%------------------------------------------------------------------------------
% 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    : SCC_Bloemen_Sequential/0000_SCC_Bloemen_Sequential/prob_00671_023266__5767694_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1484 ( 550 unt; 201 typ;   0 def)
%            Number of atoms       : 3909 (1252 equ;   0 cnn)
%            Maximal formula atoms :    9 (   3 avg)
%            Number of connectives : 12710 ( 433   ~;  54   |; 301   &;10024   @)
%                                         (   0 <=>;1898  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   7 avg)
%            Number of types       :   23 (  22 usr)
%            Number of type conns  :  693 ( 693   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  182 ( 179 usr;  13 con; 0-4 aty)
%            Number of variables   : 3733 ( 132   ^;3451   !; 150   ?;3733   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 08:50:06.601
%------------------------------------------------------------------------------
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    chains_v2: set_set_v > set_set_set_v ).

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    member3038538357316246288od_v_v: produc206430290419586791od_v_v > set_Pr2149350503807050951od_v_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
    member1739204623775007504od_v_v: produc4818592998197102311od_v_v > set_Pr4606041269135813831od_v_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__v_J_Mt__Set__Oset_Itf__v_J_J,type,
    member3828772815783460880_set_v: produc6772369698050171367_set_v > set_Pr8199228935972127175_set_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
    member7453568604450474000od_v_v: product_prod_v_v > set_Product_prod_v_v > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
    member2865299526245254384od_v_v: set_Pr2149350503807050951od_v_v > set_se5707775751431548583od_v_v > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
    member8406446414694345712od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
    member5511408251247217616od_v_v: set_se8455005133513928103od_v_v > set_se2157405561750842759od_v_v > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
    member_set_set_v: set_set_v > set_set_set_v > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__v_J,type,
    member_set_v: set_v > set_set_v > $o ).

thf(sy_c_member_001tf__v,type,
    member_v: v > set_v > $o ).

thf(sy_v_e,type,
    e: sCC_Bl1191828773336950226xt_v_a ).

thf(sy_v_successors,type,
    successors: v > set_v ).

thf(sy_v_v,type,
    v2: v ).

thf(sy_v_vertices,type,
    vertices: set_v ).

thf(sy_v_w,type,
    w: v ).

% Relevant facts (1277)
thf(fact_0_v,axiom,
    member_v @ v2 @ ( sCC_Bl1198488560823802982ed_v_a @ e ) ).

% v
thf(fact_1_w,axiom,
    sCC_Bl649662514949026229able_v @ successors @ v2 @ w ).

% w
thf(fact_2_s,axiom,
    ! [X: v] :
      ( ( member_v @ X @ ( sCC_Bl1198488560823802982ed_v_a @ e ) )
     => ! [Xa: v] :
          ( ( member_v @ Xa @ ( minus_minus_set_v @ ( successors @ X ) @ ( sCC_Bl983769552681565941cs_v_a @ e @ X ) ) )
         => ( ( sCC_Bl649662514949026229able_v @ successors @ v2 @ X )
           => ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa @ w ) ) ) ) ).

% s
thf(fact_3_e,axiom,
    sCC_Bl4124178362578471481nv_v_a @ successors @ e ).

% e
thf(fact_4_reachable_Ocases,axiom,
    ! [A1: v,A2: v] :
      ( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
     => ( ( A2 != A1 )
       => ~ ! [Y: v] :
              ( ( member_v @ Y @ ( successors @ A1 ) )
             => ~ ( sCC_Bl649662514949026229able_v @ successors @ Y @ A2 ) ) ) ) ).

% reachable.cases
thf(fact_5_reachable__refl,axiom,
    ! [X2: v] : ( sCC_Bl649662514949026229able_v @ successors @ X2 @ X2 ) ).

% reachable_refl
thf(fact_6_reachable__succ,axiom,
    ! [Y2: v,X2: v,Z: v] :
      ( ( member_v @ Y2 @ ( successors @ X2 ) )
     => ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z )
       => ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Z ) ) ) ).

% reachable_succ
thf(fact_7_reachable_Osimps,axiom,
    ! [A1: v,A2: v] :
      ( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
      = ( ? [X3: v] :
            ( ( A1 = X3 )
            & ( A2 = X3 ) )
        | ? [X3: v,Y3: v,Z2: v] :
            ( ( A1 = X3 )
            & ( A2 = Z2 )
            & ( member_v @ Y3 @ ( successors @ X3 ) )
            & ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z2 ) ) ) ) ).

% reachable.simps
thf(fact_8_reachable__edge,axiom,
    ! [Y2: v,X2: v] :
      ( ( member_v @ Y2 @ ( successors @ X2 ) )
     => ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 ) ) ).

% reachable_edge
thf(fact_9_reachable__end__induct,axiom,
    ! [X2: v,Y2: v,P: v > v > $o] :
      ( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 )
     => ( ! [X4: v] : ( P @ X4 @ X4 )
       => ( ! [X4: v,Y: v,Z3: v] :
              ( ( P @ X4 @ Y )
             => ( ( member_v @ Z3 @ ( successors @ Y ) )
               => ( P @ X4 @ Z3 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% reachable_end_induct
thf(fact_10_reachable__trans,axiom,
    ! [X2: v,Y2: v,Z: v] :
      ( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 )
     => ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z )
       => ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Z ) ) ) ).

% reachable_trans
thf(fact_11_succ__reachable,axiom,
    ! [X2: v,Y2: v,Z: v] :
      ( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 )
     => ( ( member_v @ Z @ ( successors @ Y2 ) )
       => ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Z ) ) ) ).

% succ_reachable
thf(fact_12_fold__congs_I4_J,axiom,
    ! [R: sCC_Bl1191828773336950226xt_v_a,R2: sCC_Bl1191828773336950226xt_v_a,V: set_v,F: set_v > set_v,F2: set_v > set_v] :
      ( ( R = R2 )
     => ( ( ( sCC_Bl1198488560823802982ed_v_a @ R2 )
          = V )
       => ( ! [V2: set_v] :
              ( ( V = V2 )
             => ( ( F @ V2 )
                = ( F2 @ V2 ) ) )
         => ( ( sCC_Bl5937794865345830735te_v_a @ F @ R )
            = ( sCC_Bl5937794865345830735te_v_a @ F2 @ R2 ) ) ) ) ) ).

% fold_congs(4)
thf(fact_13_unfold__congs_I4_J,axiom,
    ! [R: sCC_Bl1191828773336950226xt_v_a,R2: sCC_Bl1191828773336950226xt_v_a,V: set_v,F: set_v > set_v,F2: set_v > set_v] :
      ( ( R = R2 )
     => ( ( ( sCC_Bl1198488560823802982ed_v_a @ R2 )
          = V )
       => ( ! [V2: set_v] :
              ( ( V2 = V )
             => ( ( F @ V2 )
                = ( F2 @ V2 ) ) )
         => ( ( sCC_Bl5937794865345830735te_v_a @ F @ R )
            = ( sCC_Bl5937794865345830735te_v_a @ F2 @ R2 ) ) ) ) ) ).

% unfold_congs(4)
thf(fact_14_graph_Oreachable_Ocong,axiom,
    sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).

% graph.reachable.cong
thf(fact_15_sccE,axiom,
    ! [S: set_v,X2: v,Y2: v] :
      ( ( sCC_Bloemen_is_scc_v @ successors @ S )
     => ( ( member_v @ X2 @ S )
       => ( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 )
         => ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X2 )
           => ( member_v @ Y2 @ S ) ) ) ) ) ).

% sccE
thf(fact_16_graph_Owf__env_Ocong,axiom,
    sCC_Bl4124178362578471481nv_v_a = sCC_Bl4124178362578471481nv_v_a ).

% graph.wf_env.cong
thf(fact_17_is__subscc__def,axiom,
    ! [S: set_v] :
      ( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
      = ( ! [X3: v] :
            ( ( member_v @ X3 @ S )
           => ! [Y3: v] :
                ( ( member_v @ Y3 @ S )
               => ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y3 ) ) ) ) ) ).

% is_subscc_def
thf(fact_18_S__reflexive,axiom,
    ! [E: sCC_Bl1191828773336950226xt_v_a,N: v] :
      ( ( sCC_Bl4124178362578471481nv_v_a @ successors @ E )
     => ( member_v @ N @ ( sCC_Bloemen_S_v_a @ E @ N ) ) ) ).

% S_reflexive
thf(fact_19_re__reachable,axiom,
    ! [X2: v,Y2: v] :
      ( ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y2 )
     => ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 ) ) ).

% re_reachable
thf(fact_20_reachable__re,axiom,
    ! [X2: v,Y2: v] :
      ( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 )
     => ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y2 ) ) ).

% reachable_re
thf(fact_21_DiffI,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ A )
     => ( ~ ( member7453568604450474000od_v_v @ C @ B )
       => ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ) ).

% DiffI
thf(fact_22_DiffI,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ A )
     => ( ~ ( member_v @ C @ B )
       => ( member_v @ C @ ( minus_minus_set_v @ A @ B ) ) ) ) ).

% DiffI
thf(fact_23_Diff__iff,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) )
      = ( ( member7453568604450474000od_v_v @ C @ A )
        & ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% Diff_iff
thf(fact_24_Diff__iff,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ ( minus_minus_set_v @ A @ B ) )
      = ( ( member_v @ C @ A )
        & ~ ( member_v @ C @ B ) ) ) ).

% Diff_iff
thf(fact_25_Diff__idemp,axiom,
    ! [A: set_v,B: set_v] :
      ( ( minus_minus_set_v @ ( minus_minus_set_v @ A @ B ) @ B )
      = ( minus_minus_set_v @ A @ B ) ) ).

% Diff_idemp
thf(fact_26_ra__reachable,axiom,
    ! [X2: v,Y2: v,E2: set_Product_prod_v_v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E2 )
     => ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 ) ) ).

% ra_reachable
thf(fact_27_succ__re,axiom,
    ! [Y2: v,X2: v,Z: v] :
      ( ( member_v @ Y2 @ ( successors @ X2 ) )
     => ( ( sCC_Bl770211535891879572_end_v @ successors @ Y2 @ Z )
       => ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Z ) ) ) ).

% succ_re
thf(fact_28_reachable__end_Osimps,axiom,
    ! [A1: v,A2: v] :
      ( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
      = ( ? [X3: v] :
            ( ( A1 = X3 )
            & ( A2 = X3 ) )
        | ? [X3: v,Y3: v,Z2: v] :
            ( ( A1 = X3 )
            & ( A2 = Z2 )
            & ( sCC_Bl770211535891879572_end_v @ successors @ X3 @ Y3 )
            & ( member_v @ Z2 @ ( successors @ Y3 ) ) ) ) ) ).

% reachable_end.simps
thf(fact_29_re__succ,axiom,
    ! [X2: v,Y2: v,Z: v] :
      ( ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y2 )
     => ( ( member_v @ Z @ ( successors @ Y2 ) )
       => ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Z ) ) ) ).

% re_succ
thf(fact_30_ra__trans,axiom,
    ! [X2: v,Y2: v,E2: set_Product_prod_v_v,Z: v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E2 )
     => ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y2 @ Z @ E2 )
       => ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Z @ E2 ) ) ) ).

% ra_trans
thf(fact_31_ra__refl,axiom,
    ! [X2: v,E2: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ X2 @ E2 ) ).

% ra_refl
thf(fact_32_reachable__end_Ocases,axiom,
    ! [A1: v,A2: v] :
      ( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
     => ( ( A2 != A1 )
       => ~ ! [Y: v] :
              ( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y )
             => ~ ( member_v @ A2 @ ( successors @ Y ) ) ) ) ) ).

% reachable_end.cases
thf(fact_33_re__refl,axiom,
    ! [X2: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ X2 ) ).

% re_refl
thf(fact_34_graph_Oreachable__avoiding_Ocong,axiom,
    sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).

% graph.reachable_avoiding.cong
thf(fact_35_graph_Oreachable__end_Ocong,axiom,
    sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).

% graph.reachable_end.cong
thf(fact_36_graph_Ois__subscc_Ocong,axiom,
    sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).

% graph.is_subscc.cong
thf(fact_37_graph_Ois__scc_Ocong,axiom,
    sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).

% graph.is_scc.cong
thf(fact_38_DiffD2,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) )
     => ~ ( member7453568604450474000od_v_v @ C @ B ) ) ).

% DiffD2
thf(fact_39_DiffD2,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ ( minus_minus_set_v @ A @ B ) )
     => ~ ( member_v @ C @ B ) ) ).

% DiffD2
thf(fact_40_DiffD1,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) )
     => ( member7453568604450474000od_v_v @ C @ A ) ) ).

% DiffD1
thf(fact_41_DiffD1,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ ( minus_minus_set_v @ A @ B ) )
     => ( member_v @ C @ A ) ) ).

% DiffD1
thf(fact_42_mem__Collect__eq,axiom,
    ! [A3: v,P: v > $o] :
      ( ( member_v @ A3 @ ( collect_v @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
    ! [A3: product_prod_v_v,P: product_prod_v_v > $o] :
      ( ( member7453568604450474000od_v_v @ A3 @ ( collec140062887454715474od_v_v @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
    ! [A: set_v] :
      ( ( collect_v
        @ ^ [X3: v] : ( member_v @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( collec140062887454715474od_v_v
        @ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_46_DiffE,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A @ B ) )
     => ~ ( ( member7453568604450474000od_v_v @ C @ A )
         => ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% DiffE
thf(fact_47_DiffE,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ ( minus_minus_set_v @ A @ B ) )
     => ~ ( ( member_v @ C @ A )
         => ( member_v @ C @ B ) ) ) ).

% DiffE
thf(fact_48_ra__empty,axiom,
    ! [X2: v,Y2: v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ bot_bo723834152578015283od_v_v )
      = ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 ) ) ).

% ra_empty
thf(fact_49_subscc__add,axiom,
    ! [S: set_v,X2: v,Y2: v] :
      ( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
     => ( ( member_v @ X2 @ S )
       => ( ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y2 )
         => ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X2 )
           => ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v2 @ Y2 @ S ) ) ) ) ) ) ).

% subscc_add
thf(fact_50_scc__partition,axiom,
    ! [S: set_v,S2: set_v,X2: v] :
      ( ( sCC_Bloemen_is_scc_v @ successors @ S )
     => ( ( sCC_Bloemen_is_scc_v @ successors @ S2 )
       => ( ( member_v @ X2 @ ( inf_inf_set_v @ S @ S2 ) )
         => ( S = S2 ) ) ) ) ).

% scc_partition
thf(fact_51_ra__mono,axiom,
    ! [X2: v,Y2: v,E2: set_Product_prod_v_v,E3: set_Product_prod_v_v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E2 )
     => ( ( ord_le7336532860387713383od_v_v @ E3 @ E2 )
       => ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E3 ) ) ) ).

% ra_mono
thf(fact_52_ra__cases,axiom,
    ! [X2: v,Y2: v,E2: set_Product_prod_v_v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E2 )
     => ( ( X2 = Y2 )
        | ? [Z3: v] :
            ( ( member_v @ Z3 @ ( successors @ X2 ) )
            & ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Z3 ) @ E2 )
            & ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y2 @ E2 ) ) ) ) ).

% ra_cases
thf(fact_53_edge__ra,axiom,
    ! [Y2: v,X2: v,E2: set_Product_prod_v_v] :
      ( ( member_v @ Y2 @ ( successors @ X2 ) )
     => ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y2 ) @ E2 )
       => ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E2 ) ) ) ).

% edge_ra
thf(fact_54_reachable__avoiding_Osimps,axiom,
    ! [A1: v,A2: v,A32: set_Product_prod_v_v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
      = ( ? [X3: v,E4: set_Product_prod_v_v] :
            ( ( A1 = X3 )
            & ( A2 = X3 )
            & ( A32 = E4 ) )
        | ? [X3: v,Y3: v,E4: set_Product_prod_v_v,Z2: v] :
            ( ( A1 = X3 )
            & ( A2 = Z2 )
            & ( A32 = E4 )
            & ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y3 @ E4 )
            & ( member_v @ Z2 @ ( successors @ Y3 ) )
            & ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E4 ) ) ) ) ).

% reachable_avoiding.simps
thf(fact_55_ra__succ,axiom,
    ! [X2: v,Y2: v,E2: set_Product_prod_v_v,Z: v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E2 )
     => ( ( member_v @ Z @ ( successors @ Y2 ) )
       => ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z ) @ E2 )
         => ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Z @ E2 ) ) ) ) ).

% ra_succ
thf(fact_56_reachable__avoiding_Ocases,axiom,
    ! [A1: v,A2: v,A32: set_Product_prod_v_v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
     => ( ( A2 != A1 )
       => ~ ! [Y: v] :
              ( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y @ A32 )
             => ( ( member_v @ A2 @ ( successors @ Y ) )
               => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ A2 ) @ A32 ) ) ) ) ) ).

% reachable_avoiding.cases
thf(fact_57_graph__axioms,axiom,
    sCC_Bloemen_graph_v @ vertices @ successors ).

% graph_axioms
thf(fact_58_empty__Collect__eq,axiom,
    ! [P: product_prod_v_v > $o] :
      ( ( bot_bo723834152578015283od_v_v
        = ( collec140062887454715474od_v_v @ P ) )
      = ( ! [X3: product_prod_v_v] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_59_empty__Collect__eq,axiom,
    ! [P: v > $o] :
      ( ( bot_bot_set_v
        = ( collect_v @ P ) )
      = ( ! [X3: v] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_60_Collect__empty__eq,axiom,
    ! [P: product_prod_v_v > $o] :
      ( ( ( collec140062887454715474od_v_v @ P )
        = bot_bo723834152578015283od_v_v )
      = ( ! [X3: product_prod_v_v] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_61_Collect__empty__eq,axiom,
    ! [P: v > $o] :
      ( ( ( collect_v @ P )
        = bot_bot_set_v )
      = ( ! [X3: v] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_62_all__not__in__conv,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( ! [X3: product_prod_v_v] :
            ~ ( member7453568604450474000od_v_v @ X3 @ A ) )
      = ( A = bot_bo723834152578015283od_v_v ) ) ).

% all_not_in_conv
thf(fact_63_all__not__in__conv,axiom,
    ! [A: set_v] :
      ( ( ! [X3: v] :
            ~ ( member_v @ X3 @ A ) )
      = ( A = bot_bot_set_v ) ) ).

% all_not_in_conv
thf(fact_64_empty__iff,axiom,
    ! [C: product_prod_v_v] :
      ~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).

% empty_iff
thf(fact_65_empty__iff,axiom,
    ! [C: v] :
      ~ ( member_v @ C @ bot_bot_set_v ) ).

% empty_iff
thf(fact_66_subset__antisym,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ( ord_le7336532860387713383od_v_v @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_67_subset__antisym,axiom,
    ! [A: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ( ord_less_eq_set_v @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_68_subsetI,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ! [X4: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X4 @ A )
         => ( member7453568604450474000od_v_v @ X4 @ B ) )
     => ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).

% subsetI
thf(fact_69_subsetI,axiom,
    ! [A: set_v,B: set_v] :
      ( ! [X4: v] :
          ( ( member_v @ X4 @ A )
         => ( member_v @ X4 @ B ) )
     => ( ord_less_eq_set_v @ A @ B ) ) ).

% subsetI
thf(fact_70_insert__absorb2,axiom,
    ! [X2: v,A: set_v] :
      ( ( insert_v2 @ X2 @ ( insert_v2 @ X2 @ A ) )
      = ( insert_v2 @ X2 @ A ) ) ).

% insert_absorb2
thf(fact_71_insert__absorb2,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( insert1338601472111419319od_v_v @ X2 @ ( insert1338601472111419319od_v_v @ X2 @ A ) )
      = ( insert1338601472111419319od_v_v @ X2 @ A ) ) ).

% insert_absorb2
thf(fact_72_insert__iff,axiom,
    ! [A3: v,B2: v,A: set_v] :
      ( ( member_v @ A3 @ ( insert_v2 @ B2 @ A ) )
      = ( ( A3 = B2 )
        | ( member_v @ A3 @ A ) ) ) ).

% insert_iff
thf(fact_73_insert__iff,axiom,
    ! [A3: product_prod_v_v,B2: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ A ) )
      = ( ( A3 = B2 )
        | ( member7453568604450474000od_v_v @ A3 @ A ) ) ) ).

% insert_iff
thf(fact_74_insertCI,axiom,
    ! [A3: v,B: set_v,B2: v] :
      ( ( ~ ( member_v @ A3 @ B )
       => ( A3 = B2 ) )
     => ( member_v @ A3 @ ( insert_v2 @ B2 @ B ) ) ) ).

% insertCI
thf(fact_75_insertCI,axiom,
    ! [A3: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
      ( ( ~ ( member7453568604450474000od_v_v @ A3 @ B )
       => ( A3 = B2 ) )
     => ( member7453568604450474000od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).

% insertCI
thf(fact_76_inf__right__idem,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ Y2 )
      = ( inf_inf_set_v @ X2 @ Y2 ) ) ).

% inf_right_idem
thf(fact_77_inf_Oright__idem,axiom,
    ! [A3: set_v,B2: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ B2 )
      = ( inf_inf_set_v @ A3 @ B2 ) ) ).

% inf.right_idem
thf(fact_78_inf__left__idem,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ X2 @ Y2 ) )
      = ( inf_inf_set_v @ X2 @ Y2 ) ) ).

% inf_left_idem
thf(fact_79_inf_Oleft__idem,axiom,
    ! [A3: set_v,B2: set_v] :
      ( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ A3 @ B2 ) )
      = ( inf_inf_set_v @ A3 @ B2 ) ) ).

% inf.left_idem
thf(fact_80_inf__idem,axiom,
    ! [X2: set_v] :
      ( ( inf_inf_set_v @ X2 @ X2 )
      = X2 ) ).

% inf_idem
thf(fact_81_inf_Oidem,axiom,
    ! [A3: set_v] :
      ( ( inf_inf_set_v @ A3 @ A3 )
      = A3 ) ).

% inf.idem
thf(fact_82_Int__iff,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
      = ( ( member7453568604450474000od_v_v @ C @ A )
        & ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% Int_iff
thf(fact_83_Int__iff,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ ( inf_inf_set_v @ A @ B ) )
      = ( ( member_v @ C @ A )
        & ( member_v @ C @ B ) ) ) ).

% Int_iff
thf(fact_84_IntI,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ A )
     => ( ( member7453568604450474000od_v_v @ C @ B )
       => ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).

% IntI
thf(fact_85_IntI,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ A )
     => ( ( member_v @ C @ B )
       => ( member_v @ C @ ( inf_inf_set_v @ A @ B ) ) ) ) ).

% IntI
thf(fact_86_inf_Obounded__iff,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
      = ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
        & ( ord_le7336532860387713383od_v_v @ A3 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_87_inf_Obounded__iff,axiom,
    ! [A3: set_v,B2: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ ( inf_inf_set_v @ B2 @ C ) )
      = ( ( ord_less_eq_set_v @ A3 @ B2 )
        & ( ord_less_eq_set_v @ A3 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_88_le__inf__iff,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
      = ( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
        & ( ord_le7336532860387713383od_v_v @ X2 @ Z ) ) ) ).

% le_inf_iff
thf(fact_89_le__inf__iff,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ( ord_less_eq_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) )
      = ( ( ord_less_eq_set_v @ X2 @ Y2 )
        & ( ord_less_eq_set_v @ X2 @ Z ) ) ) ).

% le_inf_iff
thf(fact_90_empty__subsetI,axiom,
    ! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).

% empty_subsetI
thf(fact_91_empty__subsetI,axiom,
    ! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).

% empty_subsetI
thf(fact_92_subset__empty,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
      = ( A = bot_bo723834152578015283od_v_v ) ) ).

% subset_empty
thf(fact_93_subset__empty,axiom,
    ! [A: set_v] :
      ( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
      = ( A = bot_bot_set_v ) ) ).

% subset_empty
thf(fact_94_inf__bot__right,axiom,
    ! [X2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ X2 @ bot_bo723834152578015283od_v_v )
      = bot_bo723834152578015283od_v_v ) ).

% inf_bot_right
thf(fact_95_inf__bot__right,axiom,
    ! [X2: set_v] :
      ( ( inf_inf_set_v @ X2 @ bot_bot_set_v )
      = bot_bot_set_v ) ).

% inf_bot_right
thf(fact_96_inf__bot__left,axiom,
    ! [X2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X2 )
      = bot_bo723834152578015283od_v_v ) ).

% inf_bot_left
thf(fact_97_inf__bot__left,axiom,
    ! [X2: set_v] :
      ( ( inf_inf_set_v @ bot_bot_set_v @ X2 )
      = bot_bot_set_v ) ).

% inf_bot_left
thf(fact_98_singletonI,axiom,
    ! [A3: product_prod_v_v] : ( member7453568604450474000od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) ).

% singletonI
thf(fact_99_singletonI,axiom,
    ! [A3: v] : ( member_v @ A3 @ ( insert_v2 @ A3 @ bot_bot_set_v ) ) ).

% singletonI
thf(fact_100_insert__subset,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A ) @ B )
      = ( ( member7453568604450474000od_v_v @ X2 @ B )
        & ( ord_le7336532860387713383od_v_v @ A @ B ) ) ) ).

% insert_subset
thf(fact_101_insert__subset,axiom,
    ! [X2: v,A: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ ( insert_v2 @ X2 @ A ) @ B )
      = ( ( member_v @ X2 @ B )
        & ( ord_less_eq_set_v @ A @ B ) ) ) ).

% insert_subset
thf(fact_102_Diff__empty,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ A @ bot_bo723834152578015283od_v_v )
      = A ) ).

% Diff_empty
thf(fact_103_Diff__empty,axiom,
    ! [A: set_v] :
      ( ( minus_minus_set_v @ A @ bot_bot_set_v )
      = A ) ).

% Diff_empty
thf(fact_104_empty__Diff,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A )
      = bot_bo723834152578015283od_v_v ) ).

% empty_Diff
thf(fact_105_empty__Diff,axiom,
    ! [A: set_v] :
      ( ( minus_minus_set_v @ bot_bot_set_v @ A )
      = bot_bot_set_v ) ).

% empty_Diff
thf(fact_106_Diff__cancel,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ A @ A )
      = bot_bo723834152578015283od_v_v ) ).

% Diff_cancel
thf(fact_107_Diff__cancel,axiom,
    ! [A: set_v] :
      ( ( minus_minus_set_v @ A @ A )
      = bot_bot_set_v ) ).

% Diff_cancel
thf(fact_108_Int__subset__iff,axiom,
    ! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
      = ( ( ord_le7336532860387713383od_v_v @ C2 @ A )
        & ( ord_le7336532860387713383od_v_v @ C2 @ B ) ) ) ).

% Int_subset_iff
thf(fact_109_Int__subset__iff,axiom,
    ! [C2: set_v,A: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A @ B ) )
      = ( ( ord_less_eq_set_v @ C2 @ A )
        & ( ord_less_eq_set_v @ C2 @ B ) ) ) ).

% Int_subset_iff
thf(fact_110_Int__insert__right__if1,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A3 @ A )
     => ( ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ A3 @ B ) )
        = ( insert1338601472111419319od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_111_Int__insert__right__if1,axiom,
    ! [A3: v,A: set_v,B: set_v] :
      ( ( member_v @ A3 @ A )
     => ( ( inf_inf_set_v @ A @ ( insert_v2 @ A3 @ B ) )
        = ( insert_v2 @ A3 @ ( inf_inf_set_v @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_112_Int__insert__right__if0,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ A3 @ A )
     => ( ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ A3 @ B ) )
        = ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_113_Int__insert__right__if0,axiom,
    ! [A3: v,A: set_v,B: set_v] :
      ( ~ ( member_v @ A3 @ A )
     => ( ( inf_inf_set_v @ A @ ( insert_v2 @ A3 @ B ) )
        = ( inf_inf_set_v @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_114_insert__inter__insert,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ A ) @ ( insert1338601472111419319od_v_v @ A3 @ B ) )
      = ( insert1338601472111419319od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ).

% insert_inter_insert
thf(fact_115_insert__inter__insert,axiom,
    ! [A3: v,A: set_v,B: set_v] :
      ( ( inf_inf_set_v @ ( insert_v2 @ A3 @ A ) @ ( insert_v2 @ A3 @ B ) )
      = ( insert_v2 @ A3 @ ( inf_inf_set_v @ A @ B ) ) ) ).

% insert_inter_insert
thf(fact_116_Int__insert__left__if1,axiom,
    ! [A3: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A3 @ C2 )
     => ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ B ) @ C2 )
        = ( insert1338601472111419319od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_117_Int__insert__left__if1,axiom,
    ! [A3: v,C2: set_v,B: set_v] :
      ( ( member_v @ A3 @ C2 )
     => ( ( inf_inf_set_v @ ( insert_v2 @ A3 @ B ) @ C2 )
        = ( insert_v2 @ A3 @ ( inf_inf_set_v @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_118_Int__insert__left__if0,axiom,
    ! [A3: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ A3 @ C2 )
     => ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ B ) @ C2 )
        = ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_119_Int__insert__left__if0,axiom,
    ! [A3: v,C2: set_v,B: set_v] :
      ( ~ ( member_v @ A3 @ C2 )
     => ( ( inf_inf_set_v @ ( insert_v2 @ A3 @ B ) @ C2 )
        = ( inf_inf_set_v @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_120_Diff__insert0,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ X2 @ A )
     => ( ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ X2 @ B ) )
        = ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_121_Diff__insert0,axiom,
    ! [X2: v,A: set_v,B: set_v] :
      ( ~ ( member_v @ X2 @ A )
     => ( ( minus_minus_set_v @ A @ ( insert_v2 @ X2 @ B ) )
        = ( minus_minus_set_v @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_122_insert__Diff1,axiom,
    ! [X2: product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ X2 @ B )
     => ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A ) @ B )
        = ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_123_insert__Diff1,axiom,
    ! [X2: v,B: set_v,A: set_v] :
      ( ( member_v @ X2 @ B )
     => ( ( minus_minus_set_v @ ( insert_v2 @ X2 @ A ) @ B )
        = ( minus_minus_set_v @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_124_singleton__insert__inj__eq_H,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v,B2: product_prod_v_v] :
      ( ( ( insert1338601472111419319od_v_v @ A3 @ A )
        = ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
      = ( ( A3 = B2 )
        & ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_125_singleton__insert__inj__eq_H,axiom,
    ! [A3: v,A: set_v,B2: v] :
      ( ( ( insert_v2 @ A3 @ A )
        = ( insert_v2 @ B2 @ bot_bot_set_v ) )
      = ( ( A3 = B2 )
        & ( ord_less_eq_set_v @ A @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_126_singleton__insert__inj__eq,axiom,
    ! [B2: product_prod_v_v,A3: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v )
        = ( insert1338601472111419319od_v_v @ A3 @ A ) )
      = ( ( A3 = B2 )
        & ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_127_singleton__insert__inj__eq,axiom,
    ! [B2: v,A3: v,A: set_v] :
      ( ( ( insert_v2 @ B2 @ bot_bot_set_v )
        = ( insert_v2 @ A3 @ A ) )
      = ( ( A3 = B2 )
        & ( ord_less_eq_set_v @ A @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_128_insert__disjoint_I1_J,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ A ) @ B )
        = bot_bo723834152578015283od_v_v )
      = ( ~ ( member7453568604450474000od_v_v @ A3 @ B )
        & ( ( inf_in6271465464967711157od_v_v @ A @ B )
          = bot_bo723834152578015283od_v_v ) ) ) ).

% insert_disjoint(1)
thf(fact_129_insert__disjoint_I1_J,axiom,
    ! [A3: v,A: set_v,B: set_v] :
      ( ( ( inf_inf_set_v @ ( insert_v2 @ A3 @ A ) @ B )
        = bot_bot_set_v )
      = ( ~ ( member_v @ A3 @ B )
        & ( ( inf_inf_set_v @ A @ B )
          = bot_bot_set_v ) ) ) ).

% insert_disjoint(1)
thf(fact_130_insert__disjoint_I2_J,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ A ) @ B ) )
      = ( ~ ( member7453568604450474000od_v_v @ A3 @ B )
        & ( bot_bo723834152578015283od_v_v
          = ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_131_insert__disjoint_I2_J,axiom,
    ! [A3: v,A: set_v,B: set_v] :
      ( ( bot_bot_set_v
        = ( inf_inf_set_v @ ( insert_v2 @ A3 @ A ) @ B ) )
      = ( ~ ( member_v @ A3 @ B )
        & ( bot_bot_set_v
          = ( inf_inf_set_v @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_132_disjoint__insert_I1_J,axiom,
    ! [B: set_Product_prod_v_v,A3: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ( inf_in6271465464967711157od_v_v @ B @ ( insert1338601472111419319od_v_v @ A3 @ A ) )
        = bot_bo723834152578015283od_v_v )
      = ( ~ ( member7453568604450474000od_v_v @ A3 @ B )
        & ( ( inf_in6271465464967711157od_v_v @ B @ A )
          = bot_bo723834152578015283od_v_v ) ) ) ).

% disjoint_insert(1)
thf(fact_133_disjoint__insert_I1_J,axiom,
    ! [B: set_v,A3: v,A: set_v] :
      ( ( ( inf_inf_set_v @ B @ ( insert_v2 @ A3 @ A ) )
        = bot_bot_set_v )
      = ( ~ ( member_v @ A3 @ B )
        & ( ( inf_inf_set_v @ B @ A )
          = bot_bot_set_v ) ) ) ).

% disjoint_insert(1)
thf(fact_134_disjoint__insert_I2_J,axiom,
    ! [A: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) )
      = ( ~ ( member7453568604450474000od_v_v @ B2 @ A )
        & ( bot_bo723834152578015283od_v_v
          = ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_135_disjoint__insert_I2_J,axiom,
    ! [A: set_v,B2: v,B: set_v] :
      ( ( bot_bot_set_v
        = ( inf_inf_set_v @ A @ ( insert_v2 @ B2 @ B ) ) )
      = ( ~ ( member_v @ B2 @ A )
        & ( bot_bot_set_v
          = ( inf_inf_set_v @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_136_Diff__eq__empty__iff,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( minus_4183494784930505774od_v_v @ A @ B )
        = bot_bo723834152578015283od_v_v )
      = ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_137_Diff__eq__empty__iff,axiom,
    ! [A: set_v,B: set_v] :
      ( ( ( minus_minus_set_v @ A @ B )
        = bot_bot_set_v )
      = ( ord_less_eq_set_v @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_138_insert__Diff__single,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( insert1338601472111419319od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) )
      = ( insert1338601472111419319od_v_v @ A3 @ A ) ) ).

% insert_Diff_single
thf(fact_139_insert__Diff__single,axiom,
    ! [A3: v,A: set_v] :
      ( ( insert_v2 @ A3 @ ( minus_minus_set_v @ A @ ( insert_v2 @ A3 @ bot_bot_set_v ) ) )
      = ( insert_v2 @ A3 @ A ) ) ).

% insert_Diff_single
thf(fact_140_Diff__disjoint,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ A ) )
      = bot_bo723834152578015283od_v_v ) ).

% Diff_disjoint
thf(fact_141_Diff__disjoint,axiom,
    ! [A: set_v,B: set_v] :
      ( ( inf_inf_set_v @ A @ ( minus_minus_set_v @ B @ A ) )
      = bot_bot_set_v ) ).

% Diff_disjoint
thf(fact_142_subset__insert__iff,axiom,
    ! [A: set_Product_prod_v_v,X2: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ X2 @ B ) )
      = ( ( ( member7453568604450474000od_v_v @ X2 @ A )
         => ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) @ B ) )
        & ( ~ ( member7453568604450474000od_v_v @ X2 @ A )
         => ( ord_le7336532860387713383od_v_v @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_143_subset__insert__iff,axiom,
    ! [A: set_v,X2: v,B: set_v] :
      ( ( ord_less_eq_set_v @ A @ ( insert_v2 @ X2 @ B ) )
      = ( ( ( member_v @ X2 @ A )
         => ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) @ B ) )
        & ( ~ ( member_v @ X2 @ A )
         => ( ord_less_eq_set_v @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_144_Diff__single__insert,axiom,
    ! [A: set_Product_prod_v_v,X2: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) @ B )
     => ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ X2 @ B ) ) ) ).

% Diff_single_insert
thf(fact_145_Diff__single__insert,axiom,
    ! [A: set_v,X2: v,B: set_v] :
      ( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) @ B )
     => ( ord_less_eq_set_v @ A @ ( insert_v2 @ X2 @ B ) ) ) ).

% Diff_single_insert
thf(fact_146_disjoint__iff__not__equal,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( inf_in6271465464967711157od_v_v @ A @ B )
        = bot_bo723834152578015283od_v_v )
      = ( ! [X3: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ A )
           => ! [Y3: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ Y3 @ B )
               => ( X3 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_147_disjoint__iff__not__equal,axiom,
    ! [A: set_v,B: set_v] :
      ( ( ( inf_inf_set_v @ A @ B )
        = bot_bot_set_v )
      = ( ! [X3: v] :
            ( ( member_v @ X3 @ A )
           => ! [Y3: v] :
                ( ( member_v @ Y3 @ B )
               => ( X3 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_148_subset__singleton__iff,axiom,
    ! [X5: set_Product_prod_v_v,A3: product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X5 @ ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) )
      = ( ( X5 = bot_bo723834152578015283od_v_v )
        | ( X5
          = ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% subset_singleton_iff
thf(fact_149_subset__singleton__iff,axiom,
    ! [X5: set_v,A3: v] :
      ( ( ord_less_eq_set_v @ X5 @ ( insert_v2 @ A3 @ bot_bot_set_v ) )
      = ( ( X5 = bot_bot_set_v )
        | ( X5
          = ( insert_v2 @ A3 @ bot_bot_set_v ) ) ) ) ).

% subset_singleton_iff
thf(fact_150_mk__disjoint__insert,axiom,
    ! [A3: v,A: set_v] :
      ( ( member_v @ A3 @ A )
     => ? [B3: set_v] :
          ( ( A
            = ( insert_v2 @ A3 @ B3 ) )
          & ~ ( member_v @ A3 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_151_mk__disjoint__insert,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A3 @ A )
     => ? [B3: set_Product_prod_v_v] :
          ( ( A
            = ( insert1338601472111419319od_v_v @ A3 @ B3 ) )
          & ~ ( member7453568604450474000od_v_v @ A3 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_152_subset__singletonD,axiom,
    ! [A: set_Product_prod_v_v,X2: product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
     => ( ( A = bot_bo723834152578015283od_v_v )
        | ( A
          = ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% subset_singletonD
thf(fact_153_subset__singletonD,axiom,
    ! [A: set_v,X2: v] :
      ( ( ord_less_eq_set_v @ A @ ( insert_v2 @ X2 @ bot_bot_set_v ) )
     => ( ( A = bot_bot_set_v )
        | ( A
          = ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ).

% subset_singletonD
thf(fact_154_singleton__inject,axiom,
    ! [A3: product_prod_v_v,B2: product_prod_v_v] :
      ( ( ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
        = ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
     => ( A3 = B2 ) ) ).

% singleton_inject
thf(fact_155_singleton__inject,axiom,
    ! [A3: v,B2: v] :
      ( ( ( insert_v2 @ A3 @ bot_bot_set_v )
        = ( insert_v2 @ B2 @ bot_bot_set_v ) )
     => ( A3 = B2 ) ) ).

% singleton_inject
thf(fact_156_insert__not__empty,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( insert1338601472111419319od_v_v @ A3 @ A )
     != bot_bo723834152578015283od_v_v ) ).

% insert_not_empty
thf(fact_157_insert__not__empty,axiom,
    ! [A3: v,A: set_v] :
      ( ( insert_v2 @ A3 @ A )
     != bot_bot_set_v ) ).

% insert_not_empty
thf(fact_158_doubleton__eq__iff,axiom,
    ! [A3: product_prod_v_v,B2: product_prod_v_v,C: product_prod_v_v,D: product_prod_v_v] :
      ( ( ( insert1338601472111419319od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
        = ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D @ bot_bo723834152578015283od_v_v ) ) )
      = ( ( ( A3 = C )
          & ( B2 = D ) )
        | ( ( A3 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_159_doubleton__eq__iff,axiom,
    ! [A3: v,B2: v,C: v,D: v] :
      ( ( ( insert_v2 @ A3 @ ( insert_v2 @ B2 @ bot_bot_set_v ) )
        = ( insert_v2 @ C @ ( insert_v2 @ D @ bot_bot_set_v ) ) )
      = ( ( ( A3 = C )
          & ( B2 = D ) )
        | ( ( A3 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_160_Int__left__commute,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) )
      = ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A @ C2 ) ) ) ).

% Int_left_commute
thf(fact_161_Int__insert__right,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( member7453568604450474000od_v_v @ A3 @ A )
       => ( ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ A3 @ B ) )
          = ( insert1338601472111419319od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) )
      & ( ~ ( member7453568604450474000od_v_v @ A3 @ A )
       => ( ( inf_in6271465464967711157od_v_v @ A @ ( insert1338601472111419319od_v_v @ A3 @ B ) )
          = ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).

% Int_insert_right
thf(fact_162_Int__insert__right,axiom,
    ! [A3: v,A: set_v,B: set_v] :
      ( ( ( member_v @ A3 @ A )
       => ( ( inf_inf_set_v @ A @ ( insert_v2 @ A3 @ B ) )
          = ( insert_v2 @ A3 @ ( inf_inf_set_v @ A @ B ) ) ) )
      & ( ~ ( member_v @ A3 @ A )
       => ( ( inf_inf_set_v @ A @ ( insert_v2 @ A3 @ B ) )
          = ( inf_inf_set_v @ A @ B ) ) ) ) ).

% Int_insert_right
thf(fact_163_Int__Collect__mono,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ! [X4: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_164_Int__Collect__mono,axiom,
    ! [A: set_v,B: set_v,P: v > $o,Q: v > $o] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ! [X4: v] :
            ( ( member_v @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B @ ( collect_v @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_165_Collect__mono__iff,axiom,
    ! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
      ( ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) )
      = ( ! [X3: product_prod_v_v] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_166_Collect__mono__iff,axiom,
    ! [P: v > $o,Q: v > $o] :
      ( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
      = ( ! [X3: v] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_167_subset__insertI2,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_168_subset__insertI2,axiom,
    ! [A: set_v,B: set_v,B2: v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ord_less_eq_set_v @ A @ ( insert_v2 @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_169_Int__left__absorb,axiom,
    ! [A: set_v,B: set_v] :
      ( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B ) )
      = ( inf_inf_set_v @ A @ B ) ) ).

% Int_left_absorb
thf(fact_170_Int__insert__left,axiom,
    ! [A3: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( member7453568604450474000od_v_v @ A3 @ C2 )
       => ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ B ) @ C2 )
          = ( insert1338601472111419319od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) )
      & ( ~ ( member7453568604450474000od_v_v @ A3 @ C2 )
       => ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ B ) @ C2 )
          = ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_171_Int__insert__left,axiom,
    ! [A3: v,C2: set_v,B: set_v] :
      ( ( ( member_v @ A3 @ C2 )
       => ( ( inf_inf_set_v @ ( insert_v2 @ A3 @ B ) @ C2 )
          = ( insert_v2 @ A3 @ ( inf_inf_set_v @ B @ C2 ) ) ) )
      & ( ~ ( member_v @ A3 @ C2 )
       => ( ( inf_inf_set_v @ ( insert_v2 @ A3 @ B ) @ C2 )
          = ( inf_inf_set_v @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_172_Int__empty__right,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ A @ bot_bo723834152578015283od_v_v )
      = bot_bo723834152578015283od_v_v ) ).

% Int_empty_right
thf(fact_173_Int__empty__right,axiom,
    ! [A: set_v] :
      ( ( inf_inf_set_v @ A @ bot_bot_set_v )
      = bot_bot_set_v ) ).

% Int_empty_right
thf(fact_174_subset__insertI,axiom,
    ! [B: set_Product_prod_v_v,A3: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( insert1338601472111419319od_v_v @ A3 @ B ) ) ).

% subset_insertI
thf(fact_175_subset__insertI,axiom,
    ! [B: set_v,A3: v] : ( ord_less_eq_set_v @ B @ ( insert_v2 @ A3 @ B ) ) ).

% subset_insertI
thf(fact_176_insert__commute,axiom,
    ! [X2: v,Y2: v,A: set_v] :
      ( ( insert_v2 @ X2 @ ( insert_v2 @ Y2 @ A ) )
      = ( insert_v2 @ Y2 @ ( insert_v2 @ X2 @ A ) ) ) ).

% insert_commute
thf(fact_177_insert__commute,axiom,
    ! [X2: product_prod_v_v,Y2: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( insert1338601472111419319od_v_v @ X2 @ ( insert1338601472111419319od_v_v @ Y2 @ A ) )
      = ( insert1338601472111419319od_v_v @ Y2 @ ( insert1338601472111419319od_v_v @ X2 @ A ) ) ) ).

% insert_commute
thf(fact_178_Int__empty__left,axiom,
    ! [B: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B )
      = bot_bo723834152578015283od_v_v ) ).

% Int_empty_left
thf(fact_179_Int__empty__left,axiom,
    ! [B: set_v] :
      ( ( inf_inf_set_v @ bot_bot_set_v @ B )
      = bot_bot_set_v ) ).

% Int_empty_left
thf(fact_180_subset__insert,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ X2 @ A )
     => ( ( ord_le7336532860387713383od_v_v @ A @ ( insert1338601472111419319od_v_v @ X2 @ B ) )
        = ( ord_le7336532860387713383od_v_v @ A @ B ) ) ) ).

% subset_insert
thf(fact_181_subset__insert,axiom,
    ! [X2: v,A: set_v,B: set_v] :
      ( ~ ( member_v @ X2 @ A )
     => ( ( ord_less_eq_set_v @ A @ ( insert_v2 @ X2 @ B ) )
        = ( ord_less_eq_set_v @ A @ B ) ) ) ).

% subset_insert
thf(fact_182_singleton__iff,axiom,
    ! [B2: product_prod_v_v,A3: product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) )
      = ( B2 = A3 ) ) ).

% singleton_iff
thf(fact_183_singleton__iff,axiom,
    ! [B2: v,A3: v] :
      ( ( member_v @ B2 @ ( insert_v2 @ A3 @ bot_bot_set_v ) )
      = ( B2 = A3 ) ) ).

% singleton_iff
thf(fact_184_set__eq__subset,axiom,
    ( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
    = ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
          ( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
          & ( ord_le7336532860387713383od_v_v @ B4 @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_185_set__eq__subset,axiom,
    ( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
    = ( ^ [A4: set_v,B4: set_v] :
          ( ( ord_less_eq_set_v @ A4 @ B4 )
          & ( ord_less_eq_set_v @ B4 @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_186_insert__eq__iff,axiom,
    ! [A3: v,A: set_v,B2: v,B: set_v] :
      ( ~ ( member_v @ A3 @ A )
     => ( ~ ( member_v @ B2 @ B )
       => ( ( ( insert_v2 @ A3 @ A )
            = ( insert_v2 @ B2 @ B ) )
          = ( ( ( A3 = B2 )
             => ( A = B ) )
            & ( ( A3 != B2 )
             => ? [C3: set_v] :
                  ( ( A
                    = ( insert_v2 @ B2 @ C3 ) )
                  & ~ ( member_v @ B2 @ C3 )
                  & ( B
                    = ( insert_v2 @ A3 @ C3 ) )
                  & ~ ( member_v @ A3 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_187_insert__eq__iff,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ A3 @ A )
     => ( ~ ( member7453568604450474000od_v_v @ B2 @ B )
       => ( ( ( insert1338601472111419319od_v_v @ A3 @ A )
            = ( insert1338601472111419319od_v_v @ B2 @ B ) )
          = ( ( ( A3 = B2 )
             => ( A = B ) )
            & ( ( A3 != B2 )
             => ? [C3: set_Product_prod_v_v] :
                  ( ( A
                    = ( insert1338601472111419319od_v_v @ B2 @ C3 ) )
                  & ~ ( member7453568604450474000od_v_v @ B2 @ C3 )
                  & ( B
                    = ( insert1338601472111419319od_v_v @ A3 @ C3 ) )
                  & ~ ( member7453568604450474000od_v_v @ A3 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_188_insert__absorb,axiom,
    ! [A3: v,A: set_v] :
      ( ( member_v @ A3 @ A )
     => ( ( insert_v2 @ A3 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_189_insert__absorb,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A3 @ A )
     => ( ( insert1338601472111419319od_v_v @ A3 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_190_subset__trans,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
       => ( ord_le7336532860387713383od_v_v @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_191_subset__trans,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ( ord_less_eq_set_v @ B @ C2 )
       => ( ord_less_eq_set_v @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_192_insert__ident,axiom,
    ! [X2: v,A: set_v,B: set_v] :
      ( ~ ( member_v @ X2 @ A )
     => ( ~ ( member_v @ X2 @ B )
       => ( ( ( insert_v2 @ X2 @ A )
            = ( insert_v2 @ X2 @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_193_insert__ident,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ X2 @ A )
     => ( ~ ( member7453568604450474000od_v_v @ X2 @ B )
       => ( ( ( insert1338601472111419319od_v_v @ X2 @ A )
            = ( insert1338601472111419319od_v_v @ X2 @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_194_disjoint__iff,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( inf_in6271465464967711157od_v_v @ A @ B )
        = bot_bo723834152578015283od_v_v )
      = ( ! [X3: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ A )
           => ~ ( member7453568604450474000od_v_v @ X3 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_195_disjoint__iff,axiom,
    ! [A: set_v,B: set_v] :
      ( ( ( inf_inf_set_v @ A @ B )
        = bot_bot_set_v )
      = ( ! [X3: v] :
            ( ( member_v @ X3 @ A )
           => ~ ( member_v @ X3 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_196_Int__greatest,axiom,
    ! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C2 @ A )
     => ( ( ord_le7336532860387713383od_v_v @ C2 @ B )
       => ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).

% Int_greatest
thf(fact_197_Int__greatest,axiom,
    ! [C2: set_v,A: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ C2 @ A )
     => ( ( ord_less_eq_set_v @ C2 @ B )
       => ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A @ B ) ) ) ) ).

% Int_greatest
thf(fact_198_Collect__mono,axiom,
    ! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
      ( ! [X4: product_prod_v_v] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).

% Collect_mono
thf(fact_199_Collect__mono,axiom,
    ! [P: v > $o,Q: v > $o] :
      ( ! [X4: v] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).

% Collect_mono
thf(fact_200_subset__refl,axiom,
    ! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).

% subset_refl
thf(fact_201_subset__refl,axiom,
    ! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).

% subset_refl
thf(fact_202_insert__mono,axiom,
    ! [C2: set_Product_prod_v_v,D2: set_Product_prod_v_v,A3: product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C2 @ D2 )
     => ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ C2 ) @ ( insert1338601472111419319od_v_v @ A3 @ D2 ) ) ) ).

% insert_mono
thf(fact_203_insert__mono,axiom,
    ! [C2: set_v,D2: set_v,A3: v] :
      ( ( ord_less_eq_set_v @ C2 @ D2 )
     => ( ord_less_eq_set_v @ ( insert_v2 @ A3 @ C2 ) @ ( insert_v2 @ A3 @ D2 ) ) ) ).

% insert_mono
thf(fact_204_Int__commute,axiom,
    ( inf_inf_set_v
    = ( ^ [A4: set_v,B4: set_v] : ( inf_inf_set_v @ B4 @ A4 ) ) ) ).

% Int_commute
thf(fact_205_Int__absorb2,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ( inf_in6271465464967711157od_v_v @ A @ B )
        = A ) ) ).

% Int_absorb2
thf(fact_206_Int__absorb2,axiom,
    ! [A: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ( inf_inf_set_v @ A @ B )
        = A ) ) ).

% Int_absorb2
thf(fact_207_Int__absorb1,axiom,
    ! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B @ A )
     => ( ( inf_in6271465464967711157od_v_v @ A @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_208_Int__absorb1,axiom,
    ! [B: set_v,A: set_v] :
      ( ( ord_less_eq_set_v @ B @ A )
     => ( ( inf_inf_set_v @ A @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_209_subset__iff,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
        ! [T: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ T @ A4 )
         => ( member7453568604450474000od_v_v @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_210_subset__iff,axiom,
    ( ord_less_eq_set_v
    = ( ^ [A4: set_v,B4: set_v] :
        ! [T: v] :
          ( ( member_v @ T @ A4 )
         => ( member_v @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_211_singletonD,axiom,
    ! [B2: product_prod_v_v,A3: product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) )
     => ( B2 = A3 ) ) ).

% singletonD
thf(fact_212_singletonD,axiom,
    ! [B2: v,A3: v] :
      ( ( member_v @ B2 @ ( insert_v2 @ A3 @ bot_bot_set_v ) )
     => ( B2 = A3 ) ) ).

% singletonD
thf(fact_213_Set_Oset__insert,axiom,
    ! [X2: v,A: set_v] :
      ( ( member_v @ X2 @ A )
     => ~ ! [B3: set_v] :
            ( ( A
              = ( insert_v2 @ X2 @ B3 ) )
           => ( member_v @ X2 @ B3 ) ) ) ).

% Set.set_insert
thf(fact_214_Set_Oset__insert,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ X2 @ A )
     => ~ ! [B3: set_Product_prod_v_v] :
            ( ( A
              = ( insert1338601472111419319od_v_v @ X2 @ B3 ) )
           => ( member7453568604450474000od_v_v @ X2 @ B3 ) ) ) ).

% Set.set_insert
thf(fact_215_ex__in__conv,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( ? [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A ) )
      = ( A != bot_bo723834152578015283od_v_v ) ) ).

% ex_in_conv
thf(fact_216_ex__in__conv,axiom,
    ! [A: set_v] :
      ( ( ? [X3: v] : ( member_v @ X3 @ A ) )
      = ( A != bot_bot_set_v ) ) ).

% ex_in_conv
thf(fact_217_equalityD2,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( A = B )
     => ( ord_le7336532860387713383od_v_v @ B @ A ) ) ).

% equalityD2
thf(fact_218_equalityD2,axiom,
    ! [A: set_v,B: set_v] :
      ( ( A = B )
     => ( ord_less_eq_set_v @ B @ A ) ) ).

% equalityD2
thf(fact_219_equalityD1,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( A = B )
     => ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).

% equalityD1
thf(fact_220_equalityD1,axiom,
    ! [A: set_v,B: set_v] :
      ( ( A = B )
     => ( ord_less_eq_set_v @ A @ B ) ) ).

% equalityD1
thf(fact_221_Int__lower2,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ B ) ).

% Int_lower2
thf(fact_222_Int__lower2,axiom,
    ! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ B ) ).

% Int_lower2
thf(fact_223_Int__lower1,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ A ) ).

% Int_lower1
thf(fact_224_Int__lower1,axiom,
    ! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ A ) ).

% Int_lower1
thf(fact_225_Int__emptyI,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ! [X4: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X4 @ A )
         => ~ ( member7453568604450474000od_v_v @ X4 @ B ) )
     => ( ( inf_in6271465464967711157od_v_v @ A @ B )
        = bot_bo723834152578015283od_v_v ) ) ).

% Int_emptyI
thf(fact_226_Int__emptyI,axiom,
    ! [A: set_v,B: set_v] :
      ( ! [X4: v] :
          ( ( member_v @ X4 @ A )
         => ~ ( member_v @ X4 @ B ) )
     => ( ( inf_inf_set_v @ A @ B )
        = bot_bot_set_v ) ) ).

% Int_emptyI
thf(fact_227_Int__absorb,axiom,
    ! [A: set_v] :
      ( ( inf_inf_set_v @ A @ A )
      = A ) ).

% Int_absorb
thf(fact_228_subset__eq,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
        ! [X3: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X3 @ A4 )
         => ( member7453568604450474000od_v_v @ X3 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_229_subset__eq,axiom,
    ( ord_less_eq_set_v
    = ( ^ [A4: set_v,B4: set_v] :
        ! [X3: v] :
          ( ( member_v @ X3 @ A4 )
         => ( member_v @ X3 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_230_equalityE,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( A = B )
     => ~ ( ( ord_le7336532860387713383od_v_v @ A @ B )
         => ~ ( ord_le7336532860387713383od_v_v @ B @ A ) ) ) ).

% equalityE
thf(fact_231_equalityE,axiom,
    ! [A: set_v,B: set_v] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_v @ A @ B )
         => ~ ( ord_less_eq_set_v @ B @ A ) ) ) ).

% equalityE
thf(fact_232_Int__assoc,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 )
      = ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) ).

% Int_assoc
thf(fact_233_insertI2,axiom,
    ! [A3: v,B: set_v,B2: v] :
      ( ( member_v @ A3 @ B )
     => ( member_v @ A3 @ ( insert_v2 @ B2 @ B ) ) ) ).

% insertI2
thf(fact_234_insertI2,axiom,
    ! [A3: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A3 @ B )
     => ( member7453568604450474000od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).

% insertI2
thf(fact_235_insertI1,axiom,
    ! [A3: v,B: set_v] : ( member_v @ A3 @ ( insert_v2 @ A3 @ B ) ) ).

% insertI1
thf(fact_236_insertI1,axiom,
    ! [A3: product_prod_v_v,B: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A3 @ B ) ) ).

% insertI1
thf(fact_237_equals0I,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ! [Y: product_prod_v_v] :
          ~ ( member7453568604450474000od_v_v @ Y @ A )
     => ( A = bot_bo723834152578015283od_v_v ) ) ).

% equals0I
thf(fact_238_equals0I,axiom,
    ! [A: set_v] :
      ( ! [Y: v] :
          ~ ( member_v @ Y @ A )
     => ( A = bot_bot_set_v ) ) ).

% equals0I
thf(fact_239_equals0D,axiom,
    ! [A: set_Product_prod_v_v,A3: product_prod_v_v] :
      ( ( A = bot_bo723834152578015283od_v_v )
     => ~ ( member7453568604450474000od_v_v @ A3 @ A ) ) ).

% equals0D
thf(fact_240_equals0D,axiom,
    ! [A: set_v,A3: v] :
      ( ( A = bot_bot_set_v )
     => ~ ( member_v @ A3 @ A ) ) ).

% equals0D
thf(fact_241_Int__mono,axiom,
    ! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ C2 )
     => ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
       => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D2 ) ) ) ) ).

% Int_mono
thf(fact_242_Int__mono,axiom,
    ! [A: set_v,C2: set_v,B: set_v,D2: set_v] :
      ( ( ord_less_eq_set_v @ A @ C2 )
     => ( ( ord_less_eq_set_v @ B @ D2 )
       => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ C2 @ D2 ) ) ) ) ).

% Int_mono
thf(fact_243_subsetD,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ( member7453568604450474000od_v_v @ C @ A )
       => ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% subsetD
thf(fact_244_subsetD,axiom,
    ! [A: set_v,B: set_v,C: v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ( member_v @ C @ A )
       => ( member_v @ C @ B ) ) ) ).

% subsetD
thf(fact_245_insertE,axiom,
    ! [A3: v,B2: v,A: set_v] :
      ( ( member_v @ A3 @ ( insert_v2 @ B2 @ A ) )
     => ( ( A3 != B2 )
       => ( member_v @ A3 @ A ) ) ) ).

% insertE
thf(fact_246_insertE,axiom,
    ! [A3: product_prod_v_v,B2: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ A ) )
     => ( ( A3 != B2 )
       => ( member7453568604450474000od_v_v @ A3 @ A ) ) ) ).

% insertE
thf(fact_247_in__mono,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X2: product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ( member7453568604450474000od_v_v @ X2 @ A )
       => ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ).

% in_mono
thf(fact_248_in__mono,axiom,
    ! [A: set_v,B: set_v,X2: v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ( member_v @ X2 @ A )
       => ( member_v @ X2 @ B ) ) ) ).

% in_mono
thf(fact_249_emptyE,axiom,
    ! [A3: product_prod_v_v] :
      ~ ( member7453568604450474000od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ).

% emptyE
thf(fact_250_emptyE,axiom,
    ! [A3: v] :
      ~ ( member_v @ A3 @ bot_bot_set_v ) ).

% emptyE
thf(fact_251_IntD2,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
     => ( member7453568604450474000od_v_v @ C @ B ) ) ).

% IntD2
thf(fact_252_IntD2,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ ( inf_inf_set_v @ A @ B ) )
     => ( member_v @ C @ B ) ) ).

% IntD2
thf(fact_253_IntD1,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
     => ( member7453568604450474000od_v_v @ C @ A ) ) ).

% IntD1
thf(fact_254_IntD1,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ ( inf_inf_set_v @ A @ B ) )
     => ( member_v @ C @ A ) ) ).

% IntD1
thf(fact_255_IntE,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
     => ~ ( ( member7453568604450474000od_v_v @ C @ A )
         => ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% IntE
thf(fact_256_IntE,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ ( inf_inf_set_v @ A @ B ) )
     => ~ ( ( member_v @ C @ A )
         => ~ ( member_v @ C @ B ) ) ) ).

% IntE
thf(fact_257_inf__left__commute,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) )
      = ( inf_inf_set_v @ Y2 @ ( inf_inf_set_v @ X2 @ Z ) ) ) ).

% inf_left_commute
thf(fact_258_inf_Oleft__commute,axiom,
    ! [B2: set_v,A3: set_v,C: set_v] :
      ( ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A3 @ C ) )
      = ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B2 @ C ) ) ) ).

% inf.left_commute
thf(fact_259_inf_OcoboundedI2,axiom,
    ! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
     => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_260_inf_OcoboundedI2,axiom,
    ! [B2: set_v,C: set_v,A3: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ C )
     => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_261_inf_OcoboundedI1,axiom,
    ! [A3: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ C )
     => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_262_inf_OcoboundedI1,axiom,
    ! [A3: set_v,C: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ C )
     => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_263_inf_Oabsorb__iff2,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
          ( ( inf_in6271465464967711157od_v_v @ A5 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_264_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_v
    = ( ^ [B5: set_v,A5: set_v] :
          ( ( inf_inf_set_v @ A5 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_265_inf_Oabsorb__iff1,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
          ( ( inf_in6271465464967711157od_v_v @ A5 @ B5 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_266_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_v
    = ( ^ [A5: set_v,B5: set_v] :
          ( ( inf_inf_set_v @ A5 @ B5 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_267_inf_Ocobounded2,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_268_inf_Ocobounded2,axiom,
    ! [A3: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_269_inf_Ocobounded1,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ A3 ) ).

% inf.cobounded1
thf(fact_270_inf_Ocobounded1,axiom,
    ! [A3: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ A3 ) ).

% inf.cobounded1
thf(fact_271_inf_Oorder__iff,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
          ( A5
          = ( inf_in6271465464967711157od_v_v @ A5 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_272_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_v
    = ( ^ [A5: set_v,B5: set_v] :
          ( A5
          = ( inf_inf_set_v @ A5 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_273_inf__greatest,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
     => ( ( ord_le7336532860387713383od_v_v @ X2 @ Z )
       => ( ord_le7336532860387713383od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_274_inf__greatest,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ( ord_less_eq_set_v @ X2 @ Y2 )
     => ( ( ord_less_eq_set_v @ X2 @ Z )
       => ( ord_less_eq_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_275_inf_OboundedI,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
     => ( ( ord_le7336532860387713383od_v_v @ A3 @ C )
       => ( ord_le7336532860387713383od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_276_inf_OboundedI,axiom,
    ! [A3: set_v,B2: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B2 )
     => ( ( ord_less_eq_set_v @ A3 @ C )
       => ( ord_less_eq_set_v @ A3 @ ( inf_inf_set_v @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_277_inf_OboundedE,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
     => ~ ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
         => ~ ( ord_le7336532860387713383od_v_v @ A3 @ C ) ) ) ).

% inf.boundedE
thf(fact_278_inf_OboundedE,axiom,
    ! [A3: set_v,B2: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ ( inf_inf_set_v @ B2 @ C ) )
     => ~ ( ( ord_less_eq_set_v @ A3 @ B2 )
         => ~ ( ord_less_eq_set_v @ A3 @ C ) ) ) ).

% inf.boundedE
thf(fact_279_inf__commute,axiom,
    ( inf_inf_set_v
    = ( ^ [X3: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X3 ) ) ) ).

% inf_commute
thf(fact_280_inf__absorb2,axiom,
    ! [Y2: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ Y2 @ X2 )
     => ( ( inf_in6271465464967711157od_v_v @ X2 @ Y2 )
        = Y2 ) ) ).

% inf_absorb2
thf(fact_281_inf__absorb2,axiom,
    ! [Y2: set_v,X2: set_v] :
      ( ( ord_less_eq_set_v @ Y2 @ X2 )
     => ( ( inf_inf_set_v @ X2 @ Y2 )
        = Y2 ) ) ).

% inf_absorb2
thf(fact_282_inf__absorb1,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
     => ( ( inf_in6271465464967711157od_v_v @ X2 @ Y2 )
        = X2 ) ) ).

% inf_absorb1
thf(fact_283_inf__absorb1,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( ord_less_eq_set_v @ X2 @ Y2 )
     => ( ( inf_inf_set_v @ X2 @ Y2 )
        = X2 ) ) ).

% inf_absorb1
thf(fact_284_inf_Ocommute,axiom,
    ( inf_inf_set_v
    = ( ^ [A5: set_v,B5: set_v] : ( inf_inf_set_v @ B5 @ A5 ) ) ) ).

% inf.commute
thf(fact_285_inf_Oabsorb2,axiom,
    ! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
     => ( ( inf_in6271465464967711157od_v_v @ A3 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_286_inf_Oabsorb2,axiom,
    ! [B2: set_v,A3: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ A3 )
     => ( ( inf_inf_set_v @ A3 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_287_inf_Oabsorb1,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
     => ( ( inf_in6271465464967711157od_v_v @ A3 @ B2 )
        = A3 ) ) ).

% inf.absorb1
thf(fact_288_inf_Oabsorb1,axiom,
    ! [A3: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B2 )
     => ( ( inf_inf_set_v @ A3 @ B2 )
        = A3 ) ) ).

% inf.absorb1
thf(fact_289_le__iff__inf,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
          ( ( inf_in6271465464967711157od_v_v @ X3 @ Y3 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_290_le__iff__inf,axiom,
    ( ord_less_eq_set_v
    = ( ^ [X3: set_v,Y3: set_v] :
          ( ( inf_inf_set_v @ X3 @ Y3 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_291_inf__unique,axiom,
    ! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X4 @ Y ) @ X4 )
     => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X4 @ Y ) @ Y )
       => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
             => ( ( ord_le7336532860387713383od_v_v @ X4 @ Z3 )
               => ( ord_le7336532860387713383od_v_v @ X4 @ ( F @ Y @ Z3 ) ) ) )
         => ( ( inf_in6271465464967711157od_v_v @ X2 @ Y2 )
            = ( F @ X2 @ Y2 ) ) ) ) ) ).

% inf_unique
thf(fact_292_inf__unique,axiom,
    ! [F: set_v > set_v > set_v,X2: set_v,Y2: set_v] :
      ( ! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( F @ X4 @ Y ) @ X4 )
     => ( ! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( F @ X4 @ Y ) @ Y )
       => ( ! [X4: set_v,Y: set_v,Z3: set_v] :
              ( ( ord_less_eq_set_v @ X4 @ Y )
             => ( ( ord_less_eq_set_v @ X4 @ Z3 )
               => ( ord_less_eq_set_v @ X4 @ ( F @ Y @ Z3 ) ) ) )
         => ( ( inf_inf_set_v @ X2 @ Y2 )
            = ( F @ X2 @ Y2 ) ) ) ) ) ).

% inf_unique
thf(fact_293_inf_OorderI,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( A3
        = ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) )
     => ( ord_le7336532860387713383od_v_v @ A3 @ B2 ) ) ).

% inf.orderI
thf(fact_294_inf_OorderI,axiom,
    ! [A3: set_v,B2: set_v] :
      ( ( A3
        = ( inf_inf_set_v @ A3 @ B2 ) )
     => ( ord_less_eq_set_v @ A3 @ B2 ) ) ).

% inf.orderI
thf(fact_295_inf_OorderE,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
     => ( A3
        = ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) ) ) ).

% inf.orderE
thf(fact_296_inf_OorderE,axiom,
    ! [A3: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B2 )
     => ( A3
        = ( inf_inf_set_v @ A3 @ B2 ) ) ) ).

% inf.orderE
thf(fact_297_inf__assoc,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ Z )
      = ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ).

% inf_assoc
thf(fact_298_inf_Oassoc,axiom,
    ! [A3: set_v,B2: set_v,C: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ C )
      = ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B2 @ C ) ) ) ).

% inf.assoc
thf(fact_299_le__infI2,axiom,
    ! [B2: set_Product_prod_v_v,X2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ X2 )
     => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ X2 ) ) ).

% le_infI2
thf(fact_300_le__infI2,axiom,
    ! [B2: set_v,X2: set_v,A3: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ X2 )
     => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ X2 ) ) ).

% le_infI2
thf(fact_301_le__infI1,axiom,
    ! [A3: set_Product_prod_v_v,X2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ X2 )
     => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ X2 ) ) ).

% le_infI1
thf(fact_302_le__infI1,axiom,
    ! [A3: set_v,X2: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ X2 )
     => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ X2 ) ) ).

% le_infI1
thf(fact_303_inf__mono,axiom,
    ! [A3: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ C )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
       => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_304_inf__mono,axiom,
    ! [A3: set_v,C: set_v,B2: set_v,D: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ C )
     => ( ( ord_less_eq_set_v @ B2 @ D )
       => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ ( inf_inf_set_v @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_305_le__infI,axiom,
    ! [X2: set_Product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X2 @ A3 )
     => ( ( ord_le7336532860387713383od_v_v @ X2 @ B2 )
       => ( ord_le7336532860387713383od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) ) ) ) ).

% le_infI
thf(fact_306_le__infI,axiom,
    ! [X2: set_v,A3: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ X2 @ A3 )
     => ( ( ord_less_eq_set_v @ X2 @ B2 )
       => ( ord_less_eq_set_v @ X2 @ ( inf_inf_set_v @ A3 @ B2 ) ) ) ) ).

% le_infI
thf(fact_307_le__infE,axiom,
    ! [X2: set_Product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) )
     => ~ ( ( ord_le7336532860387713383od_v_v @ X2 @ A3 )
         => ~ ( ord_le7336532860387713383od_v_v @ X2 @ B2 ) ) ) ).

% le_infE
thf(fact_308_le__infE,axiom,
    ! [X2: set_v,A3: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ X2 @ ( inf_inf_set_v @ A3 @ B2 ) )
     => ~ ( ( ord_less_eq_set_v @ X2 @ A3 )
         => ~ ( ord_less_eq_set_v @ X2 @ B2 ) ) ) ).

% le_infE
thf(fact_309_inf__le2,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ Y2 ) ).

% inf_le2
thf(fact_310_inf__le2,axiom,
    ! [X2: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ Y2 ) ).

% inf_le2
thf(fact_311_inf__le1,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ X2 ) ).

% inf_le1
thf(fact_312_inf__le1,axiom,
    ! [X2: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ X2 ) ).

% inf_le1
thf(fact_313_inf__sup__ord_I1_J,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_314_inf__sup__ord_I1_J,axiom,
    ! [X2: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_315_inf__sup__ord_I2_J,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ Y2 ) ).

% inf_sup_ord(2)
thf(fact_316_inf__sup__ord_I2_J,axiom,
    ! [X2: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ Y2 ) ).

% inf_sup_ord(2)
thf(fact_317_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_v
    = ( ^ [X3: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X3 ) ) ) ).

% inf_sup_aci(1)
thf(fact_318_inf__sup__aci_I2_J,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ Z )
      = ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_319_inf__sup__aci_I3_J,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) )
      = ( inf_inf_set_v @ Y2 @ ( inf_inf_set_v @ X2 @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_320_inf__sup__aci_I4_J,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( inf_inf_set_v @ X2 @ ( inf_inf_set_v @ X2 @ Y2 ) )
      = ( inf_inf_set_v @ X2 @ Y2 ) ) ).

% inf_sup_aci(4)
thf(fact_321_graph_Osclosed,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ! [X: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X @ Vertices )
         => ( ord_le7336532860387713383od_v_v @ ( Successors @ X ) @ Vertices ) ) ) ).

% graph.sclosed
thf(fact_322_graph_Osclosed,axiom,
    ! [Vertices: set_v,Successors: v > set_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ! [X: v] :
          ( ( member_v @ X @ Vertices )
         => ( ord_less_eq_set_v @ ( Successors @ X ) @ Vertices ) ) ) ).

% graph.sclosed
thf(fact_323_subset__Diff__insert,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X2: product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ ( insert1338601472111419319od_v_v @ X2 @ C2 ) ) )
      = ( ( ord_le7336532860387713383od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ C2 ) )
        & ~ ( member7453568604450474000od_v_v @ X2 @ A ) ) ) ).

% subset_Diff_insert
thf(fact_324_subset__Diff__insert,axiom,
    ! [A: set_v,B: set_v,X2: v,C2: set_v] :
      ( ( ord_less_eq_set_v @ A @ ( minus_minus_set_v @ B @ ( insert_v2 @ X2 @ C2 ) ) )
      = ( ( ord_less_eq_set_v @ A @ ( minus_minus_set_v @ B @ C2 ) )
        & ~ ( member_v @ X2 @ A ) ) ) ).

% subset_Diff_insert
thf(fact_325_Diff__insert__absorb,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ X2 @ A )
     => ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A ) @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_326_Diff__insert__absorb,axiom,
    ! [X2: v,A: set_v] :
      ( ~ ( member_v @ X2 @ A )
     => ( ( minus_minus_set_v @ ( insert_v2 @ X2 @ A ) @ ( insert_v2 @ X2 @ bot_bot_set_v ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_327_Diff__insert2,axiom,
    ! [A: set_Product_prod_v_v,A3: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A3 @ B ) )
      = ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ).

% Diff_insert2
thf(fact_328_Diff__insert2,axiom,
    ! [A: set_v,A3: v,B: set_v] :
      ( ( minus_minus_set_v @ A @ ( insert_v2 @ A3 @ B ) )
      = ( minus_minus_set_v @ ( minus_minus_set_v @ A @ ( insert_v2 @ A3 @ bot_bot_set_v ) ) @ B ) ) ).

% Diff_insert2
thf(fact_329_insert__Diff,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A3 @ A )
     => ( ( insert1338601472111419319od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_330_insert__Diff,axiom,
    ! [A3: v,A: set_v] :
      ( ( member_v @ A3 @ A )
     => ( ( insert_v2 @ A3 @ ( minus_minus_set_v @ A @ ( insert_v2 @ A3 @ bot_bot_set_v ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_331_Diff__insert,axiom,
    ! [A: set_Product_prod_v_v,A3: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A3 @ B ) )
      = ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) ) ).

% Diff_insert
thf(fact_332_Diff__insert,axiom,
    ! [A: set_v,A3: v,B: set_v] :
      ( ( minus_minus_set_v @ A @ ( insert_v2 @ A3 @ B ) )
      = ( minus_minus_set_v @ ( minus_minus_set_v @ A @ B ) @ ( insert_v2 @ A3 @ bot_bot_set_v ) ) ) ).

% Diff_insert
thf(fact_333_Int__Diff__disjoint,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ A @ B ) )
      = bot_bo723834152578015283od_v_v ) ).

% Int_Diff_disjoint
thf(fact_334_Int__Diff__disjoint,axiom,
    ! [A: set_v,B: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ ( minus_minus_set_v @ A @ B ) )
      = bot_bot_set_v ) ).

% Int_Diff_disjoint
thf(fact_335_Diff__triv,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( inf_in6271465464967711157od_v_v @ A @ B )
        = bot_bo723834152578015283od_v_v )
     => ( ( minus_4183494784930505774od_v_v @ A @ B )
        = A ) ) ).

% Diff_triv
thf(fact_336_Diff__triv,axiom,
    ! [A: set_v,B: set_v] :
      ( ( ( inf_inf_set_v @ A @ B )
        = bot_bot_set_v )
     => ( ( minus_minus_set_v @ A @ B )
        = A ) ) ).

% Diff_triv
thf(fact_337_graph_Oedge__ra,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X2: product_prod_v_v,E2: set_Pr2149350503807050951od_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X2 ) )
       => ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ Y2 ) @ E2 )
         => ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y2 @ E2 ) ) ) ) ).

% graph.edge_ra
thf(fact_338_graph_Oedge__ra,axiom,
    ! [Vertices: set_v,Successors: v > set_v,Y2: v,X2: v,E2: set_Product_prod_v_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( member_v @ Y2 @ ( Successors @ X2 ) )
       => ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y2 ) @ E2 )
         => ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E2 ) ) ) ) ).

% graph.edge_ra
thf(fact_339_graph_Ora__cases,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v,E2: set_Pr2149350503807050951od_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y2 @ E2 )
       => ( ( X2 = Y2 )
          | ? [Z3: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ X2 ) )
              & ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ Z3 ) @ E2 )
              & ( sCC_Bl5370300055464682748od_v_v @ Successors @ Z3 @ Y2 @ E2 ) ) ) ) ) ).

% graph.ra_cases
thf(fact_340_graph_Ora__cases,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,E2: set_Product_prod_v_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E2 )
       => ( ( X2 = Y2 )
          | ? [Z3: v] :
              ( ( member_v @ Z3 @ ( Successors @ X2 ) )
              & ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Z3 ) @ E2 )
              & ( sCC_Bl4291963740693775144ding_v @ Successors @ Z3 @ Y2 @ E2 ) ) ) ) ) ).

% graph.ra_cases
thf(fact_341_graph_Oreachable__avoiding_Ocases,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A32: set_Pr2149350503807050951od_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A32 )
       => ( ( A2 != A1 )
         => ~ ! [Y: product_prod_v_v] :
                ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ Y @ A32 )
               => ( ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y ) )
                 => ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ A2 ) @ A32 ) ) ) ) ) ) ).

% graph.reachable_avoiding.cases
thf(fact_342_graph_Oreachable__avoiding_Ocases,axiom,
    ! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A32: set_Product_prod_v_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A32 )
       => ( ( A2 != A1 )
         => ~ ! [Y: v] :
                ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ Y @ A32 )
               => ( ( member_v @ A2 @ ( Successors @ Y ) )
                 => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ A2 ) @ A32 ) ) ) ) ) ) ).

% graph.reachable_avoiding.cases
thf(fact_343_graph_Oreachable__avoiding_Osimps,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A32: set_Pr2149350503807050951od_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A32 )
        = ( ? [X3: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
              ( ( A1 = X3 )
              & ( A2 = X3 )
              & ( A32 = E4 ) )
          | ? [X3: product_prod_v_v,Y3: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v,Z2: product_prod_v_v] :
              ( ( A1 = X3 )
              & ( A2 = Z2 )
              & ( A32 = E4 )
              & ( sCC_Bl5370300055464682748od_v_v @ Successors @ X3 @ Y3 @ E4 )
              & ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) )
              & ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ Z2 ) @ E4 ) ) ) ) ) ).

% graph.reachable_avoiding.simps
thf(fact_344_graph_Oreachable__avoiding_Osimps,axiom,
    ! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A32: set_Product_prod_v_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A32 )
        = ( ? [X3: v,E4: set_Product_prod_v_v] :
              ( ( A1 = X3 )
              & ( A2 = X3 )
              & ( A32 = E4 ) )
          | ? [X3: v,Y3: v,E4: set_Product_prod_v_v,Z2: v] :
              ( ( A1 = X3 )
              & ( A2 = Z2 )
              & ( A32 = E4 )
              & ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y3 @ E4 )
              & ( member_v @ Z2 @ ( Successors @ Y3 ) )
              & ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E4 ) ) ) ) ) ).

% graph.reachable_avoiding.simps
thf(fact_345_graph_Ora__succ,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v,E2: set_Pr2149350503807050951od_v_v,Z: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y2 @ E2 )
       => ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
         => ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y2 @ Z ) @ E2 )
           => ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Z @ E2 ) ) ) ) ) ).

% graph.ra_succ
thf(fact_346_graph_Ora__succ,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,E2: set_Product_prod_v_v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E2 )
       => ( ( member_v @ Z @ ( Successors @ Y2 ) )
         => ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z ) @ E2 )
           => ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Z @ E2 ) ) ) ) ) ).

% graph.ra_succ
thf(fact_347_graph_Ora__mono,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,E2: set_Product_prod_v_v,E3: set_Product_prod_v_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E2 )
       => ( ( ord_le7336532860387713383od_v_v @ E3 @ E2 )
         => ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E3 ) ) ) ) ).

% graph.ra_mono
thf(fact_348_graph_Oscc__partition,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,S2: set_Product_prod_v_v,X2: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
       => ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S2 )
         => ( ( member7453568604450474000od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ S @ S2 ) )
           => ( S = S2 ) ) ) ) ) ).

% graph.scc_partition
thf(fact_349_graph_Oscc__partition,axiom,
    ! [Vertices: set_v,Successors: v > set_v,S: set_v,S2: set_v,X2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
       => ( ( sCC_Bloemen_is_scc_v @ Successors @ S2 )
         => ( ( member_v @ X2 @ ( inf_inf_set_v @ S @ S2 ) )
           => ( S = S2 ) ) ) ) ) ).

% graph.scc_partition
thf(fact_350_graph_Ois__scc__def,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
        = ( ( S != bot_bo723834152578015283od_v_v )
          & ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
          & ! [S3: set_Product_prod_v_v] :
              ( ( ( ord_le7336532860387713383od_v_v @ S @ S3 )
                & ( sCC_Bl2301996248249672505od_v_v @ Successors @ S3 ) )
             => ( S3 = S ) ) ) ) ) ).

% graph.is_scc_def
thf(fact_351_graph_Ois__scc__def,axiom,
    ! [Vertices: set_v,Successors: v > set_v,S: set_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
        = ( ( S != bot_bot_set_v )
          & ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
          & ! [S3: set_v] :
              ( ( ( ord_less_eq_set_v @ S @ S3 )
                & ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 ) )
             => ( S3 = S ) ) ) ) ) ).

% graph.is_scc_def
thf(fact_352_graph_Ora__empty,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ bot_bo723834152578015283od_v_v )
        = ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 ) ) ) ).

% graph.ra_empty
thf(fact_353_insert__Diff__if,axiom,
    ! [X2: product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ( member7453568604450474000od_v_v @ X2 @ B )
       => ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A ) @ B )
          = ( minus_4183494784930505774od_v_v @ A @ B ) ) )
      & ( ~ ( member7453568604450474000od_v_v @ X2 @ B )
       => ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ A ) @ B )
          = ( insert1338601472111419319od_v_v @ X2 @ ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_354_insert__Diff__if,axiom,
    ! [X2: v,B: set_v,A: set_v] :
      ( ( ( member_v @ X2 @ B )
       => ( ( minus_minus_set_v @ ( insert_v2 @ X2 @ A ) @ B )
          = ( minus_minus_set_v @ A @ B ) ) )
      & ( ~ ( member_v @ X2 @ B )
       => ( ( minus_minus_set_v @ ( insert_v2 @ X2 @ A ) @ B )
          = ( insert_v2 @ X2 @ ( minus_minus_set_v @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_355_Int__Diff,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( minus_minus_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 )
      = ( inf_inf_set_v @ A @ ( minus_minus_set_v @ B @ C2 ) ) ) ).

% Int_Diff
thf(fact_356_Diff__Int2,axiom,
    ! [A: set_v,C2: set_v,B: set_v] :
      ( ( minus_minus_set_v @ ( inf_inf_set_v @ A @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) )
      = ( minus_minus_set_v @ ( inf_inf_set_v @ A @ C2 ) @ B ) ) ).

% Diff_Int2
thf(fact_357_Diff__Diff__Int,axiom,
    ! [A: set_v,B: set_v] :
      ( ( minus_minus_set_v @ A @ ( minus_minus_set_v @ A @ B ) )
      = ( inf_inf_set_v @ A @ B ) ) ).

% Diff_Diff_Int
thf(fact_358_Diff__Int__distrib,axiom,
    ! [C2: set_v,A: set_v,B: set_v] :
      ( ( inf_inf_set_v @ C2 @ ( minus_minus_set_v @ A @ B ) )
      = ( minus_minus_set_v @ ( inf_inf_set_v @ C2 @ A ) @ ( inf_inf_set_v @ C2 @ B ) ) ) ).

% Diff_Int_distrib
thf(fact_359_Diff__Int__distrib2,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( inf_inf_set_v @ ( minus_minus_set_v @ A @ B ) @ C2 )
      = ( minus_minus_set_v @ ( inf_inf_set_v @ A @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) ) ) ).

% Diff_Int_distrib2
thf(fact_360_graph_Osubscc__add,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
       => ( ( member7453568604450474000od_v_v @ X2 @ S )
         => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y2 )
           => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X2 )
             => ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y2 @ S ) ) ) ) ) ) ) ).

% graph.subscc_add
thf(fact_361_graph_Osubscc__add,axiom,
    ! [Vertices: set_v,Successors: v > set_v,S: set_v,X2: v,Y2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
       => ( ( member_v @ X2 @ S )
         => ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 )
           => ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X2 )
             => ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v2 @ Y2 @ S ) ) ) ) ) ) ) ).

% graph.subscc_add
thf(fact_362_Diff__mono,axiom,
    ! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,D2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ C2 )
     => ( ( ord_le7336532860387713383od_v_v @ D2 @ B )
       => ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ C2 @ D2 ) ) ) ) ).

% Diff_mono
thf(fact_363_Diff__mono,axiom,
    ! [A: set_v,C2: set_v,D2: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A @ C2 )
     => ( ( ord_less_eq_set_v @ D2 @ B )
       => ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ B ) @ ( minus_minus_set_v @ C2 @ D2 ) ) ) ) ).

% Diff_mono
thf(fact_364_Diff__subset,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ A ) ).

% Diff_subset
thf(fact_365_Diff__subset,axiom,
    ! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ B ) @ A ) ).

% Diff_subset
thf(fact_366_double__diff,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
       => ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C2 @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_367_double__diff,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ( ord_less_eq_set_v @ B @ C2 )
       => ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C2 @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_368_graph_Oreachable__edge,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X2: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X2 ) )
       => ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y2 ) ) ) ).

% graph.reachable_edge
thf(fact_369_graph_Oreachable__edge,axiom,
    ! [Vertices: set_v,Successors: v > set_v,Y2: v,X2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( member_v @ Y2 @ ( Successors @ X2 ) )
       => ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 ) ) ) ).

% graph.reachable_edge
thf(fact_370_graph_Osucc__reachable,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v,Z: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y2 )
       => ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
         => ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Z ) ) ) ) ).

% graph.succ_reachable
thf(fact_371_graph_Osucc__reachable,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 )
       => ( ( member_v @ Z @ ( Successors @ Y2 ) )
         => ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Z ) ) ) ) ).

% graph.succ_reachable
thf(fact_372_graph_Oreachable_Ocases,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
       => ( ( A2 != A1 )
         => ~ ! [Y: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ A1 ) )
               => ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ A2 ) ) ) ) ) ).

% graph.reachable.cases
thf(fact_373_graph_Oreachable_Ocases,axiom,
    ! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
       => ( ( A2 != A1 )
         => ~ ! [Y: v] :
                ( ( member_v @ Y @ ( Successors @ A1 ) )
               => ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y @ A2 ) ) ) ) ) ).

% graph.reachable.cases
thf(fact_374_graph_Oreachable_Osimps,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
        = ( ? [X3: product_prod_v_v] :
              ( ( A1 = X3 )
              & ( A2 = X3 ) )
          | ? [X3: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
              ( ( A1 = X3 )
              & ( A2 = Z2 )
              & ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X3 ) )
              & ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).

% graph.reachable.simps
thf(fact_375_graph_Oreachable_Osimps,axiom,
    ! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
        = ( ? [X3: v] :
              ( ( A1 = X3 )
              & ( A2 = X3 ) )
          | ? [X3: v,Y3: v,Z2: v] :
              ( ( A1 = X3 )
              & ( A2 = Z2 )
              & ( member_v @ Y3 @ ( Successors @ X3 ) )
              & ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).

% graph.reachable.simps
thf(fact_376_graph_Oreachable__trans,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 )
       => ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z )
         => ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Z ) ) ) ) ).

% graph.reachable_trans
thf(fact_377_graph_Oreachable__end__induct,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v,P: product_prod_v_v > product_prod_v_v > $o] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y2 )
       => ( ! [X4: product_prod_v_v] : ( P @ X4 @ X4 )
         => ( ! [X4: product_prod_v_v,Y: product_prod_v_v,Z3: product_prod_v_v] :
                ( ( P @ X4 @ Y )
               => ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y ) )
                 => ( P @ X4 @ Z3 ) ) )
           => ( P @ X2 @ Y2 ) ) ) ) ) ).

% graph.reachable_end_induct
thf(fact_378_graph_Oreachable__end__induct,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,P: v > v > $o] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 )
       => ( ! [X4: v] : ( P @ X4 @ X4 )
         => ( ! [X4: v,Y: v,Z3: v] :
                ( ( P @ X4 @ Y )
               => ( ( member_v @ Z3 @ ( Successors @ Y ) )
                 => ( P @ X4 @ Z3 ) ) )
           => ( P @ X2 @ Y2 ) ) ) ) ) ).

% graph.reachable_end_induct
thf(fact_379_graph_Oreachable__refl,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ X2 ) ) ).

% graph.reachable_refl
thf(fact_380_graph_Oreachable__succ,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X2: product_prod_v_v,Z: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X2 ) )
       => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ Z )
         => ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Z ) ) ) ) ).

% graph.reachable_succ
thf(fact_381_graph_Oreachable__succ,axiom,
    ! [Vertices: set_v,Successors: v > set_v,Y2: v,X2: v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( member_v @ Y2 @ ( Successors @ X2 ) )
       => ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z )
         => ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Z ) ) ) ) ).

% graph.reachable_succ
thf(fact_382_graph_Ora__trans,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,E2: set_Product_prod_v_v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E2 )
       => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y2 @ Z @ E2 )
         => ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Z @ E2 ) ) ) ) ).

% graph.ra_trans
thf(fact_383_graph_Ora__refl,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,E2: set_Product_prod_v_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ X2 @ E2 ) ) ).

% graph.ra_refl
thf(fact_384_graph_Osucc__re,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X2: product_prod_v_v,Z: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X2 ) )
       => ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y2 @ Z )
         => ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Z ) ) ) ) ).

% graph.succ_re
thf(fact_385_graph_Osucc__re,axiom,
    ! [Vertices: set_v,Successors: v > set_v,Y2: v,X2: v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( member_v @ Y2 @ ( Successors @ X2 ) )
       => ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y2 @ Z )
         => ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Z ) ) ) ) ).

% graph.succ_re
thf(fact_386_graph_Oreachable__end_Ocases,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
       => ( ( A2 != A1 )
         => ~ ! [Y: product_prod_v_v] :
                ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y )
               => ~ ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y ) ) ) ) ) ) ).

% graph.reachable_end.cases
thf(fact_387_graph_Oreachable__end_Ocases,axiom,
    ! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
       => ( ( A2 != A1 )
         => ~ ! [Y: v] :
                ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y )
               => ~ ( member_v @ A2 @ ( Successors @ Y ) ) ) ) ) ) ).

% graph.reachable_end.cases
thf(fact_388_graph_Oreachable__end_Osimps,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
        = ( ? [X3: product_prod_v_v] :
              ( ( A1 = X3 )
              & ( A2 = X3 ) )
          | ? [X3: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
              ( ( A1 = X3 )
              & ( A2 = Z2 )
              & ( sCC_Bl4714988717384592488od_v_v @ Successors @ X3 @ Y3 )
              & ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).

% graph.reachable_end.simps
thf(fact_389_graph_Oreachable__end_Osimps,axiom,
    ! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
        = ( ? [X3: v] :
              ( ( A1 = X3 )
              & ( A2 = X3 ) )
          | ? [X3: v,Y3: v,Z2: v] :
              ( ( A1 = X3 )
              & ( A2 = Z2 )
              & ( sCC_Bl770211535891879572_end_v @ Successors @ X3 @ Y3 )
              & ( member_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).

% graph.reachable_end.simps
thf(fact_390_graph_Ore__refl,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ X2 ) ) ).

% graph.re_refl
thf(fact_391_graph_Ore__succ,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v,Z: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Y2 )
       => ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
         => ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Z ) ) ) ) ).

% graph.re_succ
thf(fact_392_graph_Ore__succ,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y2 )
       => ( ( member_v @ Z @ ( Successors @ Y2 ) )
         => ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Z ) ) ) ) ).

% graph.re_succ
thf(fact_393_graph_Ora__reachable,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,E2: set_Product_prod_v_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E2 )
       => ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 ) ) ) ).

% graph.ra_reachable
thf(fact_394_graph_Ore__reachable,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y2 )
       => ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 ) ) ) ).

% graph.re_reachable
thf(fact_395_graph_Oreachable__re,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 )
       => ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y2 ) ) ) ).

% graph.reachable_re
thf(fact_396_graph_Ois__subscc__def,axiom,
    ! [Vertices: set_v,Successors: v > set_v,S: set_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
        = ( ! [X3: v] :
              ( ( member_v @ X3 @ S )
             => ! [Y3: v] :
                  ( ( member_v @ Y3 @ S )
                 => ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y3 ) ) ) ) ) ) ).

% graph.is_subscc_def
thf(fact_397_graph_OS__reflexive,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1191828773336950226xt_v_a,N: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4124178362578471481nv_v_a @ Successors @ E )
       => ( member_v @ N @ ( sCC_Bloemen_S_v_a @ E @ N ) ) ) ) ).

% graph.S_reflexive
thf(fact_398_graph_OsccE,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X2: product_prod_v_v,Y2: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
       => ( ( member7453568604450474000od_v_v @ X2 @ S )
         => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X2 @ Y2 )
           => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X2 )
             => ( member7453568604450474000od_v_v @ Y2 @ S ) ) ) ) ) ) ).

% graph.sccE
thf(fact_399_graph_OsccE,axiom,
    ! [Vertices: set_v,Successors: v > set_v,S: set_v,X2: v,Y2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
       => ( ( member_v @ X2 @ S )
         => ( ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y2 )
           => ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X2 )
             => ( member_v @ Y2 @ S ) ) ) ) ) ) ).

% graph.sccE
thf(fact_400_boolean__algebra_Oconj__zero__left,axiom,
    ! [X2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X2 )
      = bot_bo723834152578015283od_v_v ) ).

% boolean_algebra.conj_zero_left
thf(fact_401_boolean__algebra_Oconj__zero__left,axiom,
    ! [X2: set_v] :
      ( ( inf_inf_set_v @ bot_bot_set_v @ X2 )
      = bot_bot_set_v ) ).

% boolean_algebra.conj_zero_left
thf(fact_402_boolean__algebra_Oconj__zero__right,axiom,
    ! [X2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ X2 @ bot_bo723834152578015283od_v_v )
      = bot_bo723834152578015283od_v_v ) ).

% boolean_algebra.conj_zero_right
thf(fact_403_boolean__algebra_Oconj__zero__right,axiom,
    ! [X2: set_v] :
      ( ( inf_inf_set_v @ X2 @ bot_bot_set_v )
      = bot_bot_set_v ) ).

% boolean_algebra.conj_zero_right
thf(fact_404_sclosed,axiom,
    ! [X: v] :
      ( ( member_v @ X @ vertices )
     => ( ord_less_eq_set_v @ ( successors @ X ) @ vertices ) ) ).

% sclosed
thf(fact_405_diff__shunt__var,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( ( minus_4183494784930505774od_v_v @ X2 @ Y2 )
        = bot_bo723834152578015283od_v_v )
      = ( ord_le7336532860387713383od_v_v @ X2 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_406_diff__shunt__var,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( ( minus_minus_set_v @ X2 @ Y2 )
        = bot_bot_set_v )
      = ( ord_less_eq_set_v @ X2 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_407_the__elem__eq,axiom,
    ! [X2: product_prod_v_v] :
      ( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
      = X2 ) ).

% the_elem_eq
thf(fact_408_the__elem__eq,axiom,
    ! [X2: v] :
      ( ( the_elem_v @ ( insert_v2 @ X2 @ bot_bot_set_v ) )
      = X2 ) ).

% the_elem_eq
thf(fact_409_prod_Oinject,axiom,
    ! [X1: v,X22: v,Y1: v,Y22: v] :
      ( ( ( product_Pair_v_v @ X1 @ X22 )
        = ( product_Pair_v_v @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_410_old_Oprod_Oinject,axiom,
    ! [A3: v,B2: v,A6: v,B6: v] :
      ( ( ( product_Pair_v_v @ A3 @ B2 )
        = ( product_Pair_v_v @ A6 @ B6 ) )
      = ( ( A3 = A6 )
        & ( B2 = B6 ) ) ) ).

% old.prod.inject
thf(fact_411_order__refl,axiom,
    ! [X2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X2 @ X2 ) ).

% order_refl
thf(fact_412_order__refl,axiom,
    ! [X2: set_v] : ( ord_less_eq_set_v @ X2 @ X2 ) ).

% order_refl
thf(fact_413_dual__order_Orefl,axiom,
    ! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_414_dual__order_Orefl,axiom,
    ! [A3: set_v] : ( ord_less_eq_set_v @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_415_less__by__empty,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( A = bot_bo723834152578015283od_v_v )
     => ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).

% less_by_empty
thf(fact_416_is__scc__def,axiom,
    ! [S: set_v] :
      ( ( sCC_Bloemen_is_scc_v @ successors @ S )
      = ( ( S != bot_bot_set_v )
        & ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
        & ! [S3: set_v] :
            ( ( ( ord_less_eq_set_v @ S @ S3 )
              & ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 ) )
           => ( S3 = S ) ) ) ) ).

% is_scc_def
thf(fact_417_bot__set__def,axiom,
    ( bot_bo723834152578015283od_v_v
    = ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).

% bot_set_def
thf(fact_418_bot__set__def,axiom,
    ( bot_bot_set_v
    = ( collect_v @ bot_bot_v_o ) ) ).

% bot_set_def
thf(fact_419_graph_Odfss_Ocases,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: produc5741669702376414499t_unit] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ~ ! [V2: v,E5: sCC_Bl1394983891496994913t_unit] :
            ( X2
           != ( produc3862955338007567901t_unit @ V2 @ E5 ) ) ) ).

% graph.dfss.cases
thf(fact_420_order__antisym__conv,axiom,
    ! [Y2: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ Y2 @ X2 )
     => ( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_421_order__antisym__conv,axiom,
    ! [Y2: set_v,X2: set_v] :
      ( ( ord_less_eq_set_v @ Y2 @ X2 )
     => ( ( ord_less_eq_set_v @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_422_ord__le__eq__subst,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_423_ord__le__eq__subst,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_424_ord__le__eq__subst,axiom,
    ! [A3: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_less_eq_set_v @ A3 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X4: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X4 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_425_ord__le__eq__subst,axiom,
    ! [A3: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X4: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X4 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_426_ord__eq__le__subst,axiom,
    ! [A3: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( A3
        = ( F @ B2 ) )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
       => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_427_ord__eq__le__subst,axiom,
    ! [A3: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( A3
        = ( F @ B2 ) )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
       => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_428_ord__eq__le__subst,axiom,
    ! [A3: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
      ( ( A3
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_v @ B2 @ C )
       => ( ! [X4: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X4 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_429_ord__eq__le__subst,axiom,
    ! [A3: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
      ( ( A3
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_v @ B2 @ C )
       => ( ! [X4: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X4 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_430_order__eq__refl,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( X2 = Y2 )
     => ( ord_le7336532860387713383od_v_v @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_431_order__eq__refl,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( X2 = Y2 )
     => ( ord_less_eq_set_v @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_432_order__subst2,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
     => ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
       => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_433_order__subst2,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
     => ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
       => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_434_order__subst2,axiom,
    ! [A3: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_less_eq_set_v @ A3 @ B2 )
     => ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
       => ( ! [X4: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X4 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_435_order__subst2,axiom,
    ! [A3: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B2 )
     => ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
       => ( ! [X4: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X4 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_436_order__subst1,axiom,
    ! [A3: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ ( F @ B2 ) )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
       => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_437_order__subst1,axiom,
    ! [A3: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_v @ B2 @ C )
       => ( ! [X4: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X4 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_438_order__subst1,axiom,
    ! [A3: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_less_eq_set_v @ A3 @ ( F @ B2 ) )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
       => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X4 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_439_order__subst1,axiom,
    ! [A3: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_v @ B2 @ C )
       => ( ! [X4: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X4 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_440_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
    = ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
          ( ( ord_le7336532860387713383od_v_v @ A5 @ B5 )
          & ( ord_le7336532860387713383od_v_v @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_441_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
    = ( ^ [A5: set_v,B5: set_v] :
          ( ( ord_less_eq_set_v @ A5 @ B5 )
          & ( ord_less_eq_set_v @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_442_antisym,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
       => ( A3 = B2 ) ) ) ).

% antisym
thf(fact_443_antisym,axiom,
    ! [A3: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B2 )
     => ( ( ord_less_eq_set_v @ B2 @ A3 )
       => ( A3 = B2 ) ) ) ).

% antisym
thf(fact_444_dual__order_Otrans,axiom,
    ! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
     => ( ( ord_le7336532860387713383od_v_v @ C @ B2 )
       => ( ord_le7336532860387713383od_v_v @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_445_dual__order_Otrans,axiom,
    ! [B2: set_v,A3: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ A3 )
     => ( ( ord_less_eq_set_v @ C @ B2 )
       => ( ord_less_eq_set_v @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_446_dual__order_Oantisym,axiom,
    ! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
     => ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
       => ( A3 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_447_dual__order_Oantisym,axiom,
    ! [B2: set_v,A3: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ A3 )
     => ( ( ord_less_eq_set_v @ A3 @ B2 )
       => ( A3 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_448_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
    = ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
          ( ( ord_le7336532860387713383od_v_v @ B5 @ A5 )
          & ( ord_le7336532860387713383od_v_v @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_449_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
    = ( ^ [A5: set_v,B5: set_v] :
          ( ( ord_less_eq_set_v @ B5 @ A5 )
          & ( ord_less_eq_set_v @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_450_order__trans,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
     => ( ( ord_le7336532860387713383od_v_v @ Y2 @ Z )
       => ( ord_le7336532860387713383od_v_v @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_451_order__trans,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ( ord_less_eq_set_v @ X2 @ Y2 )
     => ( ( ord_less_eq_set_v @ Y2 @ Z )
       => ( ord_less_eq_set_v @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_452_order_Otrans,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
       => ( ord_le7336532860387713383od_v_v @ A3 @ C ) ) ) ).

% order.trans
thf(fact_453_order_Otrans,axiom,
    ! [A3: set_v,B2: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B2 )
     => ( ( ord_less_eq_set_v @ B2 @ C )
       => ( ord_less_eq_set_v @ A3 @ C ) ) ) ).

% order.trans
thf(fact_454_order__antisym,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
     => ( ( ord_le7336532860387713383od_v_v @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_455_order__antisym,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( ord_less_eq_set_v @ X2 @ Y2 )
     => ( ( ord_less_eq_set_v @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_456_ord__le__eq__trans,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
     => ( ( B2 = C )
       => ( ord_le7336532860387713383od_v_v @ A3 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_457_ord__le__eq__trans,axiom,
    ! [A3: set_v,B2: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_v @ A3 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_458_ord__eq__le__trans,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( A3 = B2 )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
       => ( ord_le7336532860387713383od_v_v @ A3 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_459_ord__eq__le__trans,axiom,
    ! [A3: set_v,B2: set_v,C: set_v] :
      ( ( A3 = B2 )
     => ( ( ord_less_eq_set_v @ B2 @ C )
       => ( ord_less_eq_set_v @ A3 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_460_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
    = ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
          ( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
          & ( ord_le7336532860387713383od_v_v @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_461_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
    = ( ^ [X3: set_v,Y3: set_v] :
          ( ( ord_less_eq_set_v @ X3 @ Y3 )
          & ( ord_less_eq_set_v @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_462_Pair__inject,axiom,
    ! [A3: v,B2: v,A6: v,B6: v] :
      ( ( ( product_Pair_v_v @ A3 @ B2 )
        = ( product_Pair_v_v @ A6 @ B6 ) )
     => ~ ( ( A3 = A6 )
         => ( B2 != B6 ) ) ) ).

% Pair_inject
thf(fact_463_prod__cases,axiom,
    ! [P: product_prod_v_v > $o,P2: product_prod_v_v] :
      ( ! [A7: v,B7: v] : ( P @ ( product_Pair_v_v @ A7 @ B7 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_464_surj__pair,axiom,
    ! [P2: product_prod_v_v] :
    ? [X4: v,Y: v] :
      ( P2
      = ( product_Pair_v_v @ X4 @ Y ) ) ).

% surj_pair
thf(fact_465_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_v_v] :
      ~ ! [A7: v,B7: v] :
          ( Y2
         != ( product_Pair_v_v @ A7 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_466_boolean__algebra__cancel_Oinf2,axiom,
    ! [B: set_v,K: set_v,B2: set_v,A3: set_v] :
      ( ( B
        = ( inf_inf_set_v @ K @ B2 ) )
     => ( ( inf_inf_set_v @ A3 @ B )
        = ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A3 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_467_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: set_v,K: set_v,A3: set_v,B2: set_v] :
      ( ( A
        = ( inf_inf_set_v @ K @ A3 ) )
     => ( ( inf_inf_set_v @ A @ B2 )
        = ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A3 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_468_bot_Oextremum__uniqueI,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
     => ( A3 = bot_bo723834152578015283od_v_v ) ) ).

% bot.extremum_uniqueI
thf(fact_469_bot_Oextremum__uniqueI,axiom,
    ! [A3: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ bot_bot_set_v )
     => ( A3 = bot_bot_set_v ) ) ).

% bot.extremum_uniqueI
thf(fact_470_bot_Oextremum__unique,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
      = ( A3 = bot_bo723834152578015283od_v_v ) ) ).

% bot.extremum_unique
thf(fact_471_bot_Oextremum__unique,axiom,
    ! [A3: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ bot_bot_set_v )
      = ( A3 = bot_bot_set_v ) ) ).

% bot.extremum_unique
thf(fact_472_bot_Oextremum,axiom,
    ! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A3 ) ).

% bot.extremum
thf(fact_473_bot_Oextremum,axiom,
    ! [A3: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A3 ) ).

% bot.extremum
thf(fact_474_ra__add__edge,axiom,
    ! [X2: v,Y2: v,E2: set_Product_prod_v_v,V3: v,W: v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ E2 )
     => ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y2 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
        | ( ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ V3 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
          & ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).

% ra_add_edge
thf(fact_475_is__singleton__the__elem,axiom,
    ( is_sin9198872032823709915od_v_v
    = ( ^ [A4: set_Product_prod_v_v] :
          ( A4
          = ( insert1338601472111419319od_v_v @ ( the_el5392834299063928540od_v_v @ A4 ) @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% is_singleton_the_elem
thf(fact_476_is__singleton__the__elem,axiom,
    ( is_singleton_v
    = ( ^ [A4: set_v] :
          ( A4
          = ( insert_v2 @ ( the_elem_v @ A4 ) @ bot_bot_set_v ) ) ) ) ).

% is_singleton_the_elem
thf(fact_477_is__singletonI,axiom,
    ! [X2: product_prod_v_v] : ( is_sin9198872032823709915od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ).

% is_singletonI
thf(fact_478_is__singletonI,axiom,
    ! [X2: v] : ( is_singleton_v @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) ).

% is_singletonI
thf(fact_479_vfin,axiom,
    finite_finite_v @ vertices ).

% vfin
thf(fact_480_subrelI,axiom,
    ! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
      ( ! [X4: v,Y: v] :
          ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X4 @ Y ) @ R )
         => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X4 @ Y ) @ S4 ) )
     => ( ord_le7336532860387713383od_v_v @ R @ S4 ) ) ).

% subrelI
thf(fact_481_insert__subsetI,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,X5: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ X2 @ A )
     => ( ( ord_le7336532860387713383od_v_v @ X5 @ A )
       => ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ X5 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_482_insert__subsetI,axiom,
    ! [X2: v,A: set_v,X5: set_v] :
      ( ( member_v @ X2 @ A )
     => ( ( ord_less_eq_set_v @ X5 @ A )
       => ( ord_less_eq_set_v @ ( insert_v2 @ X2 @ X5 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_483_sup_Oright__idem,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) @ B2 )
      = ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ).

% sup.right_idem
thf(fact_484_sup__left__idem,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) )
      = ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) ) ).

% sup_left_idem
thf(fact_485_sup_Oleft__idem,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) )
      = ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ).

% sup.left_idem
thf(fact_486_sup__idem,axiom,
    ! [X2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X2 @ X2 )
      = X2 ) ).

% sup_idem
thf(fact_487_sup_Oidem,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A3 @ A3 )
      = A3 ) ).

% sup.idem
thf(fact_488_Un__iff,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ ( sup_sup_set_v @ A @ B ) )
      = ( ( member_v @ C @ A )
        | ( member_v @ C @ B ) ) ) ).

% Un_iff
thf(fact_489_Un__iff,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) )
      = ( ( member7453568604450474000od_v_v @ C @ A )
        | ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% Un_iff
thf(fact_490_UnCI,axiom,
    ! [C: v,B: set_v,A: set_v] :
      ( ( ~ ( member_v @ C @ B )
       => ( member_v @ C @ A ) )
     => ( member_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).

% UnCI
thf(fact_491_UnCI,axiom,
    ! [C: product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ~ ( member7453568604450474000od_v_v @ C @ B )
       => ( member7453568604450474000od_v_v @ C @ A ) )
     => ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).

% UnCI
thf(fact_492_sup_Obounded__iff,axiom,
    ! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A3 )
      = ( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
        & ( ord_le7336532860387713383od_v_v @ C @ A3 ) ) ) ).

% sup.bounded_iff
thf(fact_493_sup_Obounded__iff,axiom,
    ! [B2: set_v,C: set_v,A3: set_v] :
      ( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A3 )
      = ( ( ord_less_eq_set_v @ B2 @ A3 )
        & ( ord_less_eq_set_v @ C @ A3 ) ) ) ).

% sup.bounded_iff
thf(fact_494_le__sup__iff,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ Z )
      = ( ( ord_le7336532860387713383od_v_v @ X2 @ Z )
        & ( ord_le7336532860387713383od_v_v @ Y2 @ Z ) ) ) ).

% le_sup_iff
thf(fact_495_le__sup__iff,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_eq_set_v @ X2 @ Z )
        & ( ord_less_eq_set_v @ Y2 @ Z ) ) ) ).

% le_sup_iff
thf(fact_496_sup__bot_Oright__neutral,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
      = A3 ) ).

% sup_bot.right_neutral
thf(fact_497_sup__bot_Oright__neutral,axiom,
    ! [A3: set_v] :
      ( ( sup_sup_set_v @ A3 @ bot_bot_set_v )
      = A3 ) ).

% sup_bot.right_neutral
thf(fact_498_sup__bot_Oneutr__eq__iff,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( sup_su414716646722978715od_v_v @ A3 @ B2 ) )
      = ( ( A3 = bot_bo723834152578015283od_v_v )
        & ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_499_sup__bot_Oneutr__eq__iff,axiom,
    ! [A3: set_v,B2: set_v] :
      ( ( bot_bot_set_v
        = ( sup_sup_set_v @ A3 @ B2 ) )
      = ( ( A3 = bot_bot_set_v )
        & ( B2 = bot_bot_set_v ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_500_sup__bot_Oleft__neutral,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ A3 )
      = A3 ) ).

% sup_bot.left_neutral
thf(fact_501_sup__bot_Oleft__neutral,axiom,
    ! [A3: set_v] :
      ( ( sup_sup_set_v @ bot_bot_set_v @ A3 )
      = A3 ) ).

% sup_bot.left_neutral
thf(fact_502_sup__bot_Oeq__neutr__iff,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ( sup_su414716646722978715od_v_v @ A3 @ B2 )
        = bot_bo723834152578015283od_v_v )
      = ( ( A3 = bot_bo723834152578015283od_v_v )
        & ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_503_sup__bot_Oeq__neutr__iff,axiom,
    ! [A3: set_v,B2: set_v] :
      ( ( ( sup_sup_set_v @ A3 @ B2 )
        = bot_bot_set_v )
      = ( ( A3 = bot_bot_set_v )
        & ( B2 = bot_bot_set_v ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_504_sup__eq__bot__iff,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( ( sup_su414716646722978715od_v_v @ X2 @ Y2 )
        = bot_bo723834152578015283od_v_v )
      = ( ( X2 = bot_bo723834152578015283od_v_v )
        & ( Y2 = bot_bo723834152578015283od_v_v ) ) ) ).

% sup_eq_bot_iff
thf(fact_505_sup__eq__bot__iff,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( ( sup_sup_set_v @ X2 @ Y2 )
        = bot_bot_set_v )
      = ( ( X2 = bot_bot_set_v )
        & ( Y2 = bot_bot_set_v ) ) ) ).

% sup_eq_bot_iff
thf(fact_506_bot__eq__sup__iff,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) )
      = ( ( X2 = bot_bo723834152578015283od_v_v )
        & ( Y2 = bot_bo723834152578015283od_v_v ) ) ) ).

% bot_eq_sup_iff
thf(fact_507_bot__eq__sup__iff,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( bot_bot_set_v
        = ( sup_sup_set_v @ X2 @ Y2 ) )
      = ( ( X2 = bot_bot_set_v )
        & ( Y2 = bot_bot_set_v ) ) ) ).

% bot_eq_sup_iff
thf(fact_508_sup__bot__right,axiom,
    ! [X2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X2 @ bot_bo723834152578015283od_v_v )
      = X2 ) ).

% sup_bot_right
thf(fact_509_sup__bot__right,axiom,
    ! [X2: set_v] :
      ( ( sup_sup_set_v @ X2 @ bot_bot_set_v )
      = X2 ) ).

% sup_bot_right
thf(fact_510_sup__bot__left,axiom,
    ! [X2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_511_sup__bot__left,axiom,
    ! [X2: set_v] :
      ( ( sup_sup_set_v @ bot_bot_set_v @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_512_sup__inf__absorb,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( sup_sup_set_v @ X2 @ ( inf_inf_set_v @ X2 @ Y2 ) )
      = X2 ) ).

% sup_inf_absorb
thf(fact_513_sup__inf__absorb,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) )
      = X2 ) ).

% sup_inf_absorb
thf(fact_514_inf__sup__absorb,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( inf_inf_set_v @ X2 @ ( sup_sup_set_v @ X2 @ Y2 ) )
      = X2 ) ).

% inf_sup_absorb
thf(fact_515_inf__sup__absorb,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) )
      = X2 ) ).

% inf_sup_absorb
thf(fact_516_Un__empty,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( sup_su414716646722978715od_v_v @ A @ B )
        = bot_bo723834152578015283od_v_v )
      = ( ( A = bot_bo723834152578015283od_v_v )
        & ( B = bot_bo723834152578015283od_v_v ) ) ) ).

% Un_empty
thf(fact_517_Un__empty,axiom,
    ! [A: set_v,B: set_v] :
      ( ( ( sup_sup_set_v @ A @ B )
        = bot_bot_set_v )
      = ( ( A = bot_bot_set_v )
        & ( B = bot_bot_set_v ) ) ) ).

% Un_empty
thf(fact_518_Un__subset__iff,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C2 )
      = ( ( ord_le7336532860387713383od_v_v @ A @ C2 )
        & ( ord_le7336532860387713383od_v_v @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_519_Un__subset__iff,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ C2 )
      = ( ( ord_less_eq_set_v @ A @ C2 )
        & ( ord_less_eq_set_v @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_520_Un__insert__right,axiom,
    ! [A: set_v,A3: v,B: set_v] :
      ( ( sup_sup_set_v @ A @ ( insert_v2 @ A3 @ B ) )
      = ( insert_v2 @ A3 @ ( sup_sup_set_v @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_521_Un__insert__right,axiom,
    ! [A: set_Product_prod_v_v,A3: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A @ ( insert1338601472111419319od_v_v @ A3 @ B ) )
      = ( insert1338601472111419319od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_522_Un__insert__left,axiom,
    ! [A3: v,B: set_v,C2: set_v] :
      ( ( sup_sup_set_v @ ( insert_v2 @ A3 @ B ) @ C2 )
      = ( insert_v2 @ A3 @ ( sup_sup_set_v @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_523_Un__insert__left,axiom,
    ! [A3: product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ B ) @ C2 )
      = ( insert1338601472111419319od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_524_Un__Int__eq_I1_J,axiom,
    ! [S: set_v,T2: set_v] :
      ( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T2 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_525_Un__Int__eq_I1_J,axiom,
    ! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_526_Un__Int__eq_I2_J,axiom,
    ! [S: set_v,T2: set_v] :
      ( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_527_Un__Int__eq_I2_J,axiom,
    ! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_528_Un__Int__eq_I3_J,axiom,
    ! [S: set_v,T2: set_v] :
      ( ( inf_inf_set_v @ S @ ( sup_sup_set_v @ S @ T2 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_529_Un__Int__eq_I3_J,axiom,
    ! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ S @ ( sup_su414716646722978715od_v_v @ S @ T2 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_530_Un__Int__eq_I4_J,axiom,
    ! [T2: set_v,S: set_v] :
      ( ( inf_inf_set_v @ T2 @ ( sup_sup_set_v @ S @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_531_Un__Int__eq_I4_J,axiom,
    ! [T2: set_Product_prod_v_v,S: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ T2 @ ( sup_su414716646722978715od_v_v @ S @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_532_Int__Un__eq_I1_J,axiom,
    ! [S: set_v,T2: set_v] :
      ( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T2 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_533_Int__Un__eq_I1_J,axiom,
    ! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_534_Int__Un__eq_I2_J,axiom,
    ! [S: set_v,T2: set_v] :
      ( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_535_Int__Un__eq_I2_J,axiom,
    ! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_536_Int__Un__eq_I3_J,axiom,
    ! [S: set_v,T2: set_v] :
      ( ( sup_sup_set_v @ S @ ( inf_inf_set_v @ S @ T2 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_537_Int__Un__eq_I3_J,axiom,
    ! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ S @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_538_Int__Un__eq_I4_J,axiom,
    ! [T2: set_v,S: set_v] :
      ( ( sup_sup_set_v @ T2 @ ( inf_inf_set_v @ S @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_539_Int__Un__eq_I4_J,axiom,
    ! [T2: set_Product_prod_v_v,S: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ T2 @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_540_Un__Diff__cancel,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ A ) )
      = ( sup_su414716646722978715od_v_v @ A @ B ) ) ).

% Un_Diff_cancel
thf(fact_541_Un__Diff__cancel,axiom,
    ! [A: set_v,B: set_v] :
      ( ( sup_sup_set_v @ A @ ( minus_minus_set_v @ B @ A ) )
      = ( sup_sup_set_v @ A @ B ) ) ).

% Un_Diff_cancel
thf(fact_542_Un__Diff__cancel2,axiom,
    ! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B @ A ) @ A )
      = ( sup_su414716646722978715od_v_v @ B @ A ) ) ).

% Un_Diff_cancel2
thf(fact_543_Un__Diff__cancel2,axiom,
    ! [B: set_v,A: set_v] :
      ( ( sup_sup_set_v @ ( minus_minus_set_v @ B @ A ) @ A )
      = ( sup_sup_set_v @ B @ A ) ) ).

% Un_Diff_cancel2
thf(fact_544_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_Product_prod_v_v,K: set_Product_prod_v_v,B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( B
        = ( sup_su414716646722978715od_v_v @ K @ B2 ) )
     => ( ( sup_su414716646722978715od_v_v @ A3 @ B )
        = ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_545_boolean__algebra__cancel_Osup1,axiom,
    ! [A: set_Product_prod_v_v,K: set_Product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( A
        = ( sup_su414716646722978715od_v_v @ K @ A3 ) )
     => ( ( sup_su414716646722978715od_v_v @ A @ B2 )
        = ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_546_Un__left__commute,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
      = ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ C2 ) ) ) ).

% Un_left_commute
thf(fact_547_Un__left__absorb,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) )
      = ( sup_su414716646722978715od_v_v @ A @ B ) ) ).

% Un_left_absorb
thf(fact_548_Un__commute,axiom,
    ( sup_su414716646722978715od_v_v
    = ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B4 @ A4 ) ) ) ).

% Un_commute
thf(fact_549_Un__absorb,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_550_Un__assoc,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C2 )
      = ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).

% Un_assoc
thf(fact_551_ball__Un,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
      ( ( ! [X3: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ A @ B ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ A )
           => ( P @ X3 ) )
        & ! [X3: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ B )
           => ( P @ X3 ) ) ) ) ).

% ball_Un
thf(fact_552_bex__Un,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
      ( ( ? [X3: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ A @ B ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ A )
            & ( P @ X3 ) )
        | ? [X3: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ B )
            & ( P @ X3 ) ) ) ) ).

% bex_Un
thf(fact_553_UnI2,axiom,
    ! [C: v,B: set_v,A: set_v] :
      ( ( member_v @ C @ B )
     => ( member_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).

% UnI2
thf(fact_554_UnI2,axiom,
    ! [C: product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ B )
     => ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).

% UnI2
thf(fact_555_UnI1,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ A )
     => ( member_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).

% UnI1
thf(fact_556_UnI1,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ A )
     => ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).

% UnI1
thf(fact_557_UnE,axiom,
    ! [C: v,A: set_v,B: set_v] :
      ( ( member_v @ C @ ( sup_sup_set_v @ A @ B ) )
     => ( ~ ( member_v @ C @ A )
       => ( member_v @ C @ B ) ) ) ).

% UnE
thf(fact_558_UnE,axiom,
    ! [C: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) )
     => ( ~ ( member7453568604450474000od_v_v @ C @ A )
       => ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% UnE
thf(fact_559_sup__left__commute,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
      = ( sup_su414716646722978715od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X2 @ Z ) ) ) ).

% sup_left_commute
thf(fact_560_sup_Oleft__commute,axiom,
    ! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A3 @ C ) )
      = ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_561_sup__commute,axiom,
    ( sup_su414716646722978715od_v_v
    = ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X3 ) ) ) ).

% sup_commute
thf(fact_562_sup_Ocommute,axiom,
    ( sup_su414716646722978715od_v_v
    = ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B5 @ A5 ) ) ) ).

% sup.commute
thf(fact_563_sup__assoc,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ Z )
      = ( sup_su414716646722978715od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).

% sup_assoc
thf(fact_564_sup_Oassoc,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) @ C )
      = ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_565_inf__sup__aci_I5_J,axiom,
    ( sup_su414716646722978715od_v_v
    = ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X3 ) ) ) ).

% inf_sup_aci(5)
thf(fact_566_inf__sup__aci_I6_J,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ Z )
      = ( sup_su414716646722978715od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_567_inf__sup__aci_I7_J,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
      = ( sup_su414716646722978715od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X2 @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_568_inf__sup__aci_I8_J,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) )
      = ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) ) ).

% inf_sup_aci(8)
thf(fact_569_sup_OcoboundedI2,axiom,
    ! [C: set_Product_prod_v_v,B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C @ B2 )
     => ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_570_sup_OcoboundedI2,axiom,
    ! [C: set_v,B2: set_v,A3: set_v] :
      ( ( ord_less_eq_set_v @ C @ B2 )
     => ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A3 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_571_sup_OcoboundedI1,axiom,
    ! [C: set_Product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C @ A3 )
     => ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_572_sup_OcoboundedI1,axiom,
    ! [C: set_v,A3: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ C @ A3 )
     => ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A3 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_573_sup_Oabsorb__iff2,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
          ( ( sup_su414716646722978715od_v_v @ A5 @ B5 )
          = B5 ) ) ) ).

% sup.absorb_iff2
thf(fact_574_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_v
    = ( ^ [A5: set_v,B5: set_v] :
          ( ( sup_sup_set_v @ A5 @ B5 )
          = B5 ) ) ) ).

% sup.absorb_iff2
thf(fact_575_sup_Oabsorb__iff1,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
          ( ( sup_su414716646722978715od_v_v @ A5 @ B5 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_576_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_v
    = ( ^ [B5: set_v,A5: set_v] :
          ( ( sup_sup_set_v @ A5 @ B5 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_577_sup_Ocobounded2,axiom,
    ! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ).

% sup.cobounded2
thf(fact_578_sup_Ocobounded2,axiom,
    ! [B2: set_v,A3: set_v] : ( ord_less_eq_set_v @ B2 @ ( sup_sup_set_v @ A3 @ B2 ) ) ).

% sup.cobounded2
thf(fact_579_sup_Ocobounded1,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ).

% sup.cobounded1
thf(fact_580_sup_Ocobounded1,axiom,
    ! [A3: set_v,B2: set_v] : ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ A3 @ B2 ) ) ).

% sup.cobounded1
thf(fact_581_sup_Oorder__iff,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
          ( A5
          = ( sup_su414716646722978715od_v_v @ A5 @ B5 ) ) ) ) ).

% sup.order_iff
thf(fact_582_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_v
    = ( ^ [B5: set_v,A5: set_v] :
          ( A5
          = ( sup_sup_set_v @ A5 @ B5 ) ) ) ) ).

% sup.order_iff
thf(fact_583_sup_OboundedI,axiom,
    ! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
     => ( ( ord_le7336532860387713383od_v_v @ C @ A3 )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A3 ) ) ) ).

% sup.boundedI
thf(fact_584_sup_OboundedI,axiom,
    ! [B2: set_v,A3: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ A3 )
     => ( ( ord_less_eq_set_v @ C @ A3 )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A3 ) ) ) ).

% sup.boundedI
thf(fact_585_sup_OboundedE,axiom,
    ! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A3 )
     => ~ ( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
         => ~ ( ord_le7336532860387713383od_v_v @ C @ A3 ) ) ) ).

% sup.boundedE
thf(fact_586_sup_OboundedE,axiom,
    ! [B2: set_v,C: set_v,A3: set_v] :
      ( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A3 )
     => ~ ( ( ord_less_eq_set_v @ B2 @ A3 )
         => ~ ( ord_less_eq_set_v @ C @ A3 ) ) ) ).

% sup.boundedE
thf(fact_587_sup__absorb2,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
     => ( ( sup_su414716646722978715od_v_v @ X2 @ Y2 )
        = Y2 ) ) ).

% sup_absorb2
thf(fact_588_sup__absorb2,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( ord_less_eq_set_v @ X2 @ Y2 )
     => ( ( sup_sup_set_v @ X2 @ Y2 )
        = Y2 ) ) ).

% sup_absorb2
thf(fact_589_sup__absorb1,axiom,
    ! [Y2: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ Y2 @ X2 )
     => ( ( sup_su414716646722978715od_v_v @ X2 @ Y2 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_590_sup__absorb1,axiom,
    ! [Y2: set_v,X2: set_v] :
      ( ( ord_less_eq_set_v @ Y2 @ X2 )
     => ( ( sup_sup_set_v @ X2 @ Y2 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_591_sup_Oabsorb2,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
     => ( ( sup_su414716646722978715od_v_v @ A3 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_592_sup_Oabsorb2,axiom,
    ! [A3: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B2 )
     => ( ( sup_sup_set_v @ A3 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_593_sup_Oabsorb1,axiom,
    ! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
     => ( ( sup_su414716646722978715od_v_v @ A3 @ B2 )
        = A3 ) ) ).

% sup.absorb1
thf(fact_594_sup_Oabsorb1,axiom,
    ! [B2: set_v,A3: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ A3 )
     => ( ( sup_sup_set_v @ A3 @ B2 )
        = A3 ) ) ).

% sup.absorb1
thf(fact_595_sup__unique,axiom,
    ! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X4 @ ( F @ X4 @ Y ) )
     => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( F @ X4 @ Y ) )
       => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ Y @ X4 )
             => ( ( ord_le7336532860387713383od_v_v @ Z3 @ X4 )
               => ( ord_le7336532860387713383od_v_v @ ( F @ Y @ Z3 ) @ X4 ) ) )
         => ( ( sup_su414716646722978715od_v_v @ X2 @ Y2 )
            = ( F @ X2 @ Y2 ) ) ) ) ) ).

% sup_unique
thf(fact_596_sup__unique,axiom,
    ! [F: set_v > set_v > set_v,X2: set_v,Y2: set_v] :
      ( ! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ X4 @ ( F @ X4 @ Y ) )
     => ( ! [X4: set_v,Y: set_v] : ( ord_less_eq_set_v @ Y @ ( F @ X4 @ Y ) )
       => ( ! [X4: set_v,Y: set_v,Z3: set_v] :
              ( ( ord_less_eq_set_v @ Y @ X4 )
             => ( ( ord_less_eq_set_v @ Z3 @ X4 )
               => ( ord_less_eq_set_v @ ( F @ Y @ Z3 ) @ X4 ) ) )
         => ( ( sup_sup_set_v @ X2 @ Y2 )
            = ( F @ X2 @ Y2 ) ) ) ) ) ).

% sup_unique
thf(fact_597_sup_OorderI,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( A3
        = ( sup_su414716646722978715od_v_v @ A3 @ B2 ) )
     => ( ord_le7336532860387713383od_v_v @ B2 @ A3 ) ) ).

% sup.orderI
thf(fact_598_sup_OorderI,axiom,
    ! [A3: set_v,B2: set_v] :
      ( ( A3
        = ( sup_sup_set_v @ A3 @ B2 ) )
     => ( ord_less_eq_set_v @ B2 @ A3 ) ) ).

% sup.orderI
thf(fact_599_sup_OorderE,axiom,
    ! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
     => ( A3
        = ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ).

% sup.orderE
thf(fact_600_sup_OorderE,axiom,
    ! [B2: set_v,A3: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ A3 )
     => ( A3
        = ( sup_sup_set_v @ A3 @ B2 ) ) ) ).

% sup.orderE
thf(fact_601_le__iff__sup,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
          ( ( sup_su414716646722978715od_v_v @ X3 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_602_le__iff__sup,axiom,
    ( ord_less_eq_set_v
    = ( ^ [X3: set_v,Y3: set_v] :
          ( ( sup_sup_set_v @ X3 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_603_sup__least,axiom,
    ! [Y2: set_Product_prod_v_v,X2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ Y2 @ X2 )
     => ( ( ord_le7336532860387713383od_v_v @ Z @ X2 )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_604_sup__least,axiom,
    ! [Y2: set_v,X2: set_v,Z: set_v] :
      ( ( ord_less_eq_set_v @ Y2 @ X2 )
     => ( ( ord_less_eq_set_v @ Z @ X2 )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_605_sup__mono,axiom,
    ! [A3: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ C )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) @ ( sup_su414716646722978715od_v_v @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_606_sup__mono,axiom,
    ! [A3: set_v,C: set_v,B2: set_v,D: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ C )
     => ( ( ord_less_eq_set_v @ B2 @ D )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B2 ) @ ( sup_sup_set_v @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_607_sup_Omono,axiom,
    ! [C: set_Product_prod_v_v,A3: set_Product_prod_v_v,D: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C @ A3 )
     => ( ( ord_le7336532860387713383od_v_v @ D @ B2 )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D ) @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_608_sup_Omono,axiom,
    ! [C: set_v,A3: set_v,D: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ C @ A3 )
     => ( ( ord_less_eq_set_v @ D @ B2 )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D ) @ ( sup_sup_set_v @ A3 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_609_le__supI2,axiom,
    ! [X2: set_Product_prod_v_v,B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X2 @ B2 )
     => ( ord_le7336532860387713383od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ).

% le_supI2
thf(fact_610_le__supI2,axiom,
    ! [X2: set_v,B2: set_v,A3: set_v] :
      ( ( ord_less_eq_set_v @ X2 @ B2 )
     => ( ord_less_eq_set_v @ X2 @ ( sup_sup_set_v @ A3 @ B2 ) ) ) ).

% le_supI2
thf(fact_611_le__supI1,axiom,
    ! [X2: set_Product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X2 @ A3 )
     => ( ord_le7336532860387713383od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ).

% le_supI1
thf(fact_612_le__supI1,axiom,
    ! [X2: set_v,A3: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ X2 @ A3 )
     => ( ord_less_eq_set_v @ X2 @ ( sup_sup_set_v @ A3 @ B2 ) ) ) ).

% le_supI1
thf(fact_613_sup__ge2,axiom,
    ! [Y2: set_Product_prod_v_v,X2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) ) ).

% sup_ge2
thf(fact_614_sup__ge2,axiom,
    ! [Y2: set_v,X2: set_v] : ( ord_less_eq_set_v @ Y2 @ ( sup_sup_set_v @ X2 @ Y2 ) ) ).

% sup_ge2
thf(fact_615_sup__ge1,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) ) ).

% sup_ge1
thf(fact_616_sup__ge1,axiom,
    ! [X2: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X2 @ ( sup_sup_set_v @ X2 @ Y2 ) ) ).

% sup_ge1
thf(fact_617_le__supI,axiom,
    ! [A3: set_Product_prod_v_v,X2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ X2 )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ X2 )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) @ X2 ) ) ) ).

% le_supI
thf(fact_618_le__supI,axiom,
    ! [A3: set_v,X2: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ X2 )
     => ( ( ord_less_eq_set_v @ B2 @ X2 )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B2 ) @ X2 ) ) ) ).

% le_supI
thf(fact_619_le__supE,axiom,
    ! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) @ X2 )
     => ~ ( ( ord_le7336532860387713383od_v_v @ A3 @ X2 )
         => ~ ( ord_le7336532860387713383od_v_v @ B2 @ X2 ) ) ) ).

% le_supE
thf(fact_620_le__supE,axiom,
    ! [A3: set_v,B2: set_v,X2: set_v] :
      ( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B2 ) @ X2 )
     => ~ ( ( ord_less_eq_set_v @ A3 @ X2 )
         => ~ ( ord_less_eq_set_v @ B2 @ X2 ) ) ) ).

% le_supE
thf(fact_621_inf__sup__ord_I3_J,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) ) ).

% inf_sup_ord(3)
thf(fact_622_inf__sup__ord_I3_J,axiom,
    ! [X2: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X2 @ ( sup_sup_set_v @ X2 @ Y2 ) ) ).

% inf_sup_ord(3)
thf(fact_623_inf__sup__ord_I4_J,axiom,
    ! [Y2: set_Product_prod_v_v,X2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) ) ).

% inf_sup_ord(4)
thf(fact_624_inf__sup__ord_I4_J,axiom,
    ! [Y2: set_v,X2: set_v] : ( ord_less_eq_set_v @ Y2 @ ( sup_sup_set_v @ X2 @ Y2 ) ) ).

% inf_sup_ord(4)
thf(fact_625_boolean__algebra_Odisj__zero__right,axiom,
    ! [X2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X2 @ bot_bo723834152578015283od_v_v )
      = X2 ) ).

% boolean_algebra.disj_zero_right
thf(fact_626_boolean__algebra_Odisj__zero__right,axiom,
    ! [X2: set_v] :
      ( ( sup_sup_set_v @ X2 @ bot_bot_set_v )
      = X2 ) ).

% boolean_algebra.disj_zero_right
thf(fact_627_sup__inf__distrib2,axiom,
    ! [Y2: set_v,Z: set_v,X2: set_v] :
      ( ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ Z ) @ X2 )
      = ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ X2 ) @ ( sup_sup_set_v @ Z @ X2 ) ) ) ).

% sup_inf_distrib2
thf(fact_628_sup__inf__distrib2,axiom,
    ! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) @ X2 )
      = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ X2 ) @ ( sup_su414716646722978715od_v_v @ Z @ X2 ) ) ) ).

% sup_inf_distrib2
thf(fact_629_sup__inf__distrib1,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ( sup_sup_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) )
      = ( inf_inf_set_v @ ( sup_sup_set_v @ X2 @ Y2 ) @ ( sup_sup_set_v @ X2 @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_630_sup__inf__distrib1,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
      = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X2 @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_631_inf__sup__distrib2,axiom,
    ! [Y2: set_v,Z: set_v,X2: set_v] :
      ( ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X2 )
      = ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ X2 ) @ ( inf_inf_set_v @ Z @ X2 ) ) ) ).

% inf_sup_distrib2
thf(fact_632_inf__sup__distrib2,axiom,
    ! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X2 )
      = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ X2 ) @ ( inf_in6271465464967711157od_v_v @ Z @ X2 ) ) ) ).

% inf_sup_distrib2
thf(fact_633_inf__sup__distrib1,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ( inf_inf_set_v @ X2 @ ( sup_sup_set_v @ Y2 @ Z ) )
      = ( sup_sup_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ ( inf_inf_set_v @ X2 @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_634_inf__sup__distrib1,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
      = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X2 @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_635_distrib__imp2,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ! [X4: set_v,Y: set_v,Z3: set_v] :
          ( ( sup_sup_set_v @ X4 @ ( inf_inf_set_v @ Y @ Z3 ) )
          = ( inf_inf_set_v @ ( sup_sup_set_v @ X4 @ Y ) @ ( sup_sup_set_v @ X4 @ Z3 ) ) )
     => ( ( inf_inf_set_v @ X2 @ ( sup_sup_set_v @ Y2 @ Z ) )
        = ( sup_sup_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ ( inf_inf_set_v @ X2 @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_636_distrib__imp2,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
          ( ( sup_su414716646722978715od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ Y @ Z3 ) )
          = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) @ ( sup_su414716646722978715od_v_v @ X4 @ Z3 ) ) )
     => ( ( inf_in6271465464967711157od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
        = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X2 @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_637_distrib__imp1,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ! [X4: set_v,Y: set_v,Z3: set_v] :
          ( ( inf_inf_set_v @ X4 @ ( sup_sup_set_v @ Y @ Z3 ) )
          = ( sup_sup_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ ( inf_inf_set_v @ X4 @ Z3 ) ) )
     => ( ( sup_sup_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) )
        = ( inf_inf_set_v @ ( sup_sup_set_v @ X2 @ Y2 ) @ ( sup_sup_set_v @ X2 @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_638_distrib__imp1,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
          ( ( inf_in6271465464967711157od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ Y @ Z3 ) )
          = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X4 @ Y ) @ ( inf_in6271465464967711157od_v_v @ X4 @ Z3 ) ) )
     => ( ( sup_su414716646722978715od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
        = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X2 @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_639_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ( inf_inf_set_v @ X2 @ ( sup_sup_set_v @ Y2 @ Z ) )
      = ( sup_sup_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ ( inf_inf_set_v @ X2 @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_640_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
      = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X2 @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_641_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] :
      ( ( sup_sup_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) )
      = ( inf_inf_set_v @ ( sup_sup_set_v @ X2 @ Y2 ) @ ( sup_sup_set_v @ X2 @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_642_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
      = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X2 @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_643_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y2: set_v,Z: set_v,X2: set_v] :
      ( ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X2 )
      = ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ X2 ) @ ( inf_inf_set_v @ Z @ X2 ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_644_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X2 )
      = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ X2 ) @ ( inf_in6271465464967711157od_v_v @ Z @ X2 ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_645_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y2: set_v,Z: set_v,X2: set_v] :
      ( ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ Z ) @ X2 )
      = ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ X2 ) @ ( sup_sup_set_v @ Z @ X2 ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_646_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) @ X2 )
      = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ X2 ) @ ( sup_su414716646722978715od_v_v @ Z @ X2 ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_647_Un__empty__right,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A @ bot_bo723834152578015283od_v_v )
      = A ) ).

% Un_empty_right
thf(fact_648_Un__empty__right,axiom,
    ! [A: set_v] :
      ( ( sup_sup_set_v @ A @ bot_bot_set_v )
      = A ) ).

% Un_empty_right
thf(fact_649_Un__empty__left,axiom,
    ! [B: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B )
      = B ) ).

% Un_empty_left
thf(fact_650_Un__empty__left,axiom,
    ! [B: set_v] :
      ( ( sup_sup_set_v @ bot_bot_set_v @ B )
      = B ) ).

% Un_empty_left
thf(fact_651_subset__Un__eq,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
          ( ( sup_su414716646722978715od_v_v @ A4 @ B4 )
          = B4 ) ) ) ).

% subset_Un_eq
thf(fact_652_subset__Un__eq,axiom,
    ( ord_less_eq_set_v
    = ( ^ [A4: set_v,B4: set_v] :
          ( ( sup_sup_set_v @ A4 @ B4 )
          = B4 ) ) ) ).

% subset_Un_eq
thf(fact_653_subset__UnE,axiom,
    ! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A @ B ) )
     => ~ ! [A8: set_Product_prod_v_v] :
            ( ( ord_le7336532860387713383od_v_v @ A8 @ A )
           => ! [B8: set_Product_prod_v_v] :
                ( ( ord_le7336532860387713383od_v_v @ B8 @ B )
               => ( C2
                 != ( sup_su414716646722978715od_v_v @ A8 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_654_subset__UnE,axiom,
    ! [C2: set_v,A: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A @ B ) )
     => ~ ! [A8: set_v] :
            ( ( ord_less_eq_set_v @ A8 @ A )
           => ! [B8: set_v] :
                ( ( ord_less_eq_set_v @ B8 @ B )
               => ( C2
                 != ( sup_sup_set_v @ A8 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_655_Un__absorb2,axiom,
    ! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B @ A )
     => ( ( sup_su414716646722978715od_v_v @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_656_Un__absorb2,axiom,
    ! [B: set_v,A: set_v] :
      ( ( ord_less_eq_set_v @ B @ A )
     => ( ( sup_sup_set_v @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_657_Un__absorb1,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ( sup_su414716646722978715od_v_v @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_658_Un__absorb1,axiom,
    ! [A: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ( sup_sup_set_v @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_659_Un__upper2,axiom,
    ! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).

% Un_upper2
thf(fact_660_Un__upper2,axiom,
    ! [B: set_v,A: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A @ B ) ) ).

% Un_upper2
thf(fact_661_Un__upper1,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).

% Un_upper1
thf(fact_662_Un__upper1,axiom,
    ! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B ) ) ).

% Un_upper1
thf(fact_663_Un__least,axiom,
    ! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ C2 )
     => ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_664_Un__least,axiom,
    ! [A: set_v,C2: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A @ C2 )
     => ( ( ord_less_eq_set_v @ B @ C2 )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_665_Un__mono,axiom,
    ! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ C2 )
     => ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ C2 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_666_Un__mono,axiom,
    ! [A: set_v,C2: set_v,B: set_v,D2: set_v] :
      ( ( ord_less_eq_set_v @ A @ C2 )
     => ( ( ord_less_eq_set_v @ B @ D2 )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ C2 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_667_Un__Int__distrib2,axiom,
    ! [B: set_v,C2: set_v,A: set_v] :
      ( ( sup_sup_set_v @ ( inf_inf_set_v @ B @ C2 ) @ A )
      = ( inf_inf_set_v @ ( sup_sup_set_v @ B @ A ) @ ( sup_sup_set_v @ C2 @ A ) ) ) ).

% Un_Int_distrib2
thf(fact_668_Un__Int__distrib2,axiom,
    ! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) @ A )
      = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ A ) @ ( sup_su414716646722978715od_v_v @ C2 @ A ) ) ) ).

% Un_Int_distrib2
thf(fact_669_Int__Un__distrib2,axiom,
    ! [B: set_v,C2: set_v,A: set_v] :
      ( ( inf_inf_set_v @ ( sup_sup_set_v @ B @ C2 ) @ A )
      = ( sup_sup_set_v @ ( inf_inf_set_v @ B @ A ) @ ( inf_inf_set_v @ C2 @ A ) ) ) ).

% Int_Un_distrib2
thf(fact_670_Int__Un__distrib2,axiom,
    ! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C2 ) @ A )
      = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ A ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A ) ) ) ).

% Int_Un_distrib2
thf(fact_671_Un__Int__distrib,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( sup_sup_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) )
      = ( inf_inf_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ A @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_672_Un__Int__distrib,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
      = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ A @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_673_Int__Un__distrib,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( inf_inf_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) )
      = ( sup_sup_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ A @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_674_Int__Un__distrib,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
      = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ A @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_675_Un__Int__crazy,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ B @ C2 ) ) @ ( inf_inf_set_v @ C2 @ A ) )
      = ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ B @ C2 ) ) @ ( sup_sup_set_v @ C2 @ A ) ) ) ).

% Un_Int_crazy
thf(fact_676_Un__Int__crazy,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A ) )
      = ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) @ ( sup_su414716646722978715od_v_v @ C2 @ A ) ) ) ).

% Un_Int_crazy
thf(fact_677_Un__Diff,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ C2 )
      = ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A @ C2 ) @ ( minus_4183494784930505774od_v_v @ B @ C2 ) ) ) ).

% Un_Diff
thf(fact_678_Un__Diff,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( minus_minus_set_v @ ( sup_sup_set_v @ A @ B ) @ C2 )
      = ( sup_sup_set_v @ ( minus_minus_set_v @ A @ C2 ) @ ( minus_minus_set_v @ B @ C2 ) ) ) ).

% Un_Diff
thf(fact_679_graph_Ovfin,axiom,
    ! [Vertices: set_v,Successors: v > set_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( finite_finite_v @ Vertices ) ) ).

% graph.vfin
thf(fact_680_distrib__sup__le,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X2 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_681_distrib__sup__le,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X2 @ ( inf_inf_set_v @ Y2 @ Z ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X2 @ Y2 ) @ ( sup_sup_set_v @ X2 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_682_distrib__inf__le,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X2 @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_683_distrib__inf__le,axiom,
    ! [X2: set_v,Y2: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X2 @ Y2 ) @ ( inf_inf_set_v @ X2 @ Z ) ) @ ( inf_inf_set_v @ X2 @ ( sup_sup_set_v @ Y2 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_684_singleton__Un__iff,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v )
        = ( sup_su414716646722978715od_v_v @ A @ B ) )
      = ( ( ( A = bot_bo723834152578015283od_v_v )
          & ( B
            = ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) )
        | ( ( A
            = ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
          & ( B = bot_bo723834152578015283od_v_v ) )
        | ( ( A
            = ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
          & ( B
            = ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_685_singleton__Un__iff,axiom,
    ! [X2: v,A: set_v,B: set_v] :
      ( ( ( insert_v2 @ X2 @ bot_bot_set_v )
        = ( sup_sup_set_v @ A @ B ) )
      = ( ( ( A = bot_bot_set_v )
          & ( B
            = ( insert_v2 @ X2 @ bot_bot_set_v ) ) )
        | ( ( A
            = ( insert_v2 @ X2 @ bot_bot_set_v ) )
          & ( B = bot_bot_set_v ) )
        | ( ( A
            = ( insert_v2 @ X2 @ bot_bot_set_v ) )
          & ( B
            = ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_686_Un__singleton__iff,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X2: product_prod_v_v] :
      ( ( ( sup_su414716646722978715od_v_v @ A @ B )
        = ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
      = ( ( ( A = bot_bo723834152578015283od_v_v )
          & ( B
            = ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) )
        | ( ( A
            = ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
          & ( B = bot_bo723834152578015283od_v_v ) )
        | ( ( A
            = ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) )
          & ( B
            = ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_687_Un__singleton__iff,axiom,
    ! [A: set_v,B: set_v,X2: v] :
      ( ( ( sup_sup_set_v @ A @ B )
        = ( insert_v2 @ X2 @ bot_bot_set_v ) )
      = ( ( ( A = bot_bot_set_v )
          & ( B
            = ( insert_v2 @ X2 @ bot_bot_set_v ) ) )
        | ( ( A
            = ( insert_v2 @ X2 @ bot_bot_set_v ) )
          & ( B = bot_bot_set_v ) )
        | ( ( A
            = ( insert_v2 @ X2 @ bot_bot_set_v ) )
          & ( B
            = ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_688_insert__is__Un,axiom,
    ( insert1338601472111419319od_v_v
    = ( ^ [A5: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A5 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% insert_is_Un
thf(fact_689_insert__is__Un,axiom,
    ( insert_v2
    = ( ^ [A5: v] : ( sup_sup_set_v @ ( insert_v2 @ A5 @ bot_bot_set_v ) ) ) ) ).

% insert_is_Un
thf(fact_690_Un__Int__assoc__eq,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C2 )
        = ( inf_in6271465464967711157od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) )
      = ( ord_le7336532860387713383od_v_v @ C2 @ A ) ) ).

% Un_Int_assoc_eq
thf(fact_691_Un__Int__assoc__eq,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 )
        = ( inf_inf_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) ) )
      = ( ord_less_eq_set_v @ C2 @ A ) ) ).

% Un_Int_assoc_eq
thf(fact_692_Diff__subset__conv,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ C2 )
      = ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_693_Diff__subset__conv,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ B ) @ C2 )
      = ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_694_Diff__partition,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ( sup_su414716646722978715od_v_v @ A @ ( minus_4183494784930505774od_v_v @ B @ A ) )
        = B ) ) ).

% Diff_partition
thf(fact_695_Diff__partition,axiom,
    ! [A: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ( sup_sup_set_v @ A @ ( minus_minus_set_v @ B @ A ) )
        = B ) ) ).

% Diff_partition
thf(fact_696_Un__Diff__Int,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
      = A ) ).

% Un_Diff_Int
thf(fact_697_Un__Diff__Int,axiom,
    ! [A: set_v,B: set_v] :
      ( ( sup_sup_set_v @ ( minus_minus_set_v @ A @ B ) @ ( inf_inf_set_v @ A @ B ) )
      = A ) ).

% Un_Diff_Int
thf(fact_698_Int__Diff__Un,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ A @ B ) )
      = A ) ).

% Int_Diff_Un
thf(fact_699_Int__Diff__Un,axiom,
    ! [A: set_v,B: set_v] :
      ( ( sup_sup_set_v @ ( inf_inf_set_v @ A @ B ) @ ( minus_minus_set_v @ A @ B ) )
      = A ) ).

% Int_Diff_Un
thf(fact_700_Diff__Int,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
      = ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ A @ C2 ) ) ) ).

% Diff_Int
thf(fact_701_Diff__Int,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( minus_minus_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) )
      = ( sup_sup_set_v @ ( minus_minus_set_v @ A @ B ) @ ( minus_minus_set_v @ A @ C2 ) ) ) ).

% Diff_Int
thf(fact_702_Diff__Un,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
      = ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ A @ C2 ) ) ) ).

% Diff_Un
thf(fact_703_Diff__Un,axiom,
    ! [A: set_v,B: set_v,C2: set_v] :
      ( ( minus_minus_set_v @ A @ ( sup_sup_set_v @ B @ C2 ) )
      = ( inf_inf_set_v @ ( minus_minus_set_v @ A @ B ) @ ( minus_minus_set_v @ A @ C2 ) ) ) ).

% Diff_Un
thf(fact_704_graph_Ointro,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ Vertices )
     => ( ! [X4: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X4 @ Vertices )
           => ( ord_le7336532860387713383od_v_v @ ( Successors @ X4 ) @ Vertices ) )
       => ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors ) ) ) ).

% graph.intro
thf(fact_705_graph_Ointro,axiom,
    ! [Vertices: set_v,Successors: v > set_v] :
      ( ( finite_finite_v @ Vertices )
     => ( ! [X4: v] :
            ( ( member_v @ X4 @ Vertices )
           => ( ord_less_eq_set_v @ ( Successors @ X4 ) @ Vertices ) )
       => ( sCC_Bloemen_graph_v @ Vertices @ Successors ) ) ) ).

% graph.intro
thf(fact_706_SCC__Bloemen__Sequential_Ograph__def,axiom,
    ( sCC_Bl8307124943676871238od_v_v
    = ( ^ [Vertices2: set_Product_prod_v_v,Successors2: product_prod_v_v > set_Product_prod_v_v] :
          ( ( finite3348123685078250256od_v_v @ Vertices2 )
          & ! [X3: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ X3 @ Vertices2 )
             => ( ord_le7336532860387713383od_v_v @ ( Successors2 @ X3 ) @ Vertices2 ) ) ) ) ) ).

% SCC_Bloemen_Sequential.graph_def
thf(fact_707_SCC__Bloemen__Sequential_Ograph__def,axiom,
    ( sCC_Bloemen_graph_v
    = ( ^ [Vertices2: set_v,Successors2: v > set_v] :
          ( ( finite_finite_v @ Vertices2 )
          & ! [X3: v] :
              ( ( member_v @ X3 @ Vertices2 )
             => ( ord_less_eq_set_v @ ( Successors2 @ X3 ) @ Vertices2 ) ) ) ) ) ).

% SCC_Bloemen_Sequential.graph_def
thf(fact_708_is__singletonI_H,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( A != bot_bo723834152578015283od_v_v )
     => ( ! [X4: product_prod_v_v,Y: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X4 @ A )
           => ( ( member7453568604450474000od_v_v @ Y @ A )
             => ( X4 = Y ) ) )
       => ( is_sin9198872032823709915od_v_v @ A ) ) ) ).

% is_singletonI'
thf(fact_709_is__singletonI_H,axiom,
    ! [A: set_v] :
      ( ( A != bot_bot_set_v )
     => ( ! [X4: v,Y: v] :
            ( ( member_v @ X4 @ A )
           => ( ( member_v @ Y @ A )
             => ( X4 = Y ) ) )
       => ( is_singleton_v @ A ) ) ) ).

% is_singletonI'
thf(fact_710_ssubst__Pair__rhs,axiom,
    ! [R: v,S4: v,R3: set_Product_prod_v_v,S5: v] :
      ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R @ S4 ) @ R3 )
     => ( ( S5 = S4 )
       => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_711_bot__empty__eq,axiom,
    ( bot_bo8461541820394803818_v_v_o
    = ( ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ).

% bot_empty_eq
thf(fact_712_bot__empty__eq,axiom,
    ( bot_bot_v_o
    = ( ^ [X3: v] : ( member_v @ X3 @ bot_bot_set_v ) ) ) ).

% bot_empty_eq
thf(fact_713_graph_Ora__add__edge,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X2: v,Y2: v,E2: set_Product_prod_v_v,V3: v,W: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ E2 )
       => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y2 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
          | ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ V3 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
            & ( sCC_Bl4291963740693775144ding_v @ Successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).

% graph.ra_add_edge
thf(fact_714_is__singleton__def,axiom,
    ( is_sin9198872032823709915od_v_v
    = ( ^ [A4: set_Product_prod_v_v] :
        ? [X3: product_prod_v_v] :
          ( A4
          = ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% is_singleton_def
thf(fact_715_is__singleton__def,axiom,
    ( is_singleton_v
    = ( ^ [A4: set_v] :
        ? [X3: v] :
          ( A4
          = ( insert_v2 @ X3 @ bot_bot_set_v ) ) ) ) ).

% is_singleton_def
thf(fact_716_is__singletonE,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( is_sin9198872032823709915od_v_v @ A )
     => ~ ! [X4: product_prod_v_v] :
            ( A
           != ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) ).

% is_singletonE
thf(fact_717_is__singletonE,axiom,
    ! [A: set_v] :
      ( ( is_singleton_v @ A )
     => ~ ! [X4: v] :
            ( A
           != ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) ).

% is_singletonE
thf(fact_718_subset__emptyI,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ! [X4: product_prod_v_v] :
          ~ ( member7453568604450474000od_v_v @ X4 @ A )
     => ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).

% subset_emptyI
thf(fact_719_subset__emptyI,axiom,
    ! [A: set_v] :
      ( ! [X4: v] :
          ~ ( member_v @ X4 @ A )
     => ( ord_less_eq_set_v @ A @ bot_bot_set_v ) ) ).

% subset_emptyI
thf(fact_720_finite__Diff__insert,axiom,
    ! [A: set_Product_prod_v_v,A3: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ A3 @ B ) ) )
      = ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ).

% finite_Diff_insert
thf(fact_721_finite__Diff__insert,axiom,
    ! [A: set_v,A3: v,B: set_v] :
      ( ( finite_finite_v @ ( minus_minus_set_v @ A @ ( insert_v2 @ A3 @ B ) ) )
      = ( finite_finite_v @ ( minus_minus_set_v @ A @ B ) ) ) ).

% finite_Diff_insert
thf(fact_722_finite__Diff2,axiom,
    ! [B: set_v,A: set_v] :
      ( ( finite_finite_v @ B )
     => ( ( finite_finite_v @ ( minus_minus_set_v @ A @ B ) )
        = ( finite_finite_v @ A ) ) ) ).

% finite_Diff2
thf(fact_723_finite__Diff,axiom,
    ! [A: set_v,B: set_v] :
      ( ( finite_finite_v @ A )
     => ( finite_finite_v @ ( minus_minus_set_v @ A @ B ) ) ) ).

% finite_Diff
thf(fact_724_finite__Un,axiom,
    ! [F3: set_v,G: set_v] :
      ( ( finite_finite_v @ ( sup_sup_set_v @ F3 @ G ) )
      = ( ( finite_finite_v @ F3 )
        & ( finite_finite_v @ G ) ) ) ).

% finite_Un
thf(fact_725_finite__Un,axiom,
    ! [F3: set_Product_prod_v_v,G: set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ F3 @ G ) )
      = ( ( finite3348123685078250256od_v_v @ F3 )
        & ( finite3348123685078250256od_v_v @ G ) ) ) ).

% finite_Un
thf(fact_726_finite__Int,axiom,
    ! [F3: set_v,G: set_v] :
      ( ( ( finite_finite_v @ F3 )
        | ( finite_finite_v @ G ) )
     => ( finite_finite_v @ ( inf_inf_set_v @ F3 @ G ) ) ) ).

% finite_Int
thf(fact_727_remove__induct,axiom,
    ! [P: set_Product_prod_v_v > $o,B: set_Product_prod_v_v] :
      ( ( P @ bot_bo723834152578015283od_v_v )
     => ( ( ~ ( finite3348123685078250256od_v_v @ B )
         => ( P @ B ) )
       => ( ! [A9: set_Product_prod_v_v] :
              ( ( finite3348123685078250256od_v_v @ A9 )
             => ( ( A9 != bot_bo723834152578015283od_v_v )
               => ( ( ord_le7336532860387713383od_v_v @ A9 @ B )
                 => ( ! [X: product_prod_v_v] :
                        ( ( member7453568604450474000od_v_v @ X @ A9 )
                       => ( P @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) )
                   => ( P @ A9 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_728_remove__induct,axiom,
    ! [P: set_v > $o,B: set_v] :
      ( ( P @ bot_bot_set_v )
     => ( ( ~ ( finite_finite_v @ B )
         => ( P @ B ) )
       => ( ! [A9: set_v] :
              ( ( finite_finite_v @ A9 )
             => ( ( A9 != bot_bot_set_v )
               => ( ( ord_less_eq_set_v @ A9 @ B )
                 => ( ! [X: v] :
                        ( ( member_v @ X @ A9 )
                       => ( P @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) )
                   => ( P @ A9 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_729_finite__insert,axiom,
    ! [A3: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ A ) )
      = ( finite3348123685078250256od_v_v @ A ) ) ).

% finite_insert
thf(fact_730_finite__insert,axiom,
    ! [A3: v,A: set_v] :
      ( ( finite_finite_v @ ( insert_v2 @ A3 @ A ) )
      = ( finite_finite_v @ A ) ) ).

% finite_insert
thf(fact_731_finite__has__maximal2,axiom,
    ! [A: set_se8455005133513928103od_v_v,A3: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( member8406446414694345712od_v_v @ A3 @ A )
       => ? [X4: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X4 @ A )
            & ( ord_le7336532860387713383od_v_v @ A3 @ X4 )
            & ! [Xa: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ Xa @ A )
               => ( ( ord_le7336532860387713383od_v_v @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_732_finite__has__maximal2,axiom,
    ! [A: set_set_v,A3: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( member_set_v @ A3 @ A )
       => ? [X4: set_v] :
            ( ( member_set_v @ X4 @ A )
            & ( ord_less_eq_set_v @ A3 @ X4 )
            & ! [Xa: set_v] :
                ( ( member_set_v @ Xa @ A )
               => ( ( ord_less_eq_set_v @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_733_finite__has__minimal2,axiom,
    ! [A: set_se8455005133513928103od_v_v,A3: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( member8406446414694345712od_v_v @ A3 @ A )
       => ? [X4: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X4 @ A )
            & ( ord_le7336532860387713383od_v_v @ X4 @ A3 )
            & ! [Xa: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ Xa @ A )
               => ( ( ord_le7336532860387713383od_v_v @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_734_finite__has__minimal2,axiom,
    ! [A: set_set_v,A3: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( member_set_v @ A3 @ A )
       => ? [X4: set_v] :
            ( ( member_set_v @ X4 @ A )
            & ( ord_less_eq_set_v @ X4 @ A3 )
            & ! [Xa: set_v] :
                ( ( member_set_v @ Xa @ A )
               => ( ( ord_less_eq_set_v @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_735_finite_OemptyI,axiom,
    finite3348123685078250256od_v_v @ bot_bo723834152578015283od_v_v ).

% finite.emptyI
thf(fact_736_finite_OemptyI,axiom,
    finite_finite_v @ bot_bot_set_v ).

% finite.emptyI
thf(fact_737_infinite__imp__nonempty,axiom,
    ! [S: set_Product_prod_v_v] :
      ( ~ ( finite3348123685078250256od_v_v @ S )
     => ( S != bot_bo723834152578015283od_v_v ) ) ).

% infinite_imp_nonempty
thf(fact_738_infinite__imp__nonempty,axiom,
    ! [S: set_v] :
      ( ~ ( finite_finite_v @ S )
     => ( S != bot_bot_set_v ) ) ).

% infinite_imp_nonempty
thf(fact_739_finite__subset,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ( finite3348123685078250256od_v_v @ B )
       => ( finite3348123685078250256od_v_v @ A ) ) ) ).

% finite_subset
thf(fact_740_finite__subset,axiom,
    ! [A: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ( finite_finite_v @ B )
       => ( finite_finite_v @ A ) ) ) ).

% finite_subset
thf(fact_741_infinite__super,axiom,
    ! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ S @ T2 )
     => ( ~ ( finite3348123685078250256od_v_v @ S )
       => ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).

% infinite_super
thf(fact_742_infinite__super,axiom,
    ! [S: set_v,T2: set_v] :
      ( ( ord_less_eq_set_v @ S @ T2 )
     => ( ~ ( finite_finite_v @ S )
       => ~ ( finite_finite_v @ T2 ) ) ) ).

% infinite_super
thf(fact_743_rev__finite__subset,axiom,
    ! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ B )
     => ( ( ord_le7336532860387713383od_v_v @ A @ B )
       => ( finite3348123685078250256od_v_v @ A ) ) ) ).

% rev_finite_subset
thf(fact_744_rev__finite__subset,axiom,
    ! [B: set_v,A: set_v] :
      ( ( finite_finite_v @ B )
     => ( ( ord_less_eq_set_v @ A @ B )
       => ( finite_finite_v @ A ) ) ) ).

% rev_finite_subset
thf(fact_745_finite_OinsertI,axiom,
    ! [A: set_Product_prod_v_v,A3: product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ A )
     => ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ A ) ) ) ).

% finite.insertI
thf(fact_746_finite_OinsertI,axiom,
    ! [A: set_v,A3: v] :
      ( ( finite_finite_v @ A )
     => ( finite_finite_v @ ( insert_v2 @ A3 @ A ) ) ) ).

% finite.insertI
thf(fact_747_infinite__Un,axiom,
    ! [S: set_v,T2: set_v] :
      ( ( ~ ( finite_finite_v @ ( sup_sup_set_v @ S @ T2 ) ) )
      = ( ~ ( finite_finite_v @ S )
        | ~ ( finite_finite_v @ T2 ) ) ) ).

% infinite_Un
thf(fact_748_infinite__Un,axiom,
    ! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) ) )
      = ( ~ ( finite3348123685078250256od_v_v @ S )
        | ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).

% infinite_Un
thf(fact_749_Un__infinite,axiom,
    ! [S: set_v,T2: set_v] :
      ( ~ ( finite_finite_v @ S )
     => ~ ( finite_finite_v @ ( sup_sup_set_v @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_750_Un__infinite,axiom,
    ! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ~ ( finite3348123685078250256od_v_v @ S )
     => ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_751_finite__UnI,axiom,
    ! [F3: set_v,G: set_v] :
      ( ( finite_finite_v @ F3 )
     => ( ( finite_finite_v @ G )
       => ( finite_finite_v @ ( sup_sup_set_v @ F3 @ G ) ) ) ) ).

% finite_UnI
thf(fact_752_finite__UnI,axiom,
    ! [F3: set_Product_prod_v_v,G: set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ F3 )
     => ( ( finite3348123685078250256od_v_v @ G )
       => ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ F3 @ G ) ) ) ) ).

% finite_UnI
thf(fact_753_Diff__infinite__finite,axiom,
    ! [T2: set_v,S: set_v] :
      ( ( finite_finite_v @ T2 )
     => ( ~ ( finite_finite_v @ S )
       => ~ ( finite_finite_v @ ( minus_minus_set_v @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_754_finite__has__minimal,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ? [X4: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X4 @ A )
            & ! [Xa: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ Xa @ A )
               => ( ( ord_le7336532860387713383od_v_v @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_755_finite__has__minimal,axiom,
    ! [A: set_set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ? [X4: set_v] :
            ( ( member_set_v @ X4 @ A )
            & ! [Xa: set_v] :
                ( ( member_set_v @ Xa @ A )
               => ( ( ord_less_eq_set_v @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_756_finite__has__maximal,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ? [X4: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X4 @ A )
            & ! [Xa: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ Xa @ A )
               => ( ( ord_le7336532860387713383od_v_v @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_757_finite__has__maximal,axiom,
    ! [A: set_set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ? [X4: set_v] :
            ( ( member_set_v @ X4 @ A )
            & ! [Xa: set_v] :
                ( ( member_set_v @ Xa @ A )
               => ( ( ord_less_eq_set_v @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_758_infinite__finite__induct,axiom,
    ! [P: set_Product_prod_v_v > $o,A: set_Product_prod_v_v] :
      ( ! [A9: set_Product_prod_v_v] :
          ( ~ ( finite3348123685078250256od_v_v @ A9 )
         => ( P @ A9 ) )
     => ( ( P @ bot_bo723834152578015283od_v_v )
       => ( ! [X4: product_prod_v_v,F4: set_Product_prod_v_v] :
              ( ( finite3348123685078250256od_v_v @ F4 )
             => ( ~ ( member7453568604450474000od_v_v @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert1338601472111419319od_v_v @ X4 @ F4 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_759_infinite__finite__induct,axiom,
    ! [P: set_v > $o,A: set_v] :
      ( ! [A9: set_v] :
          ( ~ ( finite_finite_v @ A9 )
         => ( P @ A9 ) )
     => ( ( P @ bot_bot_set_v )
       => ( ! [X4: v,F4: set_v] :
              ( ( finite_finite_v @ F4 )
             => ( ~ ( member_v @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_v2 @ X4 @ F4 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_760_finite__ne__induct,axiom,
    ! [F3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
      ( ( finite3348123685078250256od_v_v @ F3 )
     => ( ( F3 != bot_bo723834152578015283od_v_v )
       => ( ! [X4: product_prod_v_v] : ( P @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) )
         => ( ! [X4: product_prod_v_v,F4: set_Product_prod_v_v] :
                ( ( finite3348123685078250256od_v_v @ F4 )
               => ( ( F4 != bot_bo723834152578015283od_v_v )
                 => ( ~ ( member7453568604450474000od_v_v @ X4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert1338601472111419319od_v_v @ X4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_761_finite__ne__induct,axiom,
    ! [F3: set_v,P: set_v > $o] :
      ( ( finite_finite_v @ F3 )
     => ( ( F3 != bot_bot_set_v )
       => ( ! [X4: v] : ( P @ ( insert_v2 @ X4 @ bot_bot_set_v ) )
         => ( ! [X4: v,F4: set_v] :
                ( ( finite_finite_v @ F4 )
               => ( ( F4 != bot_bot_set_v )
                 => ( ~ ( member_v @ X4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_v2 @ X4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_762_finite__induct,axiom,
    ! [F3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
      ( ( finite3348123685078250256od_v_v @ F3 )
     => ( ( P @ bot_bo723834152578015283od_v_v )
       => ( ! [X4: product_prod_v_v,F4: set_Product_prod_v_v] :
              ( ( finite3348123685078250256od_v_v @ F4 )
             => ( ~ ( member7453568604450474000od_v_v @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert1338601472111419319od_v_v @ X4 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_763_finite__induct,axiom,
    ! [F3: set_v,P: set_v > $o] :
      ( ( finite_finite_v @ F3 )
     => ( ( P @ bot_bot_set_v )
       => ( ! [X4: v,F4: set_v] :
              ( ( finite_finite_v @ F4 )
             => ( ~ ( member_v @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_v2 @ X4 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_764_finite_Osimps,axiom,
    ( finite3348123685078250256od_v_v
    = ( ^ [A5: set_Product_prod_v_v] :
          ( ( A5 = bot_bo723834152578015283od_v_v )
          | ? [A4: set_Product_prod_v_v,B5: product_prod_v_v] :
              ( ( A5
                = ( insert1338601472111419319od_v_v @ B5 @ A4 ) )
              & ( finite3348123685078250256od_v_v @ A4 ) ) ) ) ) ).

% finite.simps
thf(fact_765_finite_Osimps,axiom,
    ( finite_finite_v
    = ( ^ [A5: set_v] :
          ( ( A5 = bot_bot_set_v )
          | ? [A4: set_v,B5: v] :
              ( ( A5
                = ( insert_v2 @ B5 @ A4 ) )
              & ( finite_finite_v @ A4 ) ) ) ) ) ).

% finite.simps
thf(fact_766_finite_Ocases,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ A3 )
     => ( ( A3 != bot_bo723834152578015283od_v_v )
       => ~ ! [A9: set_Product_prod_v_v] :
              ( ? [A7: product_prod_v_v] :
                  ( A3
                  = ( insert1338601472111419319od_v_v @ A7 @ A9 ) )
             => ~ ( finite3348123685078250256od_v_v @ A9 ) ) ) ) ).

% finite.cases
thf(fact_767_finite_Ocases,axiom,
    ! [A3: set_v] :
      ( ( finite_finite_v @ A3 )
     => ( ( A3 != bot_bot_set_v )
       => ~ ! [A9: set_v] :
              ( ? [A7: v] :
                  ( A3
                  = ( insert_v2 @ A7 @ A9 ) )
             => ~ ( finite_finite_v @ A9 ) ) ) ) ).

% finite.cases
thf(fact_768_finite__subset__induct_H,axiom,
    ! [F3: set_Product_prod_v_v,A: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
      ( ( finite3348123685078250256od_v_v @ F3 )
     => ( ( ord_le7336532860387713383od_v_v @ F3 @ A )
       => ( ( P @ bot_bo723834152578015283od_v_v )
         => ( ! [A7: product_prod_v_v,F4: set_Product_prod_v_v] :
                ( ( finite3348123685078250256od_v_v @ F4 )
               => ( ( member7453568604450474000od_v_v @ A7 @ A )
                 => ( ( ord_le7336532860387713383od_v_v @ F4 @ A )
                   => ( ~ ( member7453568604450474000od_v_v @ A7 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert1338601472111419319od_v_v @ A7 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_769_finite__subset__induct_H,axiom,
    ! [F3: set_v,A: set_v,P: set_v > $o] :
      ( ( finite_finite_v @ F3 )
     => ( ( ord_less_eq_set_v @ F3 @ A )
       => ( ( P @ bot_bot_set_v )
         => ( ! [A7: v,F4: set_v] :
                ( ( finite_finite_v @ F4 )
               => ( ( member_v @ A7 @ A )
                 => ( ( ord_less_eq_set_v @ F4 @ A )
                   => ( ~ ( member_v @ A7 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_v2 @ A7 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_770_finite__subset__induct,axiom,
    ! [F3: set_Product_prod_v_v,A: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
      ( ( finite3348123685078250256od_v_v @ F3 )
     => ( ( ord_le7336532860387713383od_v_v @ F3 @ A )
       => ( ( P @ bot_bo723834152578015283od_v_v )
         => ( ! [A7: product_prod_v_v,F4: set_Product_prod_v_v] :
                ( ( finite3348123685078250256od_v_v @ F4 )
               => ( ( member7453568604450474000od_v_v @ A7 @ A )
                 => ( ~ ( member7453568604450474000od_v_v @ A7 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert1338601472111419319od_v_v @ A7 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_771_finite__subset__induct,axiom,
    ! [F3: set_v,A: set_v,P: set_v > $o] :
      ( ( finite_finite_v @ F3 )
     => ( ( ord_less_eq_set_v @ F3 @ A )
       => ( ( P @ bot_bot_set_v )
         => ( ! [A7: v,F4: set_v] :
                ( ( finite_finite_v @ F4 )
               => ( ( member_v @ A7 @ A )
                 => ( ~ ( member_v @ A7 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_v2 @ A7 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_772_finite__empty__induct,axiom,
    ! [A: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
      ( ( finite3348123685078250256od_v_v @ A )
     => ( ( P @ A )
       => ( ! [A7: product_prod_v_v,A9: set_Product_prod_v_v] :
              ( ( finite3348123685078250256od_v_v @ A9 )
             => ( ( member7453568604450474000od_v_v @ A7 @ A9 )
               => ( ( P @ A9 )
                 => ( P @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ A7 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
         => ( P @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% finite_empty_induct
thf(fact_773_finite__empty__induct,axiom,
    ! [A: set_v,P: set_v > $o] :
      ( ( finite_finite_v @ A )
     => ( ( P @ A )
       => ( ! [A7: v,A9: set_v] :
              ( ( finite_finite_v @ A9 )
             => ( ( member_v @ A7 @ A9 )
               => ( ( P @ A9 )
                 => ( P @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ A7 @ bot_bot_set_v ) ) ) ) ) )
         => ( P @ bot_bot_set_v ) ) ) ) ).

% finite_empty_induct
thf(fact_774_infinite__coinduct,axiom,
    ! [X5: set_Product_prod_v_v > $o,A: set_Product_prod_v_v] :
      ( ( X5 @ A )
     => ( ! [A9: set_Product_prod_v_v] :
            ( ( X5 @ A9 )
           => ? [X: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X @ A9 )
                & ( ( X5 @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
                  | ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
       => ~ ( finite3348123685078250256od_v_v @ A ) ) ) ).

% infinite_coinduct
thf(fact_775_infinite__coinduct,axiom,
    ! [X5: set_v > $o,A: set_v] :
      ( ( X5 @ A )
     => ( ! [A9: set_v] :
            ( ( X5 @ A9 )
           => ? [X: v] :
                ( ( member_v @ X @ A9 )
                & ( ( X5 @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ X @ bot_bot_set_v ) ) )
                  | ~ ( finite_finite_v @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) )
       => ~ ( finite_finite_v @ A ) ) ) ).

% infinite_coinduct
thf(fact_776_infinite__remove,axiom,
    ! [S: set_Product_prod_v_v,A3: product_prod_v_v] :
      ( ~ ( finite3348123685078250256od_v_v @ S )
     => ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ S @ ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% infinite_remove
thf(fact_777_infinite__remove,axiom,
    ! [S: set_v,A3: v] :
      ( ~ ( finite_finite_v @ S )
     => ~ ( finite_finite_v @ ( minus_minus_set_v @ S @ ( insert_v2 @ A3 @ bot_bot_set_v ) ) ) ) ).

% infinite_remove
thf(fact_778_finite__remove__induct,axiom,
    ! [B: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
      ( ( finite3348123685078250256od_v_v @ B )
     => ( ( P @ bot_bo723834152578015283od_v_v )
       => ( ! [A9: set_Product_prod_v_v] :
              ( ( finite3348123685078250256od_v_v @ A9 )
             => ( ( A9 != bot_bo723834152578015283od_v_v )
               => ( ( ord_le7336532860387713383od_v_v @ A9 @ B )
                 => ( ! [X: product_prod_v_v] :
                        ( ( member7453568604450474000od_v_v @ X @ A9 )
                       => ( P @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) )
                   => ( P @ A9 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_779_finite__remove__induct,axiom,
    ! [B: set_v,P: set_v > $o] :
      ( ( finite_finite_v @ B )
     => ( ( P @ bot_bot_set_v )
       => ( ! [A9: set_v] :
              ( ( finite_finite_v @ A9 )
             => ( ( A9 != bot_bot_set_v )
               => ( ( ord_less_eq_set_v @ A9 @ B )
                 => ( ! [X: v] :
                        ( ( member_v @ X @ A9 )
                       => ( P @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) )
                   => ( P @ A9 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_780_Collect__empty__eq__bot,axiom,
    ! [P: product_prod_v_v > $o] :
      ( ( ( collec140062887454715474od_v_v @ P )
        = bot_bo723834152578015283od_v_v )
      = ( P = bot_bo8461541820394803818_v_v_o ) ) ).

% Collect_empty_eq_bot
thf(fact_781_Collect__empty__eq__bot,axiom,
    ! [P: v > $o] :
      ( ( ( collect_v @ P )
        = bot_bot_set_v )
      = ( P = bot_bot_v_o ) ) ).

% Collect_empty_eq_bot
thf(fact_782_Field__insert,axiom,
    ! [A3: product_prod_v_v,B2: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
      ( ( field_7153129647634986036od_v_v @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ B2 ) @ R ) )
      = ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R ) ) ) ).

% Field_insert
thf(fact_783_Field__insert,axiom,
    ! [A3: v,B2: v,R: set_Product_prod_v_v] :
      ( ( field_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R ) )
      = ( sup_sup_set_v @ ( insert_v2 @ A3 @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) @ ( field_v @ R ) ) ) ).

% Field_insert
thf(fact_784_remove__def,axiom,
    ( remove5001965847480235980od_v_v
    = ( ^ [X3: product_prod_v_v,A4: set_Product_prod_v_v] : ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% remove_def
thf(fact_785_remove__def,axiom,
    ( remove_v
    = ( ^ [X3: v,A4: set_v] : ( minus_minus_set_v @ A4 @ ( insert_v2 @ X3 @ bot_bot_set_v ) ) ) ) ).

% remove_def
thf(fact_786_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( ( ( minus_7679383599658060814od_v_v @ A @ ( insert7504383016908236695od_v_v @ X2 @ bot_bo3497076220358800403od_v_v ) )
            = bot_bo3497076220358800403od_v_v )
         => ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X2 @ A ) )
            = X2 ) )
        & ( ( ( minus_7679383599658060814od_v_v @ A @ ( insert7504383016908236695od_v_v @ X2 @ bot_bo3497076220358800403od_v_v ) )
           != bot_bo3497076220358800403od_v_v )
         => ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X2 @ A ) )
            = ( sup_su414716646722978715od_v_v @ X2 @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A @ ( insert7504383016908236695od_v_v @ X2 @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_787_member__remove,axiom,
    ! [X2: v,Y2: v,A: set_v] :
      ( ( member_v @ X2 @ ( remove_v @ Y2 @ A ) )
      = ( ( member_v @ X2 @ A )
        & ( X2 != Y2 ) ) ) ).

% member_remove
thf(fact_788_member__remove,axiom,
    ! [X2: product_prod_v_v,Y2: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ X2 @ ( remove5001965847480235980od_v_v @ Y2 @ A ) )
      = ( ( member7453568604450474000od_v_v @ X2 @ A )
        & ( X2 != Y2 ) ) ) ).

% member_remove
thf(fact_789_Field__empty,axiom,
    ( ( field_7153129647634986036od_v_v @ bot_bo3282589961317712691od_v_v )
    = bot_bo723834152578015283od_v_v ) ).

% Field_empty
thf(fact_790_Field__empty,axiom,
    ( ( field_v @ bot_bo723834152578015283od_v_v )
    = bot_bot_set_v ) ).

% Field_empty
thf(fact_791_Field__Un,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,S4: set_Pr2149350503807050951od_v_v] :
      ( ( field_7153129647634986036od_v_v @ ( sup_su1742609618068805275od_v_v @ R @ S4 ) )
      = ( sup_su414716646722978715od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ ( field_7153129647634986036od_v_v @ S4 ) ) ) ).

% Field_Un
thf(fact_792_Field__Un,axiom,
    ! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
      ( ( field_v @ ( sup_su414716646722978715od_v_v @ R @ S4 ) )
      = ( sup_sup_set_v @ ( field_v @ R ) @ ( field_v @ S4 ) ) ) ).

% Field_Un
thf(fact_793_inf__Sup__absorb,axiom,
    ! [A: set_set_v,A3: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( member_set_v @ A3 @ A )
       => ( ( inf_inf_set_v @ A3 @ ( lattic2918178447194608042_set_v @ A ) )
          = A3 ) ) ) ).

% inf_Sup_absorb
thf(fact_794_Sup__fin_Oinsert,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X2 @ A ) )
          = ( sup_su414716646722978715od_v_v @ X2 @ ( lattic5151207300795964030od_v_v @ A ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_795_finite__Field,axiom,
    ! [R: set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ R )
     => ( finite_finite_v @ ( field_v @ R ) ) ) ).

% finite_Field
thf(fact_796_FieldI2,axiom,
    ! [I: product_prod_v_v,J: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
      ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R3 )
     => ( member7453568604450474000od_v_v @ J @ ( field_7153129647634986036od_v_v @ R3 ) ) ) ).

% FieldI2
thf(fact_797_FieldI2,axiom,
    ! [I: v,J: v,R3: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R3 )
     => ( member_v @ J @ ( field_v @ R3 ) ) ) ).

% FieldI2
thf(fact_798_FieldI1,axiom,
    ! [I: product_prod_v_v,J: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
      ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R3 )
     => ( member7453568604450474000od_v_v @ I @ ( field_7153129647634986036od_v_v @ R3 ) ) ) ).

% FieldI1
thf(fact_799_FieldI1,axiom,
    ! [I: v,J: v,R3: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R3 )
     => ( member_v @ I @ ( field_v @ R3 ) ) ) ).

% FieldI1
thf(fact_800_mono__Field,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,S4: set_Pr2149350503807050951od_v_v] :
      ( ( ord_le6241436655786843239od_v_v @ R @ S4 )
     => ( ord_le7336532860387713383od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ ( field_7153129647634986036od_v_v @ S4 ) ) ) ).

% mono_Field
thf(fact_801_mono__Field,axiom,
    ! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ R @ S4 )
     => ( ord_less_eq_set_v @ ( field_v @ R ) @ ( field_v @ S4 ) ) ) ).

% mono_Field
thf(fact_802_Sup__fin_OcoboundedI,axiom,
    ! [A: set_se8455005133513928103od_v_v,A3: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( member8406446414694345712od_v_v @ A3 @ A )
       => ( ord_le7336532860387713383od_v_v @ A3 @ ( lattic5151207300795964030od_v_v @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_803_Sup__fin_OcoboundedI,axiom,
    ! [A: set_set_v,A3: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( member_set_v @ A3 @ A )
       => ( ord_less_eq_set_v @ A3 @ ( lattic2918178447194608042_set_v @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_804_Sup__fin_Oin__idem,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( member8406446414694345712od_v_v @ X2 @ A )
       => ( ( sup_su414716646722978715od_v_v @ X2 @ ( lattic5151207300795964030od_v_v @ A ) )
          = ( lattic5151207300795964030od_v_v @ A ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_805_Sup__fin_OboundedE,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A ) @ X2 )
         => ! [A10: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ A10 @ A )
             => ( ord_le7336532860387713383od_v_v @ A10 @ X2 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_806_Sup__fin_OboundedE,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A ) @ X2 )
         => ! [A10: set_v] :
              ( ( member_set_v @ A10 @ A )
             => ( ord_less_eq_set_v @ A10 @ X2 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_807_Sup__fin_OboundedI,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ! [A7: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ A7 @ A )
             => ( ord_le7336532860387713383od_v_v @ A7 @ X2 ) )
         => ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A ) @ X2 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_808_Sup__fin_OboundedI,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ( ! [A7: set_v] :
              ( ( member_set_v @ A7 @ A )
             => ( ord_less_eq_set_v @ A7 @ X2 ) )
         => ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A ) @ X2 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_809_Sup__fin_Obounded__iff,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A ) @ X2 )
          = ( ! [X3: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ X3 @ A )
               => ( ord_le7336532860387713383od_v_v @ X3 @ X2 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_810_Sup__fin_Obounded__iff,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A ) @ X2 )
          = ( ! [X3: set_v] :
                ( ( member_set_v @ X3 @ A )
               => ( ord_less_eq_set_v @ X3 @ X2 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_811_Sup__fin_Osubset__imp,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( ord_le4714265922333009223od_v_v @ A @ B )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ( finite6084192165098772208od_v_v @ B )
         => ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A ) @ ( lattic5151207300795964030od_v_v @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_812_Sup__fin_Osubset__imp,axiom,
    ! [A: set_set_v,B: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A @ B )
     => ( ( A != bot_bot_set_set_v )
       => ( ( finite_finite_set_v @ B )
         => ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A ) @ ( lattic2918178447194608042_set_v @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_813_Sup__fin_Osubset,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( B != bot_bo3497076220358800403od_v_v )
       => ( ( ord_le4714265922333009223od_v_v @ B @ A )
         => ( ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ B ) @ ( lattic5151207300795964030od_v_v @ A ) )
            = ( lattic5151207300795964030od_v_v @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_814_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ~ ( member8406446414694345712od_v_v @ X2 @ A )
       => ( ( A != bot_bo3497076220358800403od_v_v )
         => ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X2 @ A ) )
            = ( sup_su414716646722978715od_v_v @ X2 @ ( lattic5151207300795964030od_v_v @ A ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_815_Sup__fin_Oclosed,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) @ ( insert7504383016908236695od_v_v @ X4 @ ( insert7504383016908236695od_v_v @ Y @ bot_bo3497076220358800403od_v_v ) ) )
         => ( member8406446414694345712od_v_v @ ( lattic5151207300795964030od_v_v @ A ) @ A ) ) ) ) ).

% Sup_fin.closed
thf(fact_816_Sup__fin_Ounion,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ( finite6084192165098772208od_v_v @ B )
         => ( ( B != bot_bo3497076220358800403od_v_v )
           => ( ( lattic5151207300795964030od_v_v @ ( sup_su335656005089752955od_v_v @ A @ B ) )
              = ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ A ) @ ( lattic5151207300795964030od_v_v @ B ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_817_Sup__fin_Oremove,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( member8406446414694345712od_v_v @ X2 @ A )
       => ( ( ( ( minus_7679383599658060814od_v_v @ A @ ( insert7504383016908236695od_v_v @ X2 @ bot_bo3497076220358800403od_v_v ) )
              = bot_bo3497076220358800403od_v_v )
           => ( ( lattic5151207300795964030od_v_v @ A )
              = X2 ) )
          & ( ( ( minus_7679383599658060814od_v_v @ A @ ( insert7504383016908236695od_v_v @ X2 @ bot_bo3497076220358800403od_v_v ) )
             != bot_bo3497076220358800403od_v_v )
           => ( ( lattic5151207300795964030od_v_v @ A )
              = ( sup_su414716646722978715od_v_v @ X2 @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A @ ( insert7504383016908236695od_v_v @ X2 @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_818_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
            = bot_bot_set_set_v )
         => ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X2 @ A ) )
            = X2 ) )
        & ( ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
           != bot_bot_set_set_v )
         => ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X2 @ A ) )
            = ( inf_inf_set_v @ X2 @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_819_Inf__fin_Oremove,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( member_set_v @ X2 @ A )
       => ( ( ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
              = bot_bot_set_set_v )
           => ( ( lattic8209813555532694032_set_v @ A )
              = X2 ) )
          & ( ( ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) )
             != bot_bot_set_set_v )
           => ( ( lattic8209813555532694032_set_v @ A )
              = ( inf_inf_set_v @ X2 @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_820_refl__on__singleton,axiom,
    ! [X2: product_prod_v_v] : ( refl_o4548774019903118566od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ X2 ) @ bot_bo3282589961317712691od_v_v ) ) ).

% refl_on_singleton
thf(fact_821_refl__on__singleton,axiom,
    ! [X2: v] : ( refl_on_v @ ( insert_v2 @ X2 @ bot_bot_set_v ) @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ X2 @ X2 ) @ bot_bo723834152578015283od_v_v ) ) ).

% refl_on_singleton
thf(fact_822_Inf__fin_Oinsert,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X2 @ A ) )
          = ( inf_inf_set_v @ X2 @ ( lattic8209813555532694032_set_v @ A ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_823_sup__Inf__absorb,axiom,
    ! [A: set_se8455005133513928103od_v_v,A3: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( member8406446414694345712od_v_v @ A3 @ A )
       => ( ( sup_su414716646722978715od_v_v @ ( lattic4767070952889939172od_v_v @ A ) @ A3 )
          = A3 ) ) ) ).

% sup_Inf_absorb
thf(fact_824_refl__onD,axiom,
    ! [A: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v] :
      ( ( refl_o4548774019903118566od_v_v @ A @ R )
     => ( ( member7453568604450474000od_v_v @ A3 @ A )
       => ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ A3 ) @ R ) ) ) ).

% refl_onD
thf(fact_825_refl__onD,axiom,
    ! [A: set_v,R: set_Product_prod_v_v,A3: v] :
      ( ( refl_on_v @ A @ R )
     => ( ( member_v @ A3 @ A )
       => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ A3 ) @ R ) ) ) ).

% refl_onD
thf(fact_826_refl__onD1,axiom,
    ! [A: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,X2: product_prod_v_v,Y2: product_prod_v_v] :
      ( ( refl_o4548774019903118566od_v_v @ A @ R )
     => ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ Y2 ) @ R )
       => ( member7453568604450474000od_v_v @ X2 @ A ) ) ) ).

% refl_onD1
thf(fact_827_refl__onD1,axiom,
    ! [A: set_v,R: set_Product_prod_v_v,X2: v,Y2: v] :
      ( ( refl_on_v @ A @ R )
     => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y2 ) @ R )
       => ( member_v @ X2 @ A ) ) ) ).

% refl_onD1
thf(fact_828_refl__onD2,axiom,
    ! [A: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,X2: product_prod_v_v,Y2: product_prod_v_v] :
      ( ( refl_o4548774019903118566od_v_v @ A @ R )
     => ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ Y2 ) @ R )
       => ( member7453568604450474000od_v_v @ Y2 @ A ) ) ) ).

% refl_onD2
thf(fact_829_refl__onD2,axiom,
    ! [A: set_v,R: set_Product_prod_v_v,X2: v,Y2: v] :
      ( ( refl_on_v @ A @ R )
     => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y2 ) @ R )
       => ( member_v @ Y2 @ A ) ) ) ).

% refl_onD2
thf(fact_830_refl__on__Int,axiom,
    ! [A: set_v,R: set_Product_prod_v_v,B: set_v,S4: set_Product_prod_v_v] :
      ( ( refl_on_v @ A @ R )
     => ( ( refl_on_v @ B @ S4 )
       => ( refl_on_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ R @ S4 ) ) ) ) ).

% refl_on_Int
thf(fact_831_Inf__fin_OcoboundedI,axiom,
    ! [A: set_se8455005133513928103od_v_v,A3: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( member8406446414694345712od_v_v @ A3 @ A )
       => ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ A ) @ A3 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_832_Inf__fin_OcoboundedI,axiom,
    ! [A: set_set_v,A3: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( member_set_v @ A3 @ A )
       => ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ A ) @ A3 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_833_refl__on__empty,axiom,
    refl_o4548774019903118566od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).

% refl_on_empty
thf(fact_834_refl__on__empty,axiom,
    refl_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).

% refl_on_empty
thf(fact_835_Inf__fin_Oin__idem,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( member_set_v @ X2 @ A )
       => ( ( inf_inf_set_v @ X2 @ ( lattic8209813555532694032_set_v @ A ) )
          = ( lattic8209813555532694032_set_v @ A ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_836_refl__on__Un,axiom,
    ! [A: set_v,R: set_Product_prod_v_v,B: set_v,S4: set_Product_prod_v_v] :
      ( ( refl_on_v @ A @ R )
     => ( ( refl_on_v @ B @ S4 )
       => ( refl_on_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ R @ S4 ) ) ) ) ).

% refl_on_Un
thf(fact_837_refl__on__Un,axiom,
    ! [A: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v,S4: set_Pr2149350503807050951od_v_v] :
      ( ( refl_o4548774019903118566od_v_v @ A @ R )
     => ( ( refl_o4548774019903118566od_v_v @ B @ S4 )
       => ( refl_o4548774019903118566od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su1742609618068805275od_v_v @ R @ S4 ) ) ) ) ).

% refl_on_Un
thf(fact_838_Inf__fin_Obounded__iff,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ( ord_le7336532860387713383od_v_v @ X2 @ ( lattic4767070952889939172od_v_v @ A ) )
          = ( ! [X3: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ X3 @ A )
               => ( ord_le7336532860387713383od_v_v @ X2 @ X3 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_839_Inf__fin_Obounded__iff,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ( ( ord_less_eq_set_v @ X2 @ ( lattic8209813555532694032_set_v @ A ) )
          = ( ! [X3: set_v] :
                ( ( member_set_v @ X3 @ A )
               => ( ord_less_eq_set_v @ X2 @ X3 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_840_Inf__fin_OboundedI,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ! [A7: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ A7 @ A )
             => ( ord_le7336532860387713383od_v_v @ X2 @ A7 ) )
         => ( ord_le7336532860387713383od_v_v @ X2 @ ( lattic4767070952889939172od_v_v @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_841_Inf__fin_OboundedI,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ( ! [A7: set_v] :
              ( ( member_set_v @ A7 @ A )
             => ( ord_less_eq_set_v @ X2 @ A7 ) )
         => ( ord_less_eq_set_v @ X2 @ ( lattic8209813555532694032_set_v @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_842_Inf__fin_OboundedE,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ( ord_le7336532860387713383od_v_v @ X2 @ ( lattic4767070952889939172od_v_v @ A ) )
         => ! [A10: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ A10 @ A )
             => ( ord_le7336532860387713383od_v_v @ X2 @ A10 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_843_Inf__fin_OboundedE,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ( ( ord_less_eq_set_v @ X2 @ ( lattic8209813555532694032_set_v @ A ) )
         => ! [A10: set_v] :
              ( ( member_set_v @ A10 @ A )
             => ( ord_less_eq_set_v @ X2 @ A10 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_844_Inf__fin_Osubset__imp,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( ord_le4714265922333009223od_v_v @ A @ B )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ( finite6084192165098772208od_v_v @ B )
         => ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ B ) @ ( lattic4767070952889939172od_v_v @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_845_Inf__fin_Osubset__imp,axiom,
    ! [A: set_set_v,B: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A @ B )
     => ( ( A != bot_bot_set_set_v )
       => ( ( finite_finite_set_v @ B )
         => ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ B ) @ ( lattic8209813555532694032_set_v @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_846_Inf__fin_Osubset,axiom,
    ! [A: set_set_v,B: set_set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( B != bot_bot_set_set_v )
       => ( ( ord_le5216385588623774835_set_v @ B @ A )
         => ( ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ B ) @ ( lattic8209813555532694032_set_v @ A ) )
            = ( lattic8209813555532694032_set_v @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_847_Inf__fin_Oclosed,axiom,
    ! [A: set_set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ( ! [X4: set_v,Y: set_v] : ( member_set_v @ ( inf_inf_set_v @ X4 @ Y ) @ ( insert_set_v @ X4 @ ( insert_set_v @ Y @ bot_bot_set_set_v ) ) )
         => ( member_set_v @ ( lattic8209813555532694032_set_v @ A ) @ A ) ) ) ) ).

% Inf_fin.closed
thf(fact_848_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ~ ( member_set_v @ X2 @ A )
       => ( ( A != bot_bot_set_set_v )
         => ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X2 @ A ) )
            = ( inf_inf_set_v @ X2 @ ( lattic8209813555532694032_set_v @ A ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_849_Inf__fin_Ounion,axiom,
    ! [A: set_set_v,B: set_set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ( ( finite_finite_set_v @ B )
         => ( ( B != bot_bot_set_set_v )
           => ( ( lattic8209813555532694032_set_v @ ( sup_sup_set_set_v @ A @ B ) )
              = ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ A ) @ ( lattic8209813555532694032_set_v @ B ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_850_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ A ) @ ( lattic5151207300795964030od_v_v @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_851_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ A ) @ ( lattic2918178447194608042_set_v @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_852_refl__on__domain,axiom,
    ! [A: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v,B2: product_prod_v_v] :
      ( ( refl_o4548774019903118566od_v_v @ A @ R )
     => ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ B2 ) @ R )
       => ( ( member7453568604450474000od_v_v @ A3 @ A )
          & ( member7453568604450474000od_v_v @ B2 @ A ) ) ) ) ).

% refl_on_domain
thf(fact_853_refl__on__domain,axiom,
    ! [A: set_v,R: set_Product_prod_v_v,A3: v,B2: v] :
      ( ( refl_on_v @ A @ R )
     => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R )
       => ( ( member_v @ A3 @ A )
          & ( member_v @ B2 @ A ) ) ) ) ).

% refl_on_domain
thf(fact_854_Set_Ois__empty__def,axiom,
    ( is_emp8964507351669718201od_v_v
    = ( ^ [A4: set_Product_prod_v_v] : ( A4 = bot_bo723834152578015283od_v_v ) ) ) ).

% Set.is_empty_def
thf(fact_855_Set_Ois__empty__def,axiom,
    ( is_empty_v
    = ( ^ [A4: set_v] : ( A4 = bot_bot_set_v ) ) ) ).

% Set.is_empty_def
thf(fact_856_linear__order__on__singleton,axiom,
    ! [X2: product_prod_v_v] : ( order_6462556390437124636od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ X2 ) @ bot_bo3282589961317712691od_v_v ) ) ).

% linear_order_on_singleton
thf(fact_857_linear__order__on__singleton,axiom,
    ! [X2: v] : ( order_8768733634509060168r_on_v @ ( insert_v2 @ X2 @ bot_bot_set_v ) @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ X2 @ X2 ) @ bot_bo723834152578015283od_v_v ) ) ).

% linear_order_on_singleton
thf(fact_858_lnear__order__on__empty,axiom,
    order_6462556390437124636od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).

% lnear_order_on_empty
thf(fact_859_lnear__order__on__empty,axiom,
    order_8768733634509060168r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).

% lnear_order_on_empty
thf(fact_860_finite__Linear__order__induct,axiom,
    ! [R: set_Product_prod_v_v,X2: v,P: v > $o] :
      ( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
     => ( ( member_v @ X2 @ ( field_v @ R ) )
       => ( ( finite3348123685078250256od_v_v @ R )
         => ( ! [X4: v] :
                ( ( member_v @ X4 @ ( field_v @ R ) )
               => ( ! [Y5: v] :
                      ( ( member_v @ Y5 @ ( order_aboveS_v @ R @ X4 ) )
                     => ( P @ Y5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% finite_Linear_order_induct
thf(fact_861_finite__Linear__order__induct,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,X2: product_prod_v_v,P: product_prod_v_v > $o] :
      ( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
     => ( ( member7453568604450474000od_v_v @ X2 @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( finite5952053201251911184od_v_v @ R )
         => ( ! [X4: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X4 @ ( field_7153129647634986036od_v_v @ R ) )
               => ( ! [Y5: product_prod_v_v] :
                      ( ( member7453568604450474000od_v_v @ Y5 @ ( order_1156346741491923410od_v_v @ R @ X4 ) )
                     => ( P @ Y5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% finite_Linear_order_induct
thf(fact_862_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr2149350503807050951od_v_v] :
      ( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
     => ( ( order_7541072052284126853od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
        = ( ! [A4: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ A4 @ ( field_7153129647634986036od_v_v @ R ) )
             => ( ( A4 != bot_bo723834152578015283od_v_v )
               => ? [X3: product_prod_v_v] :
                    ( ( member7453568604450474000od_v_v @ X3 @ A4 )
                    & ! [Y3: product_prod_v_v] :
                        ( ( member7453568604450474000od_v_v @ Y3 @ A4 )
                       => ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_863_Linear__order__Well__order__iff,axiom,
    ! [R: set_Product_prod_v_v] :
      ( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
     => ( ( order_6972113574731384241r_on_v @ ( field_v @ R ) @ R )
        = ( ! [A4: set_v] :
              ( ( ord_less_eq_set_v @ A4 @ ( field_v @ R ) )
             => ( ( A4 != bot_bot_set_v )
               => ? [X3: v] :
                    ( ( member_v @ X3 @ A4 )
                    & ! [Y3: v] :
                        ( ( member_v @ Y3 @ A4 )
                       => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_864_underS__incl__iff,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v,B2: product_prod_v_v] :
      ( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
     => ( ( member7453568604450474000od_v_v @ A3 @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
         => ( ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A3 ) @ ( order_5211820470575790509od_v_v @ R @ B2 ) )
            = ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ B2 ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_865_underS__incl__iff,axiom,
    ! [R: set_Product_prod_v_v,A3: v,B2: v] :
      ( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
     => ( ( member_v @ A3 @ ( field_v @ R ) )
       => ( ( member_v @ B2 @ ( field_v @ R ) )
         => ( ( ord_less_eq_set_v @ ( order_underS_v @ R @ A3 ) @ ( order_underS_v @ R @ B2 ) )
            = ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_866_Range__insert,axiom,
    ! [A3: v,B2: v,R: set_Product_prod_v_v] :
      ( ( range_v_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R ) )
      = ( insert_v2 @ B2 @ ( range_v_v @ R ) ) ) ).

% Range_insert
thf(fact_867_Range__empty,axiom,
    ( ( range_v_v @ bot_bo723834152578015283od_v_v )
    = bot_bot_set_v ) ).

% Range_empty
thf(fact_868_well__order__on__domain,axiom,
    ! [A: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v,B2: product_prod_v_v] :
      ( ( order_7541072052284126853od_v_v @ A @ R )
     => ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ B2 ) @ R )
       => ( ( member7453568604450474000od_v_v @ A3 @ A )
          & ( member7453568604450474000od_v_v @ B2 @ A ) ) ) ) ).

% well_order_on_domain
thf(fact_869_well__order__on__domain,axiom,
    ! [A: set_v,R: set_Product_prod_v_v,A3: v,B2: v] :
      ( ( order_6972113574731384241r_on_v @ A @ R )
     => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R )
       => ( ( member_v @ A3 @ A )
          & ( member_v @ B2 @ A ) ) ) ) ).

% well_order_on_domain
thf(fact_870_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: v,R3: set_Product_prod_v_v,J: v] :
      ( ( member_v @ I @ ( order_underS_v @ R3 @ J ) )
     => ( member_v @ I @ ( field_v @ R3 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_871_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v,J: product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ I @ ( order_5211820470575790509od_v_v @ R3 @ J ) )
     => ( member7453568604450474000od_v_v @ I @ ( field_7153129647634986036od_v_v @ R3 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_872_underS__E,axiom,
    ! [I: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v,J: product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ I @ ( order_5211820470575790509od_v_v @ R3 @ J ) )
     => ( ( I != J )
        & ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R3 ) ) ) ).

% underS_E
thf(fact_873_underS__E,axiom,
    ! [I: v,R3: set_Product_prod_v_v,J: v] :
      ( ( member_v @ I @ ( order_underS_v @ R3 @ J ) )
     => ( ( I != J )
        & ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R3 ) ) ) ).

% underS_E
thf(fact_874_underS__I,axiom,
    ! [I: product_prod_v_v,J: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
      ( ( I != J )
     => ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R3 )
       => ( member7453568604450474000od_v_v @ I @ ( order_5211820470575790509od_v_v @ R3 @ J ) ) ) ) ).

% underS_I
thf(fact_875_underS__I,axiom,
    ! [I: v,J: v,R3: set_Product_prod_v_v] :
      ( ( I != J )
     => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R3 )
       => ( member_v @ I @ ( order_underS_v @ R3 @ J ) ) ) ) ).

% underS_I
thf(fact_876_Range_Ocases,axiom,
    ! [A3: v,R: set_Product_prod_v_v] :
      ( ( member_v @ A3 @ ( range_v_v @ R ) )
     => ~ ! [A7: v] :
            ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ A3 ) @ R ) ) ).

% Range.cases
thf(fact_877_Range_Osimps,axiom,
    ! [A3: v,R: set_Product_prod_v_v] :
      ( ( member_v @ A3 @ ( range_v_v @ R ) )
      = ( ? [A5: v,B5: v] :
            ( ( A3 = B5 )
            & ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A5 @ B5 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_878_Range_Ointros,axiom,
    ! [A3: v,B2: v,R: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R )
     => ( member_v @ B2 @ ( range_v_v @ R ) ) ) ).

% Range.intros
thf(fact_879_RangeE,axiom,
    ! [B2: v,R: set_Product_prod_v_v] :
      ( ( member_v @ B2 @ ( range_v_v @ R ) )
     => ~ ! [A7: v] :
            ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ B2 ) @ R ) ) ).

% RangeE
thf(fact_880_Range__iff,axiom,
    ! [A3: v,R: set_Product_prod_v_v] :
      ( ( member_v @ A3 @ ( range_v_v @ R ) )
      = ( ? [Y3: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ A3 ) @ R ) ) ) ).

% Range_iff
thf(fact_881_well__order__on__empty,axiom,
    order_7541072052284126853od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).

% well_order_on_empty
thf(fact_882_well__order__on__empty,axiom,
    order_6972113574731384241r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).

% well_order_on_empty
thf(fact_883_underS__empty,axiom,
    ! [A3: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ A3 @ ( field_7153129647634986036od_v_v @ R ) )
     => ( ( order_5211820470575790509od_v_v @ R @ A3 )
        = bot_bo723834152578015283od_v_v ) ) ).

% underS_empty
thf(fact_884_underS__empty,axiom,
    ! [A3: v,R: set_Product_prod_v_v] :
      ( ~ ( member_v @ A3 @ ( field_v @ R ) )
     => ( ( order_underS_v @ R @ A3 )
        = bot_bot_set_v ) ) ).

% underS_empty
thf(fact_885_Order__Relation_OunderS__Field,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A3 ) @ ( field_7153129647634986036od_v_v @ R ) ) ).

% Order_Relation.underS_Field
thf(fact_886_Order__Relation_OunderS__Field,axiom,
    ! [R: set_Product_prod_v_v,A3: v] : ( ord_less_eq_set_v @ ( order_underS_v @ R @ A3 ) @ ( field_v @ R ) ) ).

% Order_Relation.underS_Field
thf(fact_887_Range__mono,axiom,
    ! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ R @ S4 )
     => ( ord_less_eq_set_v @ ( range_v_v @ R ) @ ( range_v_v @ S4 ) ) ) ).

% Range_mono
thf(fact_888_Range__empty__iff,axiom,
    ! [R: set_Product_prod_v_v] :
      ( ( ( range_v_v @ R )
        = bot_bot_set_v )
      = ( R = bot_bo723834152578015283od_v_v ) ) ).

% Range_empty_iff
thf(fact_889_Range__Un__eq,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( range_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) )
      = ( sup_sup_set_v @ ( range_v_v @ A ) @ ( range_v_v @ B ) ) ) ).

% Range_Un_eq
thf(fact_890_Refl__under__underS,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v] :
      ( ( refl_o4548774019903118566od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
     => ( ( member7453568604450474000od_v_v @ A3 @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( order_6892855479609198156od_v_v @ R @ A3 )
          = ( sup_su414716646722978715od_v_v @ ( order_5211820470575790509od_v_v @ R @ A3 ) @ ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).

% Refl_under_underS
thf(fact_891_Refl__under__underS,axiom,
    ! [R: set_Product_prod_v_v,A3: v] :
      ( ( refl_on_v @ ( field_v @ R ) @ R )
     => ( ( member_v @ A3 @ ( field_v @ R ) )
       => ( ( order_under_v @ R @ A3 )
          = ( sup_sup_set_v @ ( order_underS_v @ R @ A3 ) @ ( insert_v2 @ A3 @ bot_bot_set_v ) ) ) ) ) ).

% Refl_under_underS
thf(fact_892_finite__Partial__order__induct,axiom,
    ! [R: set_Product_prod_v_v,X2: v,P: v > $o] :
      ( ( order_5272072345360262664r_on_v @ ( field_v @ R ) @ R )
     => ( ( member_v @ X2 @ ( field_v @ R ) )
       => ( ( finite3348123685078250256od_v_v @ R )
         => ( ! [X4: v] :
                ( ( member_v @ X4 @ ( field_v @ R ) )
               => ( ! [Y5: v] :
                      ( ( member_v @ Y5 @ ( order_aboveS_v @ R @ X4 ) )
                     => ( P @ Y5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% finite_Partial_order_induct
thf(fact_893_finite__Partial__order__induct,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,X2: product_prod_v_v,P: product_prod_v_v > $o] :
      ( ( order_4212533993404950492od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
     => ( ( member7453568604450474000od_v_v @ X2 @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( finite5952053201251911184od_v_v @ R )
         => ( ! [X4: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X4 @ ( field_7153129647634986036od_v_v @ R ) )
               => ( ! [Y5: product_prod_v_v] :
                      ( ( member7453568604450474000od_v_v @ Y5 @ ( order_1156346741491923410od_v_v @ R @ X4 ) )
                     => ( P @ Y5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% finite_Partial_order_induct
thf(fact_894_Domain__insert,axiom,
    ! [A3: v,B2: v,R: set_Product_prod_v_v] :
      ( ( domain_v_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R ) )
      = ( insert_v2 @ A3 @ ( domain_v_v @ R ) ) ) ).

% Domain_insert
thf(fact_895_stack__class,axiom,
    ! [E: sCC_Bl1191828773336950226xt_v_a,N: v,M: v] :
      ( ( sCC_Bl4124178362578471481nv_v_a @ successors @ E )
     => ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ E ) ) )
       => ( ( member_v @ M @ ( sCC_Bloemen_S_v_a @ E @ N ) )
         => ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl1198488560823802982ed_v_a @ E ) @ ( sCC_Bl6885986953353844043ed_v_a @ E ) ) ) ) ) ) ).

% stack_class
thf(fact_896_stack__visited,axiom,
    ! [E: sCC_Bl1191828773336950226xt_v_a,N: v] :
      ( ( sCC_Bl4124178362578471481nv_v_a @ successors @ E )
     => ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ E ) ) )
       => ( member_v @ N @ ( sCC_Bl1198488560823802982ed_v_a @ E ) ) ) ) ).

% stack_visited
thf(fact_897_stack__unexplored,axiom,
    ! [E: sCC_Bl1191828773336950226xt_v_a,N: v] :
      ( ( sCC_Bl4124178362578471481nv_v_a @ successors @ E )
     => ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ E ) ) )
       => ~ ( member_v @ N @ ( sCC_Bl6885986953353844043ed_v_a @ E ) ) ) ) ).

% stack_unexplored
thf(fact_898_visited__unexplored,axiom,
    ! [E: sCC_Bl1191828773336950226xt_v_a,M: v] :
      ( ( sCC_Bl4124178362578471481nv_v_a @ successors @ E )
     => ( ( member_v @ M @ ( sCC_Bl1198488560823802982ed_v_a @ E ) )
       => ( ~ ( member_v @ M @ ( sCC_Bl6885986953353844043ed_v_a @ E ) )
         => ~ ! [N2: v] :
                ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ E ) ) )
               => ~ ( member_v @ M @ ( sCC_Bloemen_S_v_a @ E @ N2 ) ) ) ) ) ) ).

% visited_unexplored
thf(fact_899_Domain__empty,axiom,
    ( ( domain_v_v @ bot_bo723834152578015283od_v_v )
    = bot_bot_set_v ) ).

% Domain_empty
thf(fact_900_Domain__iff,axiom,
    ! [A3: v,R: set_Product_prod_v_v] :
      ( ( member_v @ A3 @ ( domain_v_v @ R ) )
      = ( ? [Y3: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ Y3 ) @ R ) ) ) ).

% Domain_iff
thf(fact_901_DomainE,axiom,
    ! [A3: v,R: set_Product_prod_v_v] :
      ( ( member_v @ A3 @ ( domain_v_v @ R ) )
     => ~ ! [B7: v] :
            ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B7 ) @ R ) ) ).

% DomainE
thf(fact_902_Domain_ODomainI,axiom,
    ! [A3: v,B2: v,R: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R )
     => ( member_v @ A3 @ ( domain_v_v @ R ) ) ) ).

% Domain.DomainI
thf(fact_903_Domain_Osimps,axiom,
    ! [A3: v,R: set_Product_prod_v_v] :
      ( ( member_v @ A3 @ ( domain_v_v @ R ) )
      = ( ? [A5: v,B5: v] :
            ( ( A3 = A5 )
            & ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A5 @ B5 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_904_Domain_Ocases,axiom,
    ! [A3: v,R: set_Product_prod_v_v] :
      ( ( member_v @ A3 @ ( domain_v_v @ R ) )
     => ~ ! [B7: v] :
            ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B7 ) @ R ) ) ).

% Domain.cases
thf(fact_905_graph_Ostack__unexplored,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1191828773336950226xt_v_a,N: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4124178362578471481nv_v_a @ Successors @ E )
       => ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ E ) ) )
         => ~ ( member_v @ N @ ( sCC_Bl6885986953353844043ed_v_a @ E ) ) ) ) ) ).

% graph.stack_unexplored
thf(fact_906_graph_Ovisited__unexplored,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1191828773336950226xt_v_a,M: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4124178362578471481nv_v_a @ Successors @ E )
       => ( ( member_v @ M @ ( sCC_Bl1198488560823802982ed_v_a @ E ) )
         => ( ~ ( member_v @ M @ ( sCC_Bl6885986953353844043ed_v_a @ E ) )
           => ~ ! [N2: v] :
                  ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ E ) ) )
                 => ~ ( member_v @ M @ ( sCC_Bloemen_S_v_a @ E @ N2 ) ) ) ) ) ) ) ).

% graph.visited_unexplored
thf(fact_907_graph_Ostack__visited,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1191828773336950226xt_v_a,N: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4124178362578471481nv_v_a @ Successors @ E )
       => ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ E ) ) )
         => ( member_v @ N @ ( sCC_Bl1198488560823802982ed_v_a @ E ) ) ) ) ) ).

% graph.stack_visited
thf(fact_908_Domain__mono,axiom,
    ! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ R @ S4 )
     => ( ord_less_eq_set_v @ ( domain_v_v @ R ) @ ( domain_v_v @ S4 ) ) ) ).

% Domain_mono
thf(fact_909_Domain__empty__iff,axiom,
    ! [R: set_Product_prod_v_v] :
      ( ( ( domain_v_v @ R )
        = bot_bot_set_v )
      = ( R = bot_bo723834152578015283od_v_v ) ) ).

% Domain_empty_iff
thf(fact_910_graph_Ostack__class,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1191828773336950226xt_v_a,N: v,M: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4124178362578471481nv_v_a @ Successors @ E )
       => ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ E ) ) )
         => ( ( member_v @ M @ ( sCC_Bloemen_S_v_a @ E @ N ) )
           => ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl1198488560823802982ed_v_a @ E ) @ ( sCC_Bl6885986953353844043ed_v_a @ E ) ) ) ) ) ) ) ).

% graph.stack_class
thf(fact_911_Domain__Un__eq,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( domain_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) )
      = ( sup_sup_set_v @ ( domain_v_v @ A ) @ ( domain_v_v @ B ) ) ) ).

% Domain_Un_eq
thf(fact_912_under__Field,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_6892855479609198156od_v_v @ R @ A3 ) @ ( field_7153129647634986036od_v_v @ R ) ) ).

% under_Field
thf(fact_913_under__Field,axiom,
    ! [R: set_Product_prod_v_v,A3: v] : ( ord_less_eq_set_v @ ( order_under_v @ R @ A3 ) @ ( field_v @ R ) ) ).

% under_Field
thf(fact_914_underS__subset__under,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A3 ) @ ( order_6892855479609198156od_v_v @ R @ A3 ) ) ).

% underS_subset_under
thf(fact_915_underS__subset__under,axiom,
    ! [R: set_Product_prod_v_v,A3: v] : ( ord_less_eq_set_v @ ( order_underS_v @ R @ A3 ) @ ( order_under_v @ R @ A3 ) ) ).

% underS_subset_under
thf(fact_916_partial__order__on__empty,axiom,
    order_4212533993404950492od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).

% partial_order_on_empty
thf(fact_917_partial__order__on__empty,axiom,
    order_5272072345360262664r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).

% partial_order_on_empty
thf(fact_918_Field__def,axiom,
    ( field_7153129647634986036od_v_v
    = ( ^ [R4: set_Pr2149350503807050951od_v_v] : ( sup_su414716646722978715od_v_v @ ( domain6359000466948879308od_v_v @ R4 ) @ ( range_7878975032137371189od_v_v @ R4 ) ) ) ) ).

% Field_def
thf(fact_919_List_Ofinite__set,axiom,
    ! [Xs: list_v] : ( finite_finite_v @ ( set_v2 @ Xs ) ) ).

% List.finite_set
thf(fact_920_subset__code_I1_J,axiom,
    ! [Xs: list_P7986770385144383213od_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B )
      = ( ! [X3: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ ( set_Product_prod_v_v2 @ Xs ) )
           => ( member7453568604450474000od_v_v @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_921_subset__code_I1_J,axiom,
    ! [Xs: list_v,B: set_v] :
      ( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B )
      = ( ! [X3: v] :
            ( ( member_v @ X3 @ ( set_v2 @ Xs ) )
           => ( member_v @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_922_finite__list,axiom,
    ! [A: set_v] :
      ( ( finite_finite_v @ A )
     => ? [Xs2: list_v] :
          ( ( set_v2 @ Xs2 )
          = A ) ) ).

% finite_list
thf(fact_923_set__union,axiom,
    ! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
      ( ( set_Product_prod_v_v2 @ ( union_4602324378607836129od_v_v @ Xs @ Ys ) )
      = ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ).

% set_union
thf(fact_924_set__removeAll,axiom,
    ! [X2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
      ( ( set_Product_prod_v_v2 @ ( remove481895986417801203od_v_v @ X2 @ Xs ) )
      = ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ).

% set_removeAll
thf(fact_925_set__removeAll,axiom,
    ! [X2: v,Xs: list_v] :
      ( ( set_v2 @ ( removeAll_v @ X2 @ Xs ) )
      = ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ).

% set_removeAll
thf(fact_926_List_Oset__insert,axiom,
    ! [X2: v,Xs: list_v] :
      ( ( set_v2 @ ( insert_v @ X2 @ Xs ) )
      = ( insert_v2 @ X2 @ ( set_v2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_927_List_Oset__insert,axiom,
    ! [X2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
      ( ( set_Product_prod_v_v2 @ ( insert4539780211034306307od_v_v @ X2 @ Xs ) )
      = ( insert1338601472111419319od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_928_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v,B2: product_prod_v_v] :
      ( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
     => ( ( member7453568604450474000od_v_v @ A3 @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
         => ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ B2 ) @ R )
            = ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ A3 ) @ ( minus_5255927943254941998od_v_v @ R @ id_Product_prod_v_v ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_929_Linear__order__in__diff__Id,axiom,
    ! [R: set_Product_prod_v_v,A3: v,B2: v] :
      ( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
     => ( ( member_v @ A3 @ ( field_v @ R ) )
       => ( ( member_v @ B2 @ ( field_v @ R ) )
         => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R )
            = ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ A3 ) @ ( minus_4183494784930505774od_v_v @ R @ id_v ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_930_Sup__fin_Oeq__fold,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X2 @ A ) )
        = ( finite8952066981541671560od_v_v @ sup_su414716646722978715od_v_v @ X2 @ A ) ) ) ).

% Sup_fin.eq_fold
thf(fact_931_Inf__fin_Oeq__fold,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X2 @ A ) )
        = ( finite338946655151718280_set_v @ inf_inf_set_v @ X2 @ A ) ) ) ).

% Inf_fin.eq_fold
thf(fact_932_IdI,axiom,
    ! [A3: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ A3 ) @ id_v ) ).

% IdI
thf(fact_933_pair__in__Id__conv,axiom,
    ! [A3: v,B2: v] :
      ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ id_v )
      = ( A3 = B2 ) ) ).

% pair_in_Id_conv
thf(fact_934_IdE,axiom,
    ! [P2: product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ P2 @ id_v )
     => ~ ! [X4: v] :
            ( P2
           != ( product_Pair_v_v @ X4 @ X4 ) ) ) ).

% IdE
thf(fact_935_union__fold__insert,axiom,
    ! [A: set_v,B: set_v] :
      ( ( finite_finite_v @ A )
     => ( ( sup_sup_set_v @ A @ B )
        = ( finite_fold_v_set_v @ insert_v2 @ B @ A ) ) ) ).

% union_fold_insert
thf(fact_936_union__fold__insert,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ A )
     => ( ( sup_su414716646722978715od_v_v @ A @ B )
        = ( finite6851115414092367464od_v_v @ insert1338601472111419319od_v_v @ B @ A ) ) ) ).

% union_fold_insert
thf(fact_937_minus__fold__remove,axiom,
    ! [A: set_v,B: set_v] :
      ( ( finite_finite_v @ A )
     => ( ( minus_minus_set_v @ B @ A )
        = ( finite_fold_v_set_v @ remove_v @ B @ A ) ) ) ).

% minus_fold_remove
thf(fact_938_Total__subset__Id,axiom,
    ! [R: set_Product_prod_v_v] :
      ( ( total_on_v @ ( field_v @ R ) @ R )
     => ( ( ord_le7336532860387713383od_v_v @ R @ id_v )
       => ( ( R = bot_bo723834152578015283od_v_v )
          | ? [A7: v] :
              ( R
              = ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A7 @ A7 ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).

% Total_subset_Id
thf(fact_939_IdD,axiom,
    ! [A3: v,B2: v] :
      ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ id_v )
     => ( A3 = B2 ) ) ).

% IdD
thf(fact_940_total__on__def,axiom,
    ( total_on_v
    = ( ^ [A4: set_v,R4: set_Product_prod_v_v] :
        ! [X3: v] :
          ( ( member_v @ X3 @ A4 )
         => ! [Y3: v] :
              ( ( member_v @ Y3 @ A4 )
             => ( ( X3 != Y3 )
               => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y3 ) @ R4 )
                  | ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ X3 ) @ R4 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_941_total__onI,axiom,
    ! [A: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
      ( ! [X4: product_prod_v_v,Y: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X4 @ A )
         => ( ( member7453568604450474000od_v_v @ Y @ A )
           => ( ( X4 != Y )
             => ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X4 @ Y ) @ R )
                | ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ X4 ) @ R ) ) ) ) )
     => ( total_9075964390993782123od_v_v @ A @ R ) ) ).

% total_onI
thf(fact_942_total__onI,axiom,
    ! [A: set_v,R: set_Product_prod_v_v] :
      ( ! [X4: v,Y: v] :
          ( ( member_v @ X4 @ A )
         => ( ( member_v @ Y @ A )
           => ( ( X4 != Y )
             => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X4 @ Y ) @ R )
                | ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ X4 ) @ R ) ) ) ) )
     => ( total_on_v @ A @ R ) ) ).

% total_onI
thf(fact_943_total__on__empty,axiom,
    ! [R: set_Pr2149350503807050951od_v_v] : ( total_9075964390993782123od_v_v @ bot_bo723834152578015283od_v_v @ R ) ).

% total_on_empty
thf(fact_944_total__on__empty,axiom,
    ! [R: set_Product_prod_v_v] : ( total_on_v @ bot_bot_set_v @ R ) ).

% total_on_empty
thf(fact_945_total__on__subset,axiom,
    ! [A: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
      ( ( total_9075964390993782123od_v_v @ A @ R )
     => ( ( ord_le7336532860387713383od_v_v @ B @ A )
       => ( total_9075964390993782123od_v_v @ B @ R ) ) ) ).

% total_on_subset
thf(fact_946_total__on__subset,axiom,
    ! [A: set_v,R: set_Product_prod_v_v,B: set_v] :
      ( ( total_on_v @ A @ R )
     => ( ( ord_less_eq_set_v @ B @ A )
       => ( total_on_v @ B @ R ) ) ) ).

% total_on_subset
thf(fact_947_total__on__singleton,axiom,
    ! [X2: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] : ( total_9075964390993782123od_v_v @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) @ R ) ).

% total_on_singleton
thf(fact_948_total__on__singleton,axiom,
    ! [X2: v,R: set_Product_prod_v_v] : ( total_on_v @ ( insert_v2 @ X2 @ bot_bot_set_v ) @ R ) ).

% total_on_singleton
thf(fact_949_Total__Id__Field,axiom,
    ! [R: set_Product_prod_v_v] :
      ( ( total_on_v @ ( field_v @ R ) @ R )
     => ( ~ ( ord_le7336532860387713383od_v_v @ R @ id_v )
       => ( ( field_v @ R )
          = ( field_v @ ( minus_4183494784930505774od_v_v @ R @ id_v ) ) ) ) ) ).

% Total_Id_Field
thf(fact_950_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr2149350503807050951od_v_v] :
      ( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
     => ( ( wf_Product_prod_v_v @ ( minus_5255927943254941998od_v_v @ R @ id_Product_prod_v_v ) )
        = ( ! [A4: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ A4 @ ( field_7153129647634986036od_v_v @ R ) )
             => ( ( A4 != bot_bo723834152578015283od_v_v )
               => ? [X3: product_prod_v_v] :
                    ( ( member7453568604450474000od_v_v @ X3 @ A4 )
                    & ! [Y3: product_prod_v_v] :
                        ( ( member7453568604450474000od_v_v @ Y3 @ A4 )
                       => ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_951_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Product_prod_v_v] :
      ( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
     => ( ( wf_v @ ( minus_4183494784930505774od_v_v @ R @ id_v ) )
        = ( ! [A4: set_v] :
              ( ( ord_less_eq_set_v @ A4 @ ( field_v @ R ) )
             => ( ( A4 != bot_bot_set_v )
               => ? [X3: v] :
                    ( ( member_v @ X3 @ A4 )
                    & ! [Y3: v] :
                        ( ( member_v @ Y3 @ A4 )
                       => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_952_wf__eq__minimal2,axiom,
    ( wf_Product_prod_v_v
    = ( ^ [R4: set_Pr2149350503807050951od_v_v] :
        ! [A4: set_Product_prod_v_v] :
          ( ( ( ord_le7336532860387713383od_v_v @ A4 @ ( field_7153129647634986036od_v_v @ R4 ) )
            & ( A4 != bot_bo723834152578015283od_v_v ) )
         => ? [X3: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ X3 @ A4 )
              & ! [Y3: product_prod_v_v] :
                  ( ( member7453568604450474000od_v_v @ Y3 @ A4 )
                 => ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ X3 ) @ R4 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_953_wf__eq__minimal2,axiom,
    ( wf_v
    = ( ^ [R4: set_Product_prod_v_v] :
        ! [A4: set_v] :
          ( ( ( ord_less_eq_set_v @ A4 @ ( field_v @ R4 ) )
            & ( A4 != bot_bot_set_v ) )
         => ? [X3: v] :
              ( ( member_v @ X3 @ A4 )
              & ! [Y3: v] :
                  ( ( member_v @ Y3 @ A4 )
                 => ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ X3 ) @ R4 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_954_wf__Un,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,S4: set_Pr2149350503807050951od_v_v] :
      ( ( wf_Product_prod_v_v @ R )
     => ( ( wf_Product_prod_v_v @ S4 )
       => ( ( ( inf_in6271465464967711157od_v_v @ ( domain6359000466948879308od_v_v @ R ) @ ( range_7878975032137371189od_v_v @ S4 ) )
            = bot_bo723834152578015283od_v_v )
         => ( wf_Product_prod_v_v @ ( sup_su1742609618068805275od_v_v @ R @ S4 ) ) ) ) ) ).

% wf_Un
thf(fact_955_wf__Un,axiom,
    ! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
      ( ( wf_v @ R )
     => ( ( wf_v @ S4 )
       => ( ( ( inf_inf_set_v @ ( domain_v_v @ R ) @ ( range_v_v @ S4 ) )
            = bot_bot_set_v )
         => ( wf_v @ ( sup_su414716646722978715od_v_v @ R @ S4 ) ) ) ) ) ).

% wf_Un
thf(fact_956_wf__empty,axiom,
    wf_v @ bot_bo723834152578015283od_v_v ).

% wf_empty
thf(fact_957_wfE__min_H,axiom,
    ! [R3: set_Pr2149350503807050951od_v_v,Q: set_Product_prod_v_v] :
      ( ( wf_Product_prod_v_v @ R3 )
     => ( ( Q != bot_bo723834152578015283od_v_v )
       => ~ ! [Z3: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ Z3 @ Q )
             => ~ ! [Y5: product_prod_v_v] :
                    ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y5 @ Z3 ) @ R3 )
                   => ~ ( member7453568604450474000od_v_v @ Y5 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_958_wfE__min_H,axiom,
    ! [R3: set_Product_prod_v_v,Q: set_v] :
      ( ( wf_v @ R3 )
     => ( ( Q != bot_bot_set_v )
       => ~ ! [Z3: v] :
              ( ( member_v @ Z3 @ Q )
             => ~ ! [Y5: v] :
                    ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ Z3 ) @ R3 )
                   => ~ ( member_v @ Y5 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_959_wf__if__convertible__to__wf,axiom,
    ! [S4: set_Product_prod_v_v,R: set_Product_prod_v_v,F: v > v] :
      ( ( wf_v @ S4 )
     => ( ! [X4: v,Y: v] :
            ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X4 @ Y ) @ R )
           => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ ( F @ X4 ) @ ( F @ Y ) ) @ S4 ) )
       => ( wf_v @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_960_wf__induct__rule,axiom,
    ! [R: set_Product_prod_v_v,P: v > $o,A3: v] :
      ( ( wf_v @ R )
     => ( ! [X4: v] :
            ( ! [Y5: v] :
                ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ X4 ) @ R )
               => ( P @ Y5 ) )
           => ( P @ X4 ) )
       => ( P @ A3 ) ) ) ).

% wf_induct_rule
thf(fact_961_wf__eq__minimal,axiom,
    ( wf_Product_prod_v_v
    = ( ^ [R4: set_Pr2149350503807050951od_v_v] :
        ! [Q2: set_Product_prod_v_v] :
          ( ? [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ Q2 )
         => ? [X3: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ X3 @ Q2 )
              & ! [Y3: product_prod_v_v] :
                  ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ X3 ) @ R4 )
                 => ~ ( member7453568604450474000od_v_v @ Y3 @ Q2 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_962_wf__eq__minimal,axiom,
    ( wf_v
    = ( ^ [R4: set_Product_prod_v_v] :
        ! [Q2: set_v] :
          ( ? [X3: v] : ( member_v @ X3 @ Q2 )
         => ? [X3: v] :
              ( ( member_v @ X3 @ Q2 )
              & ! [Y3: v] :
                  ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ X3 ) @ R4 )
                 => ~ ( member_v @ Y3 @ Q2 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_963_wf__not__refl,axiom,
    ! [R: set_Product_prod_v_v,A3: v] :
      ( ( wf_v @ R )
     => ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ A3 ) @ R ) ) ).

% wf_not_refl
thf(fact_964_wf__not__sym,axiom,
    ! [R: set_Product_prod_v_v,A3: v,X2: v] :
      ( ( wf_v @ R )
     => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ X2 ) @ R )
       => ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ A3 ) @ R ) ) ) ).

% wf_not_sym
thf(fact_965_wf__irrefl,axiom,
    ! [R: set_Product_prod_v_v,A3: v] :
      ( ( wf_v @ R )
     => ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ A3 ) @ R ) ) ).

% wf_irrefl
thf(fact_966_wf__induct,axiom,
    ! [R: set_Product_prod_v_v,P: v > $o,A3: v] :
      ( ( wf_v @ R )
     => ( ! [X4: v] :
            ( ! [Y5: v] :
                ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ X4 ) @ R )
               => ( P @ Y5 ) )
           => ( P @ X4 ) )
       => ( P @ A3 ) ) ) ).

% wf_induct
thf(fact_967_wf__asym,axiom,
    ! [R: set_Product_prod_v_v,A3: v,X2: v] :
      ( ( wf_v @ R )
     => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ X2 ) @ R )
       => ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ A3 ) @ R ) ) ) ).

% wf_asym
thf(fact_968_wfUNIVI,axiom,
    ! [R: set_Product_prod_v_v] :
      ( ! [P3: v > $o,X4: v] :
          ( ! [Xa: v] :
              ( ! [Y: v] :
                  ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Xa ) @ R )
                 => ( P3 @ Y ) )
             => ( P3 @ Xa ) )
         => ( P3 @ X4 ) )
     => ( wf_v @ R ) ) ).

% wfUNIVI
thf(fact_969_wfI__min,axiom,
    ! [R3: set_Pr2149350503807050951od_v_v] :
      ( ! [X4: product_prod_v_v,Q3: set_Product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X4 @ Q3 )
         => ? [Xa: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ Xa @ Q3 )
              & ! [Y: product_prod_v_v] :
                  ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ Xa ) @ R3 )
                 => ~ ( member7453568604450474000od_v_v @ Y @ Q3 ) ) ) )
     => ( wf_Product_prod_v_v @ R3 ) ) ).

% wfI_min
thf(fact_970_wfI__min,axiom,
    ! [R3: set_Product_prod_v_v] :
      ( ! [X4: v,Q3: set_v] :
          ( ( member_v @ X4 @ Q3 )
         => ? [Xa: v] :
              ( ( member_v @ Xa @ Q3 )
              & ! [Y: v] :
                  ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Xa ) @ R3 )
                 => ~ ( member_v @ Y @ Q3 ) ) ) )
     => ( wf_v @ R3 ) ) ).

% wfI_min
thf(fact_971_wfE__min,axiom,
    ! [R3: set_Pr2149350503807050951od_v_v,X2: product_prod_v_v,Q: set_Product_prod_v_v] :
      ( ( wf_Product_prod_v_v @ R3 )
     => ( ( member7453568604450474000od_v_v @ X2 @ Q )
       => ~ ! [Z3: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ Z3 @ Q )
             => ~ ! [Y5: product_prod_v_v] :
                    ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y5 @ Z3 ) @ R3 )
                   => ~ ( member7453568604450474000od_v_v @ Y5 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_972_wfE__min,axiom,
    ! [R3: set_Product_prod_v_v,X2: v,Q: set_v] :
      ( ( wf_v @ R3 )
     => ( ( member_v @ X2 @ Q )
       => ~ ! [Z3: v] :
              ( ( member_v @ Z3 @ Q )
             => ~ ! [Y5: v] :
                    ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ Z3 ) @ R3 )
                   => ~ ( member_v @ Y5 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_973_wf__def,axiom,
    ( wf_v
    = ( ^ [R4: set_Product_prod_v_v] :
        ! [P4: v > $o] :
          ( ! [X3: v] :
              ( ! [Y3: v] :
                  ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ X3 ) @ R4 )
                 => ( P4 @ Y3 ) )
             => ( P4 @ X3 ) )
         => ! [X6: v] : ( P4 @ X6 ) ) ) ) ).

% wf_def
thf(fact_974_wf__subset,axiom,
    ! [R: set_Product_prod_v_v,P2: set_Product_prod_v_v] :
      ( ( wf_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ P2 @ R )
       => ( wf_v @ P2 ) ) ) ).

% wf_subset
thf(fact_975_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v,B2: product_prod_v_v,Phi: product_prod_v_v > product_prod_v_v > $o] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ B2 ) @ ( minus_5255927943254941998od_v_v @ R @ id_Product_prod_v_v ) )
              | ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ A3 ) @ ( minus_5255927943254941998od_v_v @ R @ id_Product_prod_v_v ) ) )
           => ( Phi @ A3 @ B2 ) )
         => ( ( ( A3 = B2 )
             => ( Phi @ A3 @ B2 ) )
           => ( Phi @ A3 @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_976_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Product_prod_v_v,A3: v,B2: v,Phi: v > v > $o] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ord_less_eq_set_v @ ( insert_v2 @ A3 @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) @ ( field_v @ R ) )
       => ( ( ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ ( minus_4183494784930505774od_v_v @ R @ id_v ) )
              | ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ A3 ) @ ( minus_4183494784930505774od_v_v @ R @ id_v ) ) )
           => ( Phi @ A3 @ B2 ) )
         => ( ( ( A3 = B2 )
             => ( Phi @ A3 @ B2 ) )
           => ( Phi @ A3 @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_977_wo__rel_OTOTALS,axiom,
    ! [R: set_Product_prod_v_v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ! [X: v] :
          ( ( member_v @ X @ ( field_v @ R ) )
         => ! [Xa: v] :
              ( ( member_v @ Xa @ ( field_v @ R ) )
             => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Xa ) @ R )
                | ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Xa @ X ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_978_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,J: product_prod_v_v,I: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member7453568604450474000od_v_v @ I @ ( field_7153129647634986036od_v_v @ R ) )
         => ( ( member7453568604450474000od_v_v @ J @ ( field_7153129647634986036od_v_v @ R ) )
           => ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_979_wo__rel_Oin__notinI,axiom,
    ! [R: set_Product_prod_v_v,J: v,I: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member_v @ I @ ( field_v @ R ) )
         => ( ( member_v @ J @ ( field_v @ R ) )
           => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_980_wo__rel_Owell__order__induct,axiom,
    ! [R: set_Product_prod_v_v,P: v > $o,A3: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ! [X4: v] :
            ( ! [Y5: v] :
                ( ( ( Y5 != X4 )
                  & ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ X4 ) @ R ) )
               => ( P @ Y5 ) )
           => ( P @ X4 ) )
       => ( P @ A3 ) ) ) ).

% wo_rel.well_order_induct
thf(fact_981_well__order__induct__imp,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,P: product_prod_v_v > $o,A3: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ! [X4: product_prod_v_v] :
            ( ! [Y5: product_prod_v_v] :
                ( ( ( Y5 != X4 )
                  & ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y5 @ X4 ) @ R ) )
               => ( ( member7453568604450474000od_v_v @ Y5 @ ( field_7153129647634986036od_v_v @ R ) )
                 => ( P @ Y5 ) ) )
           => ( ( member7453568604450474000od_v_v @ X4 @ ( field_7153129647634986036od_v_v @ R ) )
             => ( P @ X4 ) ) )
       => ( ( member7453568604450474000od_v_v @ A3 @ ( field_7153129647634986036od_v_v @ R ) )
         => ( P @ A3 ) ) ) ) ).

% well_order_induct_imp
thf(fact_982_well__order__induct__imp,axiom,
    ! [R: set_Product_prod_v_v,P: v > $o,A3: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ! [X4: v] :
            ( ! [Y5: v] :
                ( ( ( Y5 != X4 )
                  & ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ X4 ) @ R ) )
               => ( ( member_v @ Y5 @ ( field_v @ R ) )
                 => ( P @ Y5 ) ) )
           => ( ( member_v @ X4 @ ( field_v @ R ) )
             => ( P @ X4 ) ) )
       => ( ( member_v @ A3 @ ( field_v @ R ) )
         => ( P @ A3 ) ) ) ) ).

% well_order_induct_imp
thf(fact_983_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v,B2: product_prod_v_v,Phi: product_prod_v_v > product_prod_v_v > $o] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ B2 ) @ R )
           => ( Phi @ A3 @ B2 ) )
         => ( ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ A3 ) @ R )
             => ( Phi @ A3 @ B2 ) )
           => ( Phi @ A3 @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_984_wo__rel_Ocases__Total,axiom,
    ! [R: set_Product_prod_v_v,A3: v,B2: v,Phi: v > v > $o] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ord_less_eq_set_v @ ( insert_v2 @ A3 @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) @ ( field_v @ R ) )
       => ( ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R )
           => ( Phi @ A3 @ B2 ) )
         => ( ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ A3 ) @ R )
             => ( Phi @ A3 @ B2 ) )
           => ( Phi @ A3 @ B2 ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_985_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( B != bot_bo723834152578015283od_v_v )
         => ? [X_1: product_prod_v_v] : ( bNF_We6235008509051751325od_v_v @ R @ B @ X_1 ) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
thf(fact_986_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [R: set_Product_prod_v_v,B: set_v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
       => ( ( B != bot_bot_set_v )
         => ? [X_1: v] : ( bNF_We6697304935525757641inim_v @ R @ B @ X_1 ) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
thf(fact_987_wo__rel_Ominim__inField,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( B != bot_bo723834152578015283od_v_v )
         => ( member7453568604450474000od_v_v @ ( bNF_We5492458111348578227od_v_v @ R @ B ) @ ( field_7153129647634986036od_v_v @ R ) ) ) ) ) ).

% wo_rel.minim_inField
thf(fact_988_wo__rel_Ominim__inField,axiom,
    ! [R: set_Product_prod_v_v,B: set_v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
       => ( ( B != bot_bot_set_v )
         => ( member_v @ ( bNF_We5615626441682584799inim_v @ R @ B ) @ ( field_v @ R ) ) ) ) ) ).

% wo_rel.minim_inField
thf(fact_989_wo__rel_Ominim__in,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( B != bot_bo723834152578015283od_v_v )
         => ( member7453568604450474000od_v_v @ ( bNF_We5492458111348578227od_v_v @ R @ B ) @ B ) ) ) ) ).

% wo_rel.minim_in
thf(fact_990_wo__rel_Ominim__in,axiom,
    ! [R: set_Product_prod_v_v,B: set_v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
       => ( ( B != bot_bot_set_v )
         => ( member_v @ ( bNF_We5615626441682584799inim_v @ R @ B ) @ B ) ) ) ) ).

% wo_rel.minim_in
thf(fact_991_wo__rel_Ominim__isMinim,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( B != bot_bo723834152578015283od_v_v )
         => ( bNF_We6235008509051751325od_v_v @ R @ B @ ( bNF_We5492458111348578227od_v_v @ R @ B ) ) ) ) ) ).

% wo_rel.minim_isMinim
thf(fact_992_wo__rel_Ominim__isMinim,axiom,
    ! [R: set_Product_prod_v_v,B: set_v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
       => ( ( B != bot_bot_set_v )
         => ( bNF_We6697304935525757641inim_v @ R @ B @ ( bNF_We5615626441682584799inim_v @ R @ B ) ) ) ) ) ).

% wo_rel.minim_isMinim
thf(fact_993_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A: set_Product_prod_v_v,B2: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( bNF_We6235008509051751325od_v_v @ R @ A @ B2 )
        = ( ( member7453568604450474000od_v_v @ B2 @ A )
          & ! [X3: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ X3 @ A )
             => ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ X3 ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_994_wo__rel_OisMinim__def,axiom,
    ! [R: set_Product_prod_v_v,A: set_v,B2: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( bNF_We6697304935525757641inim_v @ R @ A @ B2 )
        = ( ( member_v @ B2 @ A )
          & ! [X3: v] :
              ( ( member_v @ X3 @ A )
             => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ X3 ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_995_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( member7453568604450474000od_v_v @ B2 @ B )
         => ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ ( bNF_We5492458111348578227od_v_v @ R @ B ) @ B2 ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_996_wo__rel_Ominim__least,axiom,
    ! [R: set_Product_prod_v_v,B: set_v,B2: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
       => ( ( member_v @ B2 @ B )
         => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ ( bNF_We5615626441682584799inim_v @ R @ B ) @ B2 ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_997_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v,A3: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( member7453568604450474000od_v_v @ A3 @ B )
         => ( ! [B7: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ B7 @ B )
               => ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ B7 ) @ R ) )
           => ( A3
              = ( bNF_We5492458111348578227od_v_v @ R @ B ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_998_wo__rel_Oequals__minim,axiom,
    ! [R: set_Product_prod_v_v,B: set_v,A3: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
       => ( ( member_v @ A3 @ B )
         => ( ! [B7: v] :
                ( ( member_v @ B7 @ B )
               => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B7 ) @ R ) )
           => ( A3
              = ( bNF_We5615626441682584799inim_v @ R @ B ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_999_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v,B2: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( member7453568604450474000od_v_v @ A3 @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
         => ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ ( bNF_We3854103423653685557od_v_v @ R @ A3 @ B2 ) ) @ R )
            & ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ ( bNF_We3854103423653685557od_v_v @ R @ A3 @ B2 ) ) @ R )
            & ( member7453568604450474000od_v_v @ ( bNF_We3854103423653685557od_v_v @ R @ A3 @ B2 ) @ ( insert1338601472111419319od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_1000_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Product_prod_v_v,A3: v,B2: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( member_v @ A3 @ ( field_v @ R ) )
       => ( ( member_v @ B2 @ ( field_v @ R ) )
         => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ ( bNF_We3763454674811381857max2_v @ R @ A3 @ B2 ) ) @ R )
            & ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ ( bNF_We3763454674811381857max2_v @ R @ A3 @ B2 ) ) @ R )
            & ( member_v @ ( bNF_We3763454674811381857max2_v @ R @ A3 @ B2 ) @ ( insert_v2 @ A3 @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_1001_Zorns__po__lemma,axiom,
    ! [R: set_Product_prod_v_v] :
      ( ( order_5272072345360262664r_on_v @ ( field_v @ R ) @ R )
     => ( ! [C4: set_v] :
            ( ( member_set_v @ C4 @ ( chains_v @ R ) )
           => ? [X: v] :
                ( ( member_v @ X @ ( field_v @ R ) )
                & ! [Xa2: v] :
                    ( ( member_v @ Xa2 @ C4 )
                   => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Xa2 @ X ) @ R ) ) ) )
       => ? [X4: v] :
            ( ( member_v @ X4 @ ( field_v @ R ) )
            & ! [Xa: v] :
                ( ( member_v @ Xa @ ( field_v @ R ) )
               => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X4 @ Xa ) @ R )
                 => ( Xa = X4 ) ) ) ) ) ) ).

% Zorns_po_lemma
thf(fact_1002_wo__rel_Omax2__among,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v,B2: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( member7453568604450474000od_v_v @ A3 @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
         => ( member7453568604450474000od_v_v @ ( bNF_We3854103423653685557od_v_v @ R @ A3 @ B2 ) @ ( insert1338601472111419319od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).

% wo_rel.max2_among
thf(fact_1003_wo__rel_Omax2__among,axiom,
    ! [R: set_Product_prod_v_v,A3: v,B2: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( member_v @ A3 @ ( field_v @ R ) )
       => ( ( member_v @ B2 @ ( field_v @ R ) )
         => ( member_v @ ( bNF_We3763454674811381857max2_v @ R @ A3 @ B2 ) @ ( insert_v2 @ A3 @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) ) ) ) ) ).

% wo_rel.max2_among
thf(fact_1004_mono__Chains,axiom,
    ! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ R @ S4 )
     => ( ord_le5216385588623774835_set_v @ ( chains_v @ R ) @ ( chains_v @ S4 ) ) ) ).

% mono_Chains
thf(fact_1005_wo__rel_Omax2__def,axiom,
    ! [R: set_Product_prod_v_v,A3: v,B2: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R )
         => ( ( bNF_We3763454674811381857max2_v @ R @ A3 @ B2 )
            = B2 ) )
        & ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R )
         => ( ( bNF_We3763454674811381857max2_v @ R @ A3 @ B2 )
            = A3 ) ) ) ) ).

% wo_rel.max2_def
thf(fact_1006_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v,B2: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( member7453568604450474000od_v_v @ A3 @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
         => ( ( ( bNF_We3854103423653685557od_v_v @ R @ A3 @ B2 )
              = A3 )
            = ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ A3 ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_1007_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Product_prod_v_v,A3: v,B2: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( member_v @ A3 @ ( field_v @ R ) )
       => ( ( member_v @ B2 @ ( field_v @ R ) )
         => ( ( ( bNF_We3763454674811381857max2_v @ R @ A3 @ B2 )
              = A3 )
            = ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ A3 ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_1008_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v,B2: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( member7453568604450474000od_v_v @ A3 @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
         => ( ( ( bNF_We3854103423653685557od_v_v @ R @ A3 @ B2 )
              = B2 )
            = ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ B2 ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1009_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Product_prod_v_v,A3: v,B2: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( member_v @ A3 @ ( field_v @ R ) )
       => ( ( member_v @ B2 @ ( field_v @ R ) )
         => ( ( ( bNF_We3763454674811381857max2_v @ R @ A3 @ B2 )
              = B2 )
            = ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ B2 ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1010_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v,B2: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( member7453568604450474000od_v_v @ A3 @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
         => ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ ( bNF_We3854103423653685557od_v_v @ R @ A3 @ B2 ) ) @ R )
            & ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ ( bNF_We3854103423653685557od_v_v @ R @ A3 @ B2 ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1011_wo__rel_Omax2__greater,axiom,
    ! [R: set_Product_prod_v_v,A3: v,B2: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( member_v @ A3 @ ( field_v @ R ) )
       => ( ( member_v @ B2 @ ( field_v @ R ) )
         => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ ( bNF_We3763454674811381857max2_v @ R @ A3 @ B2 ) ) @ R )
            & ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ ( bNF_We3763454674811381857max2_v @ R @ A3 @ B2 ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1012_wf__Union,axiom,
    ! [R3: set_se5707775751431548583od_v_v] :
      ( ! [X4: set_Pr2149350503807050951od_v_v] :
          ( ( member2865299526245254384od_v_v @ X4 @ R3 )
         => ( wf_Product_prod_v_v @ X4 ) )
     => ( ! [X4: set_Pr2149350503807050951od_v_v] :
            ( ( member2865299526245254384od_v_v @ X4 @ R3 )
           => ! [Xa2: set_Pr2149350503807050951od_v_v] :
                ( ( member2865299526245254384od_v_v @ Xa2 @ R3 )
               => ( ( X4 != Xa2 )
                 => ( ( inf_in6271465464967711157od_v_v @ ( domain6359000466948879308od_v_v @ X4 ) @ ( range_7878975032137371189od_v_v @ Xa2 ) )
                    = bot_bo723834152578015283od_v_v ) ) ) )
       => ( wf_Product_prod_v_v @ ( comple514088740646613812od_v_v @ R3 ) ) ) ) ).

% wf_Union
thf(fact_1013_wf__Union,axiom,
    ! [R3: set_se8455005133513928103od_v_v] :
      ( ! [X4: set_Product_prod_v_v] :
          ( ( member8406446414694345712od_v_v @ X4 @ R3 )
         => ( wf_v @ X4 ) )
     => ( ! [X4: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X4 @ R3 )
           => ! [Xa2: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ Xa2 @ R3 )
               => ( ( X4 != Xa2 )
                 => ( ( inf_inf_set_v @ ( domain_v_v @ X4 ) @ ( range_v_v @ Xa2 ) )
                    = bot_bot_set_v ) ) ) )
       => ( wf_v @ ( comple5788137035815166516od_v_v @ R3 ) ) ) ) ).

% wf_Union
thf(fact_1014_wo__rel_Oofilter__def,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A: set_Product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( order_1752595321582732413od_v_v @ R @ A )
        = ( ( ord_le7336532860387713383od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
          & ! [X3: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ X3 @ A )
             => ( ord_le7336532860387713383od_v_v @ ( order_6892855479609198156od_v_v @ R @ X3 ) @ A ) ) ) ) ) ).

% wo_rel.ofilter_def
thf(fact_1015_wo__rel_Oofilter__def,axiom,
    ! [R: set_Product_prod_v_v,A: set_v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( order_ofilter_v @ R @ A )
        = ( ( ord_less_eq_set_v @ A @ ( field_v @ R ) )
          & ! [X3: v] :
              ( ( member_v @ X3 @ A )
             => ( ord_less_eq_set_v @ ( order_under_v @ R @ X3 ) @ A ) ) ) ) ) ).

% wo_rel.ofilter_def
thf(fact_1016_max__ext_Omax__extI,axiom,
    ! [X5: set_Product_prod_v_v,Y6: set_Product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
      ( ( finite3348123685078250256od_v_v @ X5 )
     => ( ( finite3348123685078250256od_v_v @ Y6 )
       => ( ( Y6 != bot_bo723834152578015283od_v_v )
         => ( ! [X4: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X4 @ X5 )
               => ? [Xa: product_prod_v_v] :
                    ( ( member7453568604450474000od_v_v @ Xa @ Y6 )
                    & ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X4 @ Xa ) @ R3 ) ) )
           => ( member1739204623775007504od_v_v @ ( produc4078165475796654935od_v_v @ X5 @ Y6 ) @ ( max_ex4033391607822738082od_v_v @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1017_max__ext_Omax__extI,axiom,
    ! [X5: set_v,Y6: set_v,R3: set_Product_prod_v_v] :
      ( ( finite_finite_v @ X5 )
     => ( ( finite_finite_v @ Y6 )
       => ( ( Y6 != bot_bot_set_v )
         => ( ! [X4: v] :
                ( ( member_v @ X4 @ X5 )
               => ? [Xa: v] :
                    ( ( member_v @ Xa @ Y6 )
                    & ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X4 @ Xa ) @ R3 ) ) )
           => ( member3828772815783460880_set_v @ ( produc3441907479644599895_set_v @ X5 @ Y6 ) @ ( max_ext_v @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1018_finite__Union,axiom,
    ! [A: set_set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ! [M2: set_v] :
            ( ( member_set_v @ M2 @ A )
           => ( finite_finite_v @ M2 ) )
       => ( finite_finite_v @ ( comple2307003700295860064_set_v @ A ) ) ) ) ).

% finite_Union
thf(fact_1019_finite__UnionD,axiom,
    ! [A: set_set_v] :
      ( ( finite_finite_v @ ( comple2307003700295860064_set_v @ A ) )
     => ( finite_finite_set_v @ A ) ) ).

% finite_UnionD
thf(fact_1020_max__ext__additive,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,R3: set_Pr2149350503807050951od_v_v,C2: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
      ( ( member1739204623775007504od_v_v @ ( produc4078165475796654935od_v_v @ A @ B ) @ ( max_ex4033391607822738082od_v_v @ R3 ) )
     => ( ( member1739204623775007504od_v_v @ ( produc4078165475796654935od_v_v @ C2 @ D2 ) @ ( max_ex4033391607822738082od_v_v @ R3 ) )
       => ( member1739204623775007504od_v_v @ ( produc4078165475796654935od_v_v @ ( sup_su414716646722978715od_v_v @ A @ C2 ) @ ( sup_su414716646722978715od_v_v @ B @ D2 ) ) @ ( max_ex4033391607822738082od_v_v @ R3 ) ) ) ) ).

% max_ext_additive
thf(fact_1021_insert__partition,axiom,
    ! [X2: set_Product_prod_v_v,F3: set_se8455005133513928103od_v_v] :
      ( ~ ( member8406446414694345712od_v_v @ X2 @ F3 )
     => ( ! [X4: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X4 @ ( insert7504383016908236695od_v_v @ X2 @ F3 ) )
           => ! [Xa2: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ Xa2 @ ( insert7504383016908236695od_v_v @ X2 @ F3 ) )
               => ( ( X4 != Xa2 )
                 => ( ( inf_in6271465464967711157od_v_v @ X4 @ Xa2 )
                    = bot_bo723834152578015283od_v_v ) ) ) )
       => ( ( inf_in6271465464967711157od_v_v @ X2 @ ( comple5788137035815166516od_v_v @ F3 ) )
          = bot_bo723834152578015283od_v_v ) ) ) ).

% insert_partition
thf(fact_1022_insert__partition,axiom,
    ! [X2: set_v,F3: set_set_v] :
      ( ~ ( member_set_v @ X2 @ F3 )
     => ( ! [X4: set_v] :
            ( ( member_set_v @ X4 @ ( insert_set_v @ X2 @ F3 ) )
           => ! [Xa2: set_v] :
                ( ( member_set_v @ Xa2 @ ( insert_set_v @ X2 @ F3 ) )
               => ( ( X4 != Xa2 )
                 => ( ( inf_inf_set_v @ X4 @ Xa2 )
                    = bot_bot_set_v ) ) ) )
       => ( ( inf_inf_set_v @ X2 @ ( comple2307003700295860064_set_v @ F3 ) )
          = bot_bot_set_v ) ) ) ).

% insert_partition
thf(fact_1023_wo__rel_Oofilter__linord,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( order_1752595321582732413od_v_v @ R @ A )
       => ( ( order_1752595321582732413od_v_v @ R @ B )
         => ( ( ord_le7336532860387713383od_v_v @ A @ B )
            | ( ord_le7336532860387713383od_v_v @ B @ A ) ) ) ) ) ).

% wo_rel.ofilter_linord
thf(fact_1024_wo__rel_Oofilter__linord,axiom,
    ! [R: set_Product_prod_v_v,A: set_v,B: set_v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( order_ofilter_v @ R @ A )
       => ( ( order_ofilter_v @ R @ B )
         => ( ( ord_less_eq_set_v @ A @ B )
            | ( ord_less_eq_set_v @ B @ A ) ) ) ) ) ).

% wo_rel.ofilter_linord
thf(fact_1025_finite__Sup__in,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ X4 @ A )
             => ( ( member8406446414694345712od_v_v @ Y @ A )
               => ( member8406446414694345712od_v_v @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) @ A ) ) )
         => ( member8406446414694345712od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ A ) ) ) ) ).

% finite_Sup_in
thf(fact_1026_sup__Sup__fold__sup,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ B )
        = ( finite8952066981541671560od_v_v @ sup_su414716646722978715od_v_v @ B @ A ) ) ) ).

% sup_Sup_fold_sup
thf(fact_1027_ofilter__def,axiom,
    ( order_1752595321582732413od_v_v
    = ( ^ [R4: set_Pr2149350503807050951od_v_v,A4: set_Product_prod_v_v] :
          ( ( ord_le7336532860387713383od_v_v @ A4 @ ( field_7153129647634986036od_v_v @ R4 ) )
          & ! [X3: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ X3 @ A4 )
             => ( ord_le7336532860387713383od_v_v @ ( order_6892855479609198156od_v_v @ R4 @ X3 ) @ A4 ) ) ) ) ) ).

% ofilter_def
thf(fact_1028_ofilter__def,axiom,
    ( order_ofilter_v
    = ( ^ [R4: set_Product_prod_v_v,A4: set_v] :
          ( ( ord_less_eq_set_v @ A4 @ ( field_v @ R4 ) )
          & ! [X3: v] :
              ( ( member_v @ X3 @ A4 )
             => ( ord_less_eq_set_v @ ( order_under_v @ R4 @ X3 ) @ A4 ) ) ) ) ) ).

% ofilter_def
thf(fact_1029_Sup__fold__sup,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A )
     => ( ( comple5788137035815166516od_v_v @ A )
        = ( finite8952066981541671560od_v_v @ sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ A ) ) ) ).

% Sup_fold_sup
thf(fact_1030_Sup__fold__sup,axiom,
    ! [A: set_set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( comple2307003700295860064_set_v @ A )
        = ( finite338946655151718280_set_v @ sup_sup_set_v @ bot_bot_set_v @ A ) ) ) ).

% Sup_fold_sup
thf(fact_1031_max__ext_Ocases,axiom,
    ! [A1: set_Product_prod_v_v,A2: set_Product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
      ( ( member1739204623775007504od_v_v @ ( produc4078165475796654935od_v_v @ A1 @ A2 ) @ ( max_ex4033391607822738082od_v_v @ R3 ) )
     => ~ ( ( finite3348123685078250256od_v_v @ A1 )
         => ( ( finite3348123685078250256od_v_v @ A2 )
           => ( ( A2 != bot_bo723834152578015283od_v_v )
             => ~ ! [X: product_prod_v_v] :
                    ( ( member7453568604450474000od_v_v @ X @ A1 )
                   => ? [Xa2: product_prod_v_v] :
                        ( ( member7453568604450474000od_v_v @ Xa2 @ A2 )
                        & ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1032_max__ext_Ocases,axiom,
    ! [A1: set_v,A2: set_v,R3: set_Product_prod_v_v] :
      ( ( member3828772815783460880_set_v @ ( produc3441907479644599895_set_v @ A1 @ A2 ) @ ( max_ext_v @ R3 ) )
     => ~ ( ( finite_finite_v @ A1 )
         => ( ( finite_finite_v @ A2 )
           => ( ( A2 != bot_bot_set_v )
             => ~ ! [X: v] :
                    ( ( member_v @ X @ A1 )
                   => ? [Xa2: v] :
                        ( ( member_v @ Xa2 @ A2 )
                        & ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1033_max__ext_Osimps,axiom,
    ! [A1: set_Product_prod_v_v,A2: set_Product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
      ( ( member1739204623775007504od_v_v @ ( produc4078165475796654935od_v_v @ A1 @ A2 ) @ ( max_ex4033391607822738082od_v_v @ R3 ) )
      = ( ( finite3348123685078250256od_v_v @ A1 )
        & ( finite3348123685078250256od_v_v @ A2 )
        & ( A2 != bot_bo723834152578015283od_v_v )
        & ! [X3: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ A1 )
           => ? [Y3: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ Y3 @ A2 )
                & ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1034_max__ext_Osimps,axiom,
    ! [A1: set_v,A2: set_v,R3: set_Product_prod_v_v] :
      ( ( member3828772815783460880_set_v @ ( produc3441907479644599895_set_v @ A1 @ A2 ) @ ( max_ext_v @ R3 ) )
      = ( ( finite_finite_v @ A1 )
        & ( finite_finite_v @ A2 )
        & ( A2 != bot_bot_set_v )
        & ! [X3: v] :
            ( ( member_v @ X3 @ A1 )
           => ? [Y3: v] :
                ( ( member_v @ Y3 @ A2 )
                & ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1035_Sup__insert,axiom,
    ! [A3: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v] :
      ( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A3 @ A ) )
      = ( sup_su414716646722978715od_v_v @ A3 @ ( comple5788137035815166516od_v_v @ A ) ) ) ).

% Sup_insert
thf(fact_1036_Sup__bot__conv_I2_J,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( comple5788137035815166516od_v_v @ A ) )
      = ( ! [X3: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X3 @ A )
           => ( X3 = bot_bo723834152578015283od_v_v ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_1037_Sup__bot__conv_I2_J,axiom,
    ! [A: set_set_v] :
      ( ( bot_bot_set_v
        = ( comple2307003700295860064_set_v @ A ) )
      = ( ! [X3: set_v] :
            ( ( member_set_v @ X3 @ A )
           => ( X3 = bot_bot_set_v ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_1038_Sup__bot__conv_I1_J,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ( ( comple5788137035815166516od_v_v @ A )
        = bot_bo723834152578015283od_v_v )
      = ( ! [X3: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X3 @ A )
           => ( X3 = bot_bo723834152578015283od_v_v ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_1039_Sup__bot__conv_I1_J,axiom,
    ! [A: set_set_v] :
      ( ( ( comple2307003700295860064_set_v @ A )
        = bot_bot_set_v )
      = ( ! [X3: set_v] :
            ( ( member_set_v @ X3 @ A )
           => ( X3 = bot_bot_set_v ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_1040_Union__Un__distrib,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( comple5788137035815166516od_v_v @ ( sup_su335656005089752955od_v_v @ A @ B ) )
      = ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).

% Union_Un_distrib
thf(fact_1041_Sup__empty,axiom,
    ( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
    = bot_bo723834152578015283od_v_v ) ).

% Sup_empty
thf(fact_1042_Sup__empty,axiom,
    ( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
    = bot_bot_set_v ) ).

% Sup_empty
thf(fact_1043_Sup__eqI,axiom,
    ! [A: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ! [Y: set_Product_prod_v_v] :
          ( ( member8406446414694345712od_v_v @ Y @ A )
         => ( ord_le7336532860387713383od_v_v @ Y @ X2 ) )
     => ( ! [Y: set_Product_prod_v_v] :
            ( ! [Z5: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ Z5 @ A )
               => ( ord_le7336532860387713383od_v_v @ Z5 @ Y ) )
           => ( ord_le7336532860387713383od_v_v @ X2 @ Y ) )
       => ( ( comple5788137035815166516od_v_v @ A )
          = X2 ) ) ) ).

% Sup_eqI
thf(fact_1044_Sup__eqI,axiom,
    ! [A: set_set_v,X2: set_v] :
      ( ! [Y: set_v] :
          ( ( member_set_v @ Y @ A )
         => ( ord_less_eq_set_v @ Y @ X2 ) )
     => ( ! [Y: set_v] :
            ( ! [Z5: set_v] :
                ( ( member_set_v @ Z5 @ A )
               => ( ord_less_eq_set_v @ Z5 @ Y ) )
           => ( ord_less_eq_set_v @ X2 @ Y ) )
       => ( ( comple2307003700295860064_set_v @ A )
          = X2 ) ) ) ).

% Sup_eqI
thf(fact_1045_Sup__mono,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ! [A7: set_Product_prod_v_v] :
          ( ( member8406446414694345712od_v_v @ A7 @ A )
         => ? [X: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ X @ B )
              & ( ord_le7336532860387713383od_v_v @ A7 @ X ) ) )
     => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).

% Sup_mono
thf(fact_1046_Sup__mono,axiom,
    ! [A: set_set_v,B: set_set_v] :
      ( ! [A7: set_v] :
          ( ( member_set_v @ A7 @ A )
         => ? [X: set_v] :
              ( ( member_set_v @ X @ B )
              & ( ord_less_eq_set_v @ A7 @ X ) ) )
     => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).

% Sup_mono
thf(fact_1047_Sup__least,axiom,
    ! [A: set_se8455005133513928103od_v_v,Z: set_Product_prod_v_v] :
      ( ! [X4: set_Product_prod_v_v] :
          ( ( member8406446414694345712od_v_v @ X4 @ A )
         => ( ord_le7336532860387713383od_v_v @ X4 @ Z ) )
     => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ Z ) ) ).

% Sup_least
thf(fact_1048_Sup__least,axiom,
    ! [A: set_set_v,Z: set_v] :
      ( ! [X4: set_v] :
          ( ( member_set_v @ X4 @ A )
         => ( ord_less_eq_set_v @ X4 @ Z ) )
     => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ Z ) ) ).

% Sup_least
thf(fact_1049_Sup__upper,axiom,
    ! [X2: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v] :
      ( ( member8406446414694345712od_v_v @ X2 @ A )
     => ( ord_le7336532860387713383od_v_v @ X2 @ ( comple5788137035815166516od_v_v @ A ) ) ) ).

% Sup_upper
thf(fact_1050_Sup__upper,axiom,
    ! [X2: set_v,A: set_set_v] :
      ( ( member_set_v @ X2 @ A )
     => ( ord_less_eq_set_v @ X2 @ ( comple2307003700295860064_set_v @ A ) ) ) ).

% Sup_upper
thf(fact_1051_Sup__le__iff,axiom,
    ! [A: set_se8455005133513928103od_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ B2 )
      = ( ! [X3: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X3 @ A )
           => ( ord_le7336532860387713383od_v_v @ X3 @ B2 ) ) ) ) ).

% Sup_le_iff
thf(fact_1052_Sup__le__iff,axiom,
    ! [A: set_set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ B2 )
      = ( ! [X3: set_v] :
            ( ( member_set_v @ X3 @ A )
           => ( ord_less_eq_set_v @ X3 @ B2 ) ) ) ) ).

% Sup_le_iff
thf(fact_1053_Sup__upper2,axiom,
    ! [U: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v,V3: set_Product_prod_v_v] :
      ( ( member8406446414694345712od_v_v @ U @ A )
     => ( ( ord_le7336532860387713383od_v_v @ V3 @ U )
       => ( ord_le7336532860387713383od_v_v @ V3 @ ( comple5788137035815166516od_v_v @ A ) ) ) ) ).

% Sup_upper2
thf(fact_1054_Sup__upper2,axiom,
    ! [U: set_v,A: set_set_v,V3: set_v] :
      ( ( member_set_v @ U @ A )
     => ( ( ord_less_eq_set_v @ V3 @ U )
       => ( ord_less_eq_set_v @ V3 @ ( comple2307003700295860064_set_v @ A ) ) ) ) ).

% Sup_upper2
thf(fact_1055_cSup__eq__maximum,axiom,
    ! [Z: set_Product_prod_v_v,X5: set_se8455005133513928103od_v_v] :
      ( ( member8406446414694345712od_v_v @ Z @ X5 )
     => ( ! [X4: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X4 @ X5 )
           => ( ord_le7336532860387713383od_v_v @ X4 @ Z ) )
       => ( ( comple5788137035815166516od_v_v @ X5 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_1056_cSup__eq__maximum,axiom,
    ! [Z: set_v,X5: set_set_v] :
      ( ( member_set_v @ Z @ X5 )
     => ( ! [X4: set_v] :
            ( ( member_set_v @ X4 @ X5 )
           => ( ord_less_eq_set_v @ X4 @ Z ) )
       => ( ( comple2307003700295860064_set_v @ X5 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_1057_Union__empty,axiom,
    ( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
    = bot_bo723834152578015283od_v_v ) ).

% Union_empty
thf(fact_1058_Union__empty,axiom,
    ( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
    = bot_bot_set_v ) ).

% Union_empty
thf(fact_1059_Union__empty__conv,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ( ( comple5788137035815166516od_v_v @ A )
        = bot_bo723834152578015283od_v_v )
      = ( ! [X3: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X3 @ A )
           => ( X3 = bot_bo723834152578015283od_v_v ) ) ) ) ).

% Union_empty_conv
thf(fact_1060_Union__empty__conv,axiom,
    ! [A: set_set_v] :
      ( ( ( comple2307003700295860064_set_v @ A )
        = bot_bot_set_v )
      = ( ! [X3: set_v] :
            ( ( member_set_v @ X3 @ A )
           => ( X3 = bot_bot_set_v ) ) ) ) ).

% Union_empty_conv
thf(fact_1061_empty__Union__conv,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( comple5788137035815166516od_v_v @ A ) )
      = ( ! [X3: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X3 @ A )
           => ( X3 = bot_bo723834152578015283od_v_v ) ) ) ) ).

% empty_Union_conv
thf(fact_1062_empty__Union__conv,axiom,
    ! [A: set_set_v] :
      ( ( bot_bot_set_v
        = ( comple2307003700295860064_set_v @ A ) )
      = ( ! [X3: set_v] :
            ( ( member_set_v @ X3 @ A )
           => ( X3 = bot_bot_set_v ) ) ) ) ).

% empty_Union_conv
thf(fact_1063_Union__least,axiom,
    ! [A: set_se8455005133513928103od_v_v,C2: set_Product_prod_v_v] :
      ( ! [X7: set_Product_prod_v_v] :
          ( ( member8406446414694345712od_v_v @ X7 @ A )
         => ( ord_le7336532860387713383od_v_v @ X7 @ C2 ) )
     => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ C2 ) ) ).

% Union_least
thf(fact_1064_Union__least,axiom,
    ! [A: set_set_v,C2: set_v] :
      ( ! [X7: set_v] :
          ( ( member_set_v @ X7 @ A )
         => ( ord_less_eq_set_v @ X7 @ C2 ) )
     => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ C2 ) ) ).

% Union_least
thf(fact_1065_Union__upper,axiom,
    ! [B: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v] :
      ( ( member8406446414694345712od_v_v @ B @ A )
     => ( ord_le7336532860387713383od_v_v @ B @ ( comple5788137035815166516od_v_v @ A ) ) ) ).

% Union_upper
thf(fact_1066_Union__upper,axiom,
    ! [B: set_v,A: set_set_v] :
      ( ( member_set_v @ B @ A )
     => ( ord_less_eq_set_v @ B @ ( comple2307003700295860064_set_v @ A ) ) ) ).

% Union_upper
thf(fact_1067_Union__subsetI,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ! [X4: set_Product_prod_v_v] :
          ( ( member8406446414694345712od_v_v @ X4 @ A )
         => ? [Y5: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ Y5 @ B )
              & ( ord_le7336532860387713383od_v_v @ X4 @ Y5 ) ) )
     => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).

% Union_subsetI
thf(fact_1068_Union__subsetI,axiom,
    ! [A: set_set_v,B: set_set_v] :
      ( ! [X4: set_v] :
          ( ( member_set_v @ X4 @ A )
         => ? [Y5: set_v] :
              ( ( member_set_v @ Y5 @ B )
              & ( ord_less_eq_set_v @ X4 @ Y5 ) ) )
     => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).

% Union_subsetI
thf(fact_1069_Union__insert,axiom,
    ! [A3: set_Product_prod_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A3 @ B ) )
      = ( sup_su414716646722978715od_v_v @ A3 @ ( comple5788137035815166516od_v_v @ B ) ) ) ).

% Union_insert
thf(fact_1070_cSup__eq__non__empty,axiom,
    ! [X5: set_se8455005133513928103od_v_v,A3: set_Product_prod_v_v] :
      ( ( X5 != bot_bo3497076220358800403od_v_v )
     => ( ! [X4: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X4 @ X5 )
           => ( ord_le7336532860387713383od_v_v @ X4 @ A3 ) )
       => ( ! [Y: set_Product_prod_v_v] :
              ( ! [X: set_Product_prod_v_v] :
                  ( ( member8406446414694345712od_v_v @ X @ X5 )
                 => ( ord_le7336532860387713383od_v_v @ X @ Y ) )
             => ( ord_le7336532860387713383od_v_v @ A3 @ Y ) )
         => ( ( comple5788137035815166516od_v_v @ X5 )
            = A3 ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_1071_cSup__eq__non__empty,axiom,
    ! [X5: set_set_v,A3: set_v] :
      ( ( X5 != bot_bot_set_set_v )
     => ( ! [X4: set_v] :
            ( ( member_set_v @ X4 @ X5 )
           => ( ord_less_eq_set_v @ X4 @ A3 ) )
       => ( ! [Y: set_v] :
              ( ! [X: set_v] :
                  ( ( member_set_v @ X @ X5 )
                 => ( ord_less_eq_set_v @ X @ Y ) )
             => ( ord_less_eq_set_v @ A3 @ Y ) )
         => ( ( comple2307003700295860064_set_v @ X5 )
            = A3 ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_1072_cSup__least,axiom,
    ! [X5: set_se8455005133513928103od_v_v,Z: set_Product_prod_v_v] :
      ( ( X5 != bot_bo3497076220358800403od_v_v )
     => ( ! [X4: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X4 @ X5 )
           => ( ord_le7336532860387713383od_v_v @ X4 @ Z ) )
       => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ X5 ) @ Z ) ) ) ).

% cSup_least
thf(fact_1073_cSup__least,axiom,
    ! [X5: set_set_v,Z: set_v] :
      ( ( X5 != bot_bot_set_set_v )
     => ( ! [X4: set_v] :
            ( ( member_set_v @ X4 @ X5 )
           => ( ord_less_eq_set_v @ X4 @ Z ) )
       => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ X5 ) @ Z ) ) ) ).

% cSup_least
thf(fact_1074_less__eq__Sup,axiom,
    ! [A: set_se8455005133513928103od_v_v,U: set_Product_prod_v_v] :
      ( ! [V2: set_Product_prod_v_v] :
          ( ( member8406446414694345712od_v_v @ V2 @ A )
         => ( ord_le7336532860387713383od_v_v @ U @ V2 ) )
     => ( ( A != bot_bo3497076220358800403od_v_v )
       => ( ord_le7336532860387713383od_v_v @ U @ ( comple5788137035815166516od_v_v @ A ) ) ) ) ).

% less_eq_Sup
thf(fact_1075_less__eq__Sup,axiom,
    ! [A: set_set_v,U: set_v] :
      ( ! [V2: set_v] :
          ( ( member_set_v @ V2 @ A )
         => ( ord_less_eq_set_v @ U @ V2 ) )
     => ( ( A != bot_bot_set_set_v )
       => ( ord_less_eq_set_v @ U @ ( comple2307003700295860064_set_v @ A ) ) ) ) ).

% less_eq_Sup
thf(fact_1076_le__cSup__finite,axiom,
    ! [X5: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ X5 )
     => ( ( member8406446414694345712od_v_v @ X2 @ X5 )
       => ( ord_le7336532860387713383od_v_v @ X2 @ ( comple5788137035815166516od_v_v @ X5 ) ) ) ) ).

% le_cSup_finite
thf(fact_1077_le__cSup__finite,axiom,
    ! [X5: set_set_v,X2: set_v] :
      ( ( finite_finite_set_v @ X5 )
     => ( ( member_set_v @ X2 @ X5 )
       => ( ord_less_eq_set_v @ X2 @ ( comple2307003700295860064_set_v @ X5 ) ) ) ) ).

% le_cSup_finite
thf(fact_1078_Sup__subset__mono,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( ord_le4714265922333009223od_v_v @ A @ B )
     => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).

% Sup_subset_mono
thf(fact_1079_Sup__subset__mono,axiom,
    ! [A: set_set_v,B: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A @ B )
     => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).

% Sup_subset_mono
thf(fact_1080_Sup__union__distrib,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( comple5788137035815166516od_v_v @ ( sup_su335656005089752955od_v_v @ A @ B ) )
      = ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).

% Sup_union_distrib
thf(fact_1081_Union__disjoint,axiom,
    ! [C2: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
      ( ( ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ C2 ) @ A )
        = bot_bo723834152578015283od_v_v )
      = ( ! [X3: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X3 @ C2 )
           => ( ( inf_in6271465464967711157od_v_v @ X3 @ A )
              = bot_bo723834152578015283od_v_v ) ) ) ) ).

% Union_disjoint
thf(fact_1082_Union__disjoint,axiom,
    ! [C2: set_set_v,A: set_v] :
      ( ( ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ C2 ) @ A )
        = bot_bot_set_v )
      = ( ! [X3: set_v] :
            ( ( member_set_v @ X3 @ C2 )
           => ( ( inf_inf_set_v @ X3 @ A )
              = bot_bot_set_v ) ) ) ) ).

% Union_disjoint
thf(fact_1083_Union__Int__subset,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] : ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( inf_in6058586722421118357od_v_v @ A @ B ) ) @ ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).

% Union_Int_subset
thf(fact_1084_Union__Int__subset,axiom,
    ! [A: set_set_v,B: set_set_v] : ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( inf_inf_set_set_v @ A @ B ) ) @ ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).

% Union_Int_subset
thf(fact_1085_Union__mono,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( ord_le4714265922333009223od_v_v @ A @ B )
     => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).

% Union_mono
thf(fact_1086_Union__mono,axiom,
    ! [A: set_set_v,B: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A @ B )
     => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).

% Union_mono
thf(fact_1087_Sup__inter__less__eq,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] : ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( inf_in6058586722421118357od_v_v @ A @ B ) ) @ ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).

% Sup_inter_less_eq
thf(fact_1088_Sup__inter__less__eq,axiom,
    ! [A: set_set_v,B: set_set_v] : ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( inf_inf_set_set_v @ A @ B ) ) @ ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).

% Sup_inter_less_eq
thf(fact_1089_finite__subset__Union,axiom,
    ! [A: set_Product_prod_v_v,B9: set_se8455005133513928103od_v_v] :
      ( ( finite3348123685078250256od_v_v @ A )
     => ( ( ord_le7336532860387713383od_v_v @ A @ ( comple5788137035815166516od_v_v @ B9 ) )
       => ~ ! [F5: set_se8455005133513928103od_v_v] :
              ( ( finite6084192165098772208od_v_v @ F5 )
             => ( ( ord_le4714265922333009223od_v_v @ F5 @ B9 )
               => ~ ( ord_le7336532860387713383od_v_v @ A @ ( comple5788137035815166516od_v_v @ F5 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_1090_finite__subset__Union,axiom,
    ! [A: set_v,B9: set_set_v] :
      ( ( finite_finite_v @ A )
     => ( ( ord_less_eq_set_v @ A @ ( comple2307003700295860064_set_v @ B9 ) )
       => ~ ! [F5: set_set_v] :
              ( ( finite_finite_set_v @ F5 )
             => ( ( ord_le5216385588623774835_set_v @ F5 @ B9 )
               => ~ ( ord_less_eq_set_v @ A @ ( comple2307003700295860064_set_v @ F5 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_1091_Sup__inf__eq__bot__iff,axiom,
    ! [B: set_se8455005133513928103od_v_v,A3: set_Product_prod_v_v] :
      ( ( ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ B ) @ A3 )
        = bot_bo723834152578015283od_v_v )
      = ( ! [X3: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X3 @ B )
           => ( ( inf_in6271465464967711157od_v_v @ X3 @ A3 )
              = bot_bo723834152578015283od_v_v ) ) ) ) ).

% Sup_inf_eq_bot_iff
thf(fact_1092_Sup__inf__eq__bot__iff,axiom,
    ! [B: set_set_v,A3: set_v] :
      ( ( ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ B ) @ A3 )
        = bot_bot_set_v )
      = ( ! [X3: set_v] :
            ( ( member_set_v @ X3 @ B )
           => ( ( inf_inf_set_v @ X3 @ A3 )
              = bot_bot_set_v ) ) ) ) ).

% Sup_inf_eq_bot_iff
thf(fact_1093_chains__extend,axiom,
    ! [C: set_se8455005133513928103od_v_v,S: set_se8455005133513928103od_v_v,Z: set_Product_prod_v_v] :
      ( ( member5511408251247217616od_v_v @ C @ ( chains2766520108962750135od_v_v @ S ) )
     => ( ( member8406446414694345712od_v_v @ Z @ S )
       => ( ! [X4: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ X4 @ C )
             => ( ord_le7336532860387713383od_v_v @ X4 @ Z ) )
         => ( member5511408251247217616od_v_v @ ( sup_su335656005089752955od_v_v @ ( insert7504383016908236695od_v_v @ Z @ bot_bo3497076220358800403od_v_v ) @ C ) @ ( chains2766520108962750135od_v_v @ S ) ) ) ) ) ).

% chains_extend
thf(fact_1094_chains__extend,axiom,
    ! [C: set_set_v,S: set_set_v,Z: set_v] :
      ( ( member_set_set_v @ C @ ( chains_v2 @ S ) )
     => ( ( member_set_v @ Z @ S )
       => ( ! [X4: set_v] :
              ( ( member_set_v @ X4 @ C )
             => ( ord_less_eq_set_v @ X4 @ Z ) )
         => ( member_set_set_v @ ( sup_sup_set_set_v @ ( insert_set_v @ Z @ bot_bot_set_set_v ) @ C ) @ ( chains_v2 @ S ) ) ) ) ) ).

% chains_extend
thf(fact_1095_wo__rel_Oofilter__AboveS__Field,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A: set_Product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( order_1752595321582732413od_v_v @ R @ A )
       => ( ( sup_su414716646722978715od_v_v @ A @ ( order_2660712733480978866od_v_v @ R @ A ) )
          = ( field_7153129647634986036od_v_v @ R ) ) ) ) ).

% wo_rel.ofilter_AboveS_Field
thf(fact_1096_cSup__inter__less__eq,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( condit8801863763314316331od_v_v @ A )
     => ( ( condit8801863763314316331od_v_v @ B )
       => ( ( ( inf_in6058586722421118357od_v_v @ A @ B )
           != bot_bo3497076220358800403od_v_v )
         => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( inf_in6058586722421118357od_v_v @ A @ B ) ) @ ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ) ) ) ).

% cSup_inter_less_eq
thf(fact_1097_cSup__inter__less__eq,axiom,
    ! [A: set_set_v,B: set_set_v] :
      ( ( condit3373647431937589335_set_v @ A )
     => ( ( condit3373647431937589335_set_v @ B )
       => ( ( ( inf_inf_set_set_v @ A @ B )
           != bot_bot_set_set_v )
         => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( inf_inf_set_set_v @ A @ B ) ) @ ( sup_sup_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ) ) ) ).

% cSup_inter_less_eq
thf(fact_1098_bdd__above_OI,axiom,
    ! [A: set_se8455005133513928103od_v_v,M3: set_Product_prod_v_v] :
      ( ! [X4: set_Product_prod_v_v] :
          ( ( member8406446414694345712od_v_v @ X4 @ A )
         => ( ord_le7336532860387713383od_v_v @ X4 @ M3 ) )
     => ( condit8801863763314316331od_v_v @ A ) ) ).

% bdd_above.I
thf(fact_1099_bdd__above_OI,axiom,
    ! [A: set_set_v,M3: set_v] :
      ( ! [X4: set_v] :
          ( ( member_set_v @ X4 @ A )
         => ( ord_less_eq_set_v @ X4 @ M3 ) )
     => ( condit3373647431937589335_set_v @ A ) ) ).

% bdd_above.I
thf(fact_1100_bdd__above_OE,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ( condit8801863763314316331od_v_v @ A )
     => ~ ! [M2: set_Product_prod_v_v] :
            ~ ! [X: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ X @ A )
               => ( ord_le7336532860387713383od_v_v @ X @ M2 ) ) ) ).

% bdd_above.E
thf(fact_1101_bdd__above_OE,axiom,
    ! [A: set_set_v] :
      ( ( condit3373647431937589335_set_v @ A )
     => ~ ! [M2: set_v] :
            ~ ! [X: set_v] :
                ( ( member_set_v @ X @ A )
               => ( ord_less_eq_set_v @ X @ M2 ) ) ) ).

% bdd_above.E
thf(fact_1102_bdd__above_Ounfold,axiom,
    ( condit8801863763314316331od_v_v
    = ( ^ [A4: set_se8455005133513928103od_v_v] :
        ? [M4: set_Product_prod_v_v] :
        ! [X3: set_Product_prod_v_v] :
          ( ( member8406446414694345712od_v_v @ X3 @ A4 )
         => ( ord_le7336532860387713383od_v_v @ X3 @ M4 ) ) ) ) ).

% bdd_above.unfold
thf(fact_1103_bdd__above_Ounfold,axiom,
    ( condit3373647431937589335_set_v
    = ( ^ [A4: set_set_v] :
        ? [M4: set_v] :
        ! [X3: set_v] :
          ( ( member_set_v @ X3 @ A4 )
         => ( ord_less_eq_set_v @ X3 @ M4 ) ) ) ) ).

% bdd_above.unfold
thf(fact_1104_cSup__upper2,axiom,
    ! [X2: set_Product_prod_v_v,X5: set_se8455005133513928103od_v_v,Y2: set_Product_prod_v_v] :
      ( ( member8406446414694345712od_v_v @ X2 @ X5 )
     => ( ( ord_le7336532860387713383od_v_v @ Y2 @ X2 )
       => ( ( condit8801863763314316331od_v_v @ X5 )
         => ( ord_le7336532860387713383od_v_v @ Y2 @ ( comple5788137035815166516od_v_v @ X5 ) ) ) ) ) ).

% cSup_upper2
thf(fact_1105_cSup__upper2,axiom,
    ! [X2: set_v,X5: set_set_v,Y2: set_v] :
      ( ( member_set_v @ X2 @ X5 )
     => ( ( ord_less_eq_set_v @ Y2 @ X2 )
       => ( ( condit3373647431937589335_set_v @ X5 )
         => ( ord_less_eq_set_v @ Y2 @ ( comple2307003700295860064_set_v @ X5 ) ) ) ) ) ).

% cSup_upper2
thf(fact_1106_cSup__upper,axiom,
    ! [X2: set_Product_prod_v_v,X5: set_se8455005133513928103od_v_v] :
      ( ( member8406446414694345712od_v_v @ X2 @ X5 )
     => ( ( condit8801863763314316331od_v_v @ X5 )
       => ( ord_le7336532860387713383od_v_v @ X2 @ ( comple5788137035815166516od_v_v @ X5 ) ) ) ) ).

% cSup_upper
thf(fact_1107_cSup__upper,axiom,
    ! [X2: set_v,X5: set_set_v] :
      ( ( member_set_v @ X2 @ X5 )
     => ( ( condit3373647431937589335_set_v @ X5 )
       => ( ord_less_eq_set_v @ X2 @ ( comple2307003700295860064_set_v @ X5 ) ) ) ) ).

% cSup_upper
thf(fact_1108_Zorn__Lemma2,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ! [X4: set_se8455005133513928103od_v_v] :
          ( ( member5511408251247217616od_v_v @ X4 @ ( chains2766520108962750135od_v_v @ A ) )
         => ? [Xa: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ Xa @ A )
              & ! [Xb: set_Product_prod_v_v] :
                  ( ( member8406446414694345712od_v_v @ Xb @ X4 )
                 => ( ord_le7336532860387713383od_v_v @ Xb @ Xa ) ) ) )
     => ? [X4: set_Product_prod_v_v] :
          ( ( member8406446414694345712od_v_v @ X4 @ A )
          & ! [Xa: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ Xa @ A )
             => ( ( ord_le7336532860387713383od_v_v @ X4 @ Xa )
               => ( Xa = X4 ) ) ) ) ) ).

% Zorn_Lemma2
thf(fact_1109_Zorn__Lemma2,axiom,
    ! [A: set_set_v] :
      ( ! [X4: set_set_v] :
          ( ( member_set_set_v @ X4 @ ( chains_v2 @ A ) )
         => ? [Xa: set_v] :
              ( ( member_set_v @ Xa @ A )
              & ! [Xb: set_v] :
                  ( ( member_set_v @ Xb @ X4 )
                 => ( ord_less_eq_set_v @ Xb @ Xa ) ) ) )
     => ? [X4: set_v] :
          ( ( member_set_v @ X4 @ A )
          & ! [Xa: set_v] :
              ( ( member_set_v @ Xa @ A )
             => ( ( ord_less_eq_set_v @ X4 @ Xa )
               => ( Xa = X4 ) ) ) ) ) ).

% Zorn_Lemma2
thf(fact_1110_chainsD,axiom,
    ! [C: set_se8455005133513928103od_v_v,S: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( member5511408251247217616od_v_v @ C @ ( chains2766520108962750135od_v_v @ S ) )
     => ( ( member8406446414694345712od_v_v @ X2 @ C )
       => ( ( member8406446414694345712od_v_v @ Y2 @ C )
         => ( ( ord_le7336532860387713383od_v_v @ X2 @ Y2 )
            | ( ord_le7336532860387713383od_v_v @ Y2 @ X2 ) ) ) ) ) ).

% chainsD
thf(fact_1111_chainsD,axiom,
    ! [C: set_set_v,S: set_set_v,X2: set_v,Y2: set_v] :
      ( ( member_set_set_v @ C @ ( chains_v2 @ S ) )
     => ( ( member_set_v @ X2 @ C )
       => ( ( member_set_v @ Y2 @ C )
         => ( ( ord_less_eq_set_v @ X2 @ Y2 )
            | ( ord_less_eq_set_v @ Y2 @ X2 ) ) ) ) ) ).

% chainsD
thf(fact_1112_cSup__le__iff,axiom,
    ! [S: set_se8455005133513928103od_v_v,A3: set_Product_prod_v_v] :
      ( ( S != bot_bo3497076220358800403od_v_v )
     => ( ( condit8801863763314316331od_v_v @ S )
       => ( ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ S ) @ A3 )
          = ( ! [X3: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ X3 @ S )
               => ( ord_le7336532860387713383od_v_v @ X3 @ A3 ) ) ) ) ) ) ).

% cSup_le_iff
thf(fact_1113_cSup__le__iff,axiom,
    ! [S: set_set_v,A3: set_v] :
      ( ( S != bot_bot_set_set_v )
     => ( ( condit3373647431937589335_set_v @ S )
       => ( ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ S ) @ A3 )
          = ( ! [X3: set_v] :
                ( ( member_set_v @ X3 @ S )
               => ( ord_less_eq_set_v @ X3 @ A3 ) ) ) ) ) ) ).

% cSup_le_iff
thf(fact_1114_cSup__mono,axiom,
    ! [B: set_se8455005133513928103od_v_v,A: set_se8455005133513928103od_v_v] :
      ( ( B != bot_bo3497076220358800403od_v_v )
     => ( ( condit8801863763314316331od_v_v @ A )
       => ( ! [B7: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ B7 @ B )
             => ? [X: set_Product_prod_v_v] :
                  ( ( member8406446414694345712od_v_v @ X @ A )
                  & ( ord_le7336532860387713383od_v_v @ B7 @ X ) ) )
         => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ B ) @ ( comple5788137035815166516od_v_v @ A ) ) ) ) ) ).

% cSup_mono
thf(fact_1115_cSup__mono,axiom,
    ! [B: set_set_v,A: set_set_v] :
      ( ( B != bot_bot_set_set_v )
     => ( ( condit3373647431937589335_set_v @ A )
       => ( ! [B7: set_v] :
              ( ( member_set_v @ B7 @ B )
             => ? [X: set_v] :
                  ( ( member_set_v @ X @ A )
                  & ( ord_less_eq_set_v @ B7 @ X ) ) )
         => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ B ) @ ( comple2307003700295860064_set_v @ A ) ) ) ) ) ).

% cSup_mono
thf(fact_1116_AboveS__Field,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_2660712733480978866od_v_v @ R @ A ) @ ( field_7153129647634986036od_v_v @ R ) ) ).

% AboveS_Field
thf(fact_1117_AboveS__Field,axiom,
    ! [R: set_Product_prod_v_v,A: set_v] : ( ord_less_eq_set_v @ ( order_AboveS_v @ R @ A ) @ ( field_v @ R ) ) ).

% AboveS_Field
thf(fact_1118_AboveS__disjoint,axiom,
    ! [A: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ A @ ( order_2660712733480978866od_v_v @ R @ A ) )
      = bot_bo723834152578015283od_v_v ) ).

% AboveS_disjoint
thf(fact_1119_AboveS__disjoint,axiom,
    ! [A: set_v,R: set_Product_prod_v_v] :
      ( ( inf_inf_set_v @ A @ ( order_AboveS_v @ R @ A ) )
      = bot_bot_set_v ) ).

% AboveS_disjoint
thf(fact_1120_cSup__subset__mono,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( A != bot_bo3497076220358800403od_v_v )
     => ( ( condit8801863763314316331od_v_v @ B )
       => ( ( ord_le4714265922333009223od_v_v @ A @ B )
         => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ) ) ).

% cSup_subset_mono
thf(fact_1121_cSup__subset__mono,axiom,
    ! [A: set_set_v,B: set_set_v] :
      ( ( A != bot_bot_set_set_v )
     => ( ( condit3373647431937589335_set_v @ B )
       => ( ( ord_le5216385588623774835_set_v @ A @ B )
         => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ) ) ).

% cSup_subset_mono
thf(fact_1122_cSup__insert__If,axiom,
    ! [X5: set_se8455005133513928103od_v_v,A3: set_Product_prod_v_v] :
      ( ( condit8801863763314316331od_v_v @ X5 )
     => ( ( ( X5 = bot_bo3497076220358800403od_v_v )
         => ( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A3 @ X5 ) )
            = A3 ) )
        & ( ( X5 != bot_bo3497076220358800403od_v_v )
         => ( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A3 @ X5 ) )
            = ( sup_su414716646722978715od_v_v @ A3 @ ( comple5788137035815166516od_v_v @ X5 ) ) ) ) ) ) ).

% cSup_insert_If
thf(fact_1123_cSup__insert,axiom,
    ! [X5: set_se8455005133513928103od_v_v,A3: set_Product_prod_v_v] :
      ( ( X5 != bot_bo3497076220358800403od_v_v )
     => ( ( condit8801863763314316331od_v_v @ X5 )
       => ( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A3 @ X5 ) )
          = ( sup_su414716646722978715od_v_v @ A3 @ ( comple5788137035815166516od_v_v @ X5 ) ) ) ) ) ).

% cSup_insert
thf(fact_1124_cSup__union__distrib,axiom,
    ! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( A != bot_bo3497076220358800403od_v_v )
     => ( ( condit8801863763314316331od_v_v @ A )
       => ( ( B != bot_bo3497076220358800403od_v_v )
         => ( ( condit8801863763314316331od_v_v @ B )
           => ( ( comple5788137035815166516od_v_v @ ( sup_su335656005089752955od_v_v @ A @ B ) )
              = ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ) ) ) ) ).

% cSup_union_distrib
thf(fact_1125_Zorn__Lemma,axiom,
    ! [A: set_se8455005133513928103od_v_v] :
      ( ! [X4: set_se8455005133513928103od_v_v] :
          ( ( member5511408251247217616od_v_v @ X4 @ ( chains2766520108962750135od_v_v @ A ) )
         => ( member8406446414694345712od_v_v @ ( comple5788137035815166516od_v_v @ X4 ) @ A ) )
     => ? [X4: set_Product_prod_v_v] :
          ( ( member8406446414694345712od_v_v @ X4 @ A )
          & ! [Xa: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ Xa @ A )
             => ( ( ord_le7336532860387713383od_v_v @ X4 @ Xa )
               => ( Xa = X4 ) ) ) ) ) ).

% Zorn_Lemma
thf(fact_1126_Zorn__Lemma,axiom,
    ! [A: set_set_v] :
      ( ! [X4: set_set_v] :
          ( ( member_set_set_v @ X4 @ ( chains_v2 @ A ) )
         => ( member_set_v @ ( comple2307003700295860064_set_v @ X4 ) @ A ) )
     => ? [X4: set_v] :
          ( ( member_set_v @ X4 @ A )
          & ! [Xa: set_v] :
              ( ( member_set_v @ Xa @ A )
             => ( ( ord_less_eq_set_v @ X4 @ Xa )
               => ( Xa = X4 ) ) ) ) ) ).

% Zorn_Lemma
thf(fact_1127_wo__rel_Osuc__greater,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( ( order_2660712733480978866od_v_v @ R @ B )
           != bot_bo723834152578015283od_v_v )
         => ( ( member7453568604450474000od_v_v @ B2 @ B )
           => ( ( ( bNF_We8019333048942157384od_v_v @ R @ B )
               != B2 )
              & ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ ( bNF_We8019333048942157384od_v_v @ R @ B ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_1128_wo__rel_Osuc__greater,axiom,
    ! [R: set_Product_prod_v_v,B: set_v,B2: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
       => ( ( ( order_AboveS_v @ R @ B )
           != bot_bot_set_v )
         => ( ( member_v @ B2 @ B )
           => ( ( ( bNF_We6154283375207884916_suc_v @ R @ B )
               != B2 )
              & ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ ( bNF_We6154283375207884916_suc_v @ R @ B ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_1129_wo__rel_Osuc__ofilter__in,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A: set_Product_prod_v_v,B2: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( order_1752595321582732413od_v_v @ R @ A )
       => ( ( ( order_2660712733480978866od_v_v @ R @ A )
           != bot_bo723834152578015283od_v_v )
         => ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ ( bNF_We8019333048942157384od_v_v @ R @ A ) ) @ R )
           => ( ( B2
               != ( bNF_We8019333048942157384od_v_v @ R @ A ) )
             => ( member7453568604450474000od_v_v @ B2 @ A ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_1130_wo__rel_Osuc__ofilter__in,axiom,
    ! [R: set_Product_prod_v_v,A: set_v,B2: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( order_ofilter_v @ R @ A )
       => ( ( ( order_AboveS_v @ R @ A )
           != bot_bot_set_v )
         => ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ ( bNF_We6154283375207884916_suc_v @ R @ A ) ) @ R )
           => ( ( B2
               != ( bNF_We6154283375207884916_suc_v @ R @ A ) )
             => ( member_v @ B2 @ A ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_1131_wo__rel_Osuc__least__AboveS,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,A3: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( member7453568604450474000od_v_v @ A3 @ ( order_2660712733480978866od_v_v @ R @ B ) )
       => ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ ( bNF_We8019333048942157384od_v_v @ R @ B ) @ A3 ) @ R ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_1132_wo__rel_Osuc__least__AboveS,axiom,
    ! [R: set_Product_prod_v_v,A3: v,B: set_v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( member_v @ A3 @ ( order_AboveS_v @ R @ B ) )
       => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ ( bNF_We6154283375207884916_suc_v @ R @ B ) @ A3 ) @ R ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_1133_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v,A3: product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( member7453568604450474000od_v_v @ A3 @ ( order_2660712733480978866od_v_v @ R @ B ) )
         => ( ! [A11: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ A11 @ ( order_2660712733480978866od_v_v @ R @ B ) )
               => ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A3 @ A11 ) @ R ) )
           => ( A3
              = ( bNF_We8019333048942157384od_v_v @ R @ B ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_1134_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Product_prod_v_v,B: set_v,A3: v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
       => ( ( member_v @ A3 @ ( order_AboveS_v @ R @ B ) )
         => ( ! [A11: v] :
                ( ( member_v @ A11 @ ( order_AboveS_v @ R @ B ) )
               => ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A3 @ A11 ) @ R ) )
           => ( A3
              = ( bNF_We6154283375207884916_suc_v @ R @ B ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_1135_wo__rel_Osuc__inField,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( ( order_2660712733480978866od_v_v @ R @ B )
           != bot_bo723834152578015283od_v_v )
         => ( member7453568604450474000od_v_v @ ( bNF_We8019333048942157384od_v_v @ R @ B ) @ ( field_7153129647634986036od_v_v @ R ) ) ) ) ) ).

% wo_rel.suc_inField
thf(fact_1136_wo__rel_Osuc__inField,axiom,
    ! [R: set_Product_prod_v_v,B: set_v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
       => ( ( ( order_AboveS_v @ R @ B )
           != bot_bot_set_v )
         => ( member_v @ ( bNF_We6154283375207884916_suc_v @ R @ B ) @ ( field_v @ R ) ) ) ) ) ).

% wo_rel.suc_inField
thf(fact_1137_wo__rel_Osuc__AboveS,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
      ( ( bNF_We8684021896157364691od_v_v @ R )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
       => ( ( ( order_2660712733480978866od_v_v @ R @ B )
           != bot_bo723834152578015283od_v_v )
         => ( member7453568604450474000od_v_v @ ( bNF_We8019333048942157384od_v_v @ R @ B ) @ ( order_2660712733480978866od_v_v @ R @ B ) ) ) ) ) ).

% wo_rel.suc_AboveS
thf(fact_1138_wo__rel_Osuc__AboveS,axiom,
    ! [R: set_Product_prod_v_v,B: set_v] :
      ( ( bNF_We1162827675446710015_rel_v @ R )
     => ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
       => ( ( ( order_AboveS_v @ R @ B )
           != bot_bot_set_v )
         => ( member_v @ ( bNF_We6154283375207884916_suc_v @ R @ B ) @ ( order_AboveS_v @ R @ B ) ) ) ) ) ).

% wo_rel.suc_AboveS
thf(fact_1139_cSUP__union,axiom,
    ! [A: set_Product_prod_v_v,F: product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( A != bot_bo723834152578015283od_v_v )
     => ( ( condit8801863763314316331od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) )
       => ( ( B != bot_bo723834152578015283od_v_v )
         => ( ( condit8801863763314316331od_v_v @ ( image_145992280190715813od_v_v @ F @ B ) )
           => ( ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ ( sup_su414716646722978715od_v_v @ A @ B ) ) )
              = ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) ) @ ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ B ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_1140_cSUP__union,axiom,
    ! [A: set_v,F: v > set_Product_prod_v_v,B: set_v] :
      ( ( A != bot_bot_set_v )
     => ( ( condit8801863763314316331od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) )
       => ( ( B != bot_bot_set_v )
         => ( ( condit8801863763314316331od_v_v @ ( image_181480991005670265od_v_v @ F @ B ) )
           => ( ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ ( sup_sup_set_v @ A @ B ) ) )
              = ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) ) @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ B ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_1141_cSUP__insert,axiom,
    ! [A: set_Product_prod_v_v,F: product_prod_v_v > set_Product_prod_v_v,A3: product_prod_v_v] :
      ( ( A != bot_bo723834152578015283od_v_v )
     => ( ( condit8801863763314316331od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) )
       => ( ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ ( insert1338601472111419319od_v_v @ A3 @ A ) ) )
          = ( sup_su414716646722978715od_v_v @ ( F @ A3 ) @ ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_1142_cSUP__insert,axiom,
    ! [A: set_v,F: v > set_Product_prod_v_v,A3: v] :
      ( ( A != bot_bot_set_v )
     => ( ( condit8801863763314316331od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) )
       => ( ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ ( insert_v2 @ A3 @ A ) ) )
          = ( sup_su414716646722978715od_v_v @ ( F @ A3 ) @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_1143_image__eqI,axiom,
    ! [B2: v,F: v > v,X2: v,A: set_v] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_v @ X2 @ A )
       => ( member_v @ B2 @ ( image_v_v @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_1144_image__eqI,axiom,
    ! [B2: product_prod_v_v,F: v > product_prod_v_v,X2: v,A: set_v] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_v @ X2 @ A )
       => ( member7453568604450474000od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_1145_image__eqI,axiom,
    ! [B2: v,F: product_prod_v_v > v,X2: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member7453568604450474000od_v_v @ X2 @ A )
       => ( member_v @ B2 @ ( image_6152814753742948081_v_v_v @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_1146_image__eqI,axiom,
    ! [B2: product_prod_v_v,F: product_prod_v_v > product_prod_v_v,X2: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member7453568604450474000od_v_v @ X2 @ A )
       => ( member7453568604450474000od_v_v @ B2 @ ( image_781944334261467077od_v_v @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_1147_image__is__empty,axiom,
    ! [F: product_prod_v_v > product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ( image_781944334261467077od_v_v @ F @ A )
        = bot_bo723834152578015283od_v_v )
      = ( A = bot_bo723834152578015283od_v_v ) ) ).

% image_is_empty
thf(fact_1148_image__is__empty,axiom,
    ! [F: v > product_prod_v_v,A: set_v] :
      ( ( ( image_9222788639401671577od_v_v @ F @ A )
        = bot_bo723834152578015283od_v_v )
      = ( A = bot_bot_set_v ) ) ).

% image_is_empty
thf(fact_1149_image__is__empty,axiom,
    ! [F: product_prod_v_v > v,A: set_Product_prod_v_v] :
      ( ( ( image_6152814753742948081_v_v_v @ F @ A )
        = bot_bot_set_v )
      = ( A = bot_bo723834152578015283od_v_v ) ) ).

% image_is_empty
thf(fact_1150_image__is__empty,axiom,
    ! [F: v > v,A: set_v] :
      ( ( ( image_v_v @ F @ A )
        = bot_bot_set_v )
      = ( A = bot_bot_set_v ) ) ).

% image_is_empty
thf(fact_1151_empty__is__image,axiom,
    ! [F: product_prod_v_v > product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( image_781944334261467077od_v_v @ F @ A ) )
      = ( A = bot_bo723834152578015283od_v_v ) ) ).

% empty_is_image
thf(fact_1152_empty__is__image,axiom,
    ! [F: v > product_prod_v_v,A: set_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( image_9222788639401671577od_v_v @ F @ A ) )
      = ( A = bot_bot_set_v ) ) ).

% empty_is_image
thf(fact_1153_empty__is__image,axiom,
    ! [F: product_prod_v_v > v,A: set_Product_prod_v_v] :
      ( ( bot_bot_set_v
        = ( image_6152814753742948081_v_v_v @ F @ A ) )
      = ( A = bot_bo723834152578015283od_v_v ) ) ).

% empty_is_image
thf(fact_1154_empty__is__image,axiom,
    ! [F: v > v,A: set_v] :
      ( ( bot_bot_set_v
        = ( image_v_v @ F @ A ) )
      = ( A = bot_bot_set_v ) ) ).

% empty_is_image
thf(fact_1155_image__empty,axiom,
    ! [F: product_prod_v_v > product_prod_v_v] :
      ( ( image_781944334261467077od_v_v @ F @ bot_bo723834152578015283od_v_v )
      = bot_bo723834152578015283od_v_v ) ).

% image_empty
thf(fact_1156_image__empty,axiom,
    ! [F: product_prod_v_v > v] :
      ( ( image_6152814753742948081_v_v_v @ F @ bot_bo723834152578015283od_v_v )
      = bot_bot_set_v ) ).

% image_empty
thf(fact_1157_image__empty,axiom,
    ! [F: v > product_prod_v_v] :
      ( ( image_9222788639401671577od_v_v @ F @ bot_bot_set_v )
      = bot_bo723834152578015283od_v_v ) ).

% image_empty
thf(fact_1158_image__empty,axiom,
    ! [F: v > v] :
      ( ( image_v_v @ F @ bot_bot_set_v )
      = bot_bot_set_v ) ).

% image_empty
thf(fact_1159_finite__imageI,axiom,
    ! [F3: set_v,H: v > v] :
      ( ( finite_finite_v @ F3 )
     => ( finite_finite_v @ ( image_v_v @ H @ F3 ) ) ) ).

% finite_imageI
thf(fact_1160_insert__image,axiom,
    ! [X2: v,A: set_v,F: v > v] :
      ( ( member_v @ X2 @ A )
     => ( ( insert_v2 @ ( F @ X2 ) @ ( image_v_v @ F @ A ) )
        = ( image_v_v @ F @ A ) ) ) ).

% insert_image
thf(fact_1161_insert__image,axiom,
    ! [X2: v,A: set_v,F: v > product_prod_v_v] :
      ( ( member_v @ X2 @ A )
     => ( ( insert1338601472111419319od_v_v @ ( F @ X2 ) @ ( image_9222788639401671577od_v_v @ F @ A ) )
        = ( image_9222788639401671577od_v_v @ F @ A ) ) ) ).

% insert_image
thf(fact_1162_insert__image,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,F: product_prod_v_v > v] :
      ( ( member7453568604450474000od_v_v @ X2 @ A )
     => ( ( insert_v2 @ ( F @ X2 ) @ ( image_6152814753742948081_v_v_v @ F @ A ) )
        = ( image_6152814753742948081_v_v_v @ F @ A ) ) ) ).

% insert_image
thf(fact_1163_insert__image,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ X2 @ A )
     => ( ( insert1338601472111419319od_v_v @ ( F @ X2 ) @ ( image_781944334261467077od_v_v @ F @ A ) )
        = ( image_781944334261467077od_v_v @ F @ A ) ) ) ).

% insert_image
thf(fact_1164_image__insert,axiom,
    ! [F: v > v,A3: v,B: set_v] :
      ( ( image_v_v @ F @ ( insert_v2 @ A3 @ B ) )
      = ( insert_v2 @ ( F @ A3 ) @ ( image_v_v @ F @ B ) ) ) ).

% image_insert
thf(fact_1165_image__insert,axiom,
    ! [F: v > product_prod_v_v,A3: v,B: set_v] :
      ( ( image_9222788639401671577od_v_v @ F @ ( insert_v2 @ A3 @ B ) )
      = ( insert1338601472111419319od_v_v @ ( F @ A3 ) @ ( image_9222788639401671577od_v_v @ F @ B ) ) ) ).

% image_insert
thf(fact_1166_image__insert,axiom,
    ! [F: product_prod_v_v > v,A3: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( image_6152814753742948081_v_v_v @ F @ ( insert1338601472111419319od_v_v @ A3 @ B ) )
      = ( insert_v2 @ ( F @ A3 ) @ ( image_6152814753742948081_v_v_v @ F @ B ) ) ) ).

% image_insert
thf(fact_1167_image__insert,axiom,
    ! [F: product_prod_v_v > product_prod_v_v,A3: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( image_781944334261467077od_v_v @ F @ ( insert1338601472111419319od_v_v @ A3 @ B ) )
      = ( insert1338601472111419319od_v_v @ ( F @ A3 ) @ ( image_781944334261467077od_v_v @ F @ B ) ) ) ).

% image_insert
thf(fact_1168_finite__UN,axiom,
    ! [A: set_v,B: v > set_v] :
      ( ( finite_finite_v @ A )
     => ( ( finite_finite_v @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ B @ A ) ) )
        = ( ! [X3: v] :
              ( ( member_v @ X3 @ A )
             => ( finite_finite_v @ ( B @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_1169_inf__Sup,axiom,
    ! [A3: set_v,B: set_set_v] :
      ( ( inf_inf_set_v @ A3 @ ( comple2307003700295860064_set_v @ B ) )
      = ( comple2307003700295860064_set_v @ ( image_set_v_set_v @ ( inf_inf_set_v @ A3 ) @ B ) ) ) ).

% inf_Sup
thf(fact_1170_image__Un,axiom,
    ! [F: product_prod_v_v > product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( image_781944334261467077od_v_v @ F @ ( sup_su414716646722978715od_v_v @ A @ B ) )
      = ( sup_su414716646722978715od_v_v @ ( image_781944334261467077od_v_v @ F @ A ) @ ( image_781944334261467077od_v_v @ F @ B ) ) ) ).

% image_Un
thf(fact_1171_image__Int__subset,axiom,
    ! [F: v > product_prod_v_v,A: set_v,B: set_v] : ( ord_le7336532860387713383od_v_v @ ( image_9222788639401671577od_v_v @ F @ ( inf_inf_set_v @ A @ B ) ) @ ( inf_in6271465464967711157od_v_v @ ( image_9222788639401671577od_v_v @ F @ A ) @ ( image_9222788639401671577od_v_v @ F @ B ) ) ) ).

% image_Int_subset
thf(fact_1172_image__Int__subset,axiom,
    ! [F: v > v,A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( image_v_v @ F @ ( inf_inf_set_v @ A @ B ) ) @ ( inf_inf_set_v @ ( image_v_v @ F @ A ) @ ( image_v_v @ F @ B ) ) ) ).

% image_Int_subset
thf(fact_1173_subset__image__iff,axiom,
    ! [B: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B @ ( image_781944334261467077od_v_v @ F @ A ) )
      = ( ? [AA: set_Product_prod_v_v] :
            ( ( ord_le7336532860387713383od_v_v @ AA @ A )
            & ( B
              = ( image_781944334261467077od_v_v @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_1174_subset__image__iff,axiom,
    ! [B: set_Product_prod_v_v,F: v > product_prod_v_v,A: set_v] :
      ( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A ) )
      = ( ? [AA: set_v] :
            ( ( ord_less_eq_set_v @ AA @ A )
            & ( B
              = ( image_9222788639401671577od_v_v @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_1175_subset__image__iff,axiom,
    ! [B: set_v,F: product_prod_v_v > v,A: set_Product_prod_v_v] :
      ( ( ord_less_eq_set_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A ) )
      = ( ? [AA: set_Product_prod_v_v] :
            ( ( ord_le7336532860387713383od_v_v @ AA @ A )
            & ( B
              = ( image_6152814753742948081_v_v_v @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_1176_subset__image__iff,axiom,
    ! [B: set_v,F: v > v,A: set_v] :
      ( ( ord_less_eq_set_v @ B @ ( image_v_v @ F @ A ) )
      = ( ? [AA: set_v] :
            ( ( ord_less_eq_set_v @ AA @ A )
            & ( B
              = ( image_v_v @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_1177_subset__imageE,axiom,
    ! [B: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B @ ( image_781944334261467077od_v_v @ F @ A ) )
     => ~ ! [C4: set_Product_prod_v_v] :
            ( ( ord_le7336532860387713383od_v_v @ C4 @ A )
           => ( B
             != ( image_781944334261467077od_v_v @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_1178_subset__imageE,axiom,
    ! [B: set_Product_prod_v_v,F: v > product_prod_v_v,A: set_v] :
      ( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A ) )
     => ~ ! [C4: set_v] :
            ( ( ord_less_eq_set_v @ C4 @ A )
           => ( B
             != ( image_9222788639401671577od_v_v @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_1179_subset__imageE,axiom,
    ! [B: set_v,F: product_prod_v_v > v,A: set_Product_prod_v_v] :
      ( ( ord_less_eq_set_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A ) )
     => ~ ! [C4: set_Product_prod_v_v] :
            ( ( ord_le7336532860387713383od_v_v @ C4 @ A )
           => ( B
             != ( image_6152814753742948081_v_v_v @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_1180_subset__imageE,axiom,
    ! [B: set_v,F: v > v,A: set_v] :
      ( ( ord_less_eq_set_v @ B @ ( image_v_v @ F @ A ) )
     => ~ ! [C4: set_v] :
            ( ( ord_less_eq_set_v @ C4 @ A )
           => ( B
             != ( image_v_v @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_1181_image__subsetI,axiom,
    ! [A: set_v,F: v > product_prod_v_v,B: set_Product_prod_v_v] :
      ( ! [X4: v] :
          ( ( member_v @ X4 @ A )
         => ( member7453568604450474000od_v_v @ ( F @ X4 ) @ B ) )
     => ( ord_le7336532860387713383od_v_v @ ( image_9222788639401671577od_v_v @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1182_image__subsetI,axiom,
    ! [A: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,B: set_Product_prod_v_v] :
      ( ! [X4: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X4 @ A )
         => ( member7453568604450474000od_v_v @ ( F @ X4 ) @ B ) )
     => ( ord_le7336532860387713383od_v_v @ ( image_781944334261467077od_v_v @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1183_image__subsetI,axiom,
    ! [A: set_v,F: v > v,B: set_v] :
      ( ! [X4: v] :
          ( ( member_v @ X4 @ A )
         => ( member_v @ ( F @ X4 ) @ B ) )
     => ( ord_less_eq_set_v @ ( image_v_v @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1184_image__subsetI,axiom,
    ! [A: set_Product_prod_v_v,F: product_prod_v_v > v,B: set_v] :
      ( ! [X4: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X4 @ A )
         => ( member_v @ ( F @ X4 ) @ B ) )
     => ( ord_less_eq_set_v @ ( image_6152814753742948081_v_v_v @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1185_image__mono,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ord_le7336532860387713383od_v_v @ ( image_781944334261467077od_v_v @ F @ A ) @ ( image_781944334261467077od_v_v @ F @ B ) ) ) ).

% image_mono
thf(fact_1186_image__mono,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: product_prod_v_v > v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ord_less_eq_set_v @ ( image_6152814753742948081_v_v_v @ F @ A ) @ ( image_6152814753742948081_v_v_v @ F @ B ) ) ) ).

% image_mono
thf(fact_1187_image__mono,axiom,
    ! [A: set_v,B: set_v,F: v > product_prod_v_v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ord_le7336532860387713383od_v_v @ ( image_9222788639401671577od_v_v @ F @ A ) @ ( image_9222788639401671577od_v_v @ F @ B ) ) ) ).

% image_mono
thf(fact_1188_image__mono,axiom,
    ! [A: set_v,B: set_v,F: v > v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ord_less_eq_set_v @ ( image_v_v @ F @ A ) @ ( image_v_v @ F @ B ) ) ) ).

% image_mono
thf(fact_1189_all__subset__image,axiom,
    ! [F: product_prod_v_v > product_prod_v_v,A: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
      ( ( ! [B4: set_Product_prod_v_v] :
            ( ( ord_le7336532860387713383od_v_v @ B4 @ ( image_781944334261467077od_v_v @ F @ A ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_Product_prod_v_v] :
            ( ( ord_le7336532860387713383od_v_v @ B4 @ A )
           => ( P @ ( image_781944334261467077od_v_v @ F @ B4 ) ) ) ) ) ).

% all_subset_image
thf(fact_1190_all__subset__image,axiom,
    ! [F: v > product_prod_v_v,A: set_v,P: set_Product_prod_v_v > $o] :
      ( ( ! [B4: set_Product_prod_v_v] :
            ( ( ord_le7336532860387713383od_v_v @ B4 @ ( image_9222788639401671577od_v_v @ F @ A ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_v] :
            ( ( ord_less_eq_set_v @ B4 @ A )
           => ( P @ ( image_9222788639401671577od_v_v @ F @ B4 ) ) ) ) ) ).

% all_subset_image
thf(fact_1191_all__subset__image,axiom,
    ! [F: product_prod_v_v > v,A: set_Product_prod_v_v,P: set_v > $o] :
      ( ( ! [B4: set_v] :
            ( ( ord_less_eq_set_v @ B4 @ ( image_6152814753742948081_v_v_v @ F @ A ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_Product_prod_v_v] :
            ( ( ord_le7336532860387713383od_v_v @ B4 @ A )
           => ( P @ ( image_6152814753742948081_v_v_v @ F @ B4 ) ) ) ) ) ).

% all_subset_image
thf(fact_1192_all__subset__image,axiom,
    ! [F: v > v,A: set_v,P: set_v > $o] :
      ( ( ! [B4: set_v] :
            ( ( ord_less_eq_set_v @ B4 @ ( image_v_v @ F @ A ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_v] :
            ( ( ord_less_eq_set_v @ B4 @ A )
           => ( P @ ( image_v_v @ F @ B4 ) ) ) ) ) ).

% all_subset_image
thf(fact_1193_image__diff__subset,axiom,
    ! [F: v > product_prod_v_v,A: set_v,B: set_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ ( image_9222788639401671577od_v_v @ F @ A ) @ ( image_9222788639401671577od_v_v @ F @ B ) ) @ ( image_9222788639401671577od_v_v @ F @ ( minus_minus_set_v @ A @ B ) ) ) ).

% image_diff_subset
thf(fact_1194_image__diff__subset,axiom,
    ! [F: v > v,A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ ( image_v_v @ F @ A ) @ ( image_v_v @ F @ B ) ) @ ( image_v_v @ F @ ( minus_minus_set_v @ A @ B ) ) ) ).

% image_diff_subset
thf(fact_1195_rev__image__eqI,axiom,
    ! [X2: v,A: set_v,B2: v,F: v > v] :
      ( ( member_v @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_v @ B2 @ ( image_v_v @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1196_rev__image__eqI,axiom,
    ! [X2: v,A: set_v,B2: product_prod_v_v,F: v > product_prod_v_v] :
      ( ( member_v @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member7453568604450474000od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1197_rev__image__eqI,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,B2: v,F: product_prod_v_v > v] :
      ( ( member7453568604450474000od_v_v @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_v @ B2 @ ( image_6152814753742948081_v_v_v @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1198_rev__image__eqI,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,B2: product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member7453568604450474000od_v_v @ B2 @ ( image_781944334261467077od_v_v @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1199_imageI,axiom,
    ! [X2: v,A: set_v,F: v > v] :
      ( ( member_v @ X2 @ A )
     => ( member_v @ ( F @ X2 ) @ ( image_v_v @ F @ A ) ) ) ).

% imageI
thf(fact_1200_imageI,axiom,
    ! [X2: v,A: set_v,F: v > product_prod_v_v] :
      ( ( member_v @ X2 @ A )
     => ( member7453568604450474000od_v_v @ ( F @ X2 ) @ ( image_9222788639401671577od_v_v @ F @ A ) ) ) ).

% imageI
thf(fact_1201_imageI,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,F: product_prod_v_v > v] :
      ( ( member7453568604450474000od_v_v @ X2 @ A )
     => ( member_v @ ( F @ X2 ) @ ( image_6152814753742948081_v_v_v @ F @ A ) ) ) ).

% imageI
thf(fact_1202_imageI,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ X2 @ A )
     => ( member7453568604450474000od_v_v @ ( F @ X2 ) @ ( image_781944334261467077od_v_v @ F @ A ) ) ) ).

% imageI
thf(fact_1203_all__finite__subset__image,axiom,
    ! [F: product_prod_v_v > product_prod_v_v,A: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
      ( ( ! [B4: set_Product_prod_v_v] :
            ( ( ( finite3348123685078250256od_v_v @ B4 )
              & ( ord_le7336532860387713383od_v_v @ B4 @ ( image_781944334261467077od_v_v @ F @ A ) ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_Product_prod_v_v] :
            ( ( ( finite3348123685078250256od_v_v @ B4 )
              & ( ord_le7336532860387713383od_v_v @ B4 @ A ) )
           => ( P @ ( image_781944334261467077od_v_v @ F @ B4 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1204_all__finite__subset__image,axiom,
    ! [F: v > product_prod_v_v,A: set_v,P: set_Product_prod_v_v > $o] :
      ( ( ! [B4: set_Product_prod_v_v] :
            ( ( ( finite3348123685078250256od_v_v @ B4 )
              & ( ord_le7336532860387713383od_v_v @ B4 @ ( image_9222788639401671577od_v_v @ F @ A ) ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_v] :
            ( ( ( finite_finite_v @ B4 )
              & ( ord_less_eq_set_v @ B4 @ A ) )
           => ( P @ ( image_9222788639401671577od_v_v @ F @ B4 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1205_all__finite__subset__image,axiom,
    ! [F: product_prod_v_v > v,A: set_Product_prod_v_v,P: set_v > $o] :
      ( ( ! [B4: set_v] :
            ( ( ( finite_finite_v @ B4 )
              & ( ord_less_eq_set_v @ B4 @ ( image_6152814753742948081_v_v_v @ F @ A ) ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_Product_prod_v_v] :
            ( ( ( finite3348123685078250256od_v_v @ B4 )
              & ( ord_le7336532860387713383od_v_v @ B4 @ A ) )
           => ( P @ ( image_6152814753742948081_v_v_v @ F @ B4 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1206_all__finite__subset__image,axiom,
    ! [F: v > v,A: set_v,P: set_v > $o] :
      ( ( ! [B4: set_v] :
            ( ( ( finite_finite_v @ B4 )
              & ( ord_less_eq_set_v @ B4 @ ( image_v_v @ F @ A ) ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_v] :
            ( ( ( finite_finite_v @ B4 )
              & ( ord_less_eq_set_v @ B4 @ A ) )
           => ( P @ ( image_v_v @ F @ B4 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1207_ex__finite__subset__image,axiom,
    ! [F: product_prod_v_v > product_prod_v_v,A: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
      ( ( ? [B4: set_Product_prod_v_v] :
            ( ( finite3348123685078250256od_v_v @ B4 )
            & ( ord_le7336532860387713383od_v_v @ B4 @ ( image_781944334261467077od_v_v @ F @ A ) )
            & ( P @ B4 ) ) )
      = ( ? [B4: set_Product_prod_v_v] :
            ( ( finite3348123685078250256od_v_v @ B4 )
            & ( ord_le7336532860387713383od_v_v @ B4 @ A )
            & ( P @ ( image_781944334261467077od_v_v @ F @ B4 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1208_ex__finite__subset__image,axiom,
    ! [F: v > product_prod_v_v,A: set_v,P: set_Product_prod_v_v > $o] :
      ( ( ? [B4: set_Product_prod_v_v] :
            ( ( finite3348123685078250256od_v_v @ B4 )
            & ( ord_le7336532860387713383od_v_v @ B4 @ ( image_9222788639401671577od_v_v @ F @ A ) )
            & ( P @ B4 ) ) )
      = ( ? [B4: set_v] :
            ( ( finite_finite_v @ B4 )
            & ( ord_less_eq_set_v @ B4 @ A )
            & ( P @ ( image_9222788639401671577od_v_v @ F @ B4 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1209_ex__finite__subset__image,axiom,
    ! [F: product_prod_v_v > v,A: set_Product_prod_v_v,P: set_v > $o] :
      ( ( ? [B4: set_v] :
            ( ( finite_finite_v @ B4 )
            & ( ord_less_eq_set_v @ B4 @ ( image_6152814753742948081_v_v_v @ F @ A ) )
            & ( P @ B4 ) ) )
      = ( ? [B4: set_Product_prod_v_v] :
            ( ( finite3348123685078250256od_v_v @ B4 )
            & ( ord_le7336532860387713383od_v_v @ B4 @ A )
            & ( P @ ( image_6152814753742948081_v_v_v @ F @ B4 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1210_ex__finite__subset__image,axiom,
    ! [F: v > v,A: set_v,P: set_v > $o] :
      ( ( ? [B4: set_v] :
            ( ( finite_finite_v @ B4 )
            & ( ord_less_eq_set_v @ B4 @ ( image_v_v @ F @ A ) )
            & ( P @ B4 ) ) )
      = ( ? [B4: set_v] :
            ( ( finite_finite_v @ B4 )
            & ( ord_less_eq_set_v @ B4 @ A )
            & ( P @ ( image_v_v @ F @ B4 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1211_finite__subset__image,axiom,
    ! [B: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ B )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( image_781944334261467077od_v_v @ F @ A ) )
       => ? [C4: set_Product_prod_v_v] :
            ( ( ord_le7336532860387713383od_v_v @ C4 @ A )
            & ( finite3348123685078250256od_v_v @ C4 )
            & ( B
              = ( image_781944334261467077od_v_v @ F @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1212_finite__subset__image,axiom,
    ! [B: set_Product_prod_v_v,F: v > product_prod_v_v,A: set_v] :
      ( ( finite3348123685078250256od_v_v @ B )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A ) )
       => ? [C4: set_v] :
            ( ( ord_less_eq_set_v @ C4 @ A )
            & ( finite_finite_v @ C4 )
            & ( B
              = ( image_9222788639401671577od_v_v @ F @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1213_finite__subset__image,axiom,
    ! [B: set_v,F: product_prod_v_v > v,A: set_Product_prod_v_v] :
      ( ( finite_finite_v @ B )
     => ( ( ord_less_eq_set_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A ) )
       => ? [C4: set_Product_prod_v_v] :
            ( ( ord_le7336532860387713383od_v_v @ C4 @ A )
            & ( finite3348123685078250256od_v_v @ C4 )
            & ( B
              = ( image_6152814753742948081_v_v_v @ F @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1214_finite__subset__image,axiom,
    ! [B: set_v,F: v > v,A: set_v] :
      ( ( finite_finite_v @ B )
     => ( ( ord_less_eq_set_v @ B @ ( image_v_v @ F @ A ) )
       => ? [C4: set_v] :
            ( ( ord_less_eq_set_v @ C4 @ A )
            & ( finite_finite_v @ C4 )
            & ( B
              = ( image_v_v @ F @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1215_finite__surj,axiom,
    ! [A: set_v,B: set_Product_prod_v_v,F: v > product_prod_v_v] :
      ( ( finite_finite_v @ A )
     => ( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A ) )
       => ( finite3348123685078250256od_v_v @ B ) ) ) ).

% finite_surj
thf(fact_1216_finite__surj,axiom,
    ! [A: set_v,B: set_v,F: v > v] :
      ( ( finite_finite_v @ A )
     => ( ( ord_less_eq_set_v @ B @ ( image_v_v @ F @ A ) )
       => ( finite_finite_v @ B ) ) ) ).

% finite_surj
thf(fact_1217_Sup__SUP__eq,axiom,
    ( complete_Sup_Sup_v_o
    = ( ^ [S6: set_v_o,X3: v] : ( member_v @ X3 @ ( comple2307003700295860064_set_v @ ( image_v_o_set_v @ collect_v @ S6 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_1218_Sup__SUP__eq,axiom,
    ( comple4105555359904264105_v_v_o
    = ( ^ [S6: set_Pr3689507317551669004_v_v_o,X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ ( comple5788137035815166516od_v_v @ ( image_5458630951929891626od_v_v @ collec140062887454715474od_v_v @ S6 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_1219_SUP__eq,axiom,
    ! [A: set_v,B: set_v,F: v > set_Product_prod_v_v,G2: v > set_Product_prod_v_v] :
      ( ! [I2: v] :
          ( ( member_v @ I2 @ A )
         => ? [X: v] :
              ( ( member_v @ X @ B )
              & ( ord_le7336532860387713383od_v_v @ ( F @ I2 ) @ ( G2 @ X ) ) ) )
     => ( ! [J2: v] :
            ( ( member_v @ J2 @ B )
           => ? [X: v] :
                ( ( member_v @ X @ A )
                & ( ord_le7336532860387713383od_v_v @ ( G2 @ J2 ) @ ( F @ X ) ) ) )
       => ( ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) )
          = ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ G2 @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_1220_SUP__eq,axiom,
    ! [A: set_v,B: set_Product_prod_v_v,F: v > set_Product_prod_v_v,G2: product_prod_v_v > set_Product_prod_v_v] :
      ( ! [I2: v] :
          ( ( member_v @ I2 @ A )
         => ? [X: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ X @ B )
              & ( ord_le7336532860387713383od_v_v @ ( F @ I2 ) @ ( G2 @ X ) ) ) )
     => ( ! [J2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ J2 @ B )
           => ? [X: v] :
                ( ( member_v @ X @ A )
                & ( ord_le7336532860387713383od_v_v @ ( G2 @ J2 ) @ ( F @ X ) ) ) )
       => ( ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) )
          = ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ G2 @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_1221_SUP__eq,axiom,
    ! [A: set_Product_prod_v_v,B: set_v,F: product_prod_v_v > set_Product_prod_v_v,G2: v > set_Product_prod_v_v] :
      ( ! [I2: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ I2 @ A )
         => ? [X: v] :
              ( ( member_v @ X @ B )
              & ( ord_le7336532860387713383od_v_v @ ( F @ I2 ) @ ( G2 @ X ) ) ) )
     => ( ! [J2: v] :
            ( ( member_v @ J2 @ B )
           => ? [X: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X @ A )
                & ( ord_le7336532860387713383od_v_v @ ( G2 @ J2 ) @ ( F @ X ) ) ) )
       => ( ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) )
          = ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ G2 @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_1222_SUP__eq,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: product_prod_v_v > set_Product_prod_v_v,G2: product_prod_v_v > set_Product_prod_v_v] :
      ( ! [I2: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ I2 @ A )
         => ? [X: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ X @ B )
              & ( ord_le7336532860387713383od_v_v @ ( F @ I2 ) @ ( G2 @ X ) ) ) )
     => ( ! [J2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ J2 @ B )
           => ? [X: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X @ A )
                & ( ord_le7336532860387713383od_v_v @ ( G2 @ J2 ) @ ( F @ X ) ) ) )
       => ( ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) )
          = ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ G2 @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_1223_SUP__eq,axiom,
    ! [A: set_v,B: set_v,F: v > set_v,G2: v > set_v] :
      ( ! [I2: v] :
          ( ( member_v @ I2 @ A )
         => ? [X: v] :
              ( ( member_v @ X @ B )
              & ( ord_less_eq_set_v @ ( F @ I2 ) @ ( G2 @ X ) ) ) )
     => ( ! [J2: v] :
            ( ( member_v @ J2 @ B )
           => ? [X: v] :
                ( ( member_v @ X @ A )
                & ( ord_less_eq_set_v @ ( G2 @ J2 ) @ ( F @ X ) ) ) )
       => ( ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) )
          = ( comple2307003700295860064_set_v @ ( image_v_set_v @ G2 @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_1224_SUP__eq,axiom,
    ! [A: set_v,B: set_Product_prod_v_v,F: v > set_v,G2: product_prod_v_v > set_v] :
      ( ! [I2: v] :
          ( ( member_v @ I2 @ A )
         => ? [X: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ X @ B )
              & ( ord_less_eq_set_v @ ( F @ I2 ) @ ( G2 @ X ) ) ) )
     => ( ! [J2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ J2 @ B )
           => ? [X: v] :
                ( ( member_v @ X @ A )
                & ( ord_less_eq_set_v @ ( G2 @ J2 ) @ ( F @ X ) ) ) )
       => ( ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) )
          = ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ G2 @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_1225_SUP__eq,axiom,
    ! [A: set_Product_prod_v_v,B: set_v,F: product_prod_v_v > set_v,G2: v > set_v] :
      ( ! [I2: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ I2 @ A )
         => ? [X: v] :
              ( ( member_v @ X @ B )
              & ( ord_less_eq_set_v @ ( F @ I2 ) @ ( G2 @ X ) ) ) )
     => ( ! [J2: v] :
            ( ( member_v @ J2 @ B )
           => ? [X: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X @ A )
                & ( ord_less_eq_set_v @ ( G2 @ J2 ) @ ( F @ X ) ) ) )
       => ( ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ F @ A ) )
          = ( comple2307003700295860064_set_v @ ( image_v_set_v @ G2 @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_1226_SUP__eq,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: product_prod_v_v > set_v,G2: product_prod_v_v > set_v] :
      ( ! [I2: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ I2 @ A )
         => ? [X: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ X @ B )
              & ( ord_less_eq_set_v @ ( F @ I2 ) @ ( G2 @ X ) ) ) )
     => ( ! [J2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ J2 @ B )
           => ? [X: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X @ A )
                & ( ord_less_eq_set_v @ ( G2 @ J2 ) @ ( F @ X ) ) ) )
       => ( ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ F @ A ) )
          = ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ G2 @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_1227_bdd__above_OI2,axiom,
    ! [A: set_v,F: v > set_Product_prod_v_v,M3: set_Product_prod_v_v] :
      ( ! [X4: v] :
          ( ( member_v @ X4 @ A )
         => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ M3 ) )
     => ( condit8801863763314316331od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) ) ) ).

% bdd_above.I2
thf(fact_1228_bdd__above_OI2,axiom,
    ! [A: set_Product_prod_v_v,F: product_prod_v_v > set_Product_prod_v_v,M3: set_Product_prod_v_v] :
      ( ! [X4: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X4 @ A )
         => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ M3 ) )
     => ( condit8801863763314316331od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) ) ) ).

% bdd_above.I2
thf(fact_1229_bdd__above_OI2,axiom,
    ! [A: set_v,F: v > set_v,M3: set_v] :
      ( ! [X4: v] :
          ( ( member_v @ X4 @ A )
         => ( ord_less_eq_set_v @ ( F @ X4 ) @ M3 ) )
     => ( condit3373647431937589335_set_v @ ( image_v_set_v @ F @ A ) ) ) ).

% bdd_above.I2
thf(fact_1230_bdd__above_OI2,axiom,
    ! [A: set_Product_prod_v_v,F: product_prod_v_v > set_v,M3: set_v] :
      ( ! [X4: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X4 @ A )
         => ( ord_less_eq_set_v @ ( F @ X4 ) @ M3 ) )
     => ( condit3373647431937589335_set_v @ ( image_2529437795422174673_set_v @ F @ A ) ) ) ).

% bdd_above.I2
thf(fact_1231_cSUP__least,axiom,
    ! [A: set_Product_prod_v_v,F: product_prod_v_v > set_Product_prod_v_v,M3: set_Product_prod_v_v] :
      ( ( A != bot_bo723834152578015283od_v_v )
     => ( ! [X4: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X4 @ A )
           => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ M3 ) )
       => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1232_cSUP__least,axiom,
    ! [A: set_v,F: v > set_Product_prod_v_v,M3: set_Product_prod_v_v] :
      ( ( A != bot_bot_set_v )
     => ( ! [X4: v] :
            ( ( member_v @ X4 @ A )
           => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ M3 ) )
       => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1233_cSUP__least,axiom,
    ! [A: set_Product_prod_v_v,F: product_prod_v_v > set_v,M3: set_v] :
      ( ( A != bot_bo723834152578015283od_v_v )
     => ( ! [X4: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X4 @ A )
           => ( ord_less_eq_set_v @ ( F @ X4 ) @ M3 ) )
       => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ F @ A ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1234_cSUP__least,axiom,
    ! [A: set_v,F: v > set_v,M3: set_v] :
      ( ( A != bot_bot_set_v )
     => ( ! [X4: v] :
            ( ( member_v @ X4 @ A )
           => ( ord_less_eq_set_v @ ( F @ X4 ) @ M3 ) )
       => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1235_SUP__eq__iff,axiom,
    ! [I3: set_Product_prod_v_v,C: set_Product_prod_v_v,F: product_prod_v_v > set_Product_prod_v_v] :
      ( ( I3 != bot_bo723834152578015283od_v_v )
     => ( ! [I2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ I2 @ I3 )
           => ( ord_le7336532860387713383od_v_v @ C @ ( F @ I2 ) ) )
       => ( ( ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ I3 ) )
            = C )
          = ( ! [X3: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X3 @ I3 )
               => ( ( F @ X3 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_1236_SUP__eq__iff,axiom,
    ! [I3: set_v,C: set_Product_prod_v_v,F: v > set_Product_prod_v_v] :
      ( ( I3 != bot_bot_set_v )
     => ( ! [I2: v] :
            ( ( member_v @ I2 @ I3 )
           => ( ord_le7336532860387713383od_v_v @ C @ ( F @ I2 ) ) )
       => ( ( ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ I3 ) )
            = C )
          = ( ! [X3: v] :
                ( ( member_v @ X3 @ I3 )
               => ( ( F @ X3 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_1237_SUP__eq__iff,axiom,
    ! [I3: set_Product_prod_v_v,C: set_v,F: product_prod_v_v > set_v] :
      ( ( I3 != bot_bo723834152578015283od_v_v )
     => ( ! [I2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ I2 @ I3 )
           => ( ord_less_eq_set_v @ C @ ( F @ I2 ) ) )
       => ( ( ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ F @ I3 ) )
            = C )
          = ( ! [X3: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X3 @ I3 )
               => ( ( F @ X3 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_1238_SUP__eq__iff,axiom,
    ! [I3: set_v,C: set_v,F: v > set_v] :
      ( ( I3 != bot_bot_set_v )
     => ( ! [I2: v] :
            ( ( member_v @ I2 @ I3 )
           => ( ord_less_eq_set_v @ C @ ( F @ I2 ) ) )
       => ( ( ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ I3 ) )
            = C )
          = ( ! [X3: v] :
                ( ( member_v @ X3 @ I3 )
               => ( ( F @ X3 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_1239_in__image__insert__iff,axiom,
    ! [B: set_se8455005133513928103od_v_v,X2: product_prod_v_v,A: set_Product_prod_v_v] :
      ( ! [C4: set_Product_prod_v_v] :
          ( ( member8406446414694345712od_v_v @ C4 @ B )
         => ~ ( member7453568604450474000od_v_v @ X2 @ C4 ) )
     => ( ( member8406446414694345712od_v_v @ A @ ( image_5212826947168092101od_v_v @ ( insert1338601472111419319od_v_v @ X2 ) @ B ) )
        = ( ( member7453568604450474000od_v_v @ X2 @ A )
          & ( member8406446414694345712od_v_v @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ) ) ).

% in_image_insert_iff
thf(fact_1240_in__image__insert__iff,axiom,
    ! [B: set_set_v,X2: v,A: set_v] :
      ( ! [C4: set_v] :
          ( ( member_set_v @ C4 @ B )
         => ~ ( member_v @ X2 @ C4 ) )
     => ( ( member_set_v @ A @ ( image_set_v_set_v @ ( insert_v2 @ X2 ) @ B ) )
        = ( ( member_v @ X2 @ A )
          & ( member_set_v @ ( minus_minus_set_v @ A @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) @ B ) ) ) ) ).

% in_image_insert_iff
thf(fact_1241_cSUP__upper2,axiom,
    ! [F: v > set_Product_prod_v_v,A: set_v,X2: v,U: set_Product_prod_v_v] :
      ( ( condit8801863763314316331od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) )
     => ( ( member_v @ X2 @ A )
       => ( ( ord_le7336532860387713383od_v_v @ U @ ( F @ X2 ) )
         => ( ord_le7336532860387713383od_v_v @ U @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) ) ) ) ) ) ).

% cSUP_upper2
thf(fact_1242_cSUP__upper2,axiom,
    ! [F: product_prod_v_v > set_Product_prod_v_v,A: set_Product_prod_v_v,X2: product_prod_v_v,U: set_Product_prod_v_v] :
      ( ( condit8801863763314316331od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) )
     => ( ( member7453568604450474000od_v_v @ X2 @ A )
       => ( ( ord_le7336532860387713383od_v_v @ U @ ( F @ X2 ) )
         => ( ord_le7336532860387713383od_v_v @ U @ ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) ) ) ) ) ) ).

% cSUP_upper2
thf(fact_1243_cSUP__upper2,axiom,
    ! [F: v > set_v,A: set_v,X2: v,U: set_v] :
      ( ( condit3373647431937589335_set_v @ ( image_v_set_v @ F @ A ) )
     => ( ( member_v @ X2 @ A )
       => ( ( ord_less_eq_set_v @ U @ ( F @ X2 ) )
         => ( ord_less_eq_set_v @ U @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) ) ) ) ) ) ).

% cSUP_upper2
thf(fact_1244_cSUP__upper2,axiom,
    ! [F: product_prod_v_v > set_v,A: set_Product_prod_v_v,X2: product_prod_v_v,U: set_v] :
      ( ( condit3373647431937589335_set_v @ ( image_2529437795422174673_set_v @ F @ A ) )
     => ( ( member7453568604450474000od_v_v @ X2 @ A )
       => ( ( ord_less_eq_set_v @ U @ ( F @ X2 ) )
         => ( ord_less_eq_set_v @ U @ ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ F @ A ) ) ) ) ) ) ).

% cSUP_upper2
thf(fact_1245_cSUP__upper,axiom,
    ! [X2: v,A: set_v,F: v > set_Product_prod_v_v] :
      ( ( member_v @ X2 @ A )
     => ( ( condit8801863763314316331od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) )
       => ( ord_le7336532860387713383od_v_v @ ( F @ X2 ) @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) ) ) ) ) ).

% cSUP_upper
thf(fact_1246_cSUP__upper,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,F: product_prod_v_v > set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ X2 @ A )
     => ( ( condit8801863763314316331od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) )
       => ( ord_le7336532860387713383od_v_v @ ( F @ X2 ) @ ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) ) ) ) ) ).

% cSUP_upper
thf(fact_1247_cSUP__upper,axiom,
    ! [X2: v,A: set_v,F: v > set_v] :
      ( ( member_v @ X2 @ A )
     => ( ( condit3373647431937589335_set_v @ ( image_v_set_v @ F @ A ) )
       => ( ord_less_eq_set_v @ ( F @ X2 ) @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) ) ) ) ) ).

% cSUP_upper
thf(fact_1248_cSUP__upper,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,F: product_prod_v_v > set_v] :
      ( ( member7453568604450474000od_v_v @ X2 @ A )
     => ( ( condit3373647431937589335_set_v @ ( image_2529437795422174673_set_v @ F @ A ) )
       => ( ord_less_eq_set_v @ ( F @ X2 ) @ ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ F @ A ) ) ) ) ) ).

% cSUP_upper
thf(fact_1249_cSUP__le__iff,axiom,
    ! [A: set_Product_prod_v_v,F: product_prod_v_v > set_Product_prod_v_v,U: set_Product_prod_v_v] :
      ( ( A != bot_bo723834152578015283od_v_v )
     => ( ( condit8801863763314316331od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) )
       => ( ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) ) @ U )
          = ( ! [X3: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X3 @ A )
               => ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ U ) ) ) ) ) ) ).

% cSUP_le_iff
thf(fact_1250_cSUP__le__iff,axiom,
    ! [A: set_v,F: v > set_Product_prod_v_v,U: set_Product_prod_v_v] :
      ( ( A != bot_bot_set_v )
     => ( ( condit8801863763314316331od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) )
       => ( ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) ) @ U )
          = ( ! [X3: v] :
                ( ( member_v @ X3 @ A )
               => ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ U ) ) ) ) ) ) ).

% cSUP_le_iff
thf(fact_1251_cSUP__le__iff,axiom,
    ! [A: set_Product_prod_v_v,F: product_prod_v_v > set_v,U: set_v] :
      ( ( A != bot_bo723834152578015283od_v_v )
     => ( ( condit3373647431937589335_set_v @ ( image_2529437795422174673_set_v @ F @ A ) )
       => ( ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ F @ A ) ) @ U )
          = ( ! [X3: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X3 @ A )
               => ( ord_less_eq_set_v @ ( F @ X3 ) @ U ) ) ) ) ) ) ).

% cSUP_le_iff
thf(fact_1252_cSUP__le__iff,axiom,
    ! [A: set_v,F: v > set_v,U: set_v] :
      ( ( A != bot_bot_set_v )
     => ( ( condit3373647431937589335_set_v @ ( image_v_set_v @ F @ A ) )
       => ( ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) ) @ U )
          = ( ! [X3: v] :
                ( ( member_v @ X3 @ A )
               => ( ord_less_eq_set_v @ ( F @ X3 ) @ U ) ) ) ) ) ) ).

% cSUP_le_iff
thf(fact_1253_Inf__fin_Ohom__commute,axiom,
    ! [H: set_v > set_v,N3: set_set_v] :
      ( ! [X4: set_v,Y: set_v] :
          ( ( H @ ( inf_inf_set_v @ X4 @ Y ) )
          = ( inf_inf_set_v @ ( H @ X4 ) @ ( H @ Y ) ) )
     => ( ( finite_finite_set_v @ N3 )
       => ( ( N3 != bot_bot_set_set_v )
         => ( ( H @ ( lattic8209813555532694032_set_v @ N3 ) )
            = ( lattic8209813555532694032_set_v @ ( image_set_v_set_v @ H @ N3 ) ) ) ) ) ) ).

% Inf_fin.hom_commute
thf(fact_1254_Sup__fin_Ohom__commute,axiom,
    ! [H: set_Product_prod_v_v > set_Product_prod_v_v,N3: set_se8455005133513928103od_v_v] :
      ( ! [X4: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
          ( ( H @ ( sup_su414716646722978715od_v_v @ X4 @ Y ) )
          = ( sup_su414716646722978715od_v_v @ ( H @ X4 ) @ ( H @ Y ) ) )
     => ( ( finite6084192165098772208od_v_v @ N3 )
       => ( ( N3 != bot_bo3497076220358800403od_v_v )
         => ( ( H @ ( lattic5151207300795964030od_v_v @ N3 ) )
            = ( lattic5151207300795964030od_v_v @ ( image_5212826947168092101od_v_v @ H @ N3 ) ) ) ) ) ) ).

% Sup_fin.hom_commute
thf(fact_1255_cSUP__subset__mono,axiom,
    ! [A: set_Product_prod_v_v,G2: product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,F: product_prod_v_v > set_Product_prod_v_v] :
      ( ( A != bot_bo723834152578015283od_v_v )
     => ( ( condit8801863763314316331od_v_v @ ( image_145992280190715813od_v_v @ G2 @ B ) )
       => ( ( ord_le7336532860387713383od_v_v @ A @ B )
         => ( ! [X4: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X4 @ A )
               => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
           => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ A ) ) @ ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ G2 @ B ) ) ) ) ) ) ) ).

% cSUP_subset_mono
thf(fact_1256_cSUP__subset__mono,axiom,
    ! [A: set_Product_prod_v_v,G2: product_prod_v_v > set_v,B: set_Product_prod_v_v,F: product_prod_v_v > set_v] :
      ( ( A != bot_bo723834152578015283od_v_v )
     => ( ( condit3373647431937589335_set_v @ ( image_2529437795422174673_set_v @ G2 @ B ) )
       => ( ( ord_le7336532860387713383od_v_v @ A @ B )
         => ( ! [X4: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X4 @ A )
               => ( ord_less_eq_set_v @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
           => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ F @ A ) ) @ ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ G2 @ B ) ) ) ) ) ) ) ).

% cSUP_subset_mono
thf(fact_1257_cSUP__subset__mono,axiom,
    ! [A: set_v,G2: v > set_Product_prod_v_v,B: set_v,F: v > set_Product_prod_v_v] :
      ( ( A != bot_bot_set_v )
     => ( ( condit8801863763314316331od_v_v @ ( image_181480991005670265od_v_v @ G2 @ B ) )
       => ( ( ord_less_eq_set_v @ A @ B )
         => ( ! [X4: v] :
                ( ( member_v @ X4 @ A )
               => ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
           => ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) ) @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ G2 @ B ) ) ) ) ) ) ) ).

% cSUP_subset_mono
thf(fact_1258_cSUP__subset__mono,axiom,
    ! [A: set_v,G2: v > set_v,B: set_v,F: v > set_v] :
      ( ( A != bot_bot_set_v )
     => ( ( condit3373647431937589335_set_v @ ( image_v_set_v @ G2 @ B ) )
       => ( ( ord_less_eq_set_v @ A @ B )
         => ( ! [X4: v] :
                ( ( member_v @ X4 @ A )
               => ( ord_less_eq_set_v @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
           => ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) ) @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ G2 @ B ) ) ) ) ) ) ) ).

% cSUP_subset_mono
thf(fact_1259_Union__image__insert,axiom,
    ! [F: v > set_Product_prod_v_v,A3: v,B: set_v] :
      ( ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ ( insert_v2 @ A3 @ B ) ) )
      = ( sup_su414716646722978715od_v_v @ ( F @ A3 ) @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ B ) ) ) ) ).

% Union_image_insert
thf(fact_1260_Union__image__insert,axiom,
    ! [F: product_prod_v_v > set_Product_prod_v_v,A3: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ ( insert1338601472111419319od_v_v @ A3 @ B ) ) )
      = ( sup_su414716646722978715od_v_v @ ( F @ A3 ) @ ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ B ) ) ) ) ).

% Union_image_insert
thf(fact_1261_Union__image__empty,axiom,
    ! [A: set_Product_prod_v_v,F: product_prod_v_v > set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A @ ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ bot_bo723834152578015283od_v_v ) ) )
      = A ) ).

% Union_image_empty
thf(fact_1262_Union__image__empty,axiom,
    ! [A: set_Product_prod_v_v,F: v > set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ bot_bot_set_v ) ) )
      = A ) ).

% Union_image_empty
thf(fact_1263_UNION__fun__upd,axiom,
    ! [A: product_prod_v_v > set_Product_prod_v_v,I: product_prod_v_v,B: set_Product_prod_v_v,J3: set_Product_prod_v_v] :
      ( ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ ( fun_up5820869682384608285od_v_v @ A @ I @ B ) @ J3 ) )
      = ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ A @ ( minus_4183494784930505774od_v_v @ J3 @ ( insert1338601472111419319od_v_v @ I @ bot_bo723834152578015283od_v_v ) ) ) ) @ ( if_set4279007504652509325od_v_v @ ( member7453568604450474000od_v_v @ I @ J3 ) @ B @ bot_bo723834152578015283od_v_v ) ) ) ).

% UNION_fun_upd
thf(fact_1264_UNION__fun__upd,axiom,
    ! [A: product_prod_v_v > set_v,I: product_prod_v_v,B: set_v,J3: set_Product_prod_v_v] :
      ( ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ ( fun_up5282948658002250313_set_v @ A @ I @ B ) @ J3 ) )
      = ( sup_sup_set_v @ ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ A @ ( minus_4183494784930505774od_v_v @ J3 @ ( insert1338601472111419319od_v_v @ I @ bot_bo723834152578015283od_v_v ) ) ) ) @ ( if_set_v @ ( member7453568604450474000od_v_v @ I @ J3 ) @ B @ bot_bot_set_v ) ) ) ).

% UNION_fun_upd
thf(fact_1265_UNION__fun__upd,axiom,
    ! [A: v > set_Product_prod_v_v,I: v,B: set_Product_prod_v_v,J3: set_v] :
      ( ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ ( fun_up2934991853585745905od_v_v @ A @ I @ B ) @ J3 ) )
      = ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ A @ ( minus_minus_set_v @ J3 @ ( insert_v2 @ I @ bot_bot_set_v ) ) ) ) @ ( if_set4279007504652509325od_v_v @ ( member_v @ I @ J3 ) @ B @ bot_bo723834152578015283od_v_v ) ) ) ).

% UNION_fun_upd
thf(fact_1266_UNION__fun__upd,axiom,
    ! [A: v > set_v,I: v,B: set_v,J3: set_v] :
      ( ( comple2307003700295860064_set_v @ ( image_v_set_v @ ( fun_upd_v_set_v @ A @ I @ B ) @ J3 ) )
      = ( sup_sup_set_v @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ A @ ( minus_minus_set_v @ J3 @ ( insert_v2 @ I @ bot_bot_set_v ) ) ) ) @ ( if_set_v @ ( member_v @ I @ J3 ) @ B @ bot_bot_set_v ) ) ) ).

% UNION_fun_upd
thf(fact_1267_empty__in__Fpow,axiom,
    ! [A: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ bot_bo723834152578015283od_v_v @ ( finite315275465967970893od_v_v @ A ) ) ).

% empty_in_Fpow
thf(fact_1268_empty__in__Fpow,axiom,
    ! [A: set_v] : ( member_set_v @ bot_bot_set_v @ ( finite_Fpow_v @ A ) ) ).

% empty_in_Fpow
thf(fact_1269_Fpow__mono,axiom,
    ! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B )
     => ( ord_le4714265922333009223od_v_v @ ( finite315275465967970893od_v_v @ A ) @ ( finite315275465967970893od_v_v @ B ) ) ) ).

% Fpow_mono
thf(fact_1270_Fpow__mono,axiom,
    ! [A: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A @ B )
     => ( ord_le5216385588623774835_set_v @ ( finite_Fpow_v @ A ) @ ( finite_Fpow_v @ B ) ) ) ).

% Fpow_mono
thf(fact_1271_fun__upd__image,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,F: product_prod_v_v > v,Y2: v] :
      ( ( ( member7453568604450474000od_v_v @ X2 @ A )
       => ( ( image_6152814753742948081_v_v_v @ ( fun_up7876960227022812009_v_v_v @ F @ X2 @ Y2 ) @ A )
          = ( insert_v2 @ Y2 @ ( image_6152814753742948081_v_v_v @ F @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
      & ( ~ ( member7453568604450474000od_v_v @ X2 @ A )
       => ( ( image_6152814753742948081_v_v_v @ ( fun_up7876960227022812009_v_v_v @ F @ X2 @ Y2 ) @ A )
          = ( image_6152814753742948081_v_v_v @ F @ A ) ) ) ) ).

% fun_upd_image
thf(fact_1272_fun__upd__image,axiom,
    ! [X2: product_prod_v_v,A: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,Y2: product_prod_v_v] :
      ( ( ( member7453568604450474000od_v_v @ X2 @ A )
       => ( ( image_781944334261467077od_v_v @ ( fun_up2403976309316201533od_v_v @ F @ X2 @ Y2 ) @ A )
          = ( insert1338601472111419319od_v_v @ Y2 @ ( image_781944334261467077od_v_v @ F @ ( minus_4183494784930505774od_v_v @ A @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
      & ( ~ ( member7453568604450474000od_v_v @ X2 @ A )
       => ( ( image_781944334261467077od_v_v @ ( fun_up2403976309316201533od_v_v @ F @ X2 @ Y2 ) @ A )
          = ( image_781944334261467077od_v_v @ F @ A ) ) ) ) ).

% fun_upd_image
thf(fact_1273_fun__upd__image,axiom,
    ! [X2: v,A: set_v,F: v > v,Y2: v] :
      ( ( ( member_v @ X2 @ A )
       => ( ( image_v_v @ ( fun_upd_v_v @ F @ X2 @ Y2 ) @ A )
          = ( insert_v2 @ Y2 @ ( image_v_v @ F @ ( minus_minus_set_v @ A @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ) )
      & ( ~ ( member_v @ X2 @ A )
       => ( ( image_v_v @ ( fun_upd_v_v @ F @ X2 @ Y2 ) @ A )
          = ( image_v_v @ F @ A ) ) ) ) ).

% fun_upd_image
thf(fact_1274_fun__upd__image,axiom,
    ! [X2: v,A: set_v,F: v > product_prod_v_v,Y2: product_prod_v_v] :
      ( ( ( member_v @ X2 @ A )
       => ( ( image_9222788639401671577od_v_v @ ( fun_up1723562075826759697od_v_v @ F @ X2 @ Y2 ) @ A )
          = ( insert1338601472111419319od_v_v @ Y2 @ ( image_9222788639401671577od_v_v @ F @ ( minus_minus_set_v @ A @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ) )
      & ( ~ ( member_v @ X2 @ A )
       => ( ( image_9222788639401671577od_v_v @ ( fun_up1723562075826759697od_v_v @ F @ X2 @ Y2 ) @ A )
          = ( image_9222788639401671577od_v_v @ F @ A ) ) ) ) ).

% fun_upd_image
thf(fact_1275_SUP__fold__sup,axiom,
    ! [A: set_v,F: v > set_Product_prod_v_v] :
      ( ( finite_finite_v @ A )
     => ( ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) )
        = ( finite3801144243089372476od_v_v @ ( comp_s955718560886654856_v_v_v @ sup_su414716646722978715od_v_v @ F ) @ bot_bo723834152578015283od_v_v @ A ) ) ) ).

% SUP_fold_sup
thf(fact_1276_SUP__fold__sup,axiom,
    ! [A: set_v,F: v > set_v] :
      ( ( finite_finite_v @ A )
     => ( ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) )
        = ( finite_fold_v_set_v @ ( comp_s2238573166010299484et_v_v @ sup_sup_set_v @ F ) @ bot_bot_set_v @ A ) ) ) ).

% SUP_fold_sup

% Helper facts (5)
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( if_set_v @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
    ! [X2: set_v,Y2: set_v] :
      ( ( if_set_v @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_3_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( if_set4279007504652509325od_v_v @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
    ! [X2: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( if_set4279007504652509325od_v_v @ $true @ X2 @ Y2 )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    member_v @ w @ ( sCC_Bl1198488560823802982ed_v_a @ e ) ).

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