2: a > produc6972303929186420058tate_a] :
      ( ( wf_a @ R )
     => ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
       => ( wf_Pro1739528972846749379tate_a @ ( image_177899513600418373tate_a @ ( produc3303943909295981742tate_a @ F2 @ F2 ) @ R ) ) ) ) ).

% wf_map_prod_image
thf(fact_1014_wf__map__prod__image,axiom,
    ! [R: set_Pr4734257497606733575te_a_a,F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a] :
      ( ( wf_Pro9116859674037970973te_a_a @ R )
     => ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
       => ( wf_Pro1739528972846749379tate_a @ ( image_8117903017743089477tate_a @ ( produc1508824635387789378tate_a @ F2 @ F2 ) @ R ) ) ) ) ).

% wf_map_prod_image
thf(fact_1015_the__inv__into__onto,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ A5 )
     => ( ( image_4497581360776675627tate_a @ ( the_in8236540466326210573te_a_a @ A5 @ F2 ) @ ( image_71777787125004895te_a_a @ F2 @ A5 ) )
        = A5 ) ) ).

% the_inv_into_onto
thf(fact_1016_the__inv__into__onto,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ A5 )
     => ( ( image_1917819122788559109tate_a @ ( the_in859209765134988979tate_a @ A5 @ F2 ) @ ( image_1917819122788559109tate_a @ F2 @ A5 ) )
        = A5 ) ) ).

% the_inv_into_onto
thf(fact_1017_the__inv__into__onto,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( ( image_2134324207772891945te_a_a @ ( the_in5147833725387914767tate_a @ A5 @ F2 ) @ ( image_5502445033238569825tate_a @ F2 @ A5 ) )
        = A5 ) ) ).

% the_inv_into_onto
thf(fact_1018_the__inv__into__onto,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( ( image_71777787125004895te_a_a @ ( the_in3438972003123105497tate_a @ A5 @ F2 ) @ ( image_4497581360776675627tate_a @ F2 @ A5 ) )
        = A5 ) ) ).

% the_inv_into_onto
thf(fact_1019_wf__def,axiom,
    ( wf_state_a
    = ( ^ [R3: set_Pr1753757909640466247tate_a] :
        ! [P4: state_a > $o] :
          ( ! [X4: state_a] :
              ( ! [Y6: state_a] :
                  ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y6 @ X4 ) @ R3 )
                 => ( P4 @ Y6 ) )
             => ( P4 @ X4 ) )
         => ! [X7: state_a] : ( P4 @ X7 ) ) ) ) ).

% wf_def
thf(fact_1020_wf__def,axiom,
    ( wf_Pro1739528972846749379tate_a
    = ( ^ [R3: set_Pr7963165934123475591tate_a] :
        ! [P4: produc6972303929186420058tate_a > $o] :
          ( ! [X4: produc6972303929186420058tate_a] :
              ( ! [Y6: produc6972303929186420058tate_a] :
                  ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y6 @ X4 ) @ R3 )
                 => ( P4 @ Y6 ) )
             => ( P4 @ X4 ) )
         => ! [X7: produc6972303929186420058tate_a] : ( P4 @ X7 ) ) ) ) ).

% wf_def
thf(fact_1021_wf__def,axiom,
    ( wf_a
    = ( ^ [R3: set_Product_prod_a_a] :
        ! [P4: a > $o] :
          ( ! [X4: a] :
              ( ! [Y6: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y6 @ X4 ) @ R3 )
                 => ( P4 @ Y6 ) )
             => ( P4 @ X4 ) )
         => ! [X7: a] : ( P4 @ X7 ) ) ) ) ).

% wf_def
thf(fact_1022_wfE__min,axiom,
    ! [R2: set_Pr4734257497606733575te_a_a,X3: produc5126262593522865844te_a_a,Q2: set_Pr2749968112124914068te_a_a] :
      ( ( wf_Pro9116859674037970973te_a_a @ R2 )
     => ( ( member1329951143264900189te_a_a @ X3 @ Q2 )
       => ~ ! [Z3: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ Z3 @ Q2 )
             => ~ ! [Y8: produc5126262593522865844te_a_a] :
                    ( ( member2649312330005299536te_a_a @ ( produc2877451964877216791te_a_a @ Y8 @ Z3 ) @ R2 )
                   => ~ ( member1329951143264900189te_a_a @ Y8 @ Q2 ) ) ) ) ) ).

% wfE_min
thf(fact_1023_wfE__min,axiom,
    ! [R2: set_Pr2656942177312230599tate_a,X3: produc4774439622479004775tate_a,Q2: set_Pr1753757909640466247tate_a] :
      ( ( wf_Pro7444320194016922960tate_a @ R2 )
     => ( ( member3801464090328701328tate_a @ X3 @ Q2 )
       => ~ ! [Z3: produc4774439622479004775tate_a] :
              ( ( member3801464090328701328tate_a @ Z3 @ Q2 )
             => ~ ! [Y8: produc4774439622479004775tate_a] :
                    ( ( member7923945571256933136tate_a @ ( produc1588899729123023703tate_a @ Y8 @ Z3 ) @ R2 )
                   => ~ ( member3801464090328701328tate_a @ Y8 @ Q2 ) ) ) ) ) ).

% wfE_min
thf(fact_1024_wfE__min,axiom,
    ! [R2: set_Pr1690608023759948999tate_a,X3: produc5940350820068290983tate_a,Q2: set_Pr7963165934123475591tate_a] :
      ( ( wf_Pro740797393413888144tate_a @ R2 )
     => ( ( member1595334363549574352tate_a @ X3 @ Q2 )
       => ~ ! [Z3: produc5940350820068290983tate_a] :
              ( ( member1595334363549574352tate_a @ Z3 @ Q2 )
             => ~ ! [Y8: produc5940350820068290983tate_a] :
                    ( ( member1643692453717642000tate_a @ ( produc7335381768849070935tate_a @ Y8 @ Z3 ) @ R2 )
                   => ~ ( member1595334363549574352tate_a @ Y8 @ Q2 ) ) ) ) ) ).

% wfE_min
thf(fact_1025_wfE__min,axiom,
    ! [R2: set_Pr8600417178894128327od_a_a,X3: product_prod_a_a,Q2: set_Product_prod_a_a] :
      ( ( wf_Product_prod_a_a @ R2 )
     => ( ( member1426531477525435216od_a_a @ X3 @ Q2 )
       => ~ ! [Z3: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ Z3 @ Q2 )
             => ~ ! [Y8: product_prod_a_a] :
                    ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y8 @ Z3 ) @ R2 )
                   => ~ ( member1426531477525435216od_a_a @ Y8 @ Q2 ) ) ) ) ) ).

% wfE_min
thf(fact_1026_wfE__min,axiom,
    ! [R2: set_Pr1753757909640466247tate_a,X3: state_a,Q2: set_state_a] :
      ( ( wf_state_a @ R2 )
     => ( ( member_state_a @ X3 @ Q2 )
       => ~ ! [Z3: state_a] :
              ( ( member_state_a @ Z3 @ Q2 )
             => ~ ! [Y8: state_a] :
                    ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y8 @ Z3 ) @ R2 )
                   => ~ ( member_state_a @ Y8 @ Q2 ) ) ) ) ) ).

% wfE_min
thf(fact_1027_wfE__min,axiom,
    ! [R2: set_Pr7963165934123475591tate_a,X3: produc6972303929186420058tate_a,Q2: set_Pr4275752383657305402tate_a] :
      ( ( wf_Pro1739528972846749379tate_a @ R2 )
     => ( ( member3175992478928454403tate_a @ X3 @ Q2 )
       => ~ ! [Z3: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ Z3 @ Q2 )
             => ~ ! [Y8: produc6972303929186420058tate_a] :
                    ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y8 @ Z3 ) @ R2 )
                   => ~ ( member3175992478928454403tate_a @ Y8 @ Q2 ) ) ) ) ) ).

% wfE_min
thf(fact_1028_wfE__min,axiom,
    ! [R2: set_Product_prod_a_a,X3: a,Q2: set_a] :
      ( ( wf_a @ R2 )
     => ( ( member_a @ X3 @ Q2 )
       => ~ ! [Z3: a] :
              ( ( member_a @ Z3 @ Q2 )
             => ~ ! [Y8: a] :
                    ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y8 @ Z3 ) @ R2 )
                   => ~ ( member_a @ Y8 @ Q2 ) ) ) ) ) ).

% wfE_min
thf(fact_1029_wfI__min,axiom,
    ! [R2: set_Pr4734257497606733575te_a_a] :
      ( ! [X: produc5126262593522865844te_a_a,Q4: set_Pr2749968112124914068te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ Q4 )
         => ? [Xa2: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ Xa2 @ Q4 )
              & ! [Y3: produc5126262593522865844te_a_a] :
                  ( ( member2649312330005299536te_a_a @ ( produc2877451964877216791te_a_a @ Y3 @ Xa2 ) @ R2 )
                 => ~ ( member1329951143264900189te_a_a @ Y3 @ Q4 ) ) ) )
     => ( wf_Pro9116859674037970973te_a_a @ R2 ) ) ).

% wfI_min
thf(fact_1030_wfI__min,axiom,
    ! [R2: set_Pr2656942177312230599tate_a] :
      ( ! [X: produc4774439622479004775tate_a,Q4: set_Pr1753757909640466247tate_a] :
          ( ( member3801464090328701328tate_a @ X @ Q4 )
         => ? [Xa2: produc4774439622479004775tate_a] :
              ( ( member3801464090328701328tate_a @ Xa2 @ Q4 )
              & ! [Y3: produc4774439622479004775tate_a] :
                  ( ( member7923945571256933136tate_a @ ( produc1588899729123023703tate_a @ Y3 @ Xa2 ) @ R2 )
                 => ~ ( member3801464090328701328tate_a @ Y3 @ Q4 ) ) ) )
     => ( wf_Pro7444320194016922960tate_a @ R2 ) ) ).

% wfI_min
thf(fact_1031_wfI__min,axiom,
    ! [R2: set_Pr1690608023759948999tate_a] :
      ( ! [X: produc5940350820068290983tate_a,Q4: set_Pr7963165934123475591tate_a] :
          ( ( member1595334363549574352tate_a @ X @ Q4 )
         => ? [Xa2: produc5940350820068290983tate_a] :
              ( ( member1595334363549574352tate_a @ Xa2 @ Q4 )
              & ! [Y3: produc5940350820068290983tate_a] :
                  ( ( member1643692453717642000tate_a @ ( produc7335381768849070935tate_a @ Y3 @ Xa2 ) @ R2 )
                 => ~ ( member1595334363549574352tate_a @ Y3 @ Q4 ) ) ) )
     => ( wf_Pro740797393413888144tate_a @ R2 ) ) ).

% wfI_min
thf(fact_1032_wfI__min,axiom,
    ! [R2: set_Pr8600417178894128327od_a_a] :
      ( ! [X: product_prod_a_a,Q4: set_Product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ Q4 )
         => ? [Xa2: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ Xa2 @ Q4 )
              & ! [Y3: product_prod_a_a] :
                  ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Xa2 ) @ R2 )
                 => ~ ( member1426531477525435216od_a_a @ Y3 @ Q4 ) ) ) )
     => ( wf_Product_prod_a_a @ R2 ) ) ).

% wfI_min
thf(fact_1033_wfI__min,axiom,
    ! [R2: set_Pr1753757909640466247tate_a] :
      ( ! [X: state_a,Q4: set_state_a] :
          ( ( member_state_a @ X @ Q4 )
         => ? [Xa2: state_a] :
              ( ( member_state_a @ Xa2 @ Q4 )
              & ! [Y3: state_a] :
                  ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y3 @ Xa2 ) @ R2 )
                 => ~ ( member_state_a @ Y3 @ Q4 ) ) ) )
     => ( wf_state_a @ R2 ) ) ).

% wfI_min
thf(fact_1034_wfI__min,axiom,
    ! [R2: set_Pr7963165934123475591tate_a] :
      ( ! [X: produc6972303929186420058tate_a,Q4: set_Pr4275752383657305402tate_a] :
          ( ( member3175992478928454403tate_a @ X @ Q4 )
         => ? [Xa2: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ Xa2 @ Q4 )
              & ! [Y3: produc6972303929186420058tate_a] :
                  ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y3 @ Xa2 ) @ R2 )
                 => ~ ( member3175992478928454403tate_a @ Y3 @ Q4 ) ) ) )
     => ( wf_Pro1739528972846749379tate_a @ R2 ) ) ).

% wfI_min
thf(fact_1035_wfI__min,axiom,
    ! [R2: set_Product_prod_a_a] :
      ( ! [X: a,Q4: set_a] :
          ( ( member_a @ X @ Q4 )
         => ? [Xa2: a] :
              ( ( member_a @ Xa2 @ Q4 )
              & ! [Y3: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Xa2 ) @ R2 )
                 => ~ ( member_a @ Y3 @ Q4 ) ) ) )
     => ( wf_a @ R2 ) ) ).

% wfI_min
thf(fact_1036_wfUNIVI,axiom,
    ! [R: set_Pr1753757909640466247tate_a] :
      ( ! [P5: state_a > $o,X: state_a] :
          ( ! [Xa2: state_a] :
              ( ! [Y3: state_a] :
                  ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y3 @ Xa2 ) @ R )
                 => ( P5 @ Y3 ) )
             => ( P5 @ Xa2 ) )
         => ( P5 @ X ) )
     => ( wf_state_a @ R ) ) ).

% wfUNIVI
thf(fact_1037_wfUNIVI,axiom,
    ! [R: set_Pr7963165934123475591tate_a] :
      ( ! [P5: produc6972303929186420058tate_a > $o,X: produc6972303929186420058tate_a] :
          ( ! [Xa2: produc6972303929186420058tate_a] :
              ( ! [Y3: produc6972303929186420058tate_a] :
                  ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y3 @ Xa2 ) @ R )
                 => ( P5 @ Y3 ) )
             => ( P5 @ Xa2 ) )
         => ( P5 @ X ) )
     => ( wf_Pro1739528972846749379tate_a @ R ) ) ).

% wfUNIVI
thf(fact_1038_wfUNIVI,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ! [P5: a > $o,X: a] :
          ( ! [Xa2: a] :
              ( ! [Y3: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Xa2 ) @ R )
                 => ( P5 @ Y3 ) )
             => ( P5 @ Xa2 ) )
         => ( P5 @ X ) )
     => ( wf_a @ R ) ) ).

% wfUNIVI
thf(fact_1039_wf__asym,axiom,
    ! [R: set_Pr1753757909640466247tate_a,A: state_a,X3: state_a] :
      ( ( wf_state_a @ R )
     => ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ A @ X3 ) @ R )
       => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X3 @ A ) @ R ) ) ) ).

% wf_asym
thf(fact_1040_wf__asym,axiom,
    ! [R: set_Pr7963165934123475591tate_a,A: produc6972303929186420058tate_a,X3: produc6972303929186420058tate_a] :
      ( ( wf_Pro1739528972846749379tate_a @ R )
     => ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ A @ X3 ) @ R )
       => ~ ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X3 @ A ) @ R ) ) ) ).

% wf_asym
thf(fact_1041_wf__asym,axiom,
    ! [R: set_Product_prod_a_a,A: a,X3: a] :
      ( ( wf_a @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ X3 ) @ R )
       => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ A ) @ R ) ) ) ).

% wf_asym
thf(fact_1042_wf__induct,axiom,
    ! [R: set_Pr1753757909640466247tate_a,P2: state_a > $o,A: state_a] :
      ( ( wf_state_a @ R )
     => ( ! [X: state_a] :
            ( ! [Y8: state_a] :
                ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y8 @ X ) @ R )
               => ( P2 @ Y8 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wf_induct
thf(fact_1043_wf__induct,axiom,
    ! [R: set_Pr7963165934123475591tate_a,P2: produc6972303929186420058tate_a > $o,A: produc6972303929186420058tate_a] :
      ( ( wf_Pro1739528972846749379tate_a @ R )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ! [Y8: produc6972303929186420058tate_a] :
                ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y8 @ X ) @ R )
               => ( P2 @ Y8 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wf_induct
thf(fact_1044_wf__induct,axiom,
    ! [R: set_Product_prod_a_a,P2: a > $o,A: a] :
      ( ( wf_a @ R )
     => ( ! [X: a] :
            ( ! [Y8: a] :
                ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y8 @ X ) @ R )
               => ( P2 @ Y8 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wf_induct
thf(fact_1045_wf__irrefl,axiom,
    ! [R: set_Pr1753757909640466247tate_a,A: state_a] :
      ( ( wf_state_a @ R )
     => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ A @ A ) @ R ) ) ).

% wf_irrefl
thf(fact_1046_wf__irrefl,axiom,
    ! [R: set_Pr7963165934123475591tate_a,A: produc6972303929186420058tate_a] :
      ( ( wf_Pro1739528972846749379tate_a @ R )
     => ~ ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ A @ A ) @ R ) ) ).

% wf_irrefl
thf(fact_1047_wf__irrefl,axiom,
    ! [R: set_Product_prod_a_a,A: a] :
      ( ( wf_a @ R )
     => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ).

% wf_irrefl
thf(fact_1048_wf__not__sym,axiom,
    ! [R: set_Pr1753757909640466247tate_a,A: state_a,X3: state_a] :
      ( ( wf_state_a @ R )
     => ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ A @ X3 ) @ R )
       => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X3 @ A ) @ R ) ) ) ).

% wf_not_sym
thf(fact_1049_wf__not__sym,axiom,
    ! [R: set_Pr7963165934123475591tate_a,A: produc6972303929186420058tate_a,X3: produc6972303929186420058tate_a] :
      ( ( wf_Pro1739528972846749379tate_a @ R )
     => ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ A @ X3 ) @ R )
       => ~ ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X3 @ A ) @ R ) ) ) ).

% wf_not_sym
thf(fact_1050_wf__not__sym,axiom,
    ! [R: set_Product_prod_a_a,A: a,X3: a] :
      ( ( wf_a @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ X3 ) @ R )
       => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ A ) @ R ) ) ) ).

% wf_not_sym
thf(fact_1051_wf__not__refl,axiom,
    ! [R: set_Pr1753757909640466247tate_a,A: state_a] :
      ( ( wf_state_a @ R )
     => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ A @ A ) @ R ) ) ).

% wf_not_refl
thf(fact_1052_wf__not__refl,axiom,
    ! [R: set_Pr7963165934123475591tate_a,A: produc6972303929186420058tate_a] :
      ( ( wf_Pro1739528972846749379tate_a @ R )
     => ~ ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ A @ A ) @ R ) ) ).

% wf_not_refl
thf(fact_1053_wf__not__refl,axiom,
    ! [R: set_Product_prod_a_a,A: a] :
      ( ( wf_a @ R )
     => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ).

% wf_not_refl
thf(fact_1054_wf__eq__minimal,axiom,
    ( wf_Pro9116859674037970973te_a_a
    = ( ^ [R3: set_Pr4734257497606733575te_a_a] :
        ! [Q3: set_Pr2749968112124914068te_a_a] :
          ( ? [X4: produc5126262593522865844te_a_a] : ( member1329951143264900189te_a_a @ X4 @ Q3 )
         => ? [X4: produc5126262593522865844te_a_a] :
              ( ( member1329951143264900189te_a_a @ X4 @ Q3 )
              & ! [Y6: produc5126262593522865844te_a_a] :
                  ( ( member2649312330005299536te_a_a @ ( produc2877451964877216791te_a_a @ Y6 @ X4 ) @ R3 )
                 => ~ ( member1329951143264900189te_a_a @ Y6 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1055_wf__eq__minimal,axiom,
    ( wf_Pro7444320194016922960tate_a
    = ( ^ [R3: set_Pr2656942177312230599tate_a] :
        ! [Q3: set_Pr1753757909640466247tate_a] :
          ( ? [X4: produc4774439622479004775tate_a] : ( member3801464090328701328tate_a @ X4 @ Q3 )
         => ? [X4: produc4774439622479004775tate_a] :
              ( ( member3801464090328701328tate_a @ X4 @ Q3 )
              & ! [Y6: produc4774439622479004775tate_a] :
                  ( ( member7923945571256933136tate_a @ ( produc1588899729123023703tate_a @ Y6 @ X4 ) @ R3 )
                 => ~ ( member3801464090328701328tate_a @ Y6 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1056_wf__eq__minimal,axiom,
    ( wf_Pro740797393413888144tate_a
    = ( ^ [R3: set_Pr1690608023759948999tate_a] :
        ! [Q3: set_Pr7963165934123475591tate_a] :
          ( ? [X4: produc5940350820068290983tate_a] : ( member1595334363549574352tate_a @ X4 @ Q3 )
         => ? [X4: produc5940350820068290983tate_a] :
              ( ( member1595334363549574352tate_a @ X4 @ Q3 )
              & ! [Y6: produc5940350820068290983tate_a] :
                  ( ( member1643692453717642000tate_a @ ( produc7335381768849070935tate_a @ Y6 @ X4 ) @ R3 )
                 => ~ ( member1595334363549574352tate_a @ Y6 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1057_wf__eq__minimal,axiom,
    ( wf_Product_prod_a_a
    = ( ^ [R3: set_Pr8600417178894128327od_a_a] :
        ! [Q3: set_Product_prod_a_a] :
          ( ? [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ Q3 )
         => ? [X4: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X4 @ Q3 )
              & ! [Y6: product_prod_a_a] :
                  ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y6 @ X4 ) @ R3 )
                 => ~ ( member1426531477525435216od_a_a @ Y6 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1058_wf__eq__minimal,axiom,
    ( wf_state_a
    = ( ^ [R3: set_Pr1753757909640466247tate_a] :
        ! [Q3: set_state_a] :
          ( ? [X4: state_a] : ( member_state_a @ X4 @ Q3 )
         => ? [X4: state_a] :
              ( ( member_state_a @ X4 @ Q3 )
              & ! [Y6: state_a] :
                  ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y6 @ X4 ) @ R3 )
                 => ~ ( member_state_a @ Y6 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1059_wf__eq__minimal,axiom,
    ( wf_Pro1739528972846749379tate_a
    = ( ^ [R3: set_Pr7963165934123475591tate_a] :
        ! [Q3: set_Pr4275752383657305402tate_a] :
          ( ? [X4: produc6972303929186420058tate_a] : ( member3175992478928454403tate_a @ X4 @ Q3 )
         => ? [X4: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ X4 @ Q3 )
              & ! [Y6: produc6972303929186420058tate_a] :
                  ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y6 @ X4 ) @ R3 )
                 => ~ ( member3175992478928454403tate_a @ Y6 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1060_wf__eq__minimal,axiom,
    ( wf_a
    = ( ^ [R3: set_Product_prod_a_a] :
        ! [Q3: set_a] :
          ( ? [X4: a] : ( member_a @ X4 @ Q3 )
         => ? [X4: a] :
              ( ( member_a @ X4 @ Q3 )
              & ! [Y6: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y6 @ X4 ) @ R3 )
                 => ~ ( member_a @ Y6 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1061_wf__induct__rule,axiom,
    ! [R: set_Pr1753757909640466247tate_a,P2: state_a > $o,A: state_a] :
      ( ( wf_state_a @ R )
     => ( ! [X: state_a] :
            ( ! [Y8: state_a] :
                ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y8 @ X ) @ R )
               => ( P2 @ Y8 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wf_induct_rule
thf(fact_1062_wf__induct__rule,axiom,
    ! [R: set_Pr7963165934123475591tate_a,P2: produc6972303929186420058tate_a > $o,A: produc6972303929186420058tate_a] :
      ( ( wf_Pro1739528972846749379tate_a @ R )
     => ( ! [X: produc6972303929186420058tate_a] :
            ( ! [Y8: produc6972303929186420058tate_a] :
                ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y8 @ X ) @ R )
               => ( P2 @ Y8 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wf_induct_rule
thf(fact_1063_wf__induct__rule,axiom,
    ! [R: set_Product_prod_a_a,P2: a > $o,A: a] :
      ( ( wf_a @ R )
     => ( ! [X: a] :
            ( ! [Y8: a] :
                ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y8 @ X ) @ R )
               => ( P2 @ Y8 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wf_induct_rule
thf(fact_1064_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr1753757909640466247tate_a,R: set_Pr1753757909640466247tate_a,F2: state_a > state_a] :
      ( ( wf_state_a @ S )
     => ( ! [X: state_a,Y3: state_a] :
            ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X @ Y3 ) @ R )
           => ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
       => ( wf_state_a @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_1065_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr7963165934123475591tate_a,R: set_Pr1753757909640466247tate_a,F2: state_a > produc6972303929186420058tate_a] :
      ( ( wf_Pro1739528972846749379tate_a @ S )
     => ( ! [X: state_a,Y3: state_a] :
            ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X @ Y3 ) @ R )
           => ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
       => ( wf_state_a @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_1066_wf__if__convertible__to__wf,axiom,
    ! [S: set_Product_prod_a_a,R: set_Pr1753757909640466247tate_a,F2: state_a > a] :
      ( ( wf_a @ S )
     => ( ! [X: state_a,Y3: state_a] :
            ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X @ Y3 ) @ R )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
       => ( wf_state_a @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_1067_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr1753757909640466247tate_a,R: set_Pr7963165934123475591tate_a,F2: produc6972303929186420058tate_a > state_a] :
      ( ( wf_state_a @ S )
     => ( ! [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a] :
            ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X @ Y3 ) @ R )
           => ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
       => ( wf_Pro1739528972846749379tate_a @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_1068_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr7963165934123475591tate_a,R: set_Pr7963165934123475591tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( wf_Pro1739528972846749379tate_a @ S )
     => ( ! [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a] :
            ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X @ Y3 ) @ R )
           => ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
       => ( wf_Pro1739528972846749379tate_a @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_1069_wf__if__convertible__to__wf,axiom,
    ! [S: set_Product_prod_a_a,R: set_Pr7963165934123475591tate_a,F2: produc6972303929186420058tate_a > a] :
      ( ( wf_a @ S )
     => ( ! [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a] :
            ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X @ Y3 ) @ R )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
       => ( wf_Pro1739528972846749379tate_a @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_1070_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr1753757909640466247tate_a,R: set_Product_prod_a_a,F2: a > state_a] :
      ( ( wf_state_a @ S )
     => ( ! [X: a,Y3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
           => ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
       => ( wf_a @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_1071_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr7963165934123475591tate_a,R: set_Product_prod_a_a,F2: a > produc6972303929186420058tate_a] :
      ( ( wf_Pro1739528972846749379tate_a @ S )
     => ( ! [X: a,Y3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
           => ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
       => ( wf_a @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_1072_wf__if__convertible__to__wf,axiom,
    ! [S: set_Product_prod_a_a,R: set_Product_prod_a_a,F2: a > a] :
      ( ( wf_a @ S )
     => ( ! [X: a,Y3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
       => ( wf_a @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_1073_wf__same__fst,axiom,
    ! [P2: a > $o,R2: a > set_Pr1753757909640466247tate_a] :
      ( ! [X: a] :
          ( ( P2 @ X )
         => ( wf_state_a @ ( R2 @ X ) ) )
     => ( wf_Pro1739528972846749379tate_a @ ( same_fst_a_state_a @ P2 @ R2 ) ) ) ).

% wf_same_fst
thf(fact_1074_the__inv__into__f__f,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,X3: a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( ( member_a @ X3 @ A5 )
       => ( ( the_in5147833725387914767tate_a @ A5 @ F2 @ ( F2 @ X3 ) )
          = X3 ) ) ) ).

% the_inv_into_f_f
thf(fact_1075_the__inv__into__f__f,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,X3: produc5126262593522865844te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
       => ( ( the_in3438972003123105497tate_a @ A5 @ F2 @ ( F2 @ X3 ) )
          = X3 ) ) ) ).

% the_inv_into_f_f
thf(fact_1076_the__inv__into__f__eq,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,X3: a,Y: produc6972303929186420058tate_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( ( ( F2 @ X3 )
          = Y )
       => ( ( member_a @ X3 @ A5 )
         => ( ( the_in5147833725387914767tate_a @ A5 @ F2 @ Y )
            = X3 ) ) ) ) ).

% the_inv_into_f_eq
thf(fact_1077_the__inv__into__f__eq,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,X3: produc5126262593522865844te_a_a,Y: produc6972303929186420058tate_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( ( ( F2 @ X3 )
          = Y )
       => ( ( member1329951143264900189te_a_a @ X3 @ A5 )
         => ( ( the_in3438972003123105497tate_a @ A5 @ F2 @ Y )
            = X3 ) ) ) ) ).

% the_inv_into_f_eq
thf(fact_1078_wf__imp__asym,axiom,
    ! [R: set_Pr7963165934123475591tate_a] :
      ( ( wf_Pro1739528972846749379tate_a @ R )
     => ( asym_o3617238991104122252tate_a @ top_to8456965716270553738tate_a @ R ) ) ).

% wf_imp_asym
thf(fact_1079_f__the__inv__into__f,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,Y: produc6972303929186420058tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ A5 )
     => ( ( member3175992478928454403tate_a @ Y @ ( image_1917819122788559109tate_a @ F2 @ A5 ) )
       => ( ( F2 @ ( the_in859209765134988979tate_a @ A5 @ F2 @ Y ) )
          = Y ) ) ) ).

% f_the_inv_into_f
thf(fact_1080_f__the__inv__into__f,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a,Y: produc6972303929186420058tate_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( ( member3175992478928454403tate_a @ Y @ ( image_5502445033238569825tate_a @ F2 @ A5 ) )
       => ( ( F2 @ ( the_in5147833725387914767tate_a @ A5 @ F2 @ Y ) )
          = Y ) ) ) ).

% f_the_inv_into_f
thf(fact_1081_f__the__inv__into__f,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a,Y: produc6972303929186420058tate_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( ( member3175992478928454403tate_a @ Y @ ( image_4497581360776675627tate_a @ F2 @ A5 ) )
       => ( ( F2 @ ( the_in3438972003123105497tate_a @ A5 @ F2 @ Y ) )
          = Y ) ) ) ).

% f_the_inv_into_f
thf(fact_1082_f__the__inv__into__f,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a,Y: produc5126262593522865844te_a_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ A5 )
     => ( ( member1329951143264900189te_a_a @ Y @ ( image_71777787125004895te_a_a @ F2 @ A5 ) )
       => ( ( F2 @ ( the_in8236540466326210573te_a_a @ A5 @ F2 @ Y ) )
          = Y ) ) ) ).

% f_the_inv_into_f
thf(fact_1083_inj__on__the__inv__into,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ A5 )
     => ( inj_on4126798524766655601tate_a @ ( the_in859209765134988979tate_a @ A5 @ F2 ) @ ( image_1917819122788559109tate_a @ F2 @ A5 ) ) ) ).

% inj_on_the_inv_into
thf(fact_1084_inj__on__the__inv__into,axiom,
    ! [F2: produc6972303929186420058tate_a > a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on412176999605016725te_a_a @ F2 @ A5 )
     => ( inj_on3780297825070694605tate_a @ ( the_in1779712899922236887te_a_a @ A5 @ F2 ) @ ( image_2134324207772891945te_a_a @ F2 @ A5 ) ) ) ).

% inj_on_the_inv_into
thf(fact_1085_inj__on__the__inv__into,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ A5 )
     => ( inj_on6706560762754772119tate_a @ ( the_in8236540466326210573te_a_a @ A5 @ F2 ) @ ( image_71777787125004895te_a_a @ F2 @ A5 ) ) ) ).

% inj_on_the_inv_into
thf(fact_1086_inj__on__the__inv__into,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ A5 )
     => ( inj_on412176999605016725te_a_a @ ( the_in5147833725387914767tate_a @ A5 @ F2 ) @ ( image_5502445033238569825tate_a @ F2 @ A5 ) ) ) ).

% inj_on_the_inv_into
thf(fact_1087_inj__on__the__inv__into,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ A5 )
     => ( inj_on2280757189103101387te_a_a @ ( the_in3438972003123105497tate_a @ A5 @ F2 ) @ ( image_4497581360776675627tate_a @ F2 @ A5 ) ) ) ).

% inj_on_the_inv_into
thf(fact_1088_the__inv__f__f,axiom,
    ! [F2: a > produc6972303929186420058tate_a,X3: a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
     => ( ( the_in5147833725387914767tate_a @ top_top_set_a @ F2 @ ( F2 @ X3 ) )
        = X3 ) ) ).

% the_inv_f_f
thf(fact_1089_the__inv__f__f,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,X3: produc5126262593522865844te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
     => ( ( the_in3438972003123105497tate_a @ top_to6931181444738162404te_a_a @ F2 @ ( F2 @ X3 ) )
        = X3 ) ) ).

% the_inv_f_f
thf(fact_1090_total__inv__image,axiom,
    ! [F2: a > produc6972303929186420058tate_a,R: set_Pr7963165934123475591tate_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ top_top_set_a )
     => ( ( total_8559467112218031838tate_a @ top_to8456965716270553738tate_a @ R )
       => ( total_on_a @ top_top_set_a @ ( inv_im760813232939325942te_a_a @ R @ F2 ) ) ) ) ).

% total_inv_image
thf(fact_1091_total__inv__image,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,R: set_Pr7963165934123475591tate_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ top_to6931181444738162404te_a_a )
     => ( ( total_8559467112218031838tate_a @ top_to8456965716270553738tate_a @ R )
       => ( total_6713425776554477624te_a_a @ top_to6931181444738162404te_a_a @ ( inv_im2556494317396703660te_a_a @ R @ F2 ) ) ) ) ).

% total_inv_image
thf(fact_1092_total__inv__image,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,R: set_Pr7963165934123475591tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ top_to8456965716270553738tate_a )
     => ( ( total_8559467112218031838tate_a @ top_to8456965716270553738tate_a @ R )
       => ( total_8559467112218031838tate_a @ top_to8456965716270553738tate_a @ ( inv_im4402535653060257874tate_a @ R @ F2 ) ) ) ) ).

% total_inv_image
thf(fact_1093_inj__on__image,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A5: set_se5193884918812770586tate_a] :
      ( ( inj_on2280757189103101387te_a_a @ F2 @ ( comple8313209218559427367tate_a @ A5 ) )
     => ( inj_on779827642740309195te_a_a @ ( image_71777787125004895te_a_a @ F2 ) @ A5 ) ) ).

% inj_on_image
thf(fact_1094_inj__on__image,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A5: set_se5193884918812770586tate_a] :
      ( ( inj_on4126798524766655601tate_a @ F2 @ ( comple8313209218559427367tate_a @ A5 ) )
     => ( inj_on2305611914272700529tate_a @ ( image_1917819122788559109tate_a @ F2 ) @ A5 ) ) ).

% inj_on_image
thf(fact_1095_inj__on__image,axiom,
    ! [F2: a > produc6972303929186420058tate_a,A5: set_set_a] :
      ( ( inj_on3780297825070694605tate_a @ F2 @ ( comple2307003609928055243_set_a @ A5 ) )
     => ( inj_on8057499159837936461tate_a @ ( image_5502445033238569825tate_a @ F2 ) @ A5 ) ) ).

% inj_on_image
thf(fact_1096_inj__on__image,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A5: set_se8668151452852975732te_a_a] :
      ( ( inj_on6706560762754772119tate_a @ F2 @ ( comple6787424947027036033te_a_a @ A5 ) )
     => ( inj_on6204426022611600791tate_a @ ( image_4497581360776675627tate_a @ F2 ) @ A5 ) ) ).

% inj_on_image
thf(fact_1097_Pow__insert,axiom,
    ! [A: a,A5: set_a] :
      ( ( pow_a @ ( insert_a @ A @ A5 ) )
      = ( sup_sup_set_set_a @ ( pow_a @ A5 ) @ ( image_set_a_set_a @ ( insert_a @ A ) @ ( pow_a @ A5 ) ) ) ) ).

% Pow_insert
thf(fact_1098_Pow__insert,axiom,
    ! [A: state_a,A5: set_state_a] :
      ( ( pow_state_a @ ( insert_state_a @ A @ A5 ) )
      = ( sup_su6743348009228009029tate_a @ ( pow_state_a @ A5 ) @ ( image_5748707248109567301tate_a @ ( insert_state_a @ A ) @ ( pow_state_a @ A5 ) ) ) ) ).

% Pow_insert
thf(fact_1099_insert__absorb2,axiom,
    ! [X3: a,A5: set_a] :
      ( ( insert_a @ X3 @ ( insert_a @ X3 @ A5 ) )
      = ( insert_a @ X3 @ A5 ) ) ).

% insert_absorb2
thf(fact_1100_insert__absorb2,axiom,
    ! [X3: state_a,A5: set_state_a] :
      ( ( insert_state_a @ X3 @ ( insert_state_a @ X3 @ A5 ) )
      = ( insert_state_a @ X3 @ A5 ) ) ).

% insert_absorb2
thf(fact_1101_insert__iff,axiom,
    ! [A: a,B: a,A5: set_a] :
      ( ( member_a @ A @ ( insert_a @ B @ A5 ) )
      = ( ( A = B )
        | ( member_a @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_1102_insert__iff,axiom,
    ! [A: state_a,B: state_a,A5: set_state_a] :
      ( ( member_state_a @ A @ ( insert_state_a @ B @ A5 ) )
      = ( ( A = B )
        | ( member_state_a @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_1103_insert__iff,axiom,
    ! [A: produc6972303929186420058tate_a,B: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ A @ ( insert3073504663777712938tate_a @ B @ A5 ) )
      = ( ( A = B )
        | ( member3175992478928454403tate_a @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_1104_insert__iff,axiom,
    ! [A: produc5126262593522865844te_a_a,B: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ A @ ( insert1227463328114158724te_a_a @ B @ A5 ) )
      = ( ( A = B )
        | ( member1329951143264900189te_a_a @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_1105_insert__iff,axiom,
    ! [A: produc4774439622479004775tate_a,B: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ A @ ( insert6689868412790682167tate_a @ B @ A5 ) )
      = ( ( A = B )
        | ( member3801464090328701328tate_a @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_1106_insert__iff,axiom,
    ! [A: produc5940350820068290983tate_a,B: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ A @ ( insert1906009156415373175tate_a @ B @ A5 ) )
      = ( ( A = B )
        | ( member1595334363549574352tate_a @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_1107_insert__iff,axiom,
    ! [A: product_prod_a_a,B: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ A5 ) )
      = ( ( A = B )
        | ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_1108_insertCI,axiom,
    ! [A: a,B5: set_a,B: a] :
      ( ( ~ ( member_a @ A @ B5 )
       => ( A = B ) )
     => ( member_a @ A @ ( insert_a @ B @ B5 ) ) ) ).

% insertCI
thf(fact_1109_insertCI,axiom,
    ! [A: state_a,B5: set_state_a,B: state_a] :
      ( ( ~ ( member_state_a @ A @ B5 )
       => ( A = B ) )
     => ( member_state_a @ A @ ( insert_state_a @ B @ B5 ) ) ) ).

% insertCI
thf(fact_1110_insertCI,axiom,
    ! [A: produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a,B: produc6972303929186420058tate_a] :
      ( ( ~ ( member3175992478928454403tate_a @ A @ B5 )
       => ( A = B ) )
     => ( member3175992478928454403tate_a @ A @ ( insert3073504663777712938tate_a @ B @ B5 ) ) ) ).

% insertCI
thf(fact_1111_insertCI,axiom,
    ! [A: produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a,B: produc5126262593522865844te_a_a] :
      ( ( ~ ( member1329951143264900189te_a_a @ A @ B5 )
       => ( A = B ) )
     => ( member1329951143264900189te_a_a @ A @ ( insert1227463328114158724te_a_a @ B @ B5 ) ) ) ).

% insertCI
thf(fact_1112_insertCI,axiom,
    ! [A: produc4774439622479004775tate_a,B5: set_Pr1753757909640466247tate_a,B: produc4774439622479004775tate_a] :
      ( ( ~ ( member3801464090328701328tate_a @ A @ B5 )
       => ( A = B ) )
     => ( member3801464090328701328tate_a @ A @ ( insert6689868412790682167tate_a @ B @ B5 ) ) ) ).

% insertCI
thf(fact_1113_insertCI,axiom,
    ! [A: produc5940350820068290983tate_a,B5: set_Pr7963165934123475591tate_a,B: produc5940350820068290983tate_a] :
      ( ( ~ ( member1595334363549574352tate_a @ A @ B5 )
       => ( A = B ) )
     => ( member1595334363549574352tate_a @ A @ ( insert1906009156415373175tate_a @ B @ B5 ) ) ) ).

% insertCI
thf(fact_1114_insertCI,axiom,
    ! [A: product_prod_a_a,B5: set_Product_prod_a_a,B: product_prod_a_a] :
      ( ( ~ ( member1426531477525435216od_a_a @ A @ B5 )
       => ( A = B ) )
     => ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ B5 ) ) ) ).

% insertCI
thf(fact_1115_image__insert,axiom,
    ! [F2: produc5126262593522865844te_a_a > produc6972303929186420058tate_a,A: produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( image_4497581360776675627tate_a @ F2 @ ( insert1227463328114158724te_a_a @ A @ B5 ) )
      = ( insert3073504663777712938tate_a @ ( F2 @ A ) @ ( image_4497581360776675627tate_a @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_1116_image__insert,axiom,
    ! [F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a,A: produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( image_71777787125004895te_a_a @ F2 @ ( insert3073504663777712938tate_a @ A @ B5 ) )
      = ( insert1227463328114158724te_a_a @ ( F2 @ A ) @ ( image_71777787125004895te_a_a @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_1117_image__insert,axiom,
    ! [F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a,A: produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( image_1917819122788559109tate_a @ F2 @ ( insert3073504663777712938tate_a @ A @ B5 ) )
      = ( insert3073504663777712938tate_a @ ( F2 @ A ) @ ( image_1917819122788559109tate_a @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_1118_image__insert,axiom,
    ! [F2: a > a,A: a,B5: set_a] :
      ( ( image_a_a @ F2 @ ( insert_a @ A @ B5 ) )
      = ( insert_a @ ( F2 @ A ) @ ( image_a_a @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_1119_image__insert,axiom,
    ! [F2: a > state_a,A: a,B5: set_a] :
      ( ( image_a_state_a @ F2 @ ( insert_a @ A @ B5 ) )
      = ( insert_state_a @ ( F2 @ A ) @ ( image_a_state_a @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_1120_image__insert,axiom,
    ! [F2: state_a > a,A: state_a,B5: set_state_a] :
      ( ( image_state_a_a @ F2 @ ( insert_state_a @ A @ B5 ) )
      = ( insert_a @ ( F2 @ A ) @ ( image_state_a_a @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_1121_image__insert,axiom,
    ! [F2: state_a > state_a,A: state_a,B5: set_state_a] :
      ( ( image_3877308803806453061tate_a @ F2 @ ( insert_state_a @ A @ B5 ) )
      = ( insert_state_a @ ( F2 @ A ) @ ( image_3877308803806453061tate_a @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_1122_insert__image,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a,F2: product_prod_a_a > a] :
      ( ( member1426531477525435216od_a_a @ X3 @ A5 )
     => ( ( insert_a @ ( F2 @ X3 ) @ ( image_3437945252899457948_a_a_a @ F2 @ A5 ) )
        = ( image_3437945252899457948_a_a_a @ F2 @ A5 ) ) ) ).

% insert_image
thf(fact_1123_insert__image,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > a] :
      ( ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( ( insert_a @ ( F2 @ X3 ) @ ( image_2134324207772891945te_a_a @ F2 @ A5 ) )
        = ( image_2134324207772891945te_a_a @ F2 @ A5 ) ) ) ).

% insert_image
thf(fact_1124_insert__image,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > a] :
      ( ( member1329951143264900189te_a_a @ X3 @ A5 )
     => ( ( insert_a @ ( F2 @ X3 ) @ ( image_3052321768533554255_a_a_a @ F2 @ A5 ) )
        = ( image_3052321768533554255_a_a_a @ F2 @ A5 ) ) ) ).

% insert_image
thf(fact_1125_insert__image,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a,F2: product_prod_a_a > state_a] :
      ( ( member1426531477525435216od_a_a @ X3 @ A5 )
     => ( ( insert_state_a @ ( F2 @ X3 ) @ ( image_2556981004245761359tate_a @ F2 @ A5 ) )
        = ( image_2556981004245761359tate_a @ F2 @ A5 ) ) ) ).

% insert_image
thf(fact_1126_insert__image,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > state_a] :
      ( ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( ( insert_state_a @ ( F2 @ X3 ) @ ( image_2364236133221778012tate_a @ F2 @ A5 ) )
        = ( image_2364236133221778012tate_a @ F2 @ A5 ) ) ) ).

% insert_image
thf(fact_1127_insert__image,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,F2: produc5126262593522865844te_a_a > state_a] :
      ( ( member1329951143264900189te_a_a @ X3 @ A5 )
     => ( ( insert_state_a @ ( F2 @ X3 ) @ ( image_6150641562437185666tate_a @ F2 @ A5 ) )
        = ( image_6150641562437185666tate_a @ F2 @ A5 ) ) ) ).

% insert_image
thf(fact_1128_insert__image,axiom,
    ! [X3: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a,F2: produc4774439622479004775tate_a > a] :
      ( ( member3801464090328701328tate_a @ X3 @ A5 )
     => ( ( insert_a @ ( F2 @ X3 ) @ ( image_5727984765964316252te_a_a @ F2 @ A5 ) )
        = ( image_5727984765964316252te_a_a @ F2 @ A5 ) ) ) ).

% insert_image
thf(fact_1129_insert__image,axiom,
    ! [X3: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a,F2: produc4774439622479004775tate_a > state_a] :
      ( ( member3801464090328701328tate_a @ X3 @ A5 )
     => ( ( insert_state_a @ ( F2 @ X3 ) @ ( image_7566902363611725839tate_a @ F2 @ A5 ) )
        = ( image_7566902363611725839tate_a @ F2 @ A5 ) ) ) ).

% insert_image
thf(fact_1130_insert__image,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc5126262593522865844te_a_a] :
      ( ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( ( insert1227463328114158724te_a_a @ ( F2 @ X3 ) @ ( image_71777787125004895te_a_a @ F2 @ A5 ) )
        = ( image_71777787125004895te_a_a @ F2 @ A5 ) ) ) ).

% insert_image
thf(fact_1131_insert__image,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,F2: produc6972303929186420058tate_a > produc6972303929186420058tate_a] :
      ( ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( ( insert3073504663777712938tate_a @ ( F2 @ X3 ) @ ( image_1917819122788559109tate_a @ F2 @ A5 ) )
        = ( image_1917819122788559109tate_a @ F2 @ A5 ) ) ) ).

% insert_image
thf(fact_1132_insert__subset,axiom,
    ! [X3: a,A5: set_a,B5: set_a] :
      ( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A5 ) @ B5 )
      = ( ( member_a @ X3 @ B5 )
        & ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_1133_insert__subset,axiom,
    ! [X3: state_a,A5: set_state_a,B5: set_state_a] :
      ( ( ord_le7147511027547350193tate_a @ ( insert_state_a @ X3 @ A5 ) @ B5 )
      = ( ( member_state_a @ X3 @ B5 )
        & ( ord_le7147511027547350193tate_a @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_1134_insert__subset,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ( ord_le5819720210775101748te_a_a @ ( insert1227463328114158724te_a_a @ X3 @ A5 ) @ B5 )
      = ( ( member1329951143264900189te_a_a @ X3 @ B5 )
        & ( ord_le5819720210775101748te_a_a @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_1135_insert__subset,axiom,
    ! [X3: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a,B5: set_Pr1753757909640466247tate_a] :
      ( ( ord_le5561929874677720807tate_a @ ( insert6689868412790682167tate_a @ X3 @ A5 ) @ B5 )
      = ( ( member3801464090328701328tate_a @ X3 @ B5 )
        & ( ord_le5561929874677720807tate_a @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_1136_insert__subset,axiom,
    ! [X3: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a,B5: set_Pr7963165934123475591tate_a] :
      ( ( ord_le4349072690221431335tate_a @ ( insert1906009156415373175tate_a @ X3 @ A5 ) @ B5 )
      = ( ( member1595334363549574352tate_a @ X3 @ B5 )
        & ( ord_le4349072690221431335tate_a @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_1137_insert__subset,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ A5 ) @ B5 )
      = ( ( member1426531477525435216od_a_a @ X3 @ B5 )
        & ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_1138_insert__subset,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( insert3073504663777712938tate_a @ X3 @ A5 ) @ B5 )
      = ( ( member3175992478928454403tate_a @ X3 @ B5 )
        & ( ord_le7345504482307493082tate_a @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_1139_Un__insert__left,axiom,
    ! [A: a,B5: set_a,C3: set_a] :
      ( ( sup_sup_set_a @ ( insert_a @ A @ B5 ) @ C3 )
      = ( insert_a @ A @ ( sup_sup_set_a @ B5 @ C3 ) ) ) ).

% Un_insert_left
thf(fact_1140_Un__insert__left,axiom,
    ! [A: state_a,B5: set_state_a,C3: set_state_a] :
      ( ( sup_sup_set_state_a @ ( insert_state_a @ A @ B5 ) @ C3 )
      = ( insert_state_a @ A @ ( sup_sup_set_state_a @ B5 @ C3 ) ) ) ).

% Un_insert_left
thf(fact_1141_Un__insert__left,axiom,
    ! [A: produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a,C3: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ ( insert3073504663777712938tate_a @ A @ B5 ) @ C3 )
      = ( insert3073504663777712938tate_a @ A @ ( sup_su1662690461080552718tate_a @ B5 @ C3 ) ) ) ).

% Un_insert_left
thf(fact_1142_Un__insert__right,axiom,
    ! [A5: set_a,A: a,B5: set_a] :
      ( ( sup_sup_set_a @ A5 @ ( insert_a @ A @ B5 ) )
      = ( insert_a @ A @ ( sup_sup_set_a @ A5 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_1143_Un__insert__right,axiom,
    ! [A5: set_state_a,A: state_a,B5: set_state_a] :
      ( ( sup_sup_set_state_a @ A5 @ ( insert_state_a @ A @ B5 ) )
      = ( insert_state_a @ A @ ( sup_sup_set_state_a @ A5 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_1144_Un__insert__right,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,A: produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( sup_su1662690461080552718tate_a @ A5 @ ( insert3073504663777712938tate_a @ A @ B5 ) )
      = ( insert3073504663777712938tate_a @ A @ ( sup_su1662690461080552718tate_a @ A5 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_1145_Sup__UNIV,axiom,
    ( ( comple760828771110116584te_a_o @ top_to5380514444333567915te_a_o )
    = top_top_a_state_a_o ) ).

% Sup_UNIV
thf(fact_1146_Sup__UNIV,axiom,
    ( ( comple8313209218559427367tate_a @ top_to4898740227593795178tate_a )
    = top_to8456965716270553738tate_a ) ).

% Sup_UNIV
thf(fact_1147_insert__UNIV,axiom,
    ! [X3: a] :
      ( ( insert_a @ X3 @ top_top_set_a )
      = top_top_set_a ) ).

% insert_UNIV
thf(fact_1148_insert__UNIV,axiom,
    ! [X3: state_a] :
      ( ( insert_state_a @ X3 @ top_top_set_state_a )
      = top_top_set_state_a ) ).

% insert_UNIV
thf(fact_1149_insert__UNIV,axiom,
    ! [X3: produc6972303929186420058tate_a] :
      ( ( insert3073504663777712938tate_a @ X3 @ top_to8456965716270553738tate_a )
      = top_to8456965716270553738tate_a ) ).

% insert_UNIV
thf(fact_1150_Union__least,axiom,
    ! [A5: set_se5193884918812770586tate_a,C3: set_Pr4275752383657305402tate_a] :
      ( ! [X9: set_Pr4275752383657305402tate_a] :
          ( ( member4824975370526228195tate_a @ X9 @ A5 )
         => ( ord_le7345504482307493082tate_a @ X9 @ C3 ) )
     => ( ord_le7345504482307493082tate_a @ ( comple8313209218559427367tate_a @ A5 ) @ C3 ) ) ).

% Union_least
thf(fact_1151_Union__upper,axiom,
    ! [B5: set_Pr4275752383657305402tate_a,A5: set_se5193884918812770586tate_a] :
      ( ( member4824975370526228195tate_a @ B5 @ A5 )
     => ( ord_le7345504482307493082tate_a @ B5 @ ( comple8313209218559427367tate_a @ A5 ) ) ) ).

% Union_upper
thf(fact_1152_Union__subsetI,axiom,
    ! [A5: set_se5193884918812770586tate_a,B5: set_se5193884918812770586tate_a] :
      ( ! [X: set_Pr4275752383657305402tate_a] :
          ( ( member4824975370526228195tate_a @ X @ A5 )
         => ? [Y8: set_Pr4275752383657305402tate_a] :
              ( ( member4824975370526228195tate_a @ Y8 @ B5 )
              & ( ord_le7345504482307493082tate_a @ X @ Y8 ) ) )
     => ( ord_le7345504482307493082tate_a @ ( comple8313209218559427367tate_a @ A5 ) @ ( comple8313209218559427367tate_a @ B5 ) ) ) ).

% Union_subsetI
thf(fact_1153_insert__subsetI,axiom,
    ! [X3: a,A5: set_a,X6: set_a] :
      ( ( member_a @ X3 @ A5 )
     => ( ( ord_less_eq_set_a @ X6 @ A5 )
       => ( ord_less_eq_set_a @ ( insert_a @ X3 @ X6 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_1154_insert__subsetI,axiom,
    ! [X3: state_a,A5: set_state_a,X6: set_state_a] :
      ( ( member_state_a @ X3 @ A5 )
     => ( ( ord_le7147511027547350193tate_a @ X6 @ A5 )
       => ( ord_le7147511027547350193tate_a @ ( insert_state_a @ X3 @ X6 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_1155_insert__subsetI,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,X6: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ X3 @ A5 )
     => ( ( ord_le5819720210775101748te_a_a @ X6 @ A5 )
       => ( ord_le5819720210775101748te_a_a @ ( insert1227463328114158724te_a_a @ X3 @ X6 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_1156_insert__subsetI,axiom,
    ! [X3: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a,X6: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ X3 @ A5 )
     => ( ( ord_le5561929874677720807tate_a @ X6 @ A5 )
       => ( ord_le5561929874677720807tate_a @ ( insert6689868412790682167tate_a @ X3 @ X6 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_1157_insert__subsetI,axiom,
    ! [X3: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a,X6: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ X3 @ A5 )
     => ( ( ord_le4349072690221431335tate_a @ X6 @ A5 )
       => ( ord_le4349072690221431335tate_a @ ( insert1906009156415373175tate_a @ X3 @ X6 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_1158_insert__subsetI,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a,X6: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ X3 @ A5 )
     => ( ( ord_le746702958409616551od_a_a @ X6 @ A5 )
       => ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ X6 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_1159_insert__subsetI,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,X6: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( ( ord_le7345504482307493082tate_a @ X6 @ A5 )
       => ( ord_le7345504482307493082tate_a @ ( insert3073504663777712938tate_a @ X3 @ X6 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_1160_insert__mono,axiom,
    ! [C3: set_a,D: set_a,A: a] :
      ( ( ord_less_eq_set_a @ C3 @ D )
     => ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D ) ) ) ).

% insert_mono
thf(fact_1161_insert__mono,axiom,
    ! [C3: set_state_a,D: set_state_a,A: state_a] :
      ( ( ord_le7147511027547350193tate_a @ C3 @ D )
     => ( ord_le7147511027547350193tate_a @ ( insert_state_a @ A @ C3 ) @ ( insert_state_a @ A @ D ) ) ) ).

% insert_mono
thf(fact_1162_insert__mono,axiom,
    ! [C3: set_Pr4275752383657305402tate_a,D: set_Pr4275752383657305402tate_a,A: produc6972303929186420058tate_a] :
      ( ( ord_le7345504482307493082tate_a @ C3 @ D )
     => ( ord_le7345504482307493082tate_a @ ( insert3073504663777712938tate_a @ A @ C3 ) @ ( insert3073504663777712938tate_a @ A @ D ) ) ) ).

% insert_mono
thf(fact_1163_subset__insert,axiom,
    ! [X3: a,A5: set_a,B5: set_a] :
      ( ~ ( member_a @ X3 @ A5 )
     => ( ( ord_less_eq_set_a @ A5 @ ( insert_a @ X3 @ B5 ) )
        = ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_1164_subset__insert,axiom,
    ! [X3: state_a,A5: set_state_a,B5: set_state_a] :
      ( ~ ( member_state_a @ X3 @ A5 )
     => ( ( ord_le7147511027547350193tate_a @ A5 @ ( insert_state_a @ X3 @ B5 ) )
        = ( ord_le7147511027547350193tate_a @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_1165_subset__insert,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ~ ( member1329951143264900189te_a_a @ X3 @ A5 )
     => ( ( ord_le5819720210775101748te_a_a @ A5 @ ( insert1227463328114158724te_a_a @ X3 @ B5 ) )
        = ( ord_le5819720210775101748te_a_a @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_1166_subset__insert,axiom,
    ! [X3: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a,B5: set_Pr1753757909640466247tate_a] :
      ( ~ ( member3801464090328701328tate_a @ X3 @ A5 )
     => ( ( ord_le5561929874677720807tate_a @ A5 @ ( insert6689868412790682167tate_a @ X3 @ B5 ) )
        = ( ord_le5561929874677720807tate_a @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_1167_subset__insert,axiom,
    ! [X3: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a,B5: set_Pr7963165934123475591tate_a] :
      ( ~ ( member1595334363549574352tate_a @ X3 @ A5 )
     => ( ( ord_le4349072690221431335tate_a @ A5 @ ( insert1906009156415373175tate_a @ X3 @ B5 ) )
        = ( ord_le4349072690221431335tate_a @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_1168_subset__insert,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ X3 @ A5 )
     => ( ( ord_le746702958409616551od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ X3 @ B5 ) )
        = ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_1169_subset__insert,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ~ ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( ( ord_le7345504482307493082tate_a @ A5 @ ( insert3073504663777712938tate_a @ X3 @ B5 ) )
        = ( ord_le7345504482307493082tate_a @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_1170_subset__insertI,axiom,
    ! [B5: set_a,A: a] : ( ord_less_eq_set_a @ B5 @ ( insert_a @ A @ B5 ) ) ).

% subset_insertI
thf(fact_1171_subset__insertI,axiom,
    ! [B5: set_state_a,A: state_a] : ( ord_le7147511027547350193tate_a @ B5 @ ( insert_state_a @ A @ B5 ) ) ).

% subset_insertI
thf(fact_1172_subset__insertI,axiom,
    ! [B5: set_Pr4275752383657305402tate_a,A: produc6972303929186420058tate_a] : ( ord_le7345504482307493082tate_a @ B5 @ ( insert3073504663777712938tate_a @ A @ B5 ) ) ).

% subset_insertI
thf(fact_1173_subset__insertI2,axiom,
    ! [A5: set_a,B5: set_a,B: a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ord_less_eq_set_a @ A5 @ ( insert_a @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_1174_subset__insertI2,axiom,
    ! [A5: set_state_a,B5: set_state_a,B: state_a] :
      ( ( ord_le7147511027547350193tate_a @ A5 @ B5 )
     => ( ord_le7147511027547350193tate_a @ A5 @ ( insert_state_a @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_1175_subset__insertI2,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a,B: produc6972303929186420058tate_a] :
      ( ( ord_le7345504482307493082tate_a @ A5 @ B5 )
     => ( ord_le7345504482307493082tate_a @ A5 @ ( insert3073504663777712938tate_a @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_1176_mk__disjoint__insert,axiom,
    ! [A: a,A5: set_a] :
      ( ( member_a @ A @ A5 )
     => ? [B9: set_a] :
          ( ( A5
            = ( insert_a @ A @ B9 ) )
          & ~ ( member_a @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_1177_mk__disjoint__insert,axiom,
    ! [A: state_a,A5: set_state_a] :
      ( ( member_state_a @ A @ A5 )
     => ? [B9: set_state_a] :
          ( ( A5
            = ( insert_state_a @ A @ B9 ) )
          & ~ ( member_state_a @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_1178_mk__disjoint__insert,axiom,
    ! [A: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ A @ A5 )
     => ? [B9: set_Pr4275752383657305402tate_a] :
          ( ( A5
            = ( insert3073504663777712938tate_a @ A @ B9 ) )
          & ~ ( member3175992478928454403tate_a @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_1179_mk__disjoint__insert,axiom,
    ! [A: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ A @ A5 )
     => ? [B9: set_Pr2749968112124914068te_a_a] :
          ( ( A5
            = ( insert1227463328114158724te_a_a @ A @ B9 ) )
          & ~ ( member1329951143264900189te_a_a @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_1180_mk__disjoint__insert,axiom,
    ! [A: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ A @ A5 )
     => ? [B9: set_Pr1753757909640466247tate_a] :
          ( ( A5
            = ( insert6689868412790682167tate_a @ A @ B9 ) )
          & ~ ( member3801464090328701328tate_a @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_1181_mk__disjoint__insert,axiom,
    ! [A: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ A @ A5 )
     => ? [B9: set_Pr7963165934123475591tate_a] :
          ( ( A5
            = ( insert1906009156415373175tate_a @ A @ B9 ) )
          & ~ ( member1595334363549574352tate_a @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_1182_mk__disjoint__insert,axiom,
    ! [A: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A @ A5 )
     => ? [B9: set_Product_prod_a_a] :
          ( ( A5
            = ( insert4534936382041156343od_a_a @ A @ B9 ) )
          & ~ ( member1426531477525435216od_a_a @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_1183_insert__commute,axiom,
    ! [X3: a,Y: a,A5: set_a] :
      ( ( insert_a @ X3 @ ( insert_a @ Y @ A5 ) )
      = ( insert_a @ Y @ ( insert_a @ X3 @ A5 ) ) ) ).

% insert_commute
thf(fact_1184_insert__commute,axiom,
    ! [X3: state_a,Y: state_a,A5: set_state_a] :
      ( ( insert_state_a @ X3 @ ( insert_state_a @ Y @ A5 ) )
      = ( insert_state_a @ Y @ ( insert_state_a @ X3 @ A5 ) ) ) ).

% insert_commute
thf(fact_1185_insert__eq__iff,axiom,
    ! [A: a,A5: set_a,B: a,B5: set_a] :
      ( ~ ( member_a @ A @ A5 )
     => ( ~ ( member_a @ B @ B5 )
       => ( ( ( insert_a @ A @ A5 )
            = ( insert_a @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A5 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_a] :
                  ( ( A5
                    = ( insert_a @ B @ C5 ) )
                  & ~ ( member_a @ B @ C5 )
                  & ( B5
                    = ( insert_a @ A @ C5 ) )
                  & ~ ( member_a @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1186_insert__eq__iff,axiom,
    ! [A: state_a,A5: set_state_a,B: state_a,B5: set_state_a] :
      ( ~ ( member_state_a @ A @ A5 )
     => ( ~ ( member_state_a @ B @ B5 )
       => ( ( ( insert_state_a @ A @ A5 )
            = ( insert_state_a @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A5 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_state_a] :
                  ( ( A5
                    = ( insert_state_a @ B @ C5 ) )
                  & ~ ( member_state_a @ B @ C5 )
                  & ( B5
                    = ( insert_state_a @ A @ C5 ) )
                  & ~ ( member_state_a @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1187_insert__eq__iff,axiom,
    ! [A: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B: produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ~ ( member3175992478928454403tate_a @ A @ A5 )
     => ( ~ ( member3175992478928454403tate_a @ B @ B5 )
       => ( ( ( insert3073504663777712938tate_a @ A @ A5 )
            = ( insert3073504663777712938tate_a @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A5 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Pr4275752383657305402tate_a] :
                  ( ( A5
                    = ( insert3073504663777712938tate_a @ B @ C5 ) )
                  & ~ ( member3175992478928454403tate_a @ B @ C5 )
                  & ( B5
                    = ( insert3073504663777712938tate_a @ A @ C5 ) )
                  & ~ ( member3175992478928454403tate_a @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1188_insert__eq__iff,axiom,
    ! [A: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,B: produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ~ ( member1329951143264900189te_a_a @ A @ A5 )
     => ( ~ ( member1329951143264900189te_a_a @ B @ B5 )
       => ( ( ( insert1227463328114158724te_a_a @ A @ A5 )
            = ( insert1227463328114158724te_a_a @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A5 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Pr2749968112124914068te_a_a] :
                  ( ( A5
                    = ( insert1227463328114158724te_a_a @ B @ C5 ) )
                  & ~ ( member1329951143264900189te_a_a @ B @ C5 )
                  & ( B5
                    = ( insert1227463328114158724te_a_a @ A @ C5 ) )
                  & ~ ( member1329951143264900189te_a_a @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1189_insert__eq__iff,axiom,
    ! [A: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a,B: produc4774439622479004775tate_a,B5: set_Pr1753757909640466247tate_a] :
      ( ~ ( member3801464090328701328tate_a @ A @ A5 )
     => ( ~ ( member3801464090328701328tate_a @ B @ B5 )
       => ( ( ( insert6689868412790682167tate_a @ A @ A5 )
            = ( insert6689868412790682167tate_a @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A5 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Pr1753757909640466247tate_a] :
                  ( ( A5
                    = ( insert6689868412790682167tate_a @ B @ C5 ) )
                  & ~ ( member3801464090328701328tate_a @ B @ C5 )
                  & ( B5
                    = ( insert6689868412790682167tate_a @ A @ C5 ) )
                  & ~ ( member3801464090328701328tate_a @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1190_insert__eq__iff,axiom,
    ! [A: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a,B: produc5940350820068290983tate_a,B5: set_Pr7963165934123475591tate_a] :
      ( ~ ( member1595334363549574352tate_a @ A @ A5 )
     => ( ~ ( member1595334363549574352tate_a @ B @ B5 )
       => ( ( ( insert1906009156415373175tate_a @ A @ A5 )
            = ( insert1906009156415373175tate_a @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A5 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Pr7963165934123475591tate_a] :
                  ( ( A5
                    = ( insert1906009156415373175tate_a @ B @ C5 ) )
                  & ~ ( member1595334363549574352tate_a @ B @ C5 )
                  & ( B5
                    = ( insert1906009156415373175tate_a @ A @ C5 ) )
                  & ~ ( member1595334363549574352tate_a @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1191_insert__eq__iff,axiom,
    ! [A: product_prod_a_a,A5: set_Product_prod_a_a,B: product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ A @ A5 )
     => ( ~ ( member1426531477525435216od_a_a @ B @ B5 )
       => ( ( ( insert4534936382041156343od_a_a @ A @ A5 )
            = ( insert4534936382041156343od_a_a @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A5 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Product_prod_a_a] :
                  ( ( A5
                    = ( insert4534936382041156343od_a_a @ B @ C5 ) )
                  & ~ ( member1426531477525435216od_a_a @ B @ C5 )
                  & ( B5
                    = ( insert4534936382041156343od_a_a @ A @ C5 ) )
                  & ~ ( member1426531477525435216od_a_a @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1192_insert__absorb,axiom,
    ! [A: a,A5: set_a] :
      ( ( member_a @ A @ A5 )
     => ( ( insert_a @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_1193_insert__absorb,axiom,
    ! [A: state_a,A5: set_state_a] :
      ( ( member_state_a @ A @ A5 )
     => ( ( insert_state_a @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_1194_insert__absorb,axiom,
    ! [A: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ A @ A5 )
     => ( ( insert3073504663777712938tate_a @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_1195_insert__absorb,axiom,
    ! [A: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ A @ A5 )
     => ( ( insert1227463328114158724te_a_a @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_1196_insert__absorb,axiom,
    ! [A: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ A @ A5 )
     => ( ( insert6689868412790682167tate_a @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_1197_insert__absorb,axiom,
    ! [A: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ A @ A5 )
     => ( ( insert1906009156415373175tate_a @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_1198_insert__absorb,axiom,
    ! [A: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A @ A5 )
     => ( ( insert4534936382041156343od_a_a @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_1199_insert__ident,axiom,
    ! [X3: a,A5: set_a,B5: set_a] :
      ( ~ ( member_a @ X3 @ A5 )
     => ( ~ ( member_a @ X3 @ B5 )
       => ( ( ( insert_a @ X3 @ A5 )
            = ( insert_a @ X3 @ B5 ) )
          = ( A5 = B5 ) ) ) ) ).

% insert_ident
thf(fact_1200_insert__ident,axiom,
    ! [X3: state_a,A5: set_state_a,B5: set_state_a] :
      ( ~ ( member_state_a @ X3 @ A5 )
     => ( ~ ( member_state_a @ X3 @ B5 )
       => ( ( ( insert_state_a @ X3 @ A5 )
            = ( insert_state_a @ X3 @ B5 ) )
          = ( A5 = B5 ) ) ) ) ).

% insert_ident
thf(fact_1201_insert__ident,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ~ ( member3175992478928454403tate_a @ X3 @ A5 )
     => ( ~ ( member3175992478928454403tate_a @ X3 @ B5 )
       => ( ( ( insert3073504663777712938tate_a @ X3 @ A5 )
            = ( insert3073504663777712938tate_a @ X3 @ B5 ) )
          = ( A5 = B5 ) ) ) ) ).

% insert_ident
thf(fact_1202_insert__ident,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a,B5: set_Pr2749968112124914068te_a_a] :
      ( ~ ( member1329951143264900189te_a_a @ X3 @ A5 )
     => ( ~ ( member1329951143264900189te_a_a @ X3 @ B5 )
       => ( ( ( insert1227463328114158724te_a_a @ X3 @ A5 )
            = ( insert1227463328114158724te_a_a @ X3 @ B5 ) )
          = ( A5 = B5 ) ) ) ) ).

% insert_ident
thf(fact_1203_insert__ident,axiom,
    ! [X3: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a,B5: set_Pr1753757909640466247tate_a] :
      ( ~ ( member3801464090328701328tate_a @ X3 @ A5 )
     => ( ~ ( member3801464090328701328tate_a @ X3 @ B5 )
       => ( ( ( insert6689868412790682167tate_a @ X3 @ A5 )
            = ( insert6689868412790682167tate_a @ X3 @ B5 ) )
          = ( A5 = B5 ) ) ) ) ).

% insert_ident
thf(fact_1204_insert__ident,axiom,
    ! [X3: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a,B5: set_Pr7963165934123475591tate_a] :
      ( ~ ( member1595334363549574352tate_a @ X3 @ A5 )
     => ( ~ ( member1595334363549574352tate_a @ X3 @ B5 )
       => ( ( ( insert1906009156415373175tate_a @ X3 @ A5 )
            = ( insert1906009156415373175tate_a @ X3 @ B5 ) )
          = ( A5 = B5 ) ) ) ) ).

% insert_ident
thf(fact_1205_insert__ident,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ X3 @ A5 )
     => ( ~ ( member1426531477525435216od_a_a @ X3 @ B5 )
       => ( ( ( insert4534936382041156343od_a_a @ X3 @ A5 )
            = ( insert4534936382041156343od_a_a @ X3 @ B5 ) )
          = ( A5 = B5 ) ) ) ) ).

% insert_ident
thf(fact_1206_Set_Oset__insert,axiom,
    ! [X3: a,A5: set_a] :
      ( ( member_a @ X3 @ A5 )
     => ~ ! [B9: set_a] :
            ( ( A5
              = ( insert_a @ X3 @ B9 ) )
           => ( member_a @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_1207_Set_Oset__insert,axiom,
    ! [X3: state_a,A5: set_state_a] :
      ( ( member_state_a @ X3 @ A5 )
     => ~ ! [B9: set_state_a] :
            ( ( A5
              = ( insert_state_a @ X3 @ B9 ) )
           => ( member_state_a @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_1208_Set_Oset__insert,axiom,
    ! [X3: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ X3 @ A5 )
     => ~ ! [B9: set_Pr4275752383657305402tate_a] :
            ( ( A5
              = ( insert3073504663777712938tate_a @ X3 @ B9 ) )
           => ( member3175992478928454403tate_a @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_1209_Set_Oset__insert,axiom,
    ! [X3: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ X3 @ A5 )
     => ~ ! [B9: set_Pr2749968112124914068te_a_a] :
            ( ( A5
              = ( insert1227463328114158724te_a_a @ X3 @ B9 ) )
           => ( member1329951143264900189te_a_a @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_1210_Set_Oset__insert,axiom,
    ! [X3: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ X3 @ A5 )
     => ~ ! [B9: set_Pr1753757909640466247tate_a] :
            ( ( A5
              = ( insert6689868412790682167tate_a @ X3 @ B9 ) )
           => ( member3801464090328701328tate_a @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_1211_Set_Oset__insert,axiom,
    ! [X3: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ X3 @ A5 )
     => ~ ! [B9: set_Pr7963165934123475591tate_a] :
            ( ( A5
              = ( insert1906009156415373175tate_a @ X3 @ B9 ) )
           => ( member1595334363549574352tate_a @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_1212_Set_Oset__insert,axiom,
    ! [X3: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ X3 @ A5 )
     => ~ ! [B9: set_Product_prod_a_a] :
            ( ( A5
              = ( insert4534936382041156343od_a_a @ X3 @ B9 ) )
           => ( member1426531477525435216od_a_a @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_1213_insertI2,axiom,
    ! [A: a,B5: set_a,B: a] :
      ( ( member_a @ A @ B5 )
     => ( member_a @ A @ ( insert_a @ B @ B5 ) ) ) ).

% insertI2
thf(fact_1214_insertI2,axiom,
    ! [A: state_a,B5: set_state_a,B: state_a] :
      ( ( member_state_a @ A @ B5 )
     => ( member_state_a @ A @ ( insert_state_a @ B @ B5 ) ) ) ).

% insertI2
thf(fact_1215_insertI2,axiom,
    ! [A: produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a,B: produc6972303929186420058tate_a] :
      ( ( member3175992478928454403tate_a @ A @ B5 )
     => ( member3175992478928454403tate_a @ A @ ( insert3073504663777712938tate_a @ B @ B5 ) ) ) ).

% insertI2
thf(fact_1216_insertI2,axiom,
    ! [A: produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a,B: produc5126262593522865844te_a_a] :
      ( ( member1329951143264900189te_a_a @ A @ B5 )
     => ( member1329951143264900189te_a_a @ A @ ( insert1227463328114158724te_a_a @ B @ B5 ) ) ) ).

% insertI2
thf(fact_1217_insertI2,axiom,
    ! [A: produc4774439622479004775tate_a,B5: set_Pr1753757909640466247tate_a,B: produc4774439622479004775tate_a] :
      ( ( member3801464090328701328tate_a @ A @ B5 )
     => ( member3801464090328701328tate_a @ A @ ( insert6689868412790682167tate_a @ B @ B5 ) ) ) ).

% insertI2
thf(fact_1218_insertI2,axiom,
    ! [A: produc5940350820068290983tate_a,B5: set_Pr7963165934123475591tate_a,B: produc5940350820068290983tate_a] :
      ( ( member1595334363549574352tate_a @ A @ B5 )
     => ( member1595334363549574352tate_a @ A @ ( insert1906009156415373175tate_a @ B @ B5 ) ) ) ).

% insertI2
thf(fact_1219_insertI2,axiom,
    ! [A: product_prod_a_a,B5: set_Product_prod_a_a,B: product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A @ B5 )
     => ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ B5 ) ) ) ).

% insertI2
thf(fact_1220_insertI1,axiom,
    ! [A: a,B5: set_a] : ( member_a @ A @ ( insert_a @ A @ B5 ) ) ).

% insertI1
thf(fact_1221_insertI1,axiom,
    ! [A: state_a,B5: set_state_a] : ( member_state_a @ A @ ( insert_state_a @ A @ B5 ) ) ).

% insertI1
thf(fact_1222_insertI1,axiom,
    ! [A: produc6972303929186420058tate_a,B5: set_Pr4275752383657305402tate_a] : ( member3175992478928454403tate_a @ A @ ( insert3073504663777712938tate_a @ A @ B5 ) ) ).

% insertI1
thf(fact_1223_insertI1,axiom,
    ! [A: produc5126262593522865844te_a_a,B5: set_Pr2749968112124914068te_a_a] : ( member1329951143264900189te_a_a @ A @ ( insert1227463328114158724te_a_a @ A @ B5 ) ) ).

% insertI1
thf(fact_1224_insertI1,axiom,
    ! [A: produc4774439622479004775tate_a,B5: set_Pr1753757909640466247tate_a] : ( member3801464090328701328tate_a @ A @ ( insert6689868412790682167tate_a @ A @ B5 ) ) ).

% insertI1
thf(fact_1225_insertI1,axiom,
    ! [A: produc5940350820068290983tate_a,B5: set_Pr7963165934123475591tate_a] : ( member1595334363549574352tate_a @ A @ ( insert1906009156415373175tate_a @ A @ B5 ) ) ).

% insertI1
thf(fact_1226_insertI1,axiom,
    ! [A: product_prod_a_a,B5: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ B5 ) ) ).

% insertI1
thf(fact_1227_insertE,axiom,
    ! [A: a,B: a,A5: set_a] :
      ( ( member_a @ A @ ( insert_a @ B @ A5 ) )
     => ( ( A != B )
       => ( member_a @ A @ A5 ) ) ) ).

% insertE
thf(fact_1228_insertE,axiom,
    ! [A: state_a,B: state_a,A5: set_state_a] :
      ( ( member_state_a @ A @ ( insert_state_a @ B @ A5 ) )
     => ( ( A != B )
       => ( member_state_a @ A @ A5 ) ) ) ).

% insertE
thf(fact_1229_insertE,axiom,
    ! [A: produc6972303929186420058tate_a,B: produc6972303929186420058tate_a,A5: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ A @ ( insert3073504663777712938tate_a @ B @ A5 ) )
     => ( ( A != B )
       => ( member3175992478928454403tate_a @ A @ A5 ) ) ) ).

% insertE
thf(fact_1230_insertE,axiom,
    ! [A: produc5126262593522865844te_a_a,B: produc5126262593522865844te_a_a,A5: set_Pr2749968112124914068te_a_a] :
      ( ( member1329951143264900189te_a_a @ A @ ( insert1227463328114158724te_a_a @ B @ A5 ) )
     => ( ( A != B )
       => ( member1329951143264900189te_a_a @ A @ A5 ) ) ) ).

% insertE
thf(fact_1231_insertE,axiom,
    ! [A: produc4774439622479004775tate_a,B: produc4774439622479004775tate_a,A5: set_Pr1753757909640466247tate_a] :
      ( ( member3801464090328701328tate_a @ A @ ( insert6689868412790682167tate_a @ B @ A5 ) )
     => ( ( A != B )
       => ( member3801464090328701328tate_a @ A @ A5 ) ) ) ).

% insertE
thf(fact_1232_insertE,axiom,
    ! [A: produc5940350820068290983tate_a,B: produc5940350820068290983tate_a,A5: set_Pr7963165934123475591tate_a] :
      ( ( member1595334363549574352tate_a @ A @ ( insert1906009156415373175tate_a @ B @ A5 ) )
     => ( ( A != B )
       => ( member1595334363549574352tate_a @ A @ A5 ) ) ) ).

% insertE
thf(fact_1233_insertE,axiom,
    ! [A: product_prod_a_a,B: product_prod_a_a,A5: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ A5 ) )
     => ( ( A != B )
       => ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ).

% insertE
thf(fact_1234_Sup__upper2,axiom,
    ! [U2: set_Pr4275752383657305402tate_a,A5: set_se5193884918812770586tate_a,V: set_Pr4275752383657305402tate_a] :
      ( ( member4824975370526228195tate_a @ U2 @ A5 )
     => ( ( ord_le7345504482307493082tate_a @ V @ U2 )
       => ( ord_le7345504482307493082tate_a @ V @ ( comple8313209218559427367tate_a @ A5 ) ) ) ) ).

% Sup_upper2
thf(fact_1235_Sup__le__iff,axiom,
    ! [A5: set_se5193884918812770586tate_a,B: set_Pr4275752383657305402tate_a] :
      ( ( ord_le7345504482307493082tate_a @ ( comple8313209218559427367tate_a @ A5 ) @ B )
      = ( ! [X4: set_Pr4275752383657305402tate_a] :
            ( ( member4824975370526228195tate_a @ X4 @ A5 )
           => ( ord_le7345504482307493082tate_a @ X4 @ B ) ) ) ) ).

% Sup_le_iff
thf(fact_1236_Sup__upper,axiom,
    ! [X3: set_Pr4275752383657305402tate_a,A5: set_se5193884918812770586tate_a] :
      ( ( member4824975370526228195tate_a @ X3 @ A5 )
     => ( ord_le7345504482307493082tate_a @ X3 @ ( comple8313209218559427367tate_a @ A5 ) ) ) ).

% Sup_upper
thf(fact_1237_Sup__least,axiom,
    ! [A5: set_se5193884918812770586tate_a,Z2: set_Pr4275752383657305402tate_a] :
      ( ! [X: set_Pr4275752383657305402tate_a] :
          ( ( member4824975370526228195tate_a @ X @ A5 )
         => ( ord_le7345504482307493082tate_a @ X @ Z2 ) )
     => ( ord_le7345504482307493082tate_a @ ( comple8313209218559427367tate_a @ A5 ) @ Z2 ) ) ).

% Sup_least
thf(fact_1238_Sup__mono,axiom,
    ! [A5: set_se5193884918812770586tate_a,B5: set_se5193884918812770586tate_a] :
      ( ! [A3: set_Pr4275752383657305402tate_a] :
          ( ( member4824975370526228195tate_a @ A3 @ A5 )
         => ? [X5: set_Pr4275752383657305402tate_a] :
              ( ( member4824975370526228195tate_a @ X5 @ B5 )
              & ( ord_le7345504482307493082tate_a @ A3 @ X5 ) ) )
     => ( ord_le7345504482307493082tate_a @ ( comple8313209218559427367tate_a @ A5 ) @ ( comple8313209218559427367tate_a @ B5 ) ) ) ).

% Sup_mono
thf(fact_1239_Sup__eqI,axiom,
    ! [A5: set_se5193884918812770586tate_a,X3: set_Pr4275752383657305402tate_a] :
      ( ! [Y3: set_Pr4275752383657305402tate_a] :
          ( ( member4824975370526228195tate_a @ Y3 @ A5 )
         => ( ord_le7345504482307493082tate_a @ Y3 @ X3 ) )
     => ( ! [Y3: set_Pr4275752383657305402tate_a] :
            ( ! [Z5: set_Pr4275752383657305402tate_a] :
                ( ( member4824975370526228195tate_a @ Z5 @ A5 )
               => ( ord_le7345504482307493082tate_a @ Z5 @ Y3 ) )
           => ( ord_le7345504482307493082tate_a @ X3 @ Y3 ) )
       => ( ( comple8313209218559427367tate_a @ A5 )
          = X3 ) ) ) ).

% Sup_eqI
thf(fact_1240_total__on__def,axiom,
    ( total_on_state_a
    = ( ^ [A6: set_state_a,R3: set_Pr1753757909640466247tate_a] :
        ! [X4: state_a] :
          ( ( member_state_a @ X4 @ A6 )
         => ! [Y6: state_a] :
              ( ( member_state_a @ Y6 @ A6 )
             => ( ( X4 != Y6 )
               => ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X4 @ Y6 ) @ R3 )
                  | ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y6 @ X4 ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_1241_total__on__def,axiom,
    ( total_8559467112218031838tate_a
    = ( ^ [A6: set_Pr4275752383657305402tate_a,R3: set_Pr7963165934123475591tate_a] :
        ! [X4: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X4 @ A6 )
         => ! [Y6: produc6972303929186420058tate_a] :
              ( ( member3175992478928454403tate_a @ Y6 @ A6 )
             => ( ( X4 != Y6 )
               => ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X4 @ Y6 ) @ R3 )
                  | ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y6 @ X4 ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_1242_total__on__def,axiom,
    ( total_on_a
    = ( ^ [A6: set_a,R3: set_Product_prod_a_a] :
        ! [X4: a] :
          ( ( member_a @ X4 @ A6 )
         => ! [Y6: a] :
              ( ( member_a @ Y6 @ A6 )
             => ( ( X4 != Y6 )
               => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ R3 )
                  | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y6 @ X4 ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_1243_total__onI,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a,R: set_Pr4734257497606733575te_a_a] :
      ( ! [X: produc5126262593522865844te_a_a,Y3: produc5126262593522865844te_a_a] :
          ( ( member1329951143264900189te_a_a @ X @ A5 )
         => ( ( member1329951143264900189te_a_a @ Y3 @ A5 )
           => ( ( X != Y3 )
             => ( ( member2649312330005299536te_a_a @ ( produc2877451964877216791te_a_a @ X @ Y3 ) @ R )
                | ( member2649312330005299536te_a_a @ ( produc2877451964877216791te_a_a @ Y3 @ X ) @ R ) ) ) ) )
     => ( total_6713425776554477624te_a_a @ A5 @ R ) ) ).

% total_onI
thf(fact_1244_total__onI,axiom,
    ! [A5: set_Pr1753757909640466247tate_a,R: set_Pr2656942177312230599tate_a] :
      ( ! [X: produc4774439622479004775tate_a,Y3: produc4774439622479004775tate_a] :
          ( ( member3801464090328701328tate_a @ X @ A5 )
         => ( ( member3801464090328701328tate_a @ Y3 @ A5 )
           => ( ( X != Y3 )
             => ( ( member7923945571256933136tate_a @ ( produc1588899729123023703tate_a @ X @ Y3 ) @ R )
                | ( member7923945571256933136tate_a @ ( produc1588899729123023703tate_a @ Y3 @ X ) @ R ) ) ) ) )
     => ( total_5784872156199265259tate_a @ A5 @ R ) ) ).

% total_onI
thf(fact_1245_total__onI,axiom,
    ! [A5: set_Pr7963165934123475591tate_a,R: set_Pr1690608023759948999tate_a] :
      ( ! [X: produc5940350820068290983tate_a,Y3: produc5940350820068290983tate_a] :
          ( ( member1595334363549574352tate_a @ X @ A5 )
         => ( ( member1595334363549574352tate_a @ Y3 @ A5 )
           => ( ( X != Y3 )
             => ( ( member1643692453717642000tate_a @ ( produc7335381768849070935tate_a @ X @ Y3 ) @ R )
                | ( member1643692453717642000tate_a @ ( produc7335381768849070935tate_a @ Y3 @ X ) @ R ) ) ) ) )
     => ( total_2079079962484902187tate_a @ A5 @ R ) ) ).

% total_onI
thf(fact_1246_total__onI,axiom,
    ! [A5: set_Product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
      ( ! [X: product_prod_a_a,Y3: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ A5 )
         => ( ( member1426531477525435216od_a_a @ Y3 @ A5 )
           => ( ( X != Y3 )
             => ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ R )
                | ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ X ) @ R ) ) ) ) )
     => ( total_3048927264068743339od_a_a @ A5 @ R ) ) ).

% total_onI
thf(fact_1247_total__onI,axiom,
    ! [A5: set_state_a,R: set_Pr1753757909640466247tate_a] :
      ( ! [X: state_a,Y3: state_a] :
          ( ( member_state_a @ X @ A5 )
         => ( ( member_state_a @ Y3 @ A5 )
           => ( ( X != Y3 )
             => ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ X @ Y3 ) @ R )
                | ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y3 @ X ) @ R ) ) ) ) )
     => ( total_on_state_a @ A5 @ R ) ) ).

% total_onI
thf(fact_1248_total__onI,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,R: set_Pr7963165934123475591tate_a] :
      ( ! [X: produc6972303929186420058tate_a,Y3: produc6972303929186420058tate_a] :
          ( ( member3175992478928454403tate_a @ X @ A5 )
         => ( ( member3175992478928454403tate_a @ Y3 @ A5 )
           => ( ( X != Y3 )
             => ( ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ X @ Y3 ) @ R )
                | ( member1595334363549574352tate_a @ ( produc2143731062552654487tate_a @ Y3 @ X ) @ R ) ) ) ) )
     => ( total_8559467112218031838tate_a @ A5 @ R ) ) ).

% total_onI
thf(fact_1249_total__onI,axiom,
    ! [A5: set_a,R: set_Product_prod_a_a] :
      ( ! [X: a,Y3: a] :
          ( ( member_a @ X @ A5 )
         => ( ( member_a @ Y3 @ A5 )
           => ( ( X != Y3 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
                | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ R ) ) ) ) )
     => ( total_on_a @ A5 @ R ) ) ).

% total_onI
thf(fact_1250_total__on__subset,axiom,
    ! [A5: set_Pr4275752383657305402tate_a,R: set_Pr7963165934123475591tate_a,B5: set_Pr4275752383657305402tate_a] :
      ( ( total_8559467112218031838tate_a @ A5 @ R )
     => ( ( ord_le7345504482307493082tate_a @ B5 @ A5 )
       => ( total_8559467112218031838tate_a @ B5 @ R ) ) ) ).

% total_on_subset
thf(fact_1251_total__lex__prod,axiom,
    ! [R_A: set_Pr7963165934123475591tate_a,R_B: set_Pr7963165934123475591tate_a] :
      ( ( total_8559467112218031838tate_a @ top_to8456965716270553738tate_a @ R_A )
     => ( ( total_8559467112218031838tate_a @ top_to8456965716270553738tate_a @ R_B )
       => ( total_2079079962484902187tate_a @ top_to93540037818290135tate_a @ ( lex_pr5802797673463856815tate_a @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_1252_relcompEpair,axiom,
    ! [A: a,C2: state_a,R: set_Product_prod_a_a,S: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A @ C2 ) @ ( relcomp_a_a_state_a @ R @ S ) )
     => ~ ! [B3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B3 ) @ R )
           => ~ ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ B3 @ C2 ) @ S ) ) ) ).

% relcompEpair
thf(fact_1253_relcompEpair,axiom,
    ! [A: a,C2: state_a,R: set_Pr4275752383657305402tate_a,S: set_Pr1753757909640466247tate_a] :
      ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A @ C2 ) @ ( relcom5388636112749073917tate_a @ R @ S ) )
     => ~ ! [B3: state_a] :
            ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A @ B3 ) @ R )
           => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ B3 @ C2 ) @ S ) ) ) ).

% relcompEpair
thf(fact_1254_relcompE,axiom,
    ! [Xz: produc6972303929186420058tate_a,R: set_Product_prod_a_a,S: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ Xz @ ( relcomp_a_a_state_a @ R @ S ) )
     => ~ ! [X: a,Y3: a,Z3: state_a] :
            ( ( Xz
              = ( produc8641956578966763338tate_a @ X @ Z3 ) )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R )
             => ~ ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ Y3 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_1255_relcompE,axiom,
    ! [Xz: produc6972303929186420058tate_a,R: set_Pr4275752383657305402tate_a,S: set_Pr1753757909640466247tate_a] :
      ( ( member3175992478928454403tate_a @ Xz @ ( relcom5388636112749073917tate_a @ R @ S ) )
     => ~ ! [X: a,Y3: state_a,Z3: state_a] :
            ( ( Xz
              = ( produc8641956578966763338tate_a @ X @ Z3 ) )
           => ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X @ Y3 ) @ R )
             => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ Y3 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_1256_relcomp_OrelcompI,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,C2: state_a,S: set_Pr4275752383657305402tate_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
     => ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ B @ C2 ) @ S )
       => ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A @ C2 ) @ ( relcomp_a_a_state_a @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_1257_relcomp_OrelcompI,axiom,
    ! [A: a,B: state_a,R: set_Pr4275752383657305402tate_a,C2: state_a,S: set_Pr1753757909640466247tate_a] :
      ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A @ B ) @ R )
     => ( ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ B @ C2 ) @ S )
       => ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A @ C2 ) @ ( relcom5388636112749073917tate_a @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_1258_relcomp_Osimps,axiom,
    ! [A1: a,A22: state_a,R: set_Product_prod_a_a,S: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A1 @ A22 ) @ ( relcomp_a_a_state_a @ R @ S ) )
      = ( ? [A4: a,B4: a,C6: state_a] :
            ( ( A1 = A4 )
            & ( A22 = C6 )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ R )
            & ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ B4 @ C6 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_1259_relcomp_Osimps,axiom,
    ! [A1: a,A22: state_a,R: set_Pr4275752383657305402tate_a,S: set_Pr1753757909640466247tate_a] :
      ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A1 @ A22 ) @ ( relcom5388636112749073917tate_a @ R @ S ) )
      = ( ? [A4: a,B4: state_a,C6: state_a] :
            ( ( A1 = A4 )
            & ( A22 = C6 )
            & ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A4 @ B4 ) @ R )
            & ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ B4 @ C6 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_1260_relcomp_Ocases,axiom,
    ! [A1: a,A22: state_a,R: set_Product_prod_a_a,S: set_Pr4275752383657305402tate_a] :
      ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A1 @ A22 ) @ ( relcomp_a_a_state_a @ R @ S ) )
     => ~ ! [B3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B3 ) @ R )
           => ~ ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_1261_relcomp_Ocases,axiom,
    ! [A1: a,A22: state_a,R: set_Pr4275752383657305402tate_a,S: set_Pr1753757909640466247tate_a] :
      ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A1 @ A22 ) @ ( relcom5388636112749073917tate_a @ R @ S ) )
     => ~ ! [B3: state_a] :
            ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ A1 @ B3 ) @ R )
           => ~ ( member3801464090328701328tate_a @ ( produc6131900399494507223tate_a @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_1262_prod__set__simps_I1_J,axiom,
    ! [X3: a,Y: state_a] :
      ( ( basic_fsts_a_state_a @ ( produc8641956578966763338tate_a @ X3 @ Y ) )
      = ( insert_a @ X3 @ bot_bot_set_a ) ) ).

% prod_set_simps(1)
thf(fact_1263_prod__set__simps_I2_J,axiom,
    ! [X3: a,Y: state_a] :
      ( ( basic_snds_a_state_a @ ( produc8641956578966763338tate_a @ X3 @ Y ) )
      = ( insert_state_a @ Y @ bot_bot_set_state_a ) ) ).

% prod_set_simps(2)
thf(fact_1264_case__prodI2,axiom,
    ! [P: produc6972303929186420058tate_a,C2: a > state_a > $o] :
      ( ! [A3: a,B3: state_a] :
          ( ( P
            = ( produc8641956578966763338tate_a @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc3717778009801768057te_a_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_1265_case__prodI,axiom,
    ! [F2: a > state_a > $o,A: a,B: state_a] :
      ( ( F2 @ A @ B )
     => ( produc3717778009801768057te_a_o @ F2 @ ( produc8641956578966763338tate_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_1266_inf__Int__eq2,axiom,
    ! [R2: set_Pr4275752383657305402tate_a,S4: set_Pr4275752383657305402tate_a] :
      ( ( inf_inf_a_state_a_o
        @ ^ [X4: a,Y6: state_a] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: a,Y6: state_a] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y6 ) @ S4 ) )
      = ( ^ [X4: a,Y6: state_a] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y6 ) @ ( inf_in2817715555991467560tate_a @ R2 @ S4 ) ) ) ) ).

% inf_Int_eq2
thf(fact_1267_bot__empty__eq2,axiom,
    ( bot_bot_a_state_a_o
    = ( ^ [X4: a,Y6: state_a] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y6 ) @ bot_bo2006125347199251878tate_a ) ) ) ).

% bot_empty_eq2
thf(fact_1268_sup__Un__eq2,axiom,
    ! [R2: set_Pr4275752383657305402tate_a,S4: set_Pr4275752383657305402tate_a] :
      ( ( sup_sup_a_state_a_o
        @ ^ [X4: a,Y6: state_a] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y6 ) @ R2 )
        @ ^ [X4: a,Y6: state_a] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y6 ) @ S4 ) )
      = ( ^ [X4: a,Y6: state_a] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y6 ) @ ( sup_su1662690461080552718tate_a @ R2 @ S4 ) ) ) ) ).

% sup_Un_eq2
thf(fact_1269_inj__on__convol__ident,axiom,
    ! [F2: a > state_a,X6: set_a] :
      ( inj_on3780297825070694605tate_a
      @ ^ [X4: a] : ( produc8641956578966763338tate_a @ X4 @ ( F2 @ X4 ) )
      @ X6 ) ).

% inj_on_convol_ident
thf(fact_1270_swap__inj__on,axiom,
    ! [A5: set_Pr2749968112124914068te_a_a] :
      ( inj_on6706560762754772119tate_a
      @ ( produc7343868943834020833tate_a
        @ ^ [I2: state_a,J: a] : ( produc8641956578966763338tate_a @ J @ I2 ) )
      @ A5 ) ).

% swap_inj_on
thf(fact_1271_case__prodD,axiom,
    ! [F2: a > state_a > $o,A: a,B: state_a] :
      ( ( produc3717778009801768057te_a_o @ F2 @ ( produc8641956578966763338tate_a @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_1272_case__prodE,axiom,
    ! [C2: a > state_a > $o,P: produc6972303929186420058tate_a] :
      ( ( produc3717778009801768057te_a_o @ C2 @ P )
     => ~ ! [X: a,Y3: state_a] :
            ( ( P
              = ( produc8641956578966763338tate_a @ X @ Y3 ) )
           => ~ ( C2 @ X @ Y3 ) ) ) ).

% case_prodE
thf(fact_1273_top__empty__eq2,axiom,
    ( top_top_a_state_a_o
    = ( ^ [X4: a,Y6: state_a] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y6 ) @ top_to8456965716270553738tate_a ) ) ) ).

% top_empty_eq2
thf(fact_1274_case__prod__Pair__iden,axiom,
    ! [P: produc6972303929186420058tate_a] :
      ( ( produc5441864759085668411tate_a @ produc8641956578966763338tate_a @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_1275_pred__equals__eq2,axiom,
    ! [R2: set_Pr4275752383657305402tate_a,S4: set_Pr4275752383657305402tate_a] :
      ( ( ( ^ [X4: a,Y6: state_a] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y6 ) @ R2 ) )
        = ( ^ [X4: a,Y6: state_a] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y6 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [X5: a,Big: state_a] :
      ( ( ( pop_a @ big )
        = ( produc8641956578966763338tate_a @ X5 @ Big ) )
     => thesis ) ).

thf(conj_1,conjecture,
    thesis ).

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