%------------------------------------------------------------------------------
% 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_02873_098852__6507342_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1474 ( 687 unt; 189 typ;   0 def)
%            Number of atoms       : 3549 (1496 equ;   0 cnn)
%            Maximal formula atoms :   13 (   2 avg)
%            Number of connectives : 12462 ( 503   ~;  63   |; 317   &;10095   @)
%                                         (   0 <=>;1484  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   6 avg)
%            Number of types       :   23 (  22 usr)
%            Number of type conns  :  573 ( 573   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  170 ( 167 usr;  19 con; 0-9 aty)
%            Number of variables   : 3737 ( 240   ^;3347   !; 150   ?;3737   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 08:54:19.798
%------------------------------------------------------------------------------
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thf(sy_v_successors,type,
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% Relevant facts (1278)
thf(fact_0_e_H,axiom,
    e2 = ea ).

% e'
thf(fact_1_dfs__dfss__rel_Ocong,axiom,
    sCC_Bl907557413677168252_rel_v = sCC_Bl907557413677168252_rel_v ).

% dfs_dfss_rel.cong
thf(fact_2_sub__env__trans,axiom,
    ! [E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
     => ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
       => ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ).

% sub_env_trans
thf(fact_3__092_060open_062sub__env_Ae_Ae_H_H_092_060close_062,axiom,
    sCC_Bl5768913643336123637t_unit @ ea @ e ).

% \<open>sub_env e e''\<close>
thf(fact_4_True,axiom,
    member_v @ w @ ( sCC_Bl157864678168468314t_unit @ ea ) ).

% True
thf(fact_5_fold__congs_I7_J,axiom,
    ! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: list_v,F: list_v > list_v,F2: list_v > list_v] :
      ( ( R = R2 )
     => ( ( ( sCC_Bl8828226123343373779t_unit @ R2 )
          = V )
       => ( ! [V2: list_v] :
              ( ( V = V2 )
             => ( ( F @ V2 )
                = ( F2 @ V2 ) ) )
         => ( ( sCC_Bl349061681862590396t_unit @ F @ R )
            = ( sCC_Bl349061681862590396t_unit @ F2 @ R2 ) ) ) ) ) ).

% fold_congs(7)
thf(fact_6_unfold__congs_I7_J,axiom,
    ! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: list_v,F: list_v > list_v,F2: list_v > list_v] :
      ( ( R = R2 )
     => ( ( ( sCC_Bl8828226123343373779t_unit @ R2 )
          = V )
       => ( ! [V2: list_v] :
              ( ( V2 = V )
             => ( ( F @ V2 )
                = ( F2 @ V2 ) ) )
         => ( ( sCC_Bl349061681862590396t_unit @ F @ R )
            = ( sCC_Bl349061681862590396t_unit @ F2 @ R2 ) ) ) ) ) ).

% unfold_congs(7)
thf(fact_7_select__convs_I7_J,axiom,
    ! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
      ( ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
      = Stack ) ).

% select_convs(7)
thf(fact_8_dfss,axiom,
    ( ( sCC_Bloemen_dfss_v @ successors @ va @ ea )
    = ( sCC_Bloemen_dfss_v @ successors @ va @ e ) ) ).

% dfss
thf(fact_9__092_060open_062v_A_092_060notin_062_Aexplored_Ae_092_060close_062,axiom,
    ~ ( member_v @ va @ ( sCC_Bl157864678168468314t_unit @ ea ) ) ).

% \<open>v \<notin> explored e\<close>
thf(fact_10__092_060open_062v_A_092_060in_062_Avisited_Ae_092_060close_062,axiom,
    member_v @ va @ ( sCC_Bl4645233313691564917t_unit @ ea ) ).

% \<open>v \<in> visited e\<close>
thf(fact_11_calculation_I3_J,axiom,
    sCC_Bl5768913643336123637t_unit @ ea @ ( sCC_Bloemen_dfss_v @ successors @ va @ e ) ).

% calculation(3)
thf(fact_12_local_Owf,axiom,
    sCC_Bl9196236973127232072t_unit @ successors @ ea ).

% local.wf
thf(fact_13_graph_Opost__dfss_Ocong,axiom,
    sCC_Bl6082031138996704384t_unit = sCC_Bl6082031138996704384t_unit ).

% graph.post_dfss.cong
thf(fact_14_ext__inject,axiom,
    ! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit,Root2: v,S2: v > set_v,Explored2: set_v,Visited2: set_v,Vsuccs2: v > set_v,Sccs2: set_set_v,Stack2: list_v,Cstack2: list_v,More2: product_unit] :
      ( ( ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More )
        = ( sCC_Bl8064756265740546429t_unit @ Root2 @ S2 @ Explored2 @ Visited2 @ Vsuccs2 @ Sccs2 @ Stack2 @ Cstack2 @ More2 ) )
      = ( ( Root = Root2 )
        & ( S = S2 )
        & ( Explored = Explored2 )
        & ( Visited = Visited2 )
        & ( Vsuccs = Vsuccs2 )
        & ( Sccs = Sccs2 )
        & ( Stack = Stack2 )
        & ( Cstack = Cstack2 )
        & ( More = More2 ) ) ) ).

% ext_inject
thf(fact_15_post_H_H,axiom,
    sCC_Bl6082031138996704384t_unit @ successors @ va @ e @ ( sCC_Bloemen_dfss_v @ successors @ va @ e ) ).

% post''
thf(fact_16_predfss,axiom,
    sCC_Bl1748261141445803503t_unit @ successors @ va @ ea ).

% predfss
thf(fact_17_select__convs_I4_J,axiom,
    ! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
      ( ( sCC_Bl4645233313691564917t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
      = Visited ) ).

% select_convs(4)
thf(fact_18_select__convs_I3_J,axiom,
    ! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
      ( ( sCC_Bl157864678168468314t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
      = Explored ) ).

% select_convs(3)
thf(fact_19_update__convs_I7_J,axiom,
    ! [Stack2: list_v > list_v,Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
      ( ( sCC_Bl349061681862590396t_unit @ Stack2 @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
      = ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ ( Stack2 @ Stack ) @ Cstack @ More ) ) ).

% update_convs(7)
thf(fact_20_cases__scheme,axiom,
    ! [R: sCC_Bl1394983891496994913t_unit] :
      ~ ! [Root3: v,S3: v > set_v,Explored3: set_v,Visited3: set_v,Vsuccs3: v > set_v,Sccs3: set_set_v,Stack3: list_v,Cstack3: list_v,More3: product_unit] :
          ( R
         != ( sCC_Bl8064756265740546429t_unit @ Root3 @ S3 @ Explored3 @ Visited3 @ Vsuccs3 @ Sccs3 @ Stack3 @ Cstack3 @ More3 ) ) ).

% cases_scheme
thf(fact_21_graph_Owf__env_Ocong,axiom,
    sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).

% graph.wf_env.cong
thf(fact_22_graph_Odfss_Ocong,axiom,
    sCC_Bloemen_dfss_v = sCC_Bloemen_dfss_v ).

% graph.dfss.cong
thf(fact_23__092_060open_062pre__dfss_Av_Ae_H_H_092_060close_062,axiom,
    sCC_Bl1748261141445803503t_unit @ successors @ va @ e ).

% \<open>pre_dfss v e''\<close>
thf(fact_24_stack__unexplored,axiom,
    ! [E: sCC_Bl1394983891496994913t_unit,N: v] :
      ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
     => ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
       => ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ).

% stack_unexplored
thf(fact_25_stack__visited,axiom,
    ! [E: sCC_Bl1394983891496994913t_unit,N: v] :
      ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
     => ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
       => ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ).

% stack_visited
thf(fact_26_e_H__def,axiom,
    ( ( ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ ea ) )
     => ( e2 = ea ) )
    & ( ~ ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ ea ) )
     => ( ( ~ ( member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
         => ( e2
            = ( sCC_Bloemen_dfs_v @ successors @ w @ ea ) ) )
        & ( ( member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
         => ( e2
            = ( sCC_Bloemen_unite_v @ va @ w @ ea ) ) ) ) ) ) ).

% e'_def
thf(fact_27_S__reflexive,axiom,
    ! [E: sCC_Bl1394983891496994913t_unit,N: v] :
      ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
     => ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ).

% S_reflexive
thf(fact_28_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_29_re__refl,axiom,
    ! [X: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X @ X ) ).

% re_refl
thf(fact_30_re__succ,axiom,
    ! [X: v,Y2: v,Z: v] :
      ( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y2 )
     => ( ( member_v @ Z @ ( successors @ Y2 ) )
       => ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).

% re_succ
thf(fact_31_reachable__end_Osimps,axiom,
    ! [A1: v,A2: v] :
      ( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
      = ( ? [X2: v] :
            ( ( A1 = X2 )
            & ( A2 = X2 ) )
        | ? [X2: v,Y3: v,Z2: v] :
            ( ( A1 = X2 )
            & ( A2 = Z2 )
            & ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y3 )
            & ( member_v @ Z2 @ ( successors @ Y3 ) ) ) ) ) ).

% reachable_end.simps
thf(fact_32_succ__re,axiom,
    ! [Y2: v,X: v,Z: v] :
      ( ( member_v @ Y2 @ ( successors @ X ) )
     => ( ( sCC_Bl770211535891879572_end_v @ successors @ Y2 @ Z )
       => ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).

% succ_re
thf(fact_33_wvs,axiom,
    member_v @ w @ ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) ).

% wvs
thf(fact_34_visited__unexplored,axiom,
    ! [E: sCC_Bl1394983891496994913t_unit,M: v] :
      ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
     => ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
       => ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
         => ~ ! [N2: v] :
                ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
               => ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ).

% visited_unexplored
thf(fact_35_unite__wf__env,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
      ( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
     => ( ( member_v @ W @ ( successors @ V3 ) )
       => ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
         => ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
           => ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
             => ( sCC_Bl9196236973127232072t_unit @ successors @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ).

% unite_wf_env
thf(fact_36_unite__sub__env,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
      ( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
     => ( ( member_v @ W @ ( successors @ V3 ) )
       => ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
         => ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
           => ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
             => ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ).

% unite_sub_env
thf(fact_37_stack__class,axiom,
    ! [E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
      ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
     => ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
       => ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
         => ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ).

% stack_class
thf(fact_38_vs__case,axiom,
    ( ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) )
   != bot_bot_set_v ) ).

% vs_case
thf(fact_39_pre__dfss__pre__dfs,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
      ( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
     => ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
       => ( ( member_v @ W @ ( successors @ V3 ) )
         => ( sCC_Bl36166008131615352t_unit @ successors @ W @ E ) ) ) ) ).

% pre_dfss_pre_dfs
thf(fact_40_graph_Oreachable__end_Ocong,axiom,
    sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).

% graph.reachable_end.cong
thf(fact_41_mem__Collect__eq,axiom,
    ! [A: v,P: v > $o] :
      ( ( member_v @ A @ ( collect_v @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
    ! [A: product_prod_v_v,P: product_prod_v_v > $o] :
      ( ( member7453568604450474000od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
    ! [A: set_v,P: set_v > $o] :
      ( ( member_set_v @ A @ ( collect_set_v @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
    ! [A3: set_v] :
      ( ( collect_v
        @ ^ [X2: v] : ( member_v @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( collec140062887454715474od_v_v
        @ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
    ! [A3: set_set_v] :
      ( ( collect_set_v
        @ ^ [X2: set_v] : ( member_set_v @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
    ! [P: set_v > $o,Q: set_v > $o] :
      ( ! [X3: set_v] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_set_v @ P )
        = ( collect_set_v @ Q ) ) ) ).

% Collect_cong
thf(fact_48_graph_Odfs_Ocong,axiom,
    sCC_Bloemen_dfs_v = sCC_Bloemen_dfs_v ).

% graph.dfs.cong
thf(fact_49_graph_Opre__dfss_Ocong,axiom,
    sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).

% graph.pre_dfss.cong
thf(fact_50_select__convs_I2_J,axiom,
    ! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
      ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
      = S ) ).

% select_convs(2)
thf(fact_51_select__convs_I5_J,axiom,
    ! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
      ( ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
      = Vsuccs ) ).

% select_convs(5)
thf(fact_52_unite__S__tl,axiom,
    ! [E: sCC_Bl1394983891496994913t_unit,W: v,V3: v,N: v] :
      ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
     => ( ( member_v @ W @ ( successors @ V3 ) )
       => ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
         => ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
           => ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
             => ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) )
               => ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N )
                  = ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ).

% unite_S_tl
thf(fact_53_reachable__visited,axiom,
    ! [E: sCC_Bl1394983891496994913t_unit,V3: v,W: v] :
      ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
     => ( ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
       => ( ( sCC_Bl649662514949026229able_v @ successors @ V3 @ W )
         => ( ! [X3: v] :
                ( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
               => ! [Xa: v] :
                    ( ( member_v @ Xa @ ( minus_minus_set_v @ ( successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
                   => ( ( sCC_Bl649662514949026229able_v @ successors @ V3 @ X3 )
                     => ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa @ W ) ) ) )
           => ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).

% reachable_visited
thf(fact_54_calculation_I2_J,axiom,
    ! [X4: v] :
      ( ( member_v @ X4 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ ea ) @ ( insert_v @ va @ bot_bot_set_v ) ) )
     => ( ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfss_v @ successors @ va @ e ) @ X4 )
        = ( sCC_Bl3795065053823578884t_unit @ ea @ X4 ) ) ) ).

% calculation(2)
thf(fact_55_Diff__idemp,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ B ) @ B )
      = ( minus_minus_set_v @ A3 @ B ) ) ).

% Diff_idemp
thf(fact_56_Diff__iff,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
      = ( ( member7453568604450474000od_v_v @ C @ A3 )
        & ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% Diff_iff
thf(fact_57_Diff__iff,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
      = ( ( member_v @ C @ A3 )
        & ~ ( member_v @ C @ B ) ) ) ).

% Diff_iff
thf(fact_58_DiffI,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ A3 )
     => ( ~ ( member7453568604450474000od_v_v @ C @ B )
       => ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ) ).

% DiffI
thf(fact_59_DiffI,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ A3 )
     => ( ~ ( member_v @ C @ B )
       => ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) ) ) ) ).

% DiffI
thf(fact_60_unite__subscc,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
      ( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
     => ( ( member_v @ W @ ( successors @ V3 ) )
       => ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
         => ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
           => ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
             => ( sCC_Bl5398416737448265317bscc_v @ successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ) ).

% unite_subscc
thf(fact_61_graph_Ounite__sub__env,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
       => ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
         => ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
           => ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
             => ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
               => ( sCC_Bl7963838319573962697t_unit @ E @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).

% graph.unite_sub_env
thf(fact_62_graph_Ounite__sub__env,axiom,
    ! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
       => ( ( member_v @ W @ ( Successors @ V3 ) )
         => ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
           => ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
             => ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
               => ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).

% graph.unite_sub_env
thf(fact_63_graph_Ostack__class,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
       => ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
         => ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
           => ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ) ).

% graph.stack_class
thf(fact_64_graph_Ounite__wf__env,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
       => ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
         => ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
           => ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
             => ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
               => ( sCC_Bl7798947040364291444t_unit @ Successors @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).

% graph.unite_wf_env
thf(fact_65_graph_Ounite__wf__env,axiom,
    ! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
       => ( ( member_v @ W @ ( Successors @ V3 ) )
         => ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
           => ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
             => ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
               => ( sCC_Bl9196236973127232072t_unit @ Successors @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).

% graph.unite_wf_env
thf(fact_66_graph__axioms,axiom,
    sCC_Bloemen_graph_v @ vertices @ successors ).

% graph_axioms
thf(fact_67_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_68_reachable__refl,axiom,
    ! [X: v] : ( sCC_Bl649662514949026229able_v @ successors @ X @ X ) ).

% reachable_refl
thf(fact_69_reachable__succ,axiom,
    ! [Y2: v,X: v,Z: v] :
      ( ( member_v @ Y2 @ ( successors @ X ) )
     => ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z )
       => ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).

% reachable_succ
thf(fact_70_reachable_Osimps,axiom,
    ! [A1: v,A2: v] :
      ( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
      = ( ? [X2: v] :
            ( ( A1 = X2 )
            & ( A2 = X2 ) )
        | ? [X2: v,Y3: v,Z2: v] :
            ( ( A1 = X2 )
            & ( A2 = Z2 )
            & ( member_v @ Y3 @ ( successors @ X2 ) )
            & ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z2 ) ) ) ) ).

% reachable.simps
thf(fact_71_reachable__edge,axiom,
    ! [Y2: v,X: v] :
      ( ( member_v @ Y2 @ ( successors @ X ) )
     => ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).

% reachable_edge
thf(fact_72_reachable__end__induct,axiom,
    ! [X: v,Y2: v,P: v > v > $o] :
      ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
     => ( ! [X3: v] : ( P @ X3 @ X3 )
       => ( ! [X3: v,Y: v,Z3: v] :
              ( ( P @ X3 @ Y )
             => ( ( member_v @ Z3 @ ( successors @ Y ) )
               => ( P @ X3 @ Z3 ) ) )
         => ( P @ X @ Y2 ) ) ) ) ).

% reachable_end_induct
thf(fact_73_reachable__trans,axiom,
    ! [X: v,Y2: v,Z: v] :
      ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
     => ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z )
       => ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).

% reachable_trans
thf(fact_74_succ__reachable,axiom,
    ! [X: v,Y2: v,Z: v] :
      ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
     => ( ( member_v @ Z @ ( successors @ Y2 ) )
       => ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).

% succ_reachable
thf(fact_75_init__env__pre__dfs,axiom,
    ! [V3: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ ( sCC_Bl7693227186847904995_env_v @ V3 ) ) ).

% init_env_pre_dfs
thf(fact_76_re__reachable,axiom,
    ! [X: v,Y2: v] :
      ( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y2 )
     => ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).

% re_reachable
thf(fact_77_reachable__re,axiom,
    ! [X: v,Y2: v] :
      ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
     => ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y2 ) ) ).

% reachable_re
thf(fact_78_is__subscc__def,axiom,
    ! [S4: set_v] :
      ( ( sCC_Bl5398416737448265317bscc_v @ successors @ S4 )
      = ( ! [X2: v] :
            ( ( member_v @ X2 @ S4 )
           => ! [Y3: v] :
                ( ( member_v @ Y3 @ S4 )
               => ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y3 ) ) ) ) ) ).

% is_subscc_def
thf(fact_79_empty__iff,axiom,
    ! [C: v] :
      ~ ( member_v @ C @ bot_bot_set_v ) ).

% empty_iff
thf(fact_80_empty__iff,axiom,
    ! [C: product_prod_v_v] :
      ~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).

% empty_iff
thf(fact_81_empty__iff,axiom,
    ! [C: set_v] :
      ~ ( member_set_v @ C @ bot_bot_set_set_v ) ).

% empty_iff
thf(fact_82_all__not__in__conv,axiom,
    ! [A3: set_v] :
      ( ( ! [X2: v] :
            ~ ( member_v @ X2 @ A3 ) )
      = ( A3 = bot_bot_set_v ) ) ).

% all_not_in_conv
thf(fact_83_all__not__in__conv,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( ! [X2: product_prod_v_v] :
            ~ ( member7453568604450474000od_v_v @ X2 @ A3 ) )
      = ( A3 = bot_bo723834152578015283od_v_v ) ) ).

% all_not_in_conv
thf(fact_84_all__not__in__conv,axiom,
    ! [A3: set_set_v] :
      ( ( ! [X2: set_v] :
            ~ ( member_set_v @ X2 @ A3 ) )
      = ( A3 = bot_bot_set_set_v ) ) ).

% all_not_in_conv
thf(fact_85_Collect__empty__eq,axiom,
    ! [P: v > $o] :
      ( ( ( collect_v @ P )
        = bot_bot_set_v )
      = ( ! [X2: v] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_86_Collect__empty__eq,axiom,
    ! [P: product_prod_v_v > $o] :
      ( ( ( collec140062887454715474od_v_v @ P )
        = bot_bo723834152578015283od_v_v )
      = ( ! [X2: product_prod_v_v] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_87_Collect__empty__eq,axiom,
    ! [P: set_v > $o] :
      ( ( ( collect_set_v @ P )
        = bot_bot_set_set_v )
      = ( ! [X2: set_v] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_88_empty__Collect__eq,axiom,
    ! [P: v > $o] :
      ( ( bot_bot_set_v
        = ( collect_v @ P ) )
      = ( ! [X2: v] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_89_empty__Collect__eq,axiom,
    ! [P: product_prod_v_v > $o] :
      ( ( bot_bo723834152578015283od_v_v
        = ( collec140062887454715474od_v_v @ P ) )
      = ( ! [X2: product_prod_v_v] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_90_empty__Collect__eq,axiom,
    ! [P: set_v > $o] :
      ( ( bot_bot_set_set_v
        = ( collect_set_v @ P ) )
      = ( ! [X2: set_v] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_91_insertCI,axiom,
    ! [A: set_v,B: set_set_v,B2: set_v] :
      ( ( ~ ( member_set_v @ A @ B )
       => ( A = B2 ) )
     => ( member_set_v @ A @ ( insert_set_v @ B2 @ B ) ) ) ).

% insertCI
thf(fact_92_insertCI,axiom,
    ! [A: v,B: set_v,B2: v] :
      ( ( ~ ( member_v @ A @ B )
       => ( A = B2 ) )
     => ( member_v @ A @ ( insert_v @ B2 @ B ) ) ) ).

% insertCI
thf(fact_93_insertCI,axiom,
    ! [A: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
      ( ( ~ ( member7453568604450474000od_v_v @ A @ B )
       => ( A = B2 ) )
     => ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).

% insertCI
thf(fact_94_insert__iff,axiom,
    ! [A: set_v,B2: set_v,A3: set_set_v] :
      ( ( member_set_v @ A @ ( insert_set_v @ B2 @ A3 ) )
      = ( ( A = B2 )
        | ( member_set_v @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_95_insert__iff,axiom,
    ! [A: v,B2: v,A3: set_v] :
      ( ( member_v @ A @ ( insert_v @ B2 @ A3 ) )
      = ( ( A = B2 )
        | ( member_v @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_96_insert__iff,axiom,
    ! [A: product_prod_v_v,B2: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ A3 ) )
      = ( ( A = B2 )
        | ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_97_insert__absorb2,axiom,
    ! [X: v,A3: set_v] :
      ( ( insert_v @ X @ ( insert_v @ X @ A3 ) )
      = ( insert_v @ X @ A3 ) ) ).

% insert_absorb2
thf(fact_98_insert__absorb2,axiom,
    ! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ X @ A3 ) )
      = ( insert1338601472111419319od_v_v @ X @ A3 ) ) ).

% insert_absorb2
thf(fact_99_insert__absorb2,axiom,
    ! [X: set_v,A3: set_set_v] :
      ( ( insert_set_v @ X @ ( insert_set_v @ X @ A3 ) )
      = ( insert_set_v @ X @ A3 ) ) ).

% insert_absorb2
thf(fact_100_subscc__add,axiom,
    ! [S4: set_v,X: v,Y2: v] :
      ( ( sCC_Bl5398416737448265317bscc_v @ successors @ S4 )
     => ( ( member_v @ X @ S4 )
       => ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
         => ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X )
           => ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v @ Y2 @ S4 ) ) ) ) ) ) ).

% subscc_add
thf(fact_101_singletonI,axiom,
    ! [A: v] : ( member_v @ A @ ( insert_v @ A @ bot_bot_set_v ) ) ).

% singletonI
thf(fact_102_singletonI,axiom,
    ! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).

% singletonI
thf(fact_103_singletonI,axiom,
    ! [A: set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).

% singletonI
thf(fact_104_Diff__empty,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
      = A3 ) ).

% Diff_empty
thf(fact_105_Diff__empty,axiom,
    ! [A3: set_set_v] :
      ( ( minus_7228012346218142266_set_v @ A3 @ bot_bot_set_set_v )
      = A3 ) ).

% Diff_empty
thf(fact_106_Diff__empty,axiom,
    ! [A3: set_v] :
      ( ( minus_minus_set_v @ A3 @ bot_bot_set_v )
      = A3 ) ).

% Diff_empty
thf(fact_107_empty__Diff,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A3 )
      = bot_bo723834152578015283od_v_v ) ).

% empty_Diff
thf(fact_108_empty__Diff,axiom,
    ! [A3: set_set_v] :
      ( ( minus_7228012346218142266_set_v @ bot_bot_set_set_v @ A3 )
      = bot_bot_set_set_v ) ).

% empty_Diff
thf(fact_109_empty__Diff,axiom,
    ! [A3: set_v] :
      ( ( minus_minus_set_v @ bot_bot_set_v @ A3 )
      = bot_bot_set_v ) ).

% empty_Diff
thf(fact_110_Diff__cancel,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ A3 @ A3 )
      = bot_bo723834152578015283od_v_v ) ).

% Diff_cancel
thf(fact_111_Diff__cancel,axiom,
    ! [A3: set_set_v] :
      ( ( minus_7228012346218142266_set_v @ A3 @ A3 )
      = bot_bot_set_set_v ) ).

% Diff_cancel
thf(fact_112_Diff__cancel,axiom,
    ! [A3: set_v] :
      ( ( minus_minus_set_v @ A3 @ A3 )
      = bot_bot_set_v ) ).

% Diff_cancel
thf(fact_113_Diff__insert0,axiom,
    ! [X: set_v,A3: set_set_v,B: set_set_v] :
      ( ~ ( member_set_v @ X @ A3 )
     => ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ B ) )
        = ( minus_7228012346218142266_set_v @ A3 @ B ) ) ) ).

% Diff_insert0
thf(fact_114_Diff__insert0,axiom,
    ! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ X @ A3 )
     => ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) )
        = ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).

% Diff_insert0
thf(fact_115_Diff__insert0,axiom,
    ! [X: v,A3: set_v,B: set_v] :
      ( ~ ( member_v @ X @ A3 )
     => ( ( minus_minus_set_v @ A3 @ ( insert_v @ X @ B ) )
        = ( minus_minus_set_v @ A3 @ B ) ) ) ).

% Diff_insert0
thf(fact_116_insert__Diff1,axiom,
    ! [X: set_v,B: set_set_v,A3: set_set_v] :
      ( ( member_set_v @ X @ B )
     => ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A3 ) @ B )
        = ( minus_7228012346218142266_set_v @ A3 @ B ) ) ) ).

% insert_Diff1
thf(fact_117_insert__Diff1,axiom,
    ! [X: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ X @ B )
     => ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B )
        = ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).

% insert_Diff1
thf(fact_118_insert__Diff1,axiom,
    ! [X: v,B: set_v,A3: set_v] :
      ( ( member_v @ X @ B )
     => ( ( minus_minus_set_v @ ( insert_v @ X @ A3 ) @ B )
        = ( minus_minus_set_v @ A3 @ B ) ) ) ).

% insert_Diff1
thf(fact_119_insert__Diff__single,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
      = ( insert1338601472111419319od_v_v @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_120_insert__Diff__single,axiom,
    ! [A: set_v,A3: set_set_v] :
      ( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
      = ( insert_set_v @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_121_insert__Diff__single,axiom,
    ! [A: v,A3: set_v] :
      ( ( insert_v @ A @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ bot_bot_set_v ) ) )
      = ( insert_v @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_122_sccE,axiom,
    ! [S4: set_v,X: v,Y2: v] :
      ( ( sCC_Bloemen_is_scc_v @ successors @ S4 )
     => ( ( member_v @ X @ S4 )
       => ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
         => ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X )
           => ( member_v @ Y2 @ S4 ) ) ) ) ) ).

% sccE
thf(fact_123_Diff__insert__absorb,axiom,
    ! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ X @ A3 )
     => ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_124_Diff__insert__absorb,axiom,
    ! [X: set_v,A3: set_set_v] :
      ( ~ ( member_set_v @ X @ A3 )
     => ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A3 ) @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_125_Diff__insert__absorb,axiom,
    ! [X: v,A3: set_v] :
      ( ~ ( member_v @ X @ A3 )
     => ( ( minus_minus_set_v @ ( insert_v @ X @ A3 ) @ ( insert_v @ X @ bot_bot_set_v ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_126_Diff__insert2,axiom,
    ! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
      = ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ).

% Diff_insert2
thf(fact_127_Diff__insert2,axiom,
    ! [A3: set_set_v,A: set_v,B: set_set_v] :
      ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ B ) )
      = ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) @ B ) ) ).

% Diff_insert2
thf(fact_128_Diff__insert2,axiom,
    ! [A3: set_v,A: v,B: set_v] :
      ( ( minus_minus_set_v @ A3 @ ( insert_v @ A @ B ) )
      = ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ bot_bot_set_v ) ) @ B ) ) ).

% Diff_insert2
thf(fact_129_insert__Diff,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A @ A3 )
     => ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_130_insert__Diff,axiom,
    ! [A: set_v,A3: set_set_v] :
      ( ( member_set_v @ A @ A3 )
     => ( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_131_insert__Diff,axiom,
    ! [A: v,A3: set_v] :
      ( ( member_v @ A @ A3 )
     => ( ( insert_v @ A @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ bot_bot_set_v ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_132_Diff__insert,axiom,
    ! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
      = ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).

% Diff_insert
thf(fact_133_Diff__insert,axiom,
    ! [A3: set_set_v,A: set_v,B: set_set_v] :
      ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ B ) )
      = ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ).

% Diff_insert
thf(fact_134_Diff__insert,axiom,
    ! [A3: set_v,A: v,B: set_v] :
      ( ( minus_minus_set_v @ A3 @ ( insert_v @ A @ B ) )
      = ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( insert_v @ A @ bot_bot_set_v ) ) ) ).

% Diff_insert
thf(fact_135_graph_Osubscc__add,axiom,
    ! [Vertices: set_set_v,Successors: set_v > set_set_v,S4: set_set_v,X: set_v,Y2: set_v] :
      ( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
     => ( ( sCC_Bl7907073126578335045_set_v @ Successors @ S4 )
       => ( ( member_set_v @ X @ S4 )
         => ( ( sCC_Bl7354734129683093653_set_v @ Successors @ X @ Y2 )
           => ( ( sCC_Bl7354734129683093653_set_v @ Successors @ Y2 @ X )
             => ( sCC_Bl7907073126578335045_set_v @ Successors @ ( insert_set_v @ Y2 @ S4 ) ) ) ) ) ) ) ).

% graph.subscc_add
thf(fact_136_graph_Osubscc__add,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S4: set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S4 )
       => ( ( member7453568604450474000od_v_v @ X @ S4 )
         => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
           => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X )
             => ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y2 @ S4 ) ) ) ) ) ) ) ).

% graph.subscc_add
thf(fact_137_graph_Osubscc__add,axiom,
    ! [Vertices: set_v,Successors: v > set_v,S4: set_v,X: v,Y2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S4 )
       => ( ( member_v @ X @ S4 )
         => ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
           => ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X )
             => ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v @ Y2 @ S4 ) ) ) ) ) ) ) ).

% graph.subscc_add
thf(fact_138_graph_Ois__subscc__def,axiom,
    ! [Vertices: set_v,Successors: v > set_v,S4: set_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S4 )
        = ( ! [X2: v] :
              ( ( member_v @ X2 @ S4 )
             => ! [Y3: v] :
                  ( ( member_v @ Y3 @ S4 )
                 => ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y3 ) ) ) ) ) ) ).

% graph.is_subscc_def
thf(fact_139_graph_Oreachable_Ocong,axiom,
    sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).

% graph.reachable.cong
thf(fact_140_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,X: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
       => ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 ) ) ) ).

% graph.reachable_edge
thf(fact_141_graph_Oreachable__edge,axiom,
    ! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( member_v @ Y2 @ ( Successors @ X ) )
       => ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).

% graph.reachable_edge
thf(fact_142_graph_Osucc__reachable,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: 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 @ X @ Y2 )
       => ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
         => ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).

% graph.succ_reachable
thf(fact_143_graph_Osucc__reachable,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
       => ( ( member_v @ Z @ ( Successors @ Y2 ) )
         => ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).

% graph.succ_reachable
thf(fact_144_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_145_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_146_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 )
        = ( ? [X2: product_prod_v_v] :
              ( ( A1 = X2 )
              & ( A2 = X2 ) )
          | ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
              ( ( A1 = X2 )
              & ( A2 = Z2 )
              & ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X2 ) )
              & ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).

% graph.reachable.simps
thf(fact_147_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 )
        = ( ? [X2: v] :
              ( ( A1 = X2 )
              & ( A2 = X2 ) )
          | ? [X2: v,Y3: v,Z2: v] :
              ( ( A1 = X2 )
              & ( A2 = Z2 )
              & ( member_v @ Y3 @ ( Successors @ X2 ) )
              & ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).

% graph.reachable.simps
thf(fact_148_graph_Oreachable__trans,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
       => ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z )
         => ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).

% graph.reachable_trans
thf(fact_149_graph_Oreachable__end__induct,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: 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 @ X @ Y2 )
       => ( ! [X3: product_prod_v_v] : ( P @ X3 @ X3 )
         => ( ! [X3: product_prod_v_v,Y: product_prod_v_v,Z3: product_prod_v_v] :
                ( ( P @ X3 @ Y )
               => ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y ) )
                 => ( P @ X3 @ Z3 ) ) )
           => ( P @ X @ Y2 ) ) ) ) ) ).

% graph.reachable_end_induct
thf(fact_150_graph_Oreachable__end__induct,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,P: v > v > $o] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
       => ( ! [X3: v] : ( P @ X3 @ X3 )
         => ( ! [X3: v,Y: v,Z3: v] :
                ( ( P @ X3 @ Y )
               => ( ( member_v @ Z3 @ ( Successors @ Y ) )
                 => ( P @ X3 @ Z3 ) ) )
           => ( P @ X @ Y2 ) ) ) ) ) ).

% graph.reachable_end_induct
thf(fact_151_graph_Oreachable__refl,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( sCC_Bl649662514949026229able_v @ Successors @ X @ X ) ) ).

% graph.reachable_refl
thf(fact_152_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,X: product_prod_v_v,Z: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
       => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ Z )
         => ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).

% graph.reachable_succ
thf(fact_153_graph_Oreachable__succ,axiom,
    ! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( member_v @ Y2 @ ( Successors @ X ) )
       => ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z )
         => ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).

% graph.reachable_succ
thf(fact_154_emptyE,axiom,
    ! [A: v] :
      ~ ( member_v @ A @ bot_bot_set_v ) ).

% emptyE
thf(fact_155_emptyE,axiom,
    ! [A: product_prod_v_v] :
      ~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).

% emptyE
thf(fact_156_emptyE,axiom,
    ! [A: set_v] :
      ~ ( member_set_v @ A @ bot_bot_set_set_v ) ).

% emptyE
thf(fact_157_insertE,axiom,
    ! [A: set_v,B2: set_v,A3: set_set_v] :
      ( ( member_set_v @ A @ ( insert_set_v @ B2 @ A3 ) )
     => ( ( A != B2 )
       => ( member_set_v @ A @ A3 ) ) ) ).

% insertE
thf(fact_158_insertE,axiom,
    ! [A: v,B2: v,A3: set_v] :
      ( ( member_v @ A @ ( insert_v @ B2 @ A3 ) )
     => ( ( A != B2 )
       => ( member_v @ A @ A3 ) ) ) ).

% insertE
thf(fact_159_insertE,axiom,
    ! [A: product_prod_v_v,B2: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ A3 ) )
     => ( ( A != B2 )
       => ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).

% insertE
thf(fact_160_equals0D,axiom,
    ! [A3: set_v,A: v] :
      ( ( A3 = bot_bot_set_v )
     => ~ ( member_v @ A @ A3 ) ) ).

% equals0D
thf(fact_161_equals0D,axiom,
    ! [A3: set_Product_prod_v_v,A: product_prod_v_v] :
      ( ( A3 = bot_bo723834152578015283od_v_v )
     => ~ ( member7453568604450474000od_v_v @ A @ A3 ) ) ).

% equals0D
thf(fact_162_equals0D,axiom,
    ! [A3: set_set_v,A: set_v] :
      ( ( A3 = bot_bot_set_set_v )
     => ~ ( member_set_v @ A @ A3 ) ) ).

% equals0D
thf(fact_163_equals0I,axiom,
    ! [A3: set_v] :
      ( ! [Y: v] :
          ~ ( member_v @ Y @ A3 )
     => ( A3 = bot_bot_set_v ) ) ).

% equals0I
thf(fact_164_equals0I,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ! [Y: product_prod_v_v] :
          ~ ( member7453568604450474000od_v_v @ Y @ A3 )
     => ( A3 = bot_bo723834152578015283od_v_v ) ) ).

% equals0I
thf(fact_165_equals0I,axiom,
    ! [A3: set_set_v] :
      ( ! [Y: set_v] :
          ~ ( member_set_v @ Y @ A3 )
     => ( A3 = bot_bot_set_set_v ) ) ).

% equals0I
thf(fact_166_insertI1,axiom,
    ! [A: set_v,B: set_set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ B ) ) ).

% insertI1
thf(fact_167_insertI1,axiom,
    ! [A: v,B: set_v] : ( member_v @ A @ ( insert_v @ A @ B ) ) ).

% insertI1
thf(fact_168_insertI1,axiom,
    ! [A: product_prod_v_v,B: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).

% insertI1
thf(fact_169_insertI2,axiom,
    ! [A: set_v,B: set_set_v,B2: set_v] :
      ( ( member_set_v @ A @ B )
     => ( member_set_v @ A @ ( insert_set_v @ B2 @ B ) ) ) ).

% insertI2
thf(fact_170_insertI2,axiom,
    ! [A: v,B: set_v,B2: v] :
      ( ( member_v @ A @ B )
     => ( member_v @ A @ ( insert_v @ B2 @ B ) ) ) ).

% insertI2
thf(fact_171_insertI2,axiom,
    ! [A: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A @ B )
     => ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).

% insertI2
thf(fact_172_ex__in__conv,axiom,
    ! [A3: set_v] :
      ( ( ? [X2: v] : ( member_v @ X2 @ A3 ) )
      = ( A3 != bot_bot_set_v ) ) ).

% ex_in_conv
thf(fact_173_ex__in__conv,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( ? [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A3 ) )
      = ( A3 != bot_bo723834152578015283od_v_v ) ) ).

% ex_in_conv
thf(fact_174_ex__in__conv,axiom,
    ! [A3: set_set_v] :
      ( ( ? [X2: set_v] : ( member_set_v @ X2 @ A3 ) )
      = ( A3 != bot_bot_set_set_v ) ) ).

% ex_in_conv
thf(fact_175_Set_Oset__insert,axiom,
    ! [X: set_v,A3: set_set_v] :
      ( ( member_set_v @ X @ A3 )
     => ~ ! [B3: set_set_v] :
            ( ( A3
              = ( insert_set_v @ X @ B3 ) )
           => ( member_set_v @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_176_Set_Oset__insert,axiom,
    ! [X: v,A3: set_v] :
      ( ( member_v @ X @ A3 )
     => ~ ! [B3: set_v] :
            ( ( A3
              = ( insert_v @ X @ B3 ) )
           => ( member_v @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_177_Set_Oset__insert,axiom,
    ! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ X @ A3 )
     => ~ ! [B3: set_Product_prod_v_v] :
            ( ( A3
              = ( insert1338601472111419319od_v_v @ X @ B3 ) )
           => ( member7453568604450474000od_v_v @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_178_singletonD,axiom,
    ! [B2: v,A: v] :
      ( ( member_v @ B2 @ ( insert_v @ A @ bot_bot_set_v ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_179_singletonD,axiom,
    ! [B2: product_prod_v_v,A: product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_180_singletonD,axiom,
    ! [B2: set_v,A: set_v] :
      ( ( member_set_v @ B2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_181_insert__ident,axiom,
    ! [X: set_v,A3: set_set_v,B: set_set_v] :
      ( ~ ( member_set_v @ X @ A3 )
     => ( ~ ( member_set_v @ X @ B )
       => ( ( ( insert_set_v @ X @ A3 )
            = ( insert_set_v @ X @ B ) )
          = ( A3 = B ) ) ) ) ).

% insert_ident
thf(fact_182_insert__ident,axiom,
    ! [X: v,A3: set_v,B: set_v] :
      ( ~ ( member_v @ X @ A3 )
     => ( ~ ( member_v @ X @ B )
       => ( ( ( insert_v @ X @ A3 )
            = ( insert_v @ X @ B ) )
          = ( A3 = B ) ) ) ) ).

% insert_ident
thf(fact_183_insert__ident,axiom,
    ! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ X @ A3 )
     => ( ~ ( member7453568604450474000od_v_v @ X @ B )
       => ( ( ( insert1338601472111419319od_v_v @ X @ A3 )
            = ( insert1338601472111419319od_v_v @ X @ B ) )
          = ( A3 = B ) ) ) ) ).

% insert_ident
thf(fact_184_insert__absorb,axiom,
    ! [A: set_v,A3: set_set_v] :
      ( ( member_set_v @ A @ A3 )
     => ( ( insert_set_v @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_185_insert__absorb,axiom,
    ! [A: v,A3: set_v] :
      ( ( member_v @ A @ A3 )
     => ( ( insert_v @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_186_insert__absorb,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A @ A3 )
     => ( ( insert1338601472111419319od_v_v @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_187_insert__eq__iff,axiom,
    ! [A: set_v,A3: set_set_v,B2: set_v,B: set_set_v] :
      ( ~ ( member_set_v @ A @ A3 )
     => ( ~ ( member_set_v @ B2 @ B )
       => ( ( ( insert_set_v @ A @ A3 )
            = ( insert_set_v @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A3 = B ) )
            & ( ( A != B2 )
             => ? [C2: set_set_v] :
                  ( ( A3
                    = ( insert_set_v @ B2 @ C2 ) )
                  & ~ ( member_set_v @ B2 @ C2 )
                  & ( B
                    = ( insert_set_v @ A @ C2 ) )
                  & ~ ( member_set_v @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_188_insert__eq__iff,axiom,
    ! [A: v,A3: set_v,B2: v,B: set_v] :
      ( ~ ( member_v @ A @ A3 )
     => ( ~ ( member_v @ B2 @ B )
       => ( ( ( insert_v @ A @ A3 )
            = ( insert_v @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A3 = B ) )
            & ( ( A != B2 )
             => ? [C2: set_v] :
                  ( ( A3
                    = ( insert_v @ B2 @ C2 ) )
                  & ~ ( member_v @ B2 @ C2 )
                  & ( B
                    = ( insert_v @ A @ C2 ) )
                  & ~ ( member_v @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_189_insert__eq__iff,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ A @ A3 )
     => ( ~ ( member7453568604450474000od_v_v @ B2 @ B )
       => ( ( ( insert1338601472111419319od_v_v @ A @ A3 )
            = ( insert1338601472111419319od_v_v @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A3 = B ) )
            & ( ( A != B2 )
             => ? [C2: set_Product_prod_v_v] :
                  ( ( A3
                    = ( insert1338601472111419319od_v_v @ B2 @ C2 ) )
                  & ~ ( member7453568604450474000od_v_v @ B2 @ C2 )
                  & ( B
                    = ( insert1338601472111419319od_v_v @ A @ C2 ) )
                  & ~ ( member7453568604450474000od_v_v @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_190_singleton__iff,axiom,
    ! [B2: v,A: v] :
      ( ( member_v @ B2 @ ( insert_v @ A @ bot_bot_set_v ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_191_singleton__iff,axiom,
    ! [B2: product_prod_v_v,A: product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_192_singleton__iff,axiom,
    ! [B2: set_v,A: set_v] :
      ( ( member_set_v @ B2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_193_insert__commute,axiom,
    ! [X: v,Y2: v,A3: set_v] :
      ( ( insert_v @ X @ ( insert_v @ Y2 @ A3 ) )
      = ( insert_v @ Y2 @ ( insert_v @ X @ A3 ) ) ) ).

% insert_commute
thf(fact_194_insert__commute,axiom,
    ! [X: product_prod_v_v,Y2: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ Y2 @ A3 ) )
      = ( insert1338601472111419319od_v_v @ Y2 @ ( insert1338601472111419319od_v_v @ X @ A3 ) ) ) ).

% insert_commute
thf(fact_195_insert__commute,axiom,
    ! [X: set_v,Y2: set_v,A3: set_set_v] :
      ( ( insert_set_v @ X @ ( insert_set_v @ Y2 @ A3 ) )
      = ( insert_set_v @ Y2 @ ( insert_set_v @ X @ A3 ) ) ) ).

% insert_commute
thf(fact_196_doubleton__eq__iff,axiom,
    ! [A: v,B2: v,C: v,D: v] :
      ( ( ( insert_v @ A @ ( insert_v @ B2 @ bot_bot_set_v ) )
        = ( insert_v @ C @ ( insert_v @ D @ bot_bot_set_v ) ) )
      = ( ( ( A = C )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_197_doubleton__eq__iff,axiom,
    ! [A: product_prod_v_v,B2: product_prod_v_v,C: product_prod_v_v,D: product_prod_v_v] :
      ( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
        = ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D @ bot_bo723834152578015283od_v_v ) ) )
      = ( ( ( A = C )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_198_doubleton__eq__iff,axiom,
    ! [A: set_v,B2: set_v,C: set_v,D: set_v] :
      ( ( ( insert_set_v @ A @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
        = ( insert_set_v @ C @ ( insert_set_v @ D @ bot_bot_set_set_v ) ) )
      = ( ( ( A = C )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_199_insert__not__empty,axiom,
    ! [A: v,A3: set_v] :
      ( ( insert_v @ A @ A3 )
     != bot_bot_set_v ) ).

% insert_not_empty
thf(fact_200_insert__not__empty,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( insert1338601472111419319od_v_v @ A @ A3 )
     != bot_bo723834152578015283od_v_v ) ).

% insert_not_empty
thf(fact_201_insert__not__empty,axiom,
    ! [A: set_v,A3: set_set_v] :
      ( ( insert_set_v @ A @ A3 )
     != bot_bot_set_set_v ) ).

% insert_not_empty
thf(fact_202_singleton__inject,axiom,
    ! [A: v,B2: v] :
      ( ( ( insert_v @ A @ bot_bot_set_v )
        = ( insert_v @ B2 @ bot_bot_set_v ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_203_singleton__inject,axiom,
    ! [A: product_prod_v_v,B2: product_prod_v_v] :
      ( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
        = ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_204_singleton__inject,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( ( insert_set_v @ A @ bot_bot_set_set_v )
        = ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_205_mk__disjoint__insert,axiom,
    ! [A: set_v,A3: set_set_v] :
      ( ( member_set_v @ A @ A3 )
     => ? [B3: set_set_v] :
          ( ( A3
            = ( insert_set_v @ A @ B3 ) )
          & ~ ( member_set_v @ A @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_206_mk__disjoint__insert,axiom,
    ! [A: v,A3: set_v] :
      ( ( member_v @ A @ A3 )
     => ? [B3: set_v] :
          ( ( A3
            = ( insert_v @ A @ B3 ) )
          & ~ ( member_v @ A @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_207_mk__disjoint__insert,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A @ A3 )
     => ? [B3: set_Product_prod_v_v] :
          ( ( A3
            = ( insert1338601472111419319od_v_v @ A @ B3 ) )
          & ~ ( member7453568604450474000od_v_v @ A @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_208_graph_Ois__subscc_Ocong,axiom,
    sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).

% graph.is_subscc.cong
thf(fact_209_insert__Diff__if,axiom,
    ! [X: set_v,B: set_set_v,A3: set_set_v] :
      ( ( ( member_set_v @ X @ B )
       => ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A3 ) @ B )
          = ( minus_7228012346218142266_set_v @ A3 @ B ) ) )
      & ( ~ ( member_set_v @ X @ B )
       => ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A3 ) @ B )
          = ( insert_set_v @ X @ ( minus_7228012346218142266_set_v @ A3 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_210_insert__Diff__if,axiom,
    ! [X: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ( member7453568604450474000od_v_v @ X @ B )
       => ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B )
          = ( minus_4183494784930505774od_v_v @ A3 @ B ) ) )
      & ( ~ ( member7453568604450474000od_v_v @ X @ B )
       => ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B )
          = ( insert1338601472111419319od_v_v @ X @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_211_insert__Diff__if,axiom,
    ! [X: v,B: set_v,A3: set_v] :
      ( ( ( member_v @ X @ B )
       => ( ( minus_minus_set_v @ ( insert_v @ X @ A3 ) @ B )
          = ( minus_minus_set_v @ A3 @ B ) ) )
      & ( ~ ( member_v @ X @ B )
       => ( ( minus_minus_set_v @ ( insert_v @ X @ A3 ) @ B )
          = ( insert_v @ X @ ( minus_minus_set_v @ A3 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_212_graph_Ore__reachable,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y2 )
       => ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).

% graph.re_reachable
thf(fact_213_graph_Oreachable__re,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
       => ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y2 ) ) ) ).

% graph.reachable_re
thf(fact_214_graph_Opre__dfs_Ocong,axiom,
    sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).

% graph.pre_dfs.cong
thf(fact_215_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,X: product_prod_v_v,Z: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
       => ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y2 @ Z )
         => ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).

% graph.succ_re
thf(fact_216_graph_Osucc__re,axiom,
    ! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( member_v @ Y2 @ ( Successors @ X ) )
       => ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y2 @ Z )
         => ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).

% graph.succ_re
thf(fact_217_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_218_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_219_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 )
        = ( ? [X2: product_prod_v_v] :
              ( ( A1 = X2 )
              & ( A2 = X2 ) )
          | ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
              ( ( A1 = X2 )
              & ( A2 = Z2 )
              & ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Y3 )
              & ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).

% graph.reachable_end.simps
thf(fact_220_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 )
        = ( ? [X2: v] :
              ( ( A1 = X2 )
              & ( A2 = X2 ) )
          | ? [X2: v,Y3: v,Z2: v] :
              ( ( A1 = X2 )
              & ( A2 = Z2 )
              & ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y3 )
              & ( member_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).

% graph.reachable_end.simps
thf(fact_221_graph_Ore__refl,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( sCC_Bl770211535891879572_end_v @ Successors @ X @ X ) ) ).

% graph.re_refl
thf(fact_222_graph_Ore__succ,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: 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 @ X @ Y2 )
       => ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
         => ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).

% graph.re_succ
thf(fact_223_graph_Ore__succ,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y2 )
       => ( ( member_v @ Z @ ( Successors @ Y2 ) )
         => ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).

% graph.re_succ
thf(fact_224_graph_Osub__env__trans,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
       => ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
         => ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ) ).

% graph.sub_env_trans
thf(fact_225_graph_Opre__dfss__pre__dfs,axiom,
    ! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
       => ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
         => ( ( member_v @ W @ ( Successors @ V3 ) )
           => ( sCC_Bl36166008131615352t_unit @ Successors @ W @ E ) ) ) ) ) ).

% graph.pre_dfss_pre_dfs
thf(fact_226_graph_Ounite__subscc,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
       => ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
         => ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
           => ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
             => ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
               => ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ ( hd_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).

% graph.unite_subscc
thf(fact_227_graph_Ounite__subscc,axiom,
    ! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
       => ( ( member_v @ W @ ( Successors @ V3 ) )
         => ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
           => ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
             => ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
               => ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).

% graph.unite_subscc
thf(fact_228_graph_Oreachable__visited,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V3: v,W: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
       => ( ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
         => ( ( sCC_Bl649662514949026229able_v @ Successors @ V3 @ W )
           => ( ! [X3: v] :
                  ( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
                 => ! [Xa: v] :
                      ( ( member_v @ Xa @ ( minus_minus_set_v @ ( Successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
                     => ( ( sCC_Bl649662514949026229able_v @ Successors @ V3 @ X3 )
                       => ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa @ W ) ) ) )
             => ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).

% graph.reachable_visited
thf(fact_229_graph_OS__reflexive,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
       => ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ).

% graph.S_reflexive
thf(fact_230_graph_Ounite__S__tl,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V3: product_prod_v_v,N: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
       => ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
         => ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
           => ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
             => ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
               => ( ( member7453568604450474000od_v_v @ N @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) )
                 => ( ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ N )
                    = ( sCC_Bl8440648026628373538t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).

% graph.unite_S_tl
thf(fact_231_graph_Ounite__S__tl,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V3: v,N: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
       => ( ( member_v @ W @ ( Successors @ V3 ) )
         => ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
           => ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
             => ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
               => ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) )
                 => ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N )
                    = ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).

% graph.unite_S_tl
thf(fact_232_DiffE,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
     => ~ ( ( member7453568604450474000od_v_v @ C @ A3 )
         => ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% DiffE
thf(fact_233_DiffE,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
     => ~ ( ( member_v @ C @ A3 )
         => ( member_v @ C @ B ) ) ) ).

% DiffE
thf(fact_234_DiffD1,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
     => ( member7453568604450474000od_v_v @ C @ A3 ) ) ).

% DiffD1
thf(fact_235_DiffD1,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
     => ( member_v @ C @ A3 ) ) ).

% DiffD1
thf(fact_236_DiffD2,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
     => ~ ( member7453568604450474000od_v_v @ C @ B ) ) ).

% DiffD2
thf(fact_237_DiffD2,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
     => ~ ( member_v @ C @ B ) ) ).

% DiffD2
thf(fact_238_graph_Ostack__visited,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
       => ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
         => ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ).

% graph.stack_visited
thf(fact_239_graph_Ostack__unexplored,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
       => ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
         => ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ).

% graph.stack_unexplored
thf(fact_240_graph_Ovisited__unexplored,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,M: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
       => ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
         => ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
           => ~ ! [N2: v] :
                  ( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
                 => ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ).

% graph.visited_unexplored
thf(fact_241_pre__dfs__def,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ E )
      = ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
        & ~ ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
        & ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V3 )
        & ( ( sCC_Bl3795065053823578884t_unit @ E @ V3 )
          = bot_bot_set_v )
        & ! [X2: v] :
            ( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
           => ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) ) ) ) ).

% pre_dfs_def
thf(fact_242_graph_Opre__dfs__def,axiom,
    ! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl36166008131615352t_unit @ Successors @ V3 @ E )
        = ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
          & ~ ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
          & ( sCC_Bl649662514949026229able_v @ Successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V3 )
          & ( ( sCC_Bl3795065053823578884t_unit @ E @ V3 )
            = bot_bot_set_v )
          & ! [X2: v] :
              ( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
             => ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V3 ) ) ) ) ) ).

% graph.pre_dfs_def
thf(fact_243_the__elem__eq,axiom,
    ! [X: v] :
      ( ( the_elem_v @ ( insert_v @ X @ bot_bot_set_v ) )
      = X ) ).

% the_elem_eq
thf(fact_244_the__elem__eq,axiom,
    ! [X: product_prod_v_v] :
      ( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
      = X ) ).

% the_elem_eq
thf(fact_245_the__elem__eq,axiom,
    ! [X: set_v] :
      ( ( the_elem_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
      = X ) ).

% the_elem_eq
thf(fact_246_ra__reachable,axiom,
    ! [X: v,Y2: v,E4: set_Product_prod_v_v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
     => ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).

% ra_reachable
thf(fact_247_is__singletonI,axiom,
    ! [X: v] : ( is_singleton_v @ ( insert_v @ X @ bot_bot_set_v ) ) ).

% is_singletonI
thf(fact_248_is__singletonI,axiom,
    ! [X: product_prod_v_v] : ( is_sin9198872032823709915od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ).

% is_singletonI
thf(fact_249_is__singletonI,axiom,
    ! [X: set_v] : ( is_singleton_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ).

% is_singletonI
thf(fact_250_dfs__S__tl__stack_I2_J,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
     => ( ( ( sCC_Bl8828226123343373779t_unit @ E )
         != nil_v )
       => ! [X4: v] :
            ( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
           => ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
              = ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) ) ).

% dfs_S_tl_stack(2)
thf(fact_251_post__dfss__def,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bl6082031138996704384t_unit @ successors @ V3 @ E @ E2 )
      = ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
        & ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V3 )
          = ( successors @ V3 ) )
        & ! [X2: v] :
            ( ( member_v @ X2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) ) )
           => ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 )
              = ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
        & ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
        & ! [X2: v] :
            ( ( member_v @ X2 @ ( successors @ V3 ) )
           => ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E2 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) )
        & ! [X2: v] :
            ( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
           => ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) )
        & ( ( sCC_Bl8828226123343373779t_unit @ E2 )
         != nil_v )
        & ? [Ns: list_v] :
            ( ( sCC_Bl8828226123343373779t_unit @ E )
            = ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
        & ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
        & ! [X2: v] :
            ( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
           => ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
              = ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) )
        & ( ( ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
            = V3 )
         => ! [X2: v] :
              ( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
             => ~ ( sCC_Bl649662514949026229able_v @ successors @ V3 @ X2 ) ) )
        & ( ( sCC_Bl9201514103433284750t_unit @ E2 )
          = ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).

% post_dfss_def
thf(fact_252_dfs__S__hd__stack_I2_J,axiom,
    ! [E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
      ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
     => ( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
       => ( ( ( sCC_Bl8828226123343373779t_unit @ E )
           != nil_v )
         => ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
           => ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ).

% dfs_S_hd_stack(2)
thf(fact_253_dfs__S__hd__stack_I1_J,axiom,
    ! [E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
      ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
     => ( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
       => ( ( ( sCC_Bl8828226123343373779t_unit @ E )
           != nil_v )
         => ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
           => ( ( sCC_Bl8828226123343373779t_unit @ E2 )
             != nil_v ) ) ) ) ) ).

% dfs_S_hd_stack(1)
thf(fact_254_ra__trans,axiom,
    ! [X: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
     => ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y2 @ Z @ E4 )
       => ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E4 ) ) ) ).

% ra_trans
thf(fact_255_ra__refl,axiom,
    ! [X: v,E4: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X @ X @ E4 ) ).

% ra_refl
thf(fact_256_ra__empty,axiom,
    ! [X: v,Y2: v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ bot_bo723834152578015283od_v_v )
      = ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).

% ra_empty
thf(fact_257_sup_Oidem,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A @ A )
      = A ) ).

% sup.idem
thf(fact_258_sup_Oidem,axiom,
    ! [A: set_v] :
      ( ( sup_sup_set_v @ A @ A )
      = A ) ).

% sup.idem
thf(fact_259_sup_Oidem,axiom,
    ! [A: set_set_v] :
      ( ( sup_sup_set_set_v @ A @ A )
      = A ) ).

% sup.idem
thf(fact_260_sup__idem,axiom,
    ! [X: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X @ X )
      = X ) ).

% sup_idem
thf(fact_261_sup__idem,axiom,
    ! [X: set_v] :
      ( ( sup_sup_set_v @ X @ X )
      = X ) ).

% sup_idem
thf(fact_262_sup__idem,axiom,
    ! [X: set_set_v] :
      ( ( sup_sup_set_set_v @ X @ X )
      = X ) ).

% sup_idem
thf(fact_263_sup_Oleft__idem,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B2 ) )
      = ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).

% sup.left_idem
thf(fact_264_sup_Oleft__idem,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( sup_sup_set_v @ A @ ( sup_sup_set_v @ A @ B2 ) )
      = ( sup_sup_set_v @ A @ B2 ) ) ).

% sup.left_idem
thf(fact_265_sup_Oleft__idem,axiom,
    ! [A: set_set_v,B2: set_set_v] :
      ( ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ A @ B2 ) )
      = ( sup_sup_set_set_v @ A @ B2 ) ) ).

% sup.left_idem
thf(fact_266_sup__left__idem,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
      = ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).

% sup_left_idem
thf(fact_267_sup__left__idem,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) )
      = ( sup_sup_set_v @ X @ Y2 ) ) ).

% sup_left_idem
thf(fact_268_sup__left__idem,axiom,
    ! [X: set_set_v,Y2: set_set_v] :
      ( ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ X @ Y2 ) )
      = ( sup_sup_set_set_v @ X @ Y2 ) ) ).

% sup_left_idem
thf(fact_269_sup_Oright__idem,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ B2 )
      = ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).

% sup.right_idem
thf(fact_270_sup_Oright__idem,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B2 ) @ B2 )
      = ( sup_sup_set_v @ A @ B2 ) ) ).

% sup.right_idem
thf(fact_271_sup_Oright__idem,axiom,
    ! [A: set_set_v,B2: set_set_v] :
      ( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ B2 )
      = ( sup_sup_set_set_v @ A @ B2 ) ) ).

% sup.right_idem
thf(fact_272_UnCI,axiom,
    ! [C: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ~ ( member7453568604450474000od_v_v @ C @ B )
       => ( member7453568604450474000od_v_v @ C @ A3 ) )
     => ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).

% UnCI
thf(fact_273_UnCI,axiom,
    ! [C: v,B: set_v,A3: set_v] :
      ( ( ~ ( member_v @ C @ B )
       => ( member_v @ C @ A3 ) )
     => ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) ) ) ).

% UnCI
thf(fact_274_UnCI,axiom,
    ! [C: set_v,B: set_set_v,A3: set_set_v] :
      ( ( ~ ( member_set_v @ C @ B )
       => ( member_set_v @ C @ A3 ) )
     => ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B ) ) ) ).

% UnCI
thf(fact_275_Un__iff,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
      = ( ( member7453568604450474000od_v_v @ C @ A3 )
        | ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% Un_iff
thf(fact_276_Un__iff,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) )
      = ( ( member_v @ C @ A3 )
        | ( member_v @ C @ B ) ) ) ).

% Un_iff
thf(fact_277_Un__iff,axiom,
    ! [C: set_v,A3: set_set_v,B: set_set_v] :
      ( ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B ) )
      = ( ( member_set_v @ C @ A3 )
        | ( member_set_v @ C @ B ) ) ) ).

% Un_iff
thf(fact_278_dfs__S__tl__stack_I1_J,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
     => ( ( ( sCC_Bl8828226123343373779t_unit @ E )
         != nil_v )
       => ( ( sCC_Bl8828226123343373779t_unit @ E2 )
         != nil_v ) ) ) ).

% dfs_S_tl_stack(1)
thf(fact_279_sup__bot_Oright__neutral,axiom,
    ! [A: set_v] :
      ( ( sup_sup_set_v @ A @ bot_bot_set_v )
      = A ) ).

% sup_bot.right_neutral
thf(fact_280_sup__bot_Oright__neutral,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A @ bot_bo723834152578015283od_v_v )
      = A ) ).

% sup_bot.right_neutral
thf(fact_281_sup__bot_Oright__neutral,axiom,
    ! [A: set_set_v] :
      ( ( sup_sup_set_set_v @ A @ bot_bot_set_set_v )
      = A ) ).

% sup_bot.right_neutral
thf(fact_282_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( bot_bot_set_v
        = ( sup_sup_set_v @ A @ B2 ) )
      = ( ( A = bot_bot_set_v )
        & ( B2 = bot_bot_set_v ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_283_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( sup_su414716646722978715od_v_v @ A @ B2 ) )
      = ( ( A = bot_bo723834152578015283od_v_v )
        & ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_284_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_set_v,B2: set_set_v] :
      ( ( bot_bot_set_set_v
        = ( sup_sup_set_set_v @ A @ B2 ) )
      = ( ( A = bot_bot_set_set_v )
        & ( B2 = bot_bot_set_set_v ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_285_sup__bot_Oleft__neutral,axiom,
    ! [A: set_v] :
      ( ( sup_sup_set_v @ bot_bot_set_v @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_286_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_287_sup__bot_Oleft__neutral,axiom,
    ! [A: set_set_v] :
      ( ( sup_sup_set_set_v @ bot_bot_set_set_v @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_288_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( ( sup_sup_set_v @ A @ B2 )
        = bot_bot_set_v )
      = ( ( A = bot_bot_set_v )
        & ( B2 = bot_bot_set_v ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_289_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ( sup_su414716646722978715od_v_v @ A @ B2 )
        = bot_bo723834152578015283od_v_v )
      = ( ( A = bot_bo723834152578015283od_v_v )
        & ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_290_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_set_v,B2: set_set_v] :
      ( ( ( sup_sup_set_set_v @ A @ B2 )
        = bot_bot_set_set_v )
      = ( ( A = bot_bot_set_set_v )
        & ( B2 = bot_bot_set_set_v ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_291_sup__eq__bot__iff,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( ( sup_sup_set_v @ X @ Y2 )
        = bot_bot_set_v )
      = ( ( X = bot_bot_set_v )
        & ( Y2 = bot_bot_set_v ) ) ) ).

% sup_eq_bot_iff
thf(fact_292_sup__eq__bot__iff,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
        = bot_bo723834152578015283od_v_v )
      = ( ( X = bot_bo723834152578015283od_v_v )
        & ( Y2 = bot_bo723834152578015283od_v_v ) ) ) ).

% sup_eq_bot_iff
thf(fact_293_sup__eq__bot__iff,axiom,
    ! [X: set_set_v,Y2: set_set_v] :
      ( ( ( sup_sup_set_set_v @ X @ Y2 )
        = bot_bot_set_set_v )
      = ( ( X = bot_bot_set_set_v )
        & ( Y2 = bot_bot_set_set_v ) ) ) ).

% sup_eq_bot_iff
thf(fact_294_bot__eq__sup__iff,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( bot_bot_set_v
        = ( sup_sup_set_v @ X @ Y2 ) )
      = ( ( X = bot_bot_set_v )
        & ( Y2 = bot_bot_set_v ) ) ) ).

% bot_eq_sup_iff
thf(fact_295_bot__eq__sup__iff,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
      = ( ( X = bot_bo723834152578015283od_v_v )
        & ( Y2 = bot_bo723834152578015283od_v_v ) ) ) ).

% bot_eq_sup_iff
thf(fact_296_bot__eq__sup__iff,axiom,
    ! [X: set_set_v,Y2: set_set_v] :
      ( ( bot_bot_set_set_v
        = ( sup_sup_set_set_v @ X @ Y2 ) )
      = ( ( X = bot_bot_set_set_v )
        & ( Y2 = bot_bot_set_set_v ) ) ) ).

% bot_eq_sup_iff
thf(fact_297_sup__bot__right,axiom,
    ! [X: set_v] :
      ( ( sup_sup_set_v @ X @ bot_bot_set_v )
      = X ) ).

% sup_bot_right
thf(fact_298_sup__bot__right,axiom,
    ! [X: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
      = X ) ).

% sup_bot_right
thf(fact_299_sup__bot__right,axiom,
    ! [X: set_set_v] :
      ( ( sup_sup_set_set_v @ X @ bot_bot_set_set_v )
      = X ) ).

% sup_bot_right
thf(fact_300_sup__bot__left,axiom,
    ! [X: set_v] :
      ( ( sup_sup_set_v @ bot_bot_set_v @ X )
      = X ) ).

% sup_bot_left
thf(fact_301_sup__bot__left,axiom,
    ! [X: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ X )
      = X ) ).

% sup_bot_left
thf(fact_302_sup__bot__left,axiom,
    ! [X: set_set_v] :
      ( ( sup_sup_set_set_v @ bot_bot_set_set_v @ X )
      = X ) ).

% sup_bot_left
thf(fact_303_Un__empty,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( ( sup_sup_set_v @ A3 @ B )
        = bot_bot_set_v )
      = ( ( A3 = bot_bot_set_v )
        & ( B = bot_bot_set_v ) ) ) ).

% Un_empty
thf(fact_304_Un__empty,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( sup_su414716646722978715od_v_v @ A3 @ B )
        = bot_bo723834152578015283od_v_v )
      = ( ( A3 = bot_bo723834152578015283od_v_v )
        & ( B = bot_bo723834152578015283od_v_v ) ) ) ).

% Un_empty
thf(fact_305_Un__empty,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( ( sup_sup_set_set_v @ A3 @ B )
        = bot_bot_set_set_v )
      = ( ( A3 = bot_bot_set_set_v )
        & ( B = bot_bot_set_set_v ) ) ) ).

% Un_empty
thf(fact_306_Un__insert__right,axiom,
    ! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
      = ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).

% Un_insert_right
thf(fact_307_Un__insert__right,axiom,
    ! [A3: set_v,A: v,B: set_v] :
      ( ( sup_sup_set_v @ A3 @ ( insert_v @ A @ B ) )
      = ( insert_v @ A @ ( sup_sup_set_v @ A3 @ B ) ) ) ).

% Un_insert_right
thf(fact_308_Un__insert__right,axiom,
    ! [A3: set_set_v,A: set_v,B: set_set_v] :
      ( ( sup_sup_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
      = ( insert_set_v @ A @ ( sup_sup_set_set_v @ A3 @ B ) ) ) ).

% Un_insert_right
thf(fact_309_Un__insert__left,axiom,
    ! [A: product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
      = ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) ) ).

% Un_insert_left
thf(fact_310_Un__insert__left,axiom,
    ! [A: v,B: set_v,C3: set_v] :
      ( ( sup_sup_set_v @ ( insert_v @ A @ B ) @ C3 )
      = ( insert_v @ A @ ( sup_sup_set_v @ B @ C3 ) ) ) ).

% Un_insert_left
thf(fact_311_Un__insert__left,axiom,
    ! [A: set_v,B: set_set_v,C3: set_set_v] :
      ( ( sup_sup_set_set_v @ ( insert_set_v @ A @ B ) @ C3 )
      = ( insert_set_v @ A @ ( sup_sup_set_set_v @ B @ C3 ) ) ) ).

% Un_insert_left
thf(fact_312_Un__Diff__cancel2,axiom,
    ! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B @ A3 ) @ A3 )
      = ( sup_su414716646722978715od_v_v @ B @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_313_Un__Diff__cancel2,axiom,
    ! [B: set_set_v,A3: set_set_v] :
      ( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ B @ A3 ) @ A3 )
      = ( sup_sup_set_set_v @ B @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_314_Un__Diff__cancel2,axiom,
    ! [B: set_v,A3: set_v] :
      ( ( sup_sup_set_v @ ( minus_minus_set_v @ B @ A3 ) @ A3 )
      = ( sup_sup_set_v @ B @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_315_Un__Diff__cancel,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
      = ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).

% Un_Diff_cancel
thf(fact_316_Un__Diff__cancel,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( sup_sup_set_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ A3 ) )
      = ( sup_sup_set_set_v @ A3 @ B ) ) ).

% Un_Diff_cancel
thf(fact_317_Un__Diff__cancel,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( sup_sup_set_v @ A3 @ ( minus_minus_set_v @ B @ A3 ) )
      = ( sup_sup_set_v @ A3 @ B ) ) ).

% Un_Diff_cancel
thf(fact_318_post__dfs__def,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ E2 )
      = ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
        & ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
        & ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
        & ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V3 )
          = ( successors @ V3 ) )
        & ! [X2: v] :
            ( ( member_v @ X2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
           => ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 )
              = ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
        & ! [X2: v] :
            ( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
           => ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) )
        & ? [Ns: list_v] :
            ( ( sCC_Bl8828226123343373779t_unit @ E )
            = ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
        & ( ( ( member_v @ V3 @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
            & ( ( sCC_Bl8828226123343373779t_unit @ E2 )
              = ( sCC_Bl8828226123343373779t_unit @ E ) )
            & ! [X2: v] :
                ( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
               => ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
                  = ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) )
          | ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
             != nil_v )
            & ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
            & ! [X2: v] :
                ( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
               => ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
                  = ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) ) )
        & ( ( sCC_Bl9201514103433284750t_unit @ E2 )
          = ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).

% post_dfs_def
thf(fact_319_inf__sup__aci_I8_J,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
      = ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).

% inf_sup_aci(8)
thf(fact_320_inf__sup__aci_I8_J,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) )
      = ( sup_sup_set_v @ X @ Y2 ) ) ).

% inf_sup_aci(8)
thf(fact_321_inf__sup__aci_I8_J,axiom,
    ! [X: set_set_v,Y2: set_set_v] :
      ( ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ X @ Y2 ) )
      = ( sup_sup_set_set_v @ X @ Y2 ) ) ).

% inf_sup_aci(8)
thf(fact_322_inf__sup__aci_I7_J,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
      = ( sup_su414716646722978715od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_323_inf__sup__aci_I7_J,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
      = ( sup_sup_set_v @ Y2 @ ( sup_sup_set_v @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_324_inf__sup__aci_I7_J,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] :
      ( ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ Y2 @ Z ) )
      = ( sup_sup_set_set_v @ Y2 @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_325_inf__sup__aci_I6_J,axiom,
    ! [X: 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 @ X @ Y2 ) @ Z )
      = ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_326_inf__sup__aci_I6_J,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( sup_sup_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ Z )
      = ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_327_inf__sup__aci_I6_J,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] :
      ( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ X @ Y2 ) @ Z )
      = ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ Y2 @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_328_inf__sup__aci_I5_J,axiom,
    ( sup_su414716646722978715od_v_v
    = ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_329_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_v
    = ( ^ [X2: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_330_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_set_v
    = ( ^ [X2: set_set_v,Y3: set_set_v] : ( sup_sup_set_set_v @ Y3 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_331_sup_Oassoc,axiom,
    ! [A: 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 @ A @ B2 ) @ C )
      = ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_332_sup_Oassoc,axiom,
    ! [A: set_v,B2: set_v,C: set_v] :
      ( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B2 ) @ C )
      = ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_333_sup_Oassoc,axiom,
    ! [A: set_set_v,B2: set_set_v,C: set_set_v] :
      ( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ C )
      = ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_334_sup__assoc,axiom,
    ! [X: 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 @ X @ Y2 ) @ Z )
      = ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).

% sup_assoc
thf(fact_335_sup__assoc,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( sup_sup_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ Z )
      = ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) ) ) ).

% sup_assoc
thf(fact_336_sup__assoc,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] :
      ( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ X @ Y2 ) @ Z )
      = ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ Y2 @ Z ) ) ) ).

% sup_assoc
thf(fact_337_sup_Ocommute,axiom,
    ( sup_su414716646722978715od_v_v
    = ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_338_sup_Ocommute,axiom,
    ( sup_sup_set_v
    = ( ^ [A4: set_v,B4: set_v] : ( sup_sup_set_v @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_339_sup_Ocommute,axiom,
    ( sup_sup_set_set_v
    = ( ^ [A4: set_set_v,B4: set_set_v] : ( sup_sup_set_set_v @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_340_sup__commute,axiom,
    ( sup_su414716646722978715od_v_v
    = ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X2 ) ) ) ).

% sup_commute
thf(fact_341_sup__commute,axiom,
    ( sup_sup_set_v
    = ( ^ [X2: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X2 ) ) ) ).

% sup_commute
thf(fact_342_sup__commute,axiom,
    ( sup_sup_set_set_v
    = ( ^ [X2: set_set_v,Y3: set_set_v] : ( sup_sup_set_set_v @ Y3 @ X2 ) ) ) ).

% sup_commute
thf(fact_343_sup_Oleft__commute,axiom,
    ! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A @ C ) )
      = ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_344_sup_Oleft__commute,axiom,
    ! [B2: set_v,A: set_v,C: set_v] :
      ( ( sup_sup_set_v @ B2 @ ( sup_sup_set_v @ A @ C ) )
      = ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_345_sup_Oleft__commute,axiom,
    ! [B2: set_set_v,A: set_set_v,C: set_set_v] :
      ( ( sup_sup_set_set_v @ B2 @ ( sup_sup_set_set_v @ A @ C ) )
      = ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_346_sup__left__commute,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
      = ( sup_su414716646722978715od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_347_sup__left__commute,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
      = ( sup_sup_set_v @ Y2 @ ( sup_sup_set_v @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_348_sup__left__commute,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] :
      ( ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ Y2 @ Z ) )
      = ( sup_sup_set_set_v @ Y2 @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_349_UnE,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
     => ( ~ ( member7453568604450474000od_v_v @ C @ A3 )
       => ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% UnE
thf(fact_350_UnE,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) )
     => ( ~ ( member_v @ C @ A3 )
       => ( member_v @ C @ B ) ) ) ).

% UnE
thf(fact_351_UnE,axiom,
    ! [C: set_v,A3: set_set_v,B: set_set_v] :
      ( ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B ) )
     => ( ~ ( member_set_v @ C @ A3 )
       => ( member_set_v @ C @ B ) ) ) ).

% UnE
thf(fact_352_UnI1,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ A3 )
     => ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).

% UnI1
thf(fact_353_UnI1,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ A3 )
     => ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) ) ) ).

% UnI1
thf(fact_354_UnI1,axiom,
    ! [C: set_v,A3: set_set_v,B: set_set_v] :
      ( ( member_set_v @ C @ A3 )
     => ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B ) ) ) ).

% UnI1
thf(fact_355_UnI2,axiom,
    ! [C: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ B )
     => ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).

% UnI2
thf(fact_356_UnI2,axiom,
    ! [C: v,B: set_v,A3: set_v] :
      ( ( member_v @ C @ B )
     => ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) ) ) ).

% UnI2
thf(fact_357_UnI2,axiom,
    ! [C: set_v,B: set_set_v,A3: set_set_v] :
      ( ( member_set_v @ C @ B )
     => ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B ) ) ) ).

% UnI2
thf(fact_358_bex__Un,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
      ( ( ? [X2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X2 @ A3 )
            & ( P @ X2 ) )
        | ? [X2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X2 @ B )
            & ( P @ X2 ) ) ) ) ).

% bex_Un
thf(fact_359_bex__Un,axiom,
    ! [A3: set_v,B: set_v,P: v > $o] :
      ( ( ? [X2: v] :
            ( ( member_v @ X2 @ ( sup_sup_set_v @ A3 @ B ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: v] :
            ( ( member_v @ X2 @ A3 )
            & ( P @ X2 ) )
        | ? [X2: v] :
            ( ( member_v @ X2 @ B )
            & ( P @ X2 ) ) ) ) ).

% bex_Un
thf(fact_360_bex__Un,axiom,
    ! [A3: set_set_v,B: set_set_v,P: set_v > $o] :
      ( ( ? [X2: set_v] :
            ( ( member_set_v @ X2 @ ( sup_sup_set_set_v @ A3 @ B ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: set_v] :
            ( ( member_set_v @ X2 @ A3 )
            & ( P @ X2 ) )
        | ? [X2: set_v] :
            ( ( member_set_v @ X2 @ B )
            & ( P @ X2 ) ) ) ) ).

% bex_Un
thf(fact_361_ball__Un,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
      ( ( ! [X2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X2 @ A3 )
           => ( P @ X2 ) )
        & ! [X2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X2 @ B )
           => ( P @ X2 ) ) ) ) ).

% ball_Un
thf(fact_362_ball__Un,axiom,
    ! [A3: set_v,B: set_v,P: v > $o] :
      ( ( ! [X2: v] :
            ( ( member_v @ X2 @ ( sup_sup_set_v @ A3 @ B ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: v] :
            ( ( member_v @ X2 @ A3 )
           => ( P @ X2 ) )
        & ! [X2: v] :
            ( ( member_v @ X2 @ B )
           => ( P @ X2 ) ) ) ) ).

% ball_Un
thf(fact_363_ball__Un,axiom,
    ! [A3: set_set_v,B: set_set_v,P: set_v > $o] :
      ( ( ! [X2: set_v] :
            ( ( member_set_v @ X2 @ ( sup_sup_set_set_v @ A3 @ B ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: set_v] :
            ( ( member_set_v @ X2 @ A3 )
           => ( P @ X2 ) )
        & ! [X2: set_v] :
            ( ( member_set_v @ X2 @ B )
           => ( P @ X2 ) ) ) ) ).

% ball_Un
thf(fact_364_Un__assoc,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C3 )
      = ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) ) ).

% Un_assoc
thf(fact_365_Un__assoc,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( sup_sup_set_v @ ( sup_sup_set_v @ A3 @ B ) @ C3 )
      = ( sup_sup_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) ) ) ).

% Un_assoc
thf(fact_366_Un__assoc,axiom,
    ! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
      ( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ C3 )
      = ( sup_sup_set_set_v @ A3 @ ( sup_sup_set_set_v @ B @ C3 ) ) ) ).

% Un_assoc
thf(fact_367_Un__absorb,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_368_Un__absorb,axiom,
    ! [A3: set_v] :
      ( ( sup_sup_set_v @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_369_Un__absorb,axiom,
    ! [A3: set_set_v] :
      ( ( sup_sup_set_set_v @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_370_Un__commute,axiom,
    ( sup_su414716646722978715od_v_v
    = ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B5 @ A5 ) ) ) ).

% Un_commute
thf(fact_371_Un__commute,axiom,
    ( sup_sup_set_v
    = ( ^ [A5: set_v,B5: set_v] : ( sup_sup_set_v @ B5 @ A5 ) ) ) ).

% Un_commute
thf(fact_372_Un__commute,axiom,
    ( sup_sup_set_set_v
    = ( ^ [A5: set_set_v,B5: set_set_v] : ( sup_sup_set_set_v @ B5 @ A5 ) ) ) ).

% Un_commute
thf(fact_373_Un__left__absorb,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
      = ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).

% Un_left_absorb
thf(fact_374_Un__left__absorb,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( sup_sup_set_v @ A3 @ ( sup_sup_set_v @ A3 @ B ) )
      = ( sup_sup_set_v @ A3 @ B ) ) ).

% Un_left_absorb
thf(fact_375_Un__left__absorb,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( sup_sup_set_set_v @ A3 @ ( sup_sup_set_set_v @ A3 @ B ) )
      = ( sup_sup_set_set_v @ A3 @ B ) ) ).

% Un_left_absorb
thf(fact_376_Un__left__commute,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) )
      = ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A3 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_377_Un__left__commute,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( sup_sup_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) )
      = ( sup_sup_set_v @ B @ ( sup_sup_set_v @ A3 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_378_Un__left__commute,axiom,
    ! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
      ( ( sup_sup_set_set_v @ A3 @ ( sup_sup_set_set_v @ B @ C3 ) )
      = ( sup_sup_set_set_v @ B @ ( sup_sup_set_set_v @ A3 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_379_graph_Oreachable__avoiding_Ocong,axiom,
    sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).

% graph.reachable_avoiding.cong
thf(fact_380_graph_Opost__dfs_Ocong,axiom,
    sCC_Bl8953792750115413617t_unit = sCC_Bl8953792750115413617t_unit ).

% graph.post_dfs.cong
thf(fact_381_graph_Ois__scc_Ocong,axiom,
    sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).

% graph.is_scc.cong
thf(fact_382_bot__set__def,axiom,
    ( bot_bot_set_v
    = ( collect_v @ bot_bot_v_o ) ) ).

% bot_set_def
thf(fact_383_bot__set__def,axiom,
    ( bot_bo723834152578015283od_v_v
    = ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).

% bot_set_def
thf(fact_384_bot__set__def,axiom,
    ( bot_bot_set_set_v
    = ( collect_set_v @ bot_bot_set_v_o ) ) ).

% bot_set_def
thf(fact_385_Un__empty__right,axiom,
    ! [A3: set_v] :
      ( ( sup_sup_set_v @ A3 @ bot_bot_set_v )
      = A3 ) ).

% Un_empty_right
thf(fact_386_Un__empty__right,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
      = A3 ) ).

% Un_empty_right
thf(fact_387_Un__empty__right,axiom,
    ! [A3: set_set_v] :
      ( ( sup_sup_set_set_v @ A3 @ bot_bot_set_set_v )
      = A3 ) ).

% Un_empty_right
thf(fact_388_Un__empty__left,axiom,
    ! [B: set_v] :
      ( ( sup_sup_set_v @ bot_bot_set_v @ B )
      = B ) ).

% Un_empty_left
thf(fact_389_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_390_Un__empty__left,axiom,
    ! [B: set_set_v] :
      ( ( sup_sup_set_set_v @ bot_bot_set_set_v @ B )
      = B ) ).

% Un_empty_left
thf(fact_391_Un__Diff,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C3 )
      = ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ C3 ) @ ( minus_4183494784930505774od_v_v @ B @ C3 ) ) ) ).

% Un_Diff
thf(fact_392_Un__Diff,axiom,
    ! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
      ( ( minus_7228012346218142266_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ C3 )
      = ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A3 @ C3 ) @ ( minus_7228012346218142266_set_v @ B @ C3 ) ) ) ).

% Un_Diff
thf(fact_393_Un__Diff,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( minus_minus_set_v @ ( sup_sup_set_v @ A3 @ B ) @ C3 )
      = ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ C3 ) @ ( minus_minus_set_v @ B @ C3 ) ) ) ).

% Un_Diff
thf(fact_394_graph_Ora__empty,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ bot_bo723834152578015283od_v_v )
        = ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).

% graph.ra_empty
thf(fact_395_graph_Odfs__S__tl__stack_I1_J,axiom,
    ! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
       => ( ( ( sCC_Bl8828226123343373779t_unit @ E )
           != nil_v )
         => ( ( sCC_Bl8828226123343373779t_unit @ E2 )
           != nil_v ) ) ) ) ).

% graph.dfs_S_tl_stack(1)
thf(fact_396_graph_Ora__refl,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,E4: set_Product_prod_v_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ X @ E4 ) ) ).

% graph.ra_refl
thf(fact_397_graph_Ora__trans,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
       => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y2 @ Z @ E4 )
         => ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E4 ) ) ) ) ).

% graph.ra_trans
thf(fact_398_graph_Oinit__env__pre__dfs,axiom,
    ! [Vertices: set_v,Successors: v > set_v,V3: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( sCC_Bl36166008131615352t_unit @ Successors @ V3 @ ( sCC_Bl7693227186847904995_env_v @ V3 ) ) ) ).

% graph.init_env_pre_dfs
thf(fact_399_singleton__Un__iff,axiom,
    ! [X: v,A3: set_v,B: set_v] :
      ( ( ( insert_v @ X @ bot_bot_set_v )
        = ( sup_sup_set_v @ A3 @ B ) )
      = ( ( ( A3 = bot_bot_set_v )
          & ( B
            = ( insert_v @ X @ bot_bot_set_v ) ) )
        | ( ( A3
            = ( insert_v @ X @ bot_bot_set_v ) )
          & ( B = bot_bot_set_v ) )
        | ( ( A3
            = ( insert_v @ X @ bot_bot_set_v ) )
          & ( B
            = ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_400_singleton__Un__iff,axiom,
    ! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v )
        = ( sup_su414716646722978715od_v_v @ A3 @ B ) )
      = ( ( ( A3 = bot_bo723834152578015283od_v_v )
          & ( B
            = ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
        | ( ( A3
            = ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
          & ( B = bot_bo723834152578015283od_v_v ) )
        | ( ( A3
            = ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
          & ( B
            = ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_401_singleton__Un__iff,axiom,
    ! [X: set_v,A3: set_set_v,B: set_set_v] :
      ( ( ( insert_set_v @ X @ bot_bot_set_set_v )
        = ( sup_sup_set_set_v @ A3 @ B ) )
      = ( ( ( A3 = bot_bot_set_set_v )
          & ( B
            = ( insert_set_v @ X @ bot_bot_set_set_v ) ) )
        | ( ( A3
            = ( insert_set_v @ X @ bot_bot_set_set_v ) )
          & ( B = bot_bot_set_set_v ) )
        | ( ( A3
            = ( insert_set_v @ X @ bot_bot_set_set_v ) )
          & ( B
            = ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_402_Un__singleton__iff,axiom,
    ! [A3: set_v,B: set_v,X: v] :
      ( ( ( sup_sup_set_v @ A3 @ B )
        = ( insert_v @ X @ bot_bot_set_v ) )
      = ( ( ( A3 = bot_bot_set_v )
          & ( B
            = ( insert_v @ X @ bot_bot_set_v ) ) )
        | ( ( A3
            = ( insert_v @ X @ bot_bot_set_v ) )
          & ( B = bot_bot_set_v ) )
        | ( ( A3
            = ( insert_v @ X @ bot_bot_set_v ) )
          & ( B
            = ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_403_Un__singleton__iff,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
      ( ( ( sup_su414716646722978715od_v_v @ A3 @ B )
        = ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
      = ( ( ( A3 = bot_bo723834152578015283od_v_v )
          & ( B
            = ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
        | ( ( A3
            = ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
          & ( B = bot_bo723834152578015283od_v_v ) )
        | ( ( A3
            = ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
          & ( B
            = ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_404_Un__singleton__iff,axiom,
    ! [A3: set_set_v,B: set_set_v,X: set_v] :
      ( ( ( sup_sup_set_set_v @ A3 @ B )
        = ( insert_set_v @ X @ bot_bot_set_set_v ) )
      = ( ( ( A3 = bot_bot_set_set_v )
          & ( B
            = ( insert_set_v @ X @ bot_bot_set_set_v ) ) )
        | ( ( A3
            = ( insert_set_v @ X @ bot_bot_set_set_v ) )
          & ( B = bot_bot_set_set_v ) )
        | ( ( A3
            = ( insert_set_v @ X @ bot_bot_set_set_v ) )
          & ( B
            = ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_405_insert__is__Un,axiom,
    ( insert_v
    = ( ^ [A4: v] : ( sup_sup_set_v @ ( insert_v @ A4 @ bot_bot_set_v ) ) ) ) ).

% insert_is_Un
thf(fact_406_insert__is__Un,axiom,
    ( insert1338601472111419319od_v_v
    = ( ^ [A4: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% insert_is_Un
thf(fact_407_insert__is__Un,axiom,
    ( insert_set_v
    = ( ^ [A4: set_v] : ( sup_sup_set_set_v @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) ) ) ).

% insert_is_Un
thf(fact_408_select__convs_I8_J,axiom,
    ! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
      ( ( sCC_Bl9201514103433284750t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
      = Cstack ) ).

% select_convs(8)
thf(fact_409_is__singleton__the__elem,axiom,
    ( is_singleton_v
    = ( ^ [A5: set_v] :
          ( A5
          = ( insert_v @ ( the_elem_v @ A5 ) @ bot_bot_set_v ) ) ) ) ).

% is_singleton_the_elem
thf(fact_410_is__singleton__the__elem,axiom,
    ( is_sin9198872032823709915od_v_v
    = ( ^ [A5: set_Product_prod_v_v] :
          ( A5
          = ( insert1338601472111419319od_v_v @ ( the_el5392834299063928540od_v_v @ A5 ) @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% is_singleton_the_elem
thf(fact_411_is__singleton__the__elem,axiom,
    ( is_singleton_set_v
    = ( ^ [A5: set_set_v] :
          ( A5
          = ( insert_set_v @ ( the_elem_set_v @ A5 ) @ bot_bot_set_set_v ) ) ) ) ).

% is_singleton_the_elem
thf(fact_412_select__convs_I1_J,axiom,
    ! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
      ( ( sCC_Bl1090238580953940555t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
      = Root ) ).

% select_convs(1)
thf(fact_413_is__singletonI_H,axiom,
    ! [A3: set_v] :
      ( ( A3 != bot_bot_set_v )
     => ( ! [X3: v,Y: v] :
            ( ( member_v @ X3 @ A3 )
           => ( ( member_v @ Y @ A3 )
             => ( X3 = Y ) ) )
       => ( is_singleton_v @ A3 ) ) ) ).

% is_singletonI'
thf(fact_414_is__singletonI_H,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( A3 != bot_bo723834152578015283od_v_v )
     => ( ! [X3: product_prod_v_v,Y: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ A3 )
           => ( ( member7453568604450474000od_v_v @ Y @ A3 )
             => ( X3 = Y ) ) )
       => ( is_sin9198872032823709915od_v_v @ A3 ) ) ) ).

% is_singletonI'
thf(fact_415_is__singletonI_H,axiom,
    ! [A3: set_set_v] :
      ( ( A3 != bot_bot_set_set_v )
     => ( ! [X3: set_v,Y: set_v] :
            ( ( member_set_v @ X3 @ A3 )
           => ( ( member_set_v @ Y @ A3 )
             => ( X3 = Y ) ) )
       => ( is_singleton_set_v @ A3 ) ) ) ).

% is_singletonI'
thf(fact_416_graph_Ora__reachable,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
       => ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).

% graph.ra_reachable
thf(fact_417_graph_Odfs__S__tl__stack_I2_J,axiom,
    ! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
       => ( ( ( sCC_Bl8828226123343373779t_unit @ E )
           != nil_v )
         => ! [X4: v] :
              ( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
             => ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
                = ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) ) ) ).

% graph.dfs_S_tl_stack(2)
thf(fact_418_graph_Odfs__S__hd__stack_I1_J,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
       => ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
         => ( ( ( sCC_Bl8828226123343373779t_unit @ E )
             != nil_v )
           => ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
             => ( ( sCC_Bl8828226123343373779t_unit @ E2 )
               != nil_v ) ) ) ) ) ) ).

% graph.dfs_S_hd_stack(1)
thf(fact_419_graph_Odfs__S__hd__stack_I2_J,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V3: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
       => ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V3 @ E @ E2 )
         => ( ( ( sCC_Bl8828226123343373779t_unit @ E )
             != nil_v )
           => ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
             => ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ).

% graph.dfs_S_hd_stack(2)
thf(fact_420_is__singleton__def,axiom,
    ( is_singleton_v
    = ( ^ [A5: set_v] :
        ? [X2: v] :
          ( A5
          = ( insert_v @ X2 @ bot_bot_set_v ) ) ) ) ).

% is_singleton_def
thf(fact_421_is__singleton__def,axiom,
    ( is_sin9198872032823709915od_v_v
    = ( ^ [A5: set_Product_prod_v_v] :
        ? [X2: product_prod_v_v] :
          ( A5
          = ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% is_singleton_def
thf(fact_422_is__singleton__def,axiom,
    ( is_singleton_set_v
    = ( ^ [A5: set_set_v] :
        ? [X2: set_v] :
          ( A5
          = ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) ) ) ).

% is_singleton_def
thf(fact_423_is__singletonE,axiom,
    ! [A3: set_v] :
      ( ( is_singleton_v @ A3 )
     => ~ ! [X3: v] :
            ( A3
           != ( insert_v @ X3 @ bot_bot_set_v ) ) ) ).

% is_singletonE
thf(fact_424_is__singletonE,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( is_sin9198872032823709915od_v_v @ A3 )
     => ~ ! [X3: product_prod_v_v] :
            ( A3
           != ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ).

% is_singletonE
thf(fact_425_is__singletonE,axiom,
    ! [A3: set_set_v] :
      ( ( is_singleton_set_v @ A3 )
     => ~ ! [X3: set_v] :
            ( A3
           != ( insert_set_v @ X3 @ bot_bot_set_set_v ) ) ) ).

% is_singletonE
thf(fact_426_graph_OsccE,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S4: set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S4 )
       => ( ( member7453568604450474000od_v_v @ X @ S4 )
         => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
           => ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X )
             => ( member7453568604450474000od_v_v @ Y2 @ S4 ) ) ) ) ) ) ).

% graph.sccE
thf(fact_427_graph_OsccE,axiom,
    ! [Vertices: set_v,Successors: v > set_v,S4: set_v,X: v,Y2: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bloemen_is_scc_v @ Successors @ S4 )
       => ( ( member_v @ X @ S4 )
         => ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
           => ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X )
             => ( member_v @ Y2 @ S4 ) ) ) ) ) ) ).

% graph.sccE
thf(fact_428_pre__dfss__def,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
      = ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
        & ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
        & ( ( sCC_Bl8828226123343373779t_unit @ E )
         != nil_v )
        & ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
        & ! [X2: v] :
            ( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
           => ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
        & ! [X2: v] :
            ( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
           => ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) )
        & ? [Ns: list_v] :
            ( ( sCC_Bl9201514103433284750t_unit @ E )
            = ( cons_v @ V3 @ Ns ) ) ) ) ).

% pre_dfss_def
thf(fact_429_tl__append2,axiom,
    ! [Xs: list_v,Ys: list_v] :
      ( ( Xs != nil_v )
     => ( ( tl_v @ ( append_v @ Xs @ Ys ) )
        = ( append_v @ ( tl_v @ Xs ) @ Ys ) ) ) ).

% tl_append2
thf(fact_430_hd__append2,axiom,
    ! [Xs: list_v,Ys: list_v] :
      ( ( Xs != nil_v )
     => ( ( hd_v @ ( append_v @ Xs @ Ys ) )
        = ( hd_v @ Xs ) ) ) ).

% hd_append2
thf(fact_431_set__append,axiom,
    ! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
      ( ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
      = ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ).

% set_append
thf(fact_432_set__append,axiom,
    ! [Xs: list_v,Ys: list_v] :
      ( ( set_v2 @ ( append_v @ Xs @ Ys ) )
      = ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ).

% set_append
thf(fact_433_set__append,axiom,
    ! [Xs: list_set_v,Ys: list_set_v] :
      ( ( set_set_v2 @ ( append_set_v @ Xs @ Ys ) )
      = ( sup_sup_set_set_v @ ( set_set_v2 @ Xs ) @ ( set_set_v2 @ Ys ) ) ) ).

% set_append
thf(fact_434_set__empty,axiom,
    ! [Xs: list_v] :
      ( ( ( set_v2 @ Xs )
        = bot_bot_set_v )
      = ( Xs = nil_v ) ) ).

% set_empty
thf(fact_435_set__empty,axiom,
    ! [Xs: list_P7986770385144383213od_v_v] :
      ( ( ( set_Product_prod_v_v2 @ Xs )
        = bot_bo723834152578015283od_v_v )
      = ( Xs = nil_Product_prod_v_v ) ) ).

% set_empty
thf(fact_436_set__empty,axiom,
    ! [Xs: list_set_v] :
      ( ( ( set_set_v2 @ Xs )
        = bot_bot_set_set_v )
      = ( Xs = nil_set_v ) ) ).

% set_empty
thf(fact_437_set__empty2,axiom,
    ! [Xs: list_v] :
      ( ( bot_bot_set_v
        = ( set_v2 @ Xs ) )
      = ( Xs = nil_v ) ) ).

% set_empty2
thf(fact_438_set__empty2,axiom,
    ! [Xs: list_P7986770385144383213od_v_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( set_Product_prod_v_v2 @ Xs ) )
      = ( Xs = nil_Product_prod_v_v ) ) ).

% set_empty2
thf(fact_439_set__empty2,axiom,
    ! [Xs: list_set_v] :
      ( ( bot_bot_set_set_v
        = ( set_set_v2 @ Xs ) )
      = ( Xs = nil_set_v ) ) ).

% set_empty2
thf(fact_440_list_Oinject,axiom,
    ! [X21: v,X22: list_v,Y21: v,Y22: list_v] :
      ( ( ( cons_v @ X21 @ X22 )
        = ( cons_v @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_441_list_Osimps_I15_J,axiom,
    ! [X21: product_prod_v_v,X22: list_P7986770385144383213od_v_v] :
      ( ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) )
      = ( insert1338601472111419319od_v_v @ X21 @ ( set_Product_prod_v_v2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_442_list_Osimps_I15_J,axiom,
    ! [X21: set_v,X22: list_set_v] :
      ( ( set_set_v2 @ ( cons_set_v @ X21 @ X22 ) )
      = ( insert_set_v @ X21 @ ( set_set_v2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_443_list_Osimps_I15_J,axiom,
    ! [X21: v,X22: list_v] :
      ( ( set_v2 @ ( cons_v @ X21 @ X22 ) )
      = ( insert_v @ X21 @ ( set_v2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_444_append1__eq__conv,axiom,
    ! [Xs: list_v,X: v,Ys: list_v,Y2: v] :
      ( ( ( append_v @ Xs @ ( cons_v @ X @ nil_v ) )
        = ( append_v @ Ys @ ( cons_v @ Y2 @ nil_v ) ) )
      = ( ( Xs = Ys )
        & ( X = Y2 ) ) ) ).

% append1_eq_conv
thf(fact_445_list_Ocollapse,axiom,
    ! [List: list_v] :
      ( ( List != nil_v )
     => ( ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) )
        = List ) ) ).

% list.collapse
thf(fact_446_hd__Cons__tl,axiom,
    ! [Xs: list_v] :
      ( ( Xs != nil_v )
     => ( ( cons_v @ ( hd_v @ Xs ) @ ( tl_v @ Xs ) )
        = Xs ) ) ).

% hd_Cons_tl
thf(fact_447_list__nonempty__induct,axiom,
    ! [Xs: list_v,P: list_v > $o] :
      ( ( Xs != nil_v )
     => ( ! [X3: v] : ( P @ ( cons_v @ X3 @ nil_v ) )
       => ( ! [X3: v,Xs2: list_v] :
              ( ( Xs2 != nil_v )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_v @ X3 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_448_Cons__eq__appendI,axiom,
    ! [X: v,Xs1: list_v,Ys: list_v,Xs: list_v,Zs: list_v] :
      ( ( ( cons_v @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_v @ Xs1 @ Zs ) )
       => ( ( cons_v @ X @ Xs )
          = ( append_v @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_449_list__induct2_H,axiom,
    ! [P: list_v > list_v > $o,Xs: list_v,Ys: list_v] :
      ( ( P @ nil_v @ nil_v )
     => ( ! [X3: v,Xs2: list_v] : ( P @ ( cons_v @ X3 @ Xs2 ) @ nil_v )
       => ( ! [Y: v,Ys2: list_v] : ( P @ nil_v @ ( cons_v @ Y @ Ys2 ) )
         => ( ! [X3: v,Xs2: list_v,Y: v,Ys2: list_v] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_v @ X3 @ Xs2 ) @ ( cons_v @ Y @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_450_neq__Nil__conv,axiom,
    ! [Xs: list_v] :
      ( ( Xs != nil_v )
      = ( ? [Y3: v,Ys3: list_v] :
            ( Xs
            = ( cons_v @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_451_append__Cons,axiom,
    ! [X: v,Xs: list_v,Ys: list_v] :
      ( ( append_v @ ( cons_v @ X @ Xs ) @ Ys )
      = ( cons_v @ X @ ( append_v @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_452_remdups__adj_Ocases,axiom,
    ! [X: list_v] :
      ( ( X != nil_v )
     => ( ! [X3: v] :
            ( X
           != ( cons_v @ X3 @ nil_v ) )
       => ~ ! [X3: v,Y: v,Xs2: list_v] :
              ( X
             != ( cons_v @ X3 @ ( cons_v @ Y @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_453_transpose_Ocases,axiom,
    ! [X: list_list_v] :
      ( ( X != nil_list_v )
     => ( ! [Xss: list_list_v] :
            ( X
           != ( cons_list_v @ nil_v @ Xss ) )
       => ~ ! [X3: v,Xs2: list_v,Xss: list_list_v] :
              ( X
             != ( cons_list_v @ ( cons_v @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_454_list_Oexhaust,axiom,
    ! [Y2: list_v] :
      ( ( Y2 != nil_v )
     => ~ ! [X212: v,X222: list_v] :
            ( Y2
           != ( cons_v @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_455_list_OdiscI,axiom,
    ! [List: list_v,X21: v,X22: list_v] :
      ( ( List
        = ( cons_v @ X21 @ X22 ) )
     => ( List != nil_v ) ) ).

% list.discI
thf(fact_456_list_Odistinct_I1_J,axiom,
    ! [X21: v,X22: list_v] :
      ( nil_v
     != ( cons_v @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_457_set__ConsD,axiom,
    ! [Y2: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
      ( ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
     => ( ( Y2 = X )
        | ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_458_set__ConsD,axiom,
    ! [Y2: v,X: v,Xs: list_v] :
      ( ( member_v @ Y2 @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
     => ( ( Y2 = X )
        | ( member_v @ Y2 @ ( set_v2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_459_list_Oset__cases,axiom,
    ! [E: product_prod_v_v,A: list_P7986770385144383213od_v_v] :
      ( ( member7453568604450474000od_v_v @ E @ ( set_Product_prod_v_v2 @ A ) )
     => ( ! [Z22: list_P7986770385144383213od_v_v] :
            ( A
           != ( cons_P4120604216776828829od_v_v @ E @ Z22 ) )
       => ~ ! [Z1: product_prod_v_v,Z22: list_P7986770385144383213od_v_v] :
              ( ( A
                = ( cons_P4120604216776828829od_v_v @ Z1 @ Z22 ) )
             => ~ ( member7453568604450474000od_v_v @ E @ ( set_Product_prod_v_v2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_460_list_Oset__cases,axiom,
    ! [E: v,A: list_v] :
      ( ( member_v @ E @ ( set_v2 @ A ) )
     => ( ! [Z22: list_v] :
            ( A
           != ( cons_v @ E @ Z22 ) )
       => ~ ! [Z1: v,Z22: list_v] :
              ( ( A
                = ( cons_v @ Z1 @ Z22 ) )
             => ~ ( member_v @ E @ ( set_v2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_461_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_v_v,X22: list_P7986770385144383213od_v_v] : ( member7453568604450474000od_v_v @ X21 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_462_list_Oset__intros_I1_J,axiom,
    ! [X21: v,X22: list_v] : ( member_v @ X21 @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_463_list_Oset__intros_I2_J,axiom,
    ! [Y2: product_prod_v_v,X22: list_P7986770385144383213od_v_v,X21: product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ X22 ) )
     => ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_464_list_Oset__intros_I2_J,axiom,
    ! [Y2: v,X22: list_v,X21: v] :
      ( ( member_v @ Y2 @ ( set_v2 @ X22 ) )
     => ( member_v @ Y2 @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_465_not__Cons__self2,axiom,
    ! [X: v,Xs: list_v] :
      ( ( cons_v @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_466_list_Osel_I1_J,axiom,
    ! [X21: v,X22: list_v] :
      ( ( hd_v @ ( cons_v @ X21 @ X22 ) )
      = X21 ) ).

% list.sel(1)
thf(fact_467_list_Osel_I3_J,axiom,
    ! [X21: v,X22: list_v] :
      ( ( tl_v @ ( cons_v @ X21 @ X22 ) )
      = X22 ) ).

% list.sel(3)
thf(fact_468_rev__induct,axiom,
    ! [P: list_v > $o,Xs: list_v] :
      ( ( P @ nil_v )
     => ( ! [X3: v,Xs2: list_v] :
            ( ( P @ Xs2 )
           => ( P @ ( append_v @ Xs2 @ ( cons_v @ X3 @ nil_v ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_469_rev__exhaust,axiom,
    ! [Xs: list_v] :
      ( ( Xs != nil_v )
     => ~ ! [Ys2: list_v,Y: v] :
            ( Xs
           != ( append_v @ Ys2 @ ( cons_v @ Y @ nil_v ) ) ) ) ).

% rev_exhaust
thf(fact_470_Cons__eq__append__conv,axiom,
    ! [X: v,Xs: list_v,Ys: list_v,Zs: list_v] :
      ( ( ( cons_v @ X @ Xs )
        = ( append_v @ Ys @ Zs ) )
      = ( ( ( Ys = nil_v )
          & ( ( cons_v @ X @ Xs )
            = Zs ) )
        | ? [Ys4: list_v] :
            ( ( ( cons_v @ X @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_v @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_471_append__eq__Cons__conv,axiom,
    ! [Ys: list_v,Zs: list_v,X: v,Xs: list_v] :
      ( ( ( append_v @ Ys @ Zs )
        = ( cons_v @ X @ Xs ) )
      = ( ( ( Ys = nil_v )
          & ( Zs
            = ( cons_v @ X @ Xs ) ) )
        | ? [Ys4: list_v] :
            ( ( Ys
              = ( cons_v @ X @ Ys4 ) )
            & ( ( append_v @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_472_rev__nonempty__induct,axiom,
    ! [Xs: list_v,P: list_v > $o] :
      ( ( Xs != nil_v )
     => ( ! [X3: v] : ( P @ ( cons_v @ X3 @ nil_v ) )
       => ( ! [X3: v,Xs2: list_v] :
              ( ( Xs2 != nil_v )
             => ( ( P @ Xs2 )
               => ( P @ ( append_v @ Xs2 @ ( cons_v @ X3 @ nil_v ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_473_split__list,axiom,
    ! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
      ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
     => ? [Ys2: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
          ( Xs
          = ( append2138873909117096322od_v_v @ Ys2 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_474_split__list,axiom,
    ! [X: v,Xs: list_v] :
      ( ( member_v @ X @ ( set_v2 @ Xs ) )
     => ? [Ys2: list_v,Zs2: list_v] :
          ( Xs
          = ( append_v @ Ys2 @ ( cons_v @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_475_split__list__last,axiom,
    ! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
      ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
     => ? [Ys2: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
          ( ( Xs
            = ( append2138873909117096322od_v_v @ Ys2 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) )
          & ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_476_split__list__last,axiom,
    ! [X: v,Xs: list_v] :
      ( ( member_v @ X @ ( set_v2 @ Xs ) )
     => ? [Ys2: list_v,Zs2: list_v] :
          ( ( Xs
            = ( append_v @ Ys2 @ ( cons_v @ X @ Zs2 ) ) )
          & ~ ( member_v @ X @ ( set_v2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_477_split__list__prop,axiom,
    ! [Xs: list_v,P: v > $o] :
      ( ? [X4: v] :
          ( ( member_v @ X4 @ ( set_v2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys2: list_v,X3: v] :
          ( ? [Zs2: list_v] :
              ( Xs
              = ( append_v @ Ys2 @ ( cons_v @ X3 @ Zs2 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_478_split__list__first,axiom,
    ! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
      ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
     => ? [Ys2: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
          ( ( Xs
            = ( append2138873909117096322od_v_v @ Ys2 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) )
          & ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_479_split__list__first,axiom,
    ! [X: v,Xs: list_v] :
      ( ( member_v @ X @ ( set_v2 @ Xs ) )
     => ? [Ys2: list_v,Zs2: list_v] :
          ( ( Xs
            = ( append_v @ Ys2 @ ( cons_v @ X @ Zs2 ) ) )
          & ~ ( member_v @ X @ ( set_v2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_480_split__list__propE,axiom,
    ! [Xs: list_v,P: v > $o] :
      ( ? [X4: v] :
          ( ( member_v @ X4 @ ( set_v2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys2: list_v,X3: v] :
            ( ? [Zs2: list_v] :
                ( Xs
                = ( append_v @ Ys2 @ ( cons_v @ X3 @ Zs2 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_481_append__Cons__eq__iff,axiom,
    ! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v,Xs3: list_P7986770385144383213od_v_v,Ys5: list_P7986770385144383213od_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
     => ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
       => ( ( ( append2138873909117096322od_v_v @ Xs @ ( cons_P4120604216776828829od_v_v @ X @ Ys ) )
            = ( append2138873909117096322od_v_v @ Xs3 @ ( cons_P4120604216776828829od_v_v @ X @ Ys5 ) ) )
          = ( ( Xs = Xs3 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_482_append__Cons__eq__iff,axiom,
    ! [X: v,Xs: list_v,Ys: list_v,Xs3: list_v,Ys5: list_v] :
      ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
     => ( ~ ( member_v @ X @ ( set_v2 @ Ys ) )
       => ( ( ( append_v @ Xs @ ( cons_v @ X @ Ys ) )
            = ( append_v @ Xs3 @ ( cons_v @ X @ Ys5 ) ) )
          = ( ( Xs = Xs3 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_483_in__set__conv__decomp,axiom,
    ! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
      ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
      = ( ? [Ys3: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
            ( Xs
            = ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_484_in__set__conv__decomp,axiom,
    ! [X: v,Xs: list_v] :
      ( ( member_v @ X @ ( set_v2 @ Xs ) )
      = ( ? [Ys3: list_v,Zs3: list_v] :
            ( Xs
            = ( append_v @ Ys3 @ ( cons_v @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_485_split__list__last__prop,axiom,
    ! [Xs: list_v,P: v > $o] :
      ( ? [X4: v] :
          ( ( member_v @ X4 @ ( set_v2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys2: list_v,X3: v,Zs2: list_v] :
          ( ( Xs
            = ( append_v @ Ys2 @ ( cons_v @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: v] :
              ( ( member_v @ Xa2 @ ( set_v2 @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_486_split__list__first__prop,axiom,
    ! [Xs: list_v,P: v > $o] :
      ( ? [X4: v] :
          ( ( member_v @ X4 @ ( set_v2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys2: list_v,X3: v] :
          ( ? [Zs2: list_v] :
              ( Xs
              = ( append_v @ Ys2 @ ( cons_v @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: v] :
              ( ( member_v @ Xa2 @ ( set_v2 @ Ys2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_487_split__list__last__propE,axiom,
    ! [Xs: list_v,P: v > $o] :
      ( ? [X4: v] :
          ( ( member_v @ X4 @ ( set_v2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys2: list_v,X3: v,Zs2: list_v] :
            ( ( Xs
              = ( append_v @ Ys2 @ ( cons_v @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: v] :
                    ( ( member_v @ Xa2 @ ( set_v2 @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_488_split__list__first__propE,axiom,
    ! [Xs: list_v,P: v > $o] :
      ( ? [X4: v] :
          ( ( member_v @ X4 @ ( set_v2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys2: list_v,X3: v] :
            ( ? [Zs2: list_v] :
                ( Xs
                = ( append_v @ Ys2 @ ( cons_v @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: v] :
                    ( ( member_v @ Xa2 @ ( set_v2 @ Ys2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_489_in__set__conv__decomp__last,axiom,
    ! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
      ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
      = ( ? [Ys3: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
            ( ( Xs
              = ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) )
            & ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_490_in__set__conv__decomp__last,axiom,
    ! [X: v,Xs: list_v] :
      ( ( member_v @ X @ ( set_v2 @ Xs ) )
      = ( ? [Ys3: list_v,Zs3: list_v] :
            ( ( Xs
              = ( append_v @ Ys3 @ ( cons_v @ X @ Zs3 ) ) )
            & ~ ( member_v @ X @ ( set_v2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_491_in__set__conv__decomp__first,axiom,
    ! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
      ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
      = ( ? [Ys3: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
            ( ( Xs
              = ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) )
            & ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_492_in__set__conv__decomp__first,axiom,
    ! [X: v,Xs: list_v] :
      ( ( member_v @ X @ ( set_v2 @ Xs ) )
      = ( ? [Ys3: list_v,Zs3: list_v] :
            ( ( Xs
              = ( append_v @ Ys3 @ ( cons_v @ X @ Zs3 ) ) )
            & ~ ( member_v @ X @ ( set_v2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_493_split__list__last__prop__iff,axiom,
    ! [Xs: list_v,P: v > $o] :
      ( ( ? [X2: v] :
            ( ( member_v @ X2 @ ( set_v2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [Ys3: list_v,X2: v,Zs3: list_v] :
            ( ( Xs
              = ( append_v @ Ys3 @ ( cons_v @ X2 @ Zs3 ) ) )
            & ( P @ X2 )
            & ! [Y3: v] :
                ( ( member_v @ Y3 @ ( set_v2 @ Zs3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_494_split__list__first__prop__iff,axiom,
    ! [Xs: list_v,P: v > $o] :
      ( ( ? [X2: v] :
            ( ( member_v @ X2 @ ( set_v2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [Ys3: list_v,X2: v] :
            ( ? [Zs3: list_v] :
                ( Xs
                = ( append_v @ Ys3 @ ( cons_v @ X2 @ Zs3 ) ) )
            & ( P @ X2 )
            & ! [Y3: v] :
                ( ( member_v @ Y3 @ ( set_v2 @ Ys3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_495_tl__Nil,axiom,
    ! [Xs: list_v] :
      ( ( ( tl_v @ Xs )
        = nil_v )
      = ( ( Xs = nil_v )
        | ? [X2: v] :
            ( Xs
            = ( cons_v @ X2 @ nil_v ) ) ) ) ).

% tl_Nil
thf(fact_496_Nil__tl,axiom,
    ! [Xs: list_v] :
      ( ( nil_v
        = ( tl_v @ Xs ) )
      = ( ( Xs = nil_v )
        | ? [X2: v] :
            ( Xs
            = ( cons_v @ X2 @ nil_v ) ) ) ) ).

% Nil_tl
thf(fact_497_list_Oexhaust__sel,axiom,
    ! [List: list_v] :
      ( ( List != nil_v )
     => ( List
        = ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) ) ) ) ).

% list.exhaust_sel
thf(fact_498_the__elem__set,axiom,
    ! [X: v] :
      ( ( the_elem_v @ ( set_v2 @ ( cons_v @ X @ nil_v ) ) )
      = X ) ).

% the_elem_set
thf(fact_499_list_Osel_I2_J,axiom,
    ( ( tl_v @ nil_v )
    = nil_v ) ).

% list.sel(2)
thf(fact_500_empty__set,axiom,
    ( bot_bot_set_v
    = ( set_v2 @ nil_v ) ) ).

% empty_set
thf(fact_501_empty__set,axiom,
    ( bot_bo723834152578015283od_v_v
    = ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).

% empty_set
thf(fact_502_empty__set,axiom,
    ( bot_bot_set_set_v
    = ( set_set_v2 @ nil_set_v ) ) ).

% empty_set
thf(fact_503_hd__in__set,axiom,
    ! [Xs: list_P7986770385144383213od_v_v] :
      ( ( Xs != nil_Product_prod_v_v )
     => ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ Xs ) @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).

% hd_in_set
thf(fact_504_hd__in__set,axiom,
    ! [Xs: list_v] :
      ( ( Xs != nil_v )
     => ( member_v @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).

% hd_in_set
thf(fact_505_list_Oset__sel_I1_J,axiom,
    ! [A: list_P7986770385144383213od_v_v] :
      ( ( A != nil_Product_prod_v_v )
     => ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ A ) @ ( set_Product_prod_v_v2 @ A ) ) ) ).

% list.set_sel(1)
thf(fact_506_list_Oset__sel_I1_J,axiom,
    ! [A: list_v] :
      ( ( A != nil_v )
     => ( member_v @ ( hd_v @ A ) @ ( set_v2 @ A ) ) ) ).

% list.set_sel(1)
thf(fact_507_list_Oset__sel_I2_J,axiom,
    ! [A: list_P7986770385144383213od_v_v,X: product_prod_v_v] :
      ( ( A != nil_Product_prod_v_v )
     => ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ A ) ) )
       => ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ A ) ) ) ) ).

% list.set_sel(2)
thf(fact_508_list_Oset__sel_I2_J,axiom,
    ! [A: list_v,X: v] :
      ( ( A != nil_v )
     => ( ( member_v @ X @ ( set_v2 @ ( tl_v @ A ) ) )
       => ( member_v @ X @ ( set_v2 @ A ) ) ) ) ).

% list.set_sel(2)
thf(fact_509_longest__common__prefix,axiom,
    ! [Xs: list_v,Ys: list_v] :
    ? [Ps: list_v,Xs4: list_v,Ys6: list_v] :
      ( ( Xs
        = ( append_v @ Ps @ Xs4 ) )
      & ( Ys
        = ( append_v @ Ps @ Ys6 ) )
      & ( ( Xs4 = nil_v )
        | ( Ys6 = nil_v )
        | ( ( hd_v @ Xs4 )
         != ( hd_v @ Ys6 ) ) ) ) ).

% longest_common_prefix
thf(fact_510_hd__append,axiom,
    ! [Xs: list_v,Ys: list_v] :
      ( ( ( Xs = nil_v )
       => ( ( hd_v @ ( append_v @ Xs @ Ys ) )
          = ( hd_v @ Ys ) ) )
      & ( ( Xs != nil_v )
       => ( ( hd_v @ ( append_v @ Xs @ Ys ) )
          = ( hd_v @ Xs ) ) ) ) ).

% hd_append
thf(fact_511_tl__append__if,axiom,
    ! [Xs: list_v,Ys: list_v] :
      ( ( ( Xs = nil_v )
       => ( ( tl_v @ ( append_v @ Xs @ Ys ) )
          = ( tl_v @ Ys ) ) )
      & ( ( Xs != nil_v )
       => ( ( tl_v @ ( append_v @ Xs @ Ys ) )
          = ( append_v @ ( tl_v @ Xs ) @ Ys ) ) ) ) ).

% tl_append_if
thf(fact_512_list_Oexpand,axiom,
    ! [List: list_v,List2: list_v] :
      ( ( ( List = nil_v )
        = ( List2 = nil_v ) )
     => ( ( ( List != nil_v )
         => ( ( List2 != nil_v )
           => ( ( ( hd_v @ List )
                = ( hd_v @ List2 ) )
              & ( ( tl_v @ List )
                = ( tl_v @ List2 ) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
thf(fact_513_graph_Opre__dfss__def,axiom,
    ! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
        = ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
          & ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
          & ( ( sCC_Bl8828226123343373779t_unit @ E )
           != nil_v )
          & ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
          & ! [X2: v] :
              ( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
             => ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
          & ! [X2: v] :
              ( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
             => ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V3 ) )
          & ? [Ns: list_v] :
              ( ( sCC_Bl9201514103433284750t_unit @ E )
              = ( cons_v @ V3 @ Ns ) ) ) ) ) ).

% graph.pre_dfss_def
thf(fact_514_avoiding__explored,axiom,
    ! [E: sCC_Bl1394983891496994913t_unit,X: v,Y2: v,E4: set_Product_prod_v_v,W: v,V3: v] :
      ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
     => ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
       => ( ~ ( member_v @ Y2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
         => ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
           => ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).

% avoiding_explored
thf(fact_515_is__scc__def,axiom,
    ! [S4: set_v] :
      ( ( sCC_Bloemen_is_scc_v @ successors @ S4 )
      = ( ( S4 != bot_bot_set_v )
        & ( sCC_Bl5398416737448265317bscc_v @ successors @ S4 )
        & ! [S5: set_v] :
            ( ( ( ord_less_eq_set_v @ S4 @ S5 )
              & ( sCC_Bl5398416737448265317bscc_v @ successors @ S5 ) )
           => ( S5 = S4 ) ) ) ) ).

% is_scc_def
thf(fact_516_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_517_set__union,axiom,
    ! [Xs: list_v,Ys: list_v] :
      ( ( set_v2 @ ( union_v @ Xs @ Ys ) )
      = ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ).

% set_union
thf(fact_518_set__union,axiom,
    ! [Xs: list_set_v,Ys: list_set_v] :
      ( ( set_set_v2 @ ( union_set_v @ Xs @ Ys ) )
      = ( sup_sup_set_set_v @ ( set_set_v2 @ Xs ) @ ( set_set_v2 @ Ys ) ) ) ).

% set_union
thf(fact_519_sclosed,axiom,
    ! [X4: v] :
      ( ( member_v @ X4 @ vertices )
     => ( ord_less_eq_set_v @ ( successors @ X4 ) @ vertices ) ) ).

% sclosed
thf(fact_520_ra__add__edge,axiom,
    ! [X: v,Y2: v,E4: set_Product_prod_v_v,V3: v,W: v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
     => ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
        | ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ V3 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
          & ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).

% ra_add_edge
thf(fact_521_surjective,axiom,
    ! [R: sCC_Bl1394983891496994913t_unit] :
      ( R
      = ( sCC_Bl8064756265740546429t_unit @ ( sCC_Bl1090238580953940555t_unit @ R ) @ ( sCC_Bl1280885523602775798t_unit @ R ) @ ( sCC_Bl157864678168468314t_unit @ R ) @ ( sCC_Bl4645233313691564917t_unit @ R ) @ ( sCC_Bl3795065053823578884t_unit @ R ) @ ( sCC_Bl2536197123907397897t_unit @ R ) @ ( sCC_Bl8828226123343373779t_unit @ R ) @ ( sCC_Bl9201514103433284750t_unit @ R ) @ ( sCC_Bl3567736435408124606t_unit @ R ) ) ) ).

% surjective
thf(fact_522_equality,axiom,
    ! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit] :
      ( ( ( sCC_Bl1090238580953940555t_unit @ R )
        = ( sCC_Bl1090238580953940555t_unit @ R2 ) )
     => ( ( ( sCC_Bl1280885523602775798t_unit @ R )
          = ( sCC_Bl1280885523602775798t_unit @ R2 ) )
       => ( ( ( sCC_Bl157864678168468314t_unit @ R )
            = ( sCC_Bl157864678168468314t_unit @ R2 ) )
         => ( ( ( sCC_Bl4645233313691564917t_unit @ R )
              = ( sCC_Bl4645233313691564917t_unit @ R2 ) )
           => ( ( ( sCC_Bl3795065053823578884t_unit @ R )
                = ( sCC_Bl3795065053823578884t_unit @ R2 ) )
             => ( ( ( sCC_Bl2536197123907397897t_unit @ R )
                  = ( sCC_Bl2536197123907397897t_unit @ R2 ) )
               => ( ( ( sCC_Bl8828226123343373779t_unit @ R )
                    = ( sCC_Bl8828226123343373779t_unit @ R2 ) )
                 => ( ( ( sCC_Bl9201514103433284750t_unit @ R )
                      = ( sCC_Bl9201514103433284750t_unit @ R2 ) )
                   => ( ( ( sCC_Bl3567736435408124606t_unit @ R )
                        = ( sCC_Bl3567736435408124606t_unit @ R2 ) )
                     => ( R = R2 ) ) ) ) ) ) ) ) ) ) ).

% equality
thf(fact_523_ra__cases,axiom,
    ! [X: v,Y2: v,E4: set_Product_prod_v_v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
     => ( ( X = Y2 )
        | ? [Z3: v] :
            ( ( member_v @ Z3 @ ( successors @ X ) )
            & ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E4 )
            & ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y2 @ E4 ) ) ) ) ).

% ra_cases
thf(fact_524_edge__ra,axiom,
    ! [Y2: v,X: v,E4: set_Product_prod_v_v] :
      ( ( member_v @ Y2 @ ( successors @ X ) )
     => ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ E4 )
       => ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 ) ) ) ).

% edge_ra
thf(fact_525_reachable__avoiding_Osimps,axiom,
    ! [A1: v,A2: v,A32: set_Product_prod_v_v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
      = ( ? [X2: v,E5: set_Product_prod_v_v] :
            ( ( A1 = X2 )
            & ( A2 = X2 )
            & ( A32 = E5 ) )
        | ? [X2: v,Y3: v,E5: set_Product_prod_v_v,Z2: v] :
            ( ( A1 = X2 )
            & ( A2 = Z2 )
            & ( A32 = E5 )
            & ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y3 @ E5 )
            & ( member_v @ Z2 @ ( successors @ Y3 ) )
            & ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E5 ) ) ) ) ).

% reachable_avoiding.simps
thf(fact_526_ra__succ,axiom,
    ! [X: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
     => ( ( member_v @ Z @ ( successors @ Y2 ) )
       => ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z ) @ E4 )
         => ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E4 ) ) ) ) ).

% ra_succ
thf(fact_527_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_528_order__refl,axiom,
    ! [X: set_v] : ( ord_less_eq_set_v @ X @ X ) ).

% order_refl
thf(fact_529_order__refl,axiom,
    ! [X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ X ) ).

% order_refl
thf(fact_530_dual__order_Orefl,axiom,
    ! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).

% dual_order.refl
thf(fact_531_dual__order_Orefl,axiom,
    ! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).

% dual_order.refl
thf(fact_532_subsetI,axiom,
    ! [A3: set_v,B: set_v] :
      ( ! [X3: v] :
          ( ( member_v @ X3 @ A3 )
         => ( member_v @ X3 @ B ) )
     => ( ord_less_eq_set_v @ A3 @ B ) ) ).

% subsetI
thf(fact_533_subsetI,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ! [X3: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X3 @ A3 )
         => ( member7453568604450474000od_v_v @ X3 @ B ) )
     => ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).

% subsetI
thf(fact_534_subset__antisym,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B )
     => ( ( ord_less_eq_set_v @ B @ A3 )
       => ( A3 = B ) ) ) ).

% subset_antisym
thf(fact_535_subset__antisym,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
     => ( ( ord_le7336532860387713383od_v_v @ B @ A3 )
       => ( A3 = B ) ) ) ).

% subset_antisym
thf(fact_536_le__sup__iff,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X @ Y2 ) @ Z )
      = ( ( ord_le5216385588623774835_set_v @ X @ Z )
        & ( ord_le5216385588623774835_set_v @ Y2 @ Z ) ) ) ).

% le_sup_iff
thf(fact_537_le__sup__iff,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ Z )
      = ( ( ord_less_eq_set_v @ X @ Z )
        & ( ord_less_eq_set_v @ Y2 @ Z ) ) ) ).

% le_sup_iff
thf(fact_538_le__sup__iff,axiom,
    ! [X: 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 @ X @ Y2 ) @ Z )
      = ( ( ord_le7336532860387713383od_v_v @ X @ Z )
        & ( ord_le7336532860387713383od_v_v @ Y2 @ Z ) ) ) ).

% le_sup_iff
thf(fact_539_sup_Obounded__iff,axiom,
    ! [B2: set_set_v,C: set_set_v,A: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B2 @ C ) @ A )
      = ( ( ord_le5216385588623774835_set_v @ B2 @ A )
        & ( ord_le5216385588623774835_set_v @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_540_sup_Obounded__iff,axiom,
    ! [B2: set_v,C: set_v,A: set_v] :
      ( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A )
      = ( ( ord_less_eq_set_v @ B2 @ A )
        & ( ord_less_eq_set_v @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_541_sup_Obounded__iff,axiom,
    ! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A )
      = ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
        & ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_542_empty__subsetI,axiom,
    ! [A3: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A3 ) ).

% empty_subsetI
thf(fact_543_empty__subsetI,axiom,
    ! [A3: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A3 ) ).

% empty_subsetI
thf(fact_544_empty__subsetI,axiom,
    ! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A3 ) ).

% empty_subsetI
thf(fact_545_subset__empty,axiom,
    ! [A3: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A3 @ bot_bot_set_set_v )
      = ( A3 = bot_bot_set_set_v ) ) ).

% subset_empty
thf(fact_546_subset__empty,axiom,
    ! [A3: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ bot_bot_set_v )
      = ( A3 = bot_bot_set_v ) ) ).

% subset_empty
thf(fact_547_subset__empty,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
      = ( A3 = bot_bo723834152578015283od_v_v ) ) ).

% subset_empty
thf(fact_548_insert__subset,axiom,
    ! [X: set_v,A3: set_set_v,B: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ ( insert_set_v @ X @ A3 ) @ B )
      = ( ( member_set_v @ X @ B )
        & ( ord_le5216385588623774835_set_v @ A3 @ B ) ) ) ).

% insert_subset
thf(fact_549_insert__subset,axiom,
    ! [X: v,A3: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ ( insert_v @ X @ A3 ) @ B )
      = ( ( member_v @ X @ B )
        & ( ord_less_eq_set_v @ A3 @ B ) ) ) ).

% insert_subset
thf(fact_550_insert__subset,axiom,
    ! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B )
      = ( ( member7453568604450474000od_v_v @ X @ B )
        & ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ).

% insert_subset
thf(fact_551_Un__subset__iff,axiom,
    ! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ C3 )
      = ( ( ord_le5216385588623774835_set_v @ A3 @ C3 )
        & ( ord_le5216385588623774835_set_v @ B @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_552_Un__subset__iff,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B ) @ C3 )
      = ( ( ord_less_eq_set_v @ A3 @ C3 )
        & ( ord_less_eq_set_v @ B @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_553_Un__subset__iff,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C3 )
      = ( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
        & ( ord_le7336532860387713383od_v_v @ B @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_554_singleton__insert__inj__eq_H,axiom,
    ! [A: set_v,A3: set_set_v,B2: set_v] :
      ( ( ( insert_set_v @ A @ A3 )
        = ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
      = ( ( A = B2 )
        & ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_555_singleton__insert__inj__eq_H,axiom,
    ! [A: v,A3: set_v,B2: v] :
      ( ( ( insert_v @ A @ A3 )
        = ( insert_v @ B2 @ bot_bot_set_v ) )
      = ( ( A = B2 )
        & ( ord_less_eq_set_v @ A3 @ ( insert_v @ B2 @ bot_bot_set_v ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_556_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: product_prod_v_v] :
      ( ( ( insert1338601472111419319od_v_v @ A @ A3 )
        = ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
      = ( ( A = B2 )
        & ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_557_singleton__insert__inj__eq,axiom,
    ! [B2: set_v,A: set_v,A3: set_set_v] :
      ( ( ( insert_set_v @ B2 @ bot_bot_set_set_v )
        = ( insert_set_v @ A @ A3 ) )
      = ( ( A = B2 )
        & ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_558_singleton__insert__inj__eq,axiom,
    ! [B2: v,A: v,A3: set_v] :
      ( ( ( insert_v @ B2 @ bot_bot_set_v )
        = ( insert_v @ A @ A3 ) )
      = ( ( A = B2 )
        & ( ord_less_eq_set_v @ A3 @ ( insert_v @ B2 @ bot_bot_set_v ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_559_singleton__insert__inj__eq,axiom,
    ! [B2: product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v )
        = ( insert1338601472111419319od_v_v @ A @ A3 ) )
      = ( ( A = B2 )
        & ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_560_Diff__eq__empty__iff,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( ( minus_7228012346218142266_set_v @ A3 @ B )
        = bot_bot_set_set_v )
      = ( ord_le5216385588623774835_set_v @ A3 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_561_Diff__eq__empty__iff,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( ( minus_minus_set_v @ A3 @ B )
        = bot_bot_set_v )
      = ( ord_less_eq_set_v @ A3 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_562_Diff__eq__empty__iff,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( minus_4183494784930505774od_v_v @ A3 @ B )
        = bot_bo723834152578015283od_v_v )
      = ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_563_in__mono,axiom,
    ! [A3: set_v,B: set_v,X: v] :
      ( ( ord_less_eq_set_v @ A3 @ B )
     => ( ( member_v @ X @ A3 )
       => ( member_v @ X @ B ) ) ) ).

% in_mono
thf(fact_564_in__mono,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
     => ( ( member7453568604450474000od_v_v @ X @ A3 )
       => ( member7453568604450474000od_v_v @ X @ B ) ) ) ).

% in_mono
thf(fact_565_subsetD,axiom,
    ! [A3: set_v,B: set_v,C: v] :
      ( ( ord_less_eq_set_v @ A3 @ B )
     => ( ( member_v @ C @ A3 )
       => ( member_v @ C @ B ) ) ) ).

% subsetD
thf(fact_566_subsetD,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C: product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
     => ( ( member7453568604450474000od_v_v @ C @ A3 )
       => ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% subsetD
thf(fact_567_equalityE,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( A3 = B )
     => ~ ( ( ord_less_eq_set_v @ A3 @ B )
         => ~ ( ord_less_eq_set_v @ B @ A3 ) ) ) ).

% equalityE
thf(fact_568_equalityE,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( A3 = B )
     => ~ ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
         => ~ ( ord_le7336532860387713383od_v_v @ B @ A3 ) ) ) ).

% equalityE
thf(fact_569_subset__eq,axiom,
    ( ord_less_eq_set_v
    = ( ^ [A5: set_v,B5: set_v] :
        ! [X2: v] :
          ( ( member_v @ X2 @ A5 )
         => ( member_v @ X2 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_570_subset__eq,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
        ! [X2: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X2 @ A5 )
         => ( member7453568604450474000od_v_v @ X2 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_571_equalityD1,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( A3 = B )
     => ( ord_less_eq_set_v @ A3 @ B ) ) ).

% equalityD1
thf(fact_572_equalityD1,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( A3 = B )
     => ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).

% equalityD1
thf(fact_573_equalityD2,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( A3 = B )
     => ( ord_less_eq_set_v @ B @ A3 ) ) ).

% equalityD2
thf(fact_574_equalityD2,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( A3 = B )
     => ( ord_le7336532860387713383od_v_v @ B @ A3 ) ) ).

% equalityD2
thf(fact_575_subset__iff,axiom,
    ( ord_less_eq_set_v
    = ( ^ [A5: set_v,B5: set_v] :
        ! [T: v] :
          ( ( member_v @ T @ A5 )
         => ( member_v @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_576_subset__iff,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
        ! [T: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ T @ A5 )
         => ( member7453568604450474000od_v_v @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_577_subset__refl,axiom,
    ! [A3: set_v] : ( ord_less_eq_set_v @ A3 @ A3 ) ).

% subset_refl
thf(fact_578_subset__refl,axiom,
    ! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ A3 ) ).

% subset_refl
thf(fact_579_Collect__mono,axiom,
    ! [P: set_v > $o,Q: set_v > $o] :
      ( ! [X3: set_v] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).

% Collect_mono
thf(fact_580_Collect__mono,axiom,
    ! [P: v > $o,Q: v > $o] :
      ( ! [X3: v] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).

% Collect_mono
thf(fact_581_Collect__mono,axiom,
    ! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
      ( ! [X3: product_prod_v_v] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).

% Collect_mono
thf(fact_582_subset__trans,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B )
     => ( ( ord_less_eq_set_v @ B @ C3 )
       => ( ord_less_eq_set_v @ A3 @ C3 ) ) ) ).

% subset_trans
thf(fact_583_subset__trans,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
     => ( ( ord_le7336532860387713383od_v_v @ B @ C3 )
       => ( ord_le7336532860387713383od_v_v @ A3 @ C3 ) ) ) ).

% subset_trans
thf(fact_584_set__eq__subset,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 ) ) ) ) ).

% set_eq_subset
thf(fact_585_set__eq__subset,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 ) ) ) ) ).

% set_eq_subset
thf(fact_586_Collect__mono__iff,axiom,
    ! [P: set_v > $o,Q: set_v > $o] :
      ( ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) )
      = ( ! [X2: set_v] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_587_Collect__mono__iff,axiom,
    ! [P: v > $o,Q: v > $o] :
      ( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
      = ( ! [X2: v] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_588_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 ) )
      = ( ! [X2: product_prod_v_v] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_589_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
    = ( ^ [X2: set_v,Y3: set_v] :
          ( ( ord_less_eq_set_v @ X2 @ Y3 )
          & ( ord_less_eq_set_v @ Y3 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_590_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
    = ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
          ( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
          & ( ord_le7336532860387713383od_v_v @ Y3 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_591_ord__eq__le__trans,axiom,
    ! [A: set_v,B2: set_v,C: set_v] :
      ( ( A = B2 )
     => ( ( ord_less_eq_set_v @ B2 @ C )
       => ( ord_less_eq_set_v @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_592_ord__eq__le__trans,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( A = B2 )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
       => ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_593_ord__le__eq__trans,axiom,
    ! [A: set_v,B2: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_v @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_594_ord__le__eq__trans,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_595_order__antisym,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( ord_less_eq_set_v @ X @ Y2 )
     => ( ( ord_less_eq_set_v @ Y2 @ X )
       => ( X = Y2 ) ) ) ).

% order_antisym
thf(fact_596_order__antisym,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
     => ( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
       => ( X = Y2 ) ) ) ).

% order_antisym
thf(fact_597_order_Otrans,axiom,
    ! [A: set_v,B2: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A @ B2 )
     => ( ( ord_less_eq_set_v @ B2 @ C )
       => ( ord_less_eq_set_v @ A @ C ) ) ) ).

% order.trans
thf(fact_598_order_Otrans,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
       => ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).

% order.trans
thf(fact_599_order__trans,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( ord_less_eq_set_v @ X @ Y2 )
     => ( ( ord_less_eq_set_v @ Y2 @ Z )
       => ( ord_less_eq_set_v @ X @ Z ) ) ) ).

% order_trans
thf(fact_600_order__trans,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
     => ( ( ord_le7336532860387713383od_v_v @ Y2 @ Z )
       => ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).

% order_trans
thf(fact_601_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
    = ( ^ [A4: set_v,B4: set_v] :
          ( ( ord_less_eq_set_v @ B4 @ A4 )
          & ( ord_less_eq_set_v @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_602_dual__order_Oeq__iff,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 @ B4 @ A4 )
          & ( ord_le7336532860387713383od_v_v @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_603_dual__order_Oantisym,axiom,
    ! [B2: set_v,A: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ A )
     => ( ( ord_less_eq_set_v @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_604_dual__order_Oantisym,axiom,
    ! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
     => ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_605_dual__order_Otrans,axiom,
    ! [B2: set_v,A: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ A )
     => ( ( ord_less_eq_set_v @ C @ B2 )
       => ( ord_less_eq_set_v @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_606_dual__order_Otrans,axiom,
    ! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
     => ( ( ord_le7336532860387713383od_v_v @ C @ B2 )
       => ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_607_antisym,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A @ B2 )
     => ( ( ord_less_eq_set_v @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_608_antisym,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_609_Orderings_Oorder__eq__iff,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 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_610_Orderings_Oorder__eq__iff,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 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_611_order__subst1,axiom,
    ! [A: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_v @ B2 @ C )
       => ( ! [X3: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X3 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_612_order__subst1,axiom,
    ! [A: 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 @ A @ ( F @ B2 ) )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
       => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_613_order__subst1,axiom,
    ! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_v @ B2 @ C )
       => ( ! [X3: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X3 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_614_order__subst1,axiom,
    ! [A: 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 @ A @ ( F @ B2 ) )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
       => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_615_order__subst2,axiom,
    ! [A: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A @ B2 )
     => ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
       => ( ! [X3: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X3 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_616_order__subst2,axiom,
    ! [A: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_less_eq_set_v @ A @ B2 )
     => ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
       => ( ! [X3: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X3 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_617_order__subst2,axiom,
    ! [A: 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 @ A @ B2 )
     => ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
       => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_618_order__subst2,axiom,
    ! [A: 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 @ A @ B2 )
     => ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
       => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_619_order__eq__refl,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( X = Y2 )
     => ( ord_less_eq_set_v @ X @ Y2 ) ) ).

% order_eq_refl
thf(fact_620_order__eq__refl,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( X = Y2 )
     => ( ord_le7336532860387713383od_v_v @ X @ Y2 ) ) ).

% order_eq_refl
thf(fact_621_ord__eq__le__subst,axiom,
    ! [A: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_v @ B2 @ C )
       => ( ! [X3: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X3 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_622_ord__eq__le__subst,axiom,
    ! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_v @ B2 @ C )
       => ( ! [X3: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X3 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_623_ord__eq__le__subst,axiom,
    ! [A: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
       => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_624_ord__eq__le__subst,axiom,
    ! [A: 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] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
       => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_625_ord__le__eq__subst,axiom,
    ! [A: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X3 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_626_ord__le__eq__subst,axiom,
    ! [A: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_less_eq_set_v @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: set_v,Y: set_v] :
              ( ( ord_less_eq_set_v @ X3 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_627_ord__le__eq__subst,axiom,
    ! [A: 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 @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
             => ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_628_ord__le__eq__subst,axiom,
    ! [A: 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 @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
             => ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
         => ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_629_order__antisym__conv,axiom,
    ! [Y2: set_v,X: set_v] :
      ( ( ord_less_eq_set_v @ Y2 @ X )
     => ( ( ord_less_eq_set_v @ X @ Y2 )
        = ( X = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_630_order__antisym__conv,axiom,
    ! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
     => ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
        = ( X = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_631_bot_Oextremum__uniqueI,axiom,
    ! [A: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v )
     => ( A = bot_bot_set_set_v ) ) ).

% bot.extremum_uniqueI
thf(fact_632_bot_Oextremum__uniqueI,axiom,
    ! [A: set_v] :
      ( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
     => ( A = bot_bot_set_v ) ) ).

% bot.extremum_uniqueI
thf(fact_633_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
     => ( A = bot_bo723834152578015283od_v_v ) ) ).

% bot.extremum_uniqueI
thf(fact_634_bot_Oextremum__unique,axiom,
    ! [A: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v )
      = ( A = bot_bot_set_set_v ) ) ).

% bot.extremum_unique
thf(fact_635_bot_Oextremum__unique,axiom,
    ! [A: set_v] :
      ( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
      = ( A = bot_bot_set_v ) ) ).

% bot.extremum_unique
thf(fact_636_bot_Oextremum__unique,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
      = ( A = bot_bo723834152578015283od_v_v ) ) ).

% bot.extremum_unique
thf(fact_637_bot_Oextremum,axiom,
    ! [A: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A ) ).

% bot.extremum
thf(fact_638_bot_Oextremum,axiom,
    ! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).

% bot.extremum
thf(fact_639_bot_Oextremum,axiom,
    ! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).

% bot.extremum
thf(fact_640_inf__sup__ord_I4_J,axiom,
    ! [Y2: set_set_v,X: set_set_v] : ( ord_le5216385588623774835_set_v @ Y2 @ ( sup_sup_set_set_v @ X @ Y2 ) ) ).

% inf_sup_ord(4)
thf(fact_641_inf__sup__ord_I4_J,axiom,
    ! [Y2: set_v,X: set_v] : ( ord_less_eq_set_v @ Y2 @ ( sup_sup_set_v @ X @ Y2 ) ) ).

% inf_sup_ord(4)
thf(fact_642_inf__sup__ord_I4_J,axiom,
    ! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).

% inf_sup_ord(4)
thf(fact_643_inf__sup__ord_I3_J,axiom,
    ! [X: set_set_v,Y2: set_set_v] : ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ X @ Y2 ) ) ).

% inf_sup_ord(3)
thf(fact_644_inf__sup__ord_I3_J,axiom,
    ! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) ) ).

% inf_sup_ord(3)
thf(fact_645_inf__sup__ord_I3_J,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).

% inf_sup_ord(3)
thf(fact_646_le__supE,axiom,
    ! [A: set_set_v,B2: set_set_v,X: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ X )
     => ~ ( ( ord_le5216385588623774835_set_v @ A @ X )
         => ~ ( ord_le5216385588623774835_set_v @ B2 @ X ) ) ) ).

% le_supE
thf(fact_647_le__supE,axiom,
    ! [A: set_v,B2: set_v,X: set_v] :
      ( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ X )
     => ~ ( ( ord_less_eq_set_v @ A @ X )
         => ~ ( ord_less_eq_set_v @ B2 @ X ) ) ) ).

% le_supE
thf(fact_648_le__supE,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ X )
     => ~ ( ( ord_le7336532860387713383od_v_v @ A @ X )
         => ~ ( ord_le7336532860387713383od_v_v @ B2 @ X ) ) ) ).

% le_supE
thf(fact_649_le__supI,axiom,
    ! [A: set_set_v,X: set_set_v,B2: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A @ X )
     => ( ( ord_le5216385588623774835_set_v @ B2 @ X )
       => ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_650_le__supI,axiom,
    ! [A: set_v,X: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A @ X )
     => ( ( ord_less_eq_set_v @ B2 @ X )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_651_le__supI,axiom,
    ! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ X )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ X )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_652_sup__ge1,axiom,
    ! [X: set_set_v,Y2: set_set_v] : ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ X @ Y2 ) ) ).

% sup_ge1
thf(fact_653_sup__ge1,axiom,
    ! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) ) ).

% sup_ge1
thf(fact_654_sup__ge1,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).

% sup_ge1
thf(fact_655_sup__ge2,axiom,
    ! [Y2: set_set_v,X: set_set_v] : ( ord_le5216385588623774835_set_v @ Y2 @ ( sup_sup_set_set_v @ X @ Y2 ) ) ).

% sup_ge2
thf(fact_656_sup__ge2,axiom,
    ! [Y2: set_v,X: set_v] : ( ord_less_eq_set_v @ Y2 @ ( sup_sup_set_v @ X @ Y2 ) ) ).

% sup_ge2
thf(fact_657_sup__ge2,axiom,
    ! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).

% sup_ge2
thf(fact_658_le__supI1,axiom,
    ! [X: set_set_v,A: set_set_v,B2: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ X @ A )
     => ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ).

% le_supI1
thf(fact_659_le__supI1,axiom,
    ! [X: set_v,A: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ X @ A )
     => ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B2 ) ) ) ).

% le_supI1
thf(fact_660_le__supI1,axiom,
    ! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X @ A )
     => ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).

% le_supI1
thf(fact_661_le__supI2,axiom,
    ! [X: set_set_v,B2: set_set_v,A: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ X @ B2 )
     => ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ).

% le_supI2
thf(fact_662_le__supI2,axiom,
    ! [X: set_v,B2: set_v,A: set_v] :
      ( ( ord_less_eq_set_v @ X @ B2 )
     => ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B2 ) ) ) ).

% le_supI2
thf(fact_663_le__supI2,axiom,
    ! [X: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X @ B2 )
     => ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).

% le_supI2
thf(fact_664_sup_Omono,axiom,
    ! [C: set_set_v,A: set_set_v,D: set_set_v,B2: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ C @ A )
     => ( ( ord_le5216385588623774835_set_v @ D @ B2 )
       => ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ C @ D ) @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ) ).

% sup.mono
thf(fact_665_sup_Omono,axiom,
    ! [C: set_v,A: set_v,D: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ C @ A )
     => ( ( ord_less_eq_set_v @ D @ B2 )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D ) @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).

% sup.mono
thf(fact_666_sup_Omono,axiom,
    ! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,D: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C @ A )
     => ( ( ord_le7336532860387713383od_v_v @ D @ B2 )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D ) @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).

% sup.mono
thf(fact_667_sup__mono,axiom,
    ! [A: set_set_v,C: set_set_v,B2: set_set_v,D: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A @ C )
     => ( ( ord_le5216385588623774835_set_v @ B2 @ D )
       => ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ ( sup_sup_set_set_v @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_668_sup__mono,axiom,
    ! [A: set_v,C: set_v,B2: set_v,D: set_v] :
      ( ( ord_less_eq_set_v @ A @ C )
     => ( ( ord_less_eq_set_v @ B2 @ D )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ ( sup_sup_set_v @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_669_sup__mono,axiom,
    ! [A: 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 @ A @ C )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ ( sup_su414716646722978715od_v_v @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_670_sup__least,axiom,
    ! [Y2: set_set_v,X: set_set_v,Z: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ Y2 @ X )
     => ( ( ord_le5216385588623774835_set_v @ Z @ X )
       => ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ Y2 @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_671_sup__least,axiom,
    ! [Y2: set_v,X: set_v,Z: set_v] :
      ( ( ord_less_eq_set_v @ Y2 @ X )
     => ( ( ord_less_eq_set_v @ Z @ X )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_672_sup__least,axiom,
    ! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
     => ( ( ord_le7336532860387713383od_v_v @ Z @ X )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_673_le__iff__sup,axiom,
    ( ord_le5216385588623774835_set_v
    = ( ^ [X2: set_set_v,Y3: set_set_v] :
          ( ( sup_sup_set_set_v @ X2 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_674_le__iff__sup,axiom,
    ( ord_less_eq_set_v
    = ( ^ [X2: set_v,Y3: set_v] :
          ( ( sup_sup_set_v @ X2 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_675_le__iff__sup,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
          ( ( sup_su414716646722978715od_v_v @ X2 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_676_sup_OorderE,axiom,
    ! [B2: set_set_v,A: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ B2 @ A )
     => ( A
        = ( sup_sup_set_set_v @ A @ B2 ) ) ) ).

% sup.orderE
thf(fact_677_sup_OorderE,axiom,
    ! [B2: set_v,A: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ A )
     => ( A
        = ( sup_sup_set_v @ A @ B2 ) ) ) ).

% sup.orderE
thf(fact_678_sup_OorderE,axiom,
    ! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
     => ( A
        = ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).

% sup.orderE
thf(fact_679_sup_OorderI,axiom,
    ! [A: set_set_v,B2: set_set_v] :
      ( ( A
        = ( sup_sup_set_set_v @ A @ B2 ) )
     => ( ord_le5216385588623774835_set_v @ B2 @ A ) ) ).

% sup.orderI
thf(fact_680_sup_OorderI,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( A
        = ( sup_sup_set_v @ A @ B2 ) )
     => ( ord_less_eq_set_v @ B2 @ A ) ) ).

% sup.orderI
thf(fact_681_sup_OorderI,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( A
        = ( sup_su414716646722978715od_v_v @ A @ B2 ) )
     => ( ord_le7336532860387713383od_v_v @ B2 @ A ) ) ).

% sup.orderI
thf(fact_682_sup__unique,axiom,
    ! [F: set_set_v > set_set_v > set_set_v,X: set_set_v,Y2: set_set_v] :
      ( ! [X3: set_set_v,Y: set_set_v] : ( ord_le5216385588623774835_set_v @ X3 @ ( F @ X3 @ Y ) )
     => ( ! [X3: set_set_v,Y: set_set_v] : ( ord_le5216385588623774835_set_v @ Y @ ( F @ X3 @ Y ) )
       => ( ! [X3: set_set_v,Y: set_set_v,Z3: set_set_v] :
              ( ( ord_le5216385588623774835_set_v @ Y @ X3 )
             => ( ( ord_le5216385588623774835_set_v @ Z3 @ X3 )
               => ( ord_le5216385588623774835_set_v @ ( F @ Y @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_set_set_v @ X @ Y2 )
            = ( F @ X @ Y2 ) ) ) ) ) ).

% sup_unique
thf(fact_683_sup__unique,axiom,
    ! [F: set_v > set_v > set_v,X: set_v,Y2: set_v] :
      ( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ X3 @ ( F @ X3 @ Y ) )
     => ( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ Y @ ( F @ X3 @ Y ) )
       => ( ! [X3: set_v,Y: set_v,Z3: set_v] :
              ( ( ord_less_eq_set_v @ Y @ X3 )
             => ( ( ord_less_eq_set_v @ Z3 @ X3 )
               => ( ord_less_eq_set_v @ ( F @ Y @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_set_v @ X @ Y2 )
            = ( F @ X @ Y2 ) ) ) ) ) ).

% sup_unique
thf(fact_684_sup__unique,axiom,
    ! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ X3 @ Y ) )
     => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( F @ X3 @ Y ) )
       => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ Y @ X3 )
             => ( ( ord_le7336532860387713383od_v_v @ Z3 @ X3 )
               => ( ord_le7336532860387713383od_v_v @ ( F @ Y @ Z3 ) @ X3 ) ) )
         => ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
            = ( F @ X @ Y2 ) ) ) ) ) ).

% sup_unique
thf(fact_685_sup_Oabsorb1,axiom,
    ! [B2: set_set_v,A: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ B2 @ A )
     => ( ( sup_sup_set_set_v @ A @ B2 )
        = A ) ) ).

% sup.absorb1
thf(fact_686_sup_Oabsorb1,axiom,
    ! [B2: set_v,A: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ A )
     => ( ( sup_sup_set_v @ A @ B2 )
        = A ) ) ).

% sup.absorb1
thf(fact_687_sup_Oabsorb1,axiom,
    ! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
     => ( ( sup_su414716646722978715od_v_v @ A @ B2 )
        = A ) ) ).

% sup.absorb1
thf(fact_688_sup_Oabsorb2,axiom,
    ! [A: set_set_v,B2: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A @ B2 )
     => ( ( sup_sup_set_set_v @ A @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_689_sup_Oabsorb2,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A @ B2 )
     => ( ( sup_sup_set_v @ A @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_690_sup_Oabsorb2,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
     => ( ( sup_su414716646722978715od_v_v @ A @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_691_sup__absorb1,axiom,
    ! [Y2: set_set_v,X: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ Y2 @ X )
     => ( ( sup_sup_set_set_v @ X @ Y2 )
        = X ) ) ).

% sup_absorb1
thf(fact_692_sup__absorb1,axiom,
    ! [Y2: set_v,X: set_v] :
      ( ( ord_less_eq_set_v @ Y2 @ X )
     => ( ( sup_sup_set_v @ X @ Y2 )
        = X ) ) ).

% sup_absorb1
thf(fact_693_sup__absorb1,axiom,
    ! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
     => ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
        = X ) ) ).

% sup_absorb1
thf(fact_694_sup__absorb2,axiom,
    ! [X: set_set_v,Y2: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ X @ Y2 )
     => ( ( sup_sup_set_set_v @ X @ Y2 )
        = Y2 ) ) ).

% sup_absorb2
thf(fact_695_sup__absorb2,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( ord_less_eq_set_v @ X @ Y2 )
     => ( ( sup_sup_set_v @ X @ Y2 )
        = Y2 ) ) ).

% sup_absorb2
thf(fact_696_sup__absorb2,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
     => ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
        = Y2 ) ) ).

% sup_absorb2
thf(fact_697_sup_OboundedE,axiom,
    ! [B2: set_set_v,C: set_set_v,A: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B2 @ C ) @ A )
     => ~ ( ( ord_le5216385588623774835_set_v @ B2 @ A )
         => ~ ( ord_le5216385588623774835_set_v @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_698_sup_OboundedE,axiom,
    ! [B2: set_v,C: set_v,A: set_v] :
      ( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A )
     => ~ ( ( ord_less_eq_set_v @ B2 @ A )
         => ~ ( ord_less_eq_set_v @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_699_sup_OboundedE,axiom,
    ! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A )
     => ~ ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
         => ~ ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_700_sup_OboundedI,axiom,
    ! [B2: set_set_v,A: set_set_v,C: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ B2 @ A )
     => ( ( ord_le5216385588623774835_set_v @ C @ A )
       => ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B2 @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_701_sup_OboundedI,axiom,
    ! [B2: set_v,A: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ A )
     => ( ( ord_less_eq_set_v @ C @ A )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_702_sup_OboundedI,axiom,
    ! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
     => ( ( ord_le7336532860387713383od_v_v @ C @ A )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_703_sup_Oorder__iff,axiom,
    ( ord_le5216385588623774835_set_v
    = ( ^ [B4: set_set_v,A4: set_set_v] :
          ( A4
          = ( sup_sup_set_set_v @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_704_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_v
    = ( ^ [B4: set_v,A4: set_v] :
          ( A4
          = ( sup_sup_set_v @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_705_sup_Oorder__iff,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
          ( A4
          = ( sup_su414716646722978715od_v_v @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_706_sup_Ocobounded1,axiom,
    ! [A: set_set_v,B2: set_set_v] : ( ord_le5216385588623774835_set_v @ A @ ( sup_sup_set_set_v @ A @ B2 ) ) ).

% sup.cobounded1
thf(fact_707_sup_Ocobounded1,axiom,
    ! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B2 ) ) ).

% sup.cobounded1
thf(fact_708_sup_Ocobounded1,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).

% sup.cobounded1
thf(fact_709_sup_Ocobounded2,axiom,
    ! [B2: set_set_v,A: set_set_v] : ( ord_le5216385588623774835_set_v @ B2 @ ( sup_sup_set_set_v @ A @ B2 ) ) ).

% sup.cobounded2
thf(fact_710_sup_Ocobounded2,axiom,
    ! [B2: set_v,A: set_v] : ( ord_less_eq_set_v @ B2 @ ( sup_sup_set_v @ A @ B2 ) ) ).

% sup.cobounded2
thf(fact_711_sup_Ocobounded2,axiom,
    ! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).

% sup.cobounded2
thf(fact_712_sup_Oabsorb__iff1,axiom,
    ( ord_le5216385588623774835_set_v
    = ( ^ [B4: set_set_v,A4: set_set_v] :
          ( ( sup_sup_set_set_v @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_713_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_v
    = ( ^ [B4: set_v,A4: set_v] :
          ( ( sup_sup_set_v @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_714_sup_Oabsorb__iff1,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
          ( ( sup_su414716646722978715od_v_v @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_715_sup_Oabsorb__iff2,axiom,
    ( ord_le5216385588623774835_set_v
    = ( ^ [A4: set_set_v,B4: set_set_v] :
          ( ( sup_sup_set_set_v @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_716_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_v
    = ( ^ [A4: set_v,B4: set_v] :
          ( ( sup_sup_set_v @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_717_sup_Oabsorb__iff2,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
          ( ( sup_su414716646722978715od_v_v @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_718_sup_OcoboundedI1,axiom,
    ! [C: set_set_v,A: set_set_v,B2: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ C @ A )
     => ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_719_sup_OcoboundedI1,axiom,
    ! [C: set_v,A: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ C @ A )
     => ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_720_sup_OcoboundedI1,axiom,
    ! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C @ A )
     => ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_721_sup_OcoboundedI2,axiom,
    ! [C: set_set_v,B2: set_set_v,A: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ C @ B2 )
     => ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_722_sup_OcoboundedI2,axiom,
    ! [C: set_v,B2: set_v,A: set_v] :
      ( ( ord_less_eq_set_v @ C @ B2 )
     => ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_723_sup_OcoboundedI2,axiom,
    ! [C: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C @ B2 )
     => ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_724_subset__insertI2,axiom,
    ! [A3: set_set_v,B: set_set_v,B2: set_v] :
      ( ( ord_le5216385588623774835_set_v @ A3 @ B )
     => ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_725_subset__insertI2,axiom,
    ! [A3: set_v,B: set_v,B2: v] :
      ( ( ord_less_eq_set_v @ A3 @ B )
     => ( ord_less_eq_set_v @ A3 @ ( insert_v @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_726_subset__insertI2,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
     => ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_727_subset__insertI,axiom,
    ! [B: set_set_v,A: set_v] : ( ord_le5216385588623774835_set_v @ B @ ( insert_set_v @ A @ B ) ) ).

% subset_insertI
thf(fact_728_subset__insertI,axiom,
    ! [B: set_v,A: v] : ( ord_less_eq_set_v @ B @ ( insert_v @ A @ B ) ) ).

% subset_insertI
thf(fact_729_subset__insertI,axiom,
    ! [B: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).

% subset_insertI
thf(fact_730_subset__insert,axiom,
    ! [X: set_v,A3: set_set_v,B: set_set_v] :
      ( ~ ( member_set_v @ X @ A3 )
     => ( ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X @ B ) )
        = ( ord_le5216385588623774835_set_v @ A3 @ B ) ) ) ).

% subset_insert
thf(fact_731_subset__insert,axiom,
    ! [X: v,A3: set_v,B: set_v] :
      ( ~ ( member_v @ X @ A3 )
     => ( ( ord_less_eq_set_v @ A3 @ ( insert_v @ X @ B ) )
        = ( ord_less_eq_set_v @ A3 @ B ) ) ) ).

% subset_insert
thf(fact_732_subset__insert,axiom,
    ! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ X @ A3 )
     => ( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) )
        = ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ).

% subset_insert
thf(fact_733_insert__mono,axiom,
    ! [C3: set_set_v,D2: set_set_v,A: set_v] :
      ( ( ord_le5216385588623774835_set_v @ C3 @ D2 )
     => ( ord_le5216385588623774835_set_v @ ( insert_set_v @ A @ C3 ) @ ( insert_set_v @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_734_insert__mono,axiom,
    ! [C3: set_v,D2: set_v,A: v] :
      ( ( ord_less_eq_set_v @ C3 @ D2 )
     => ( ord_less_eq_set_v @ ( insert_v @ A @ C3 ) @ ( insert_v @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_735_insert__mono,axiom,
    ! [C3: set_Product_prod_v_v,D2: set_Product_prod_v_v,A: product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C3 @ D2 )
     => ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C3 ) @ ( insert1338601472111419319od_v_v @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_736_subset__code_I1_J,axiom,
    ! [Xs: list_v,B: set_v] :
      ( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B )
      = ( ! [X2: v] :
            ( ( member_v @ X2 @ ( set_v2 @ Xs ) )
           => ( member_v @ X2 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_737_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 )
      = ( ! [X2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
           => ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_738_Un__mono,axiom,
    ! [A3: set_set_v,C3: set_set_v,B: set_set_v,D2: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A3 @ C3 )
     => ( ( ord_le5216385588623774835_set_v @ B @ D2 )
       => ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ ( sup_sup_set_set_v @ C3 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_739_Un__mono,axiom,
    ! [A3: set_v,C3: set_v,B: set_v,D2: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ C3 )
     => ( ( ord_less_eq_set_v @ B @ D2 )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_sup_set_v @ C3 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_740_Un__mono,axiom,
    ! [A3: set_Product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
     => ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ C3 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_741_Un__least,axiom,
    ! [A3: set_set_v,C3: set_set_v,B: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A3 @ C3 )
     => ( ( ord_le5216385588623774835_set_v @ B @ C3 )
       => ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ C3 ) ) ) ).

% Un_least
thf(fact_742_Un__least,axiom,
    ! [A3: set_v,C3: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ C3 )
     => ( ( ord_less_eq_set_v @ B @ C3 )
       => ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B ) @ C3 ) ) ) ).

% Un_least
thf(fact_743_Un__least,axiom,
    ! [A3: set_Product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
     => ( ( ord_le7336532860387713383od_v_v @ B @ C3 )
       => ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C3 ) ) ) ).

% Un_least
thf(fact_744_Un__upper1,axiom,
    ! [A3: set_set_v,B: set_set_v] : ( ord_le5216385588623774835_set_v @ A3 @ ( sup_sup_set_set_v @ A3 @ B ) ) ).

% Un_upper1
thf(fact_745_Un__upper1,axiom,
    ! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ A3 @ B ) ) ).

% Un_upper1
thf(fact_746_Un__upper1,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).

% Un_upper1
thf(fact_747_Un__upper2,axiom,
    ! [B: set_set_v,A3: set_set_v] : ( ord_le5216385588623774835_set_v @ B @ ( sup_sup_set_set_v @ A3 @ B ) ) ).

% Un_upper2
thf(fact_748_Un__upper2,axiom,
    ! [B: set_v,A3: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A3 @ B ) ) ).

% Un_upper2
thf(fact_749_Un__upper2,axiom,
    ! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).

% Un_upper2
thf(fact_750_Un__absorb1,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A3 @ B )
     => ( ( sup_sup_set_set_v @ A3 @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_751_Un__absorb1,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B )
     => ( ( sup_sup_set_v @ A3 @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_752_Un__absorb1,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
     => ( ( sup_su414716646722978715od_v_v @ A3 @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_753_Un__absorb2,axiom,
    ! [B: set_set_v,A3: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ B @ A3 )
     => ( ( sup_sup_set_set_v @ A3 @ B )
        = A3 ) ) ).

% Un_absorb2
thf(fact_754_Un__absorb2,axiom,
    ! [B: set_v,A3: set_v] :
      ( ( ord_less_eq_set_v @ B @ A3 )
     => ( ( sup_sup_set_v @ A3 @ B )
        = A3 ) ) ).

% Un_absorb2
thf(fact_755_Un__absorb2,axiom,
    ! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B @ A3 )
     => ( ( sup_su414716646722978715od_v_v @ A3 @ B )
        = A3 ) ) ).

% Un_absorb2
thf(fact_756_subset__UnE,axiom,
    ! [C3: set_set_v,A3: set_set_v,B: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ C3 @ ( sup_sup_set_set_v @ A3 @ B ) )
     => ~ ! [A6: set_set_v] :
            ( ( ord_le5216385588623774835_set_v @ A6 @ A3 )
           => ! [B6: set_set_v] :
                ( ( ord_le5216385588623774835_set_v @ B6 @ B )
               => ( C3
                 != ( sup_sup_set_set_v @ A6 @ B6 ) ) ) ) ) ).

% subset_UnE
thf(fact_757_subset__UnE,axiom,
    ! [C3: set_v,A3: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ C3 @ ( sup_sup_set_v @ A3 @ B ) )
     => ~ ! [A6: set_v] :
            ( ( ord_less_eq_set_v @ A6 @ A3 )
           => ! [B6: set_v] :
                ( ( ord_less_eq_set_v @ B6 @ B )
               => ( C3
                 != ( sup_sup_set_v @ A6 @ B6 ) ) ) ) ) ).

% subset_UnE
thf(fact_758_subset__UnE,axiom,
    ! [C3: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
     => ~ ! [A6: set_Product_prod_v_v] :
            ( ( ord_le7336532860387713383od_v_v @ A6 @ A3 )
           => ! [B6: set_Product_prod_v_v] :
                ( ( ord_le7336532860387713383od_v_v @ B6 @ B )
               => ( C3
                 != ( sup_su414716646722978715od_v_v @ A6 @ B6 ) ) ) ) ) ).

% subset_UnE
thf(fact_759_subset__Un__eq,axiom,
    ( ord_le5216385588623774835_set_v
    = ( ^ [A5: set_set_v,B5: set_set_v] :
          ( ( sup_sup_set_set_v @ A5 @ B5 )
          = B5 ) ) ) ).

% subset_Un_eq
thf(fact_760_subset__Un__eq,axiom,
    ( ord_less_eq_set_v
    = ( ^ [A5: set_v,B5: set_v] :
          ( ( sup_sup_set_v @ A5 @ B5 )
          = B5 ) ) ) ).

% subset_Un_eq
thf(fact_761_subset__Un__eq,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
          ( ( sup_su414716646722978715od_v_v @ A5 @ B5 )
          = B5 ) ) ) ).

% subset_Un_eq
thf(fact_762_double__diff,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B )
     => ( ( ord_less_eq_set_v @ B @ C3 )
       => ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C3 @ A3 ) )
          = A3 ) ) ) ).

% double_diff
thf(fact_763_double__diff,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
     => ( ( ord_le7336532860387713383od_v_v @ B @ C3 )
       => ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C3 @ A3 ) )
          = A3 ) ) ) ).

% double_diff
thf(fact_764_Diff__subset,axiom,
    ! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ A3 ) ).

% Diff_subset
thf(fact_765_Diff__subset,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ A3 ) ).

% Diff_subset
thf(fact_766_Diff__mono,axiom,
    ! [A3: set_v,C3: set_v,D2: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ C3 )
     => ( ( ord_less_eq_set_v @ D2 @ B )
       => ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ C3 @ D2 ) ) ) ) ).

% Diff_mono
thf(fact_767_Diff__mono,axiom,
    ! [A3: set_Product_prod_v_v,C3: set_Product_prod_v_v,D2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
     => ( ( ord_le7336532860387713383od_v_v @ D2 @ B )
       => ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ C3 @ D2 ) ) ) ) ).

% Diff_mono
thf(fact_768_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 )
     => ! [X4: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X4 @ Vertices )
         => ( ord_le7336532860387713383od_v_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).

% graph.sclosed
thf(fact_769_graph_Osclosed,axiom,
    ! [Vertices: set_v,Successors: v > set_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ! [X4: v] :
          ( ( member_v @ X4 @ Vertices )
         => ( ord_less_eq_set_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).

% graph.sclosed
thf(fact_770_subset__singleton__iff,axiom,
    ! [X5: set_set_v,A: set_v] :
      ( ( ord_le5216385588623774835_set_v @ X5 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
      = ( ( X5 = bot_bot_set_set_v )
        | ( X5
          = ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ) ).

% subset_singleton_iff
thf(fact_771_subset__singleton__iff,axiom,
    ! [X5: set_v,A: v] :
      ( ( ord_less_eq_set_v @ X5 @ ( insert_v @ A @ bot_bot_set_v ) )
      = ( ( X5 = bot_bot_set_v )
        | ( X5
          = ( insert_v @ A @ bot_bot_set_v ) ) ) ) ).

% subset_singleton_iff
thf(fact_772_subset__singleton__iff,axiom,
    ! [X5: set_Product_prod_v_v,A: product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X5 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
      = ( ( X5 = bot_bo723834152578015283od_v_v )
        | ( X5
          = ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% subset_singleton_iff
thf(fact_773_subset__singletonD,axiom,
    ! [A3: set_set_v,X: set_v] :
      ( ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
     => ( ( A3 = bot_bot_set_set_v )
        | ( A3
          = ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ).

% subset_singletonD
thf(fact_774_subset__singletonD,axiom,
    ! [A3: set_v,X: v] :
      ( ( ord_less_eq_set_v @ A3 @ ( insert_v @ X @ bot_bot_set_v ) )
     => ( ( A3 = bot_bot_set_v )
        | ( A3
          = ( insert_v @ X @ bot_bot_set_v ) ) ) ) ).

% subset_singletonD
thf(fact_775_subset__singletonD,axiom,
    ! [A3: set_Product_prod_v_v,X: product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
     => ( ( A3 = bot_bo723834152578015283od_v_v )
        | ( A3
          = ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% subset_singletonD
thf(fact_776_set__subset__Cons,axiom,
    ! [Xs: list_v,X: v] : ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ ( cons_v @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_777_set__subset__Cons,axiom,
    ! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_778_subset__Diff__insert,axiom,
    ! [A3: set_set_v,B: set_set_v,X: set_v,C3: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ ( insert_set_v @ X @ C3 ) ) )
      = ( ( ord_le5216385588623774835_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ C3 ) )
        & ~ ( member_set_v @ X @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_779_subset__Diff__insert,axiom,
    ! [A3: set_v,B: set_v,X: v,C3: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B @ ( insert_v @ X @ C3 ) ) )
      = ( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B @ C3 ) )
        & ~ ( member_v @ X @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_780_subset__Diff__insert,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ ( insert1338601472111419319od_v_v @ X @ C3 ) ) )
      = ( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ C3 ) )
        & ~ ( member7453568604450474000od_v_v @ X @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_781_Diff__partition,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A3 @ B )
     => ( ( sup_sup_set_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ A3 ) )
        = B ) ) ).

% Diff_partition
thf(fact_782_Diff__partition,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B )
     => ( ( sup_sup_set_v @ A3 @ ( minus_minus_set_v @ B @ A3 ) )
        = B ) ) ).

% Diff_partition
thf(fact_783_Diff__partition,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
     => ( ( sup_su414716646722978715od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
        = B ) ) ).

% Diff_partition
thf(fact_784_Diff__subset__conv,axiom,
    ! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) @ C3 )
      = ( ord_le5216385588623774835_set_v @ A3 @ ( sup_sup_set_set_v @ B @ C3 ) ) ) ).

% Diff_subset_conv
thf(fact_785_Diff__subset__conv,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ C3 )
      = ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) ) ) ).

% Diff_subset_conv
thf(fact_786_Diff__subset__conv,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ C3 )
      = ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) ) ).

% Diff_subset_conv
thf(fact_787_graph_Ora__succ,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v,Z: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y2 @ E4 )
       => ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
         => ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y2 @ Z ) @ E4 )
           => ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Z @ E4 ) ) ) ) ) ).

% graph.ra_succ
thf(fact_788_graph_Ora__succ,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
       => ( ( member_v @ Z @ ( Successors @ Y2 ) )
         => ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z ) @ E4 )
           => ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E4 ) ) ) ) ) ).

% graph.ra_succ
thf(fact_789_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 )
        = ( ? [X2: product_prod_v_v,E5: set_Pr2149350503807050951od_v_v] :
              ( ( A1 = X2 )
              & ( A2 = X2 )
              & ( A32 = E5 ) )
          | ? [X2: product_prod_v_v,Y3: product_prod_v_v,E5: set_Pr2149350503807050951od_v_v,Z2: product_prod_v_v] :
              ( ( A1 = X2 )
              & ( A2 = Z2 )
              & ( A32 = E5 )
              & ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y3 @ E5 )
              & ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) )
              & ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ Z2 ) @ E5 ) ) ) ) ) ).

% graph.reachable_avoiding.simps
thf(fact_790_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 )
        = ( ? [X2: v,E5: set_Product_prod_v_v] :
              ( ( A1 = X2 )
              & ( A2 = X2 )
              & ( A32 = E5 ) )
          | ? [X2: v,Y3: v,E5: set_Product_prod_v_v,Z2: v] :
              ( ( A1 = X2 )
              & ( A2 = Z2 )
              & ( A32 = E5 )
              & ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y3 @ E5 )
              & ( member_v @ Z2 @ ( Successors @ Y3 ) )
              & ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E5 ) ) ) ) ) ).

% graph.reachable_avoiding.simps
thf(fact_791_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_792_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_793_graph_Ora__cases,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y2 @ E4 )
       => ( ( X = Y2 )
          | ? [Z3: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ X ) )
              & ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Z3 ) @ E4 )
              & ( sCC_Bl5370300055464682748od_v_v @ Successors @ Z3 @ Y2 @ E4 ) ) ) ) ) ).

% graph.ra_cases
thf(fact_794_graph_Ora__cases,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
       => ( ( X = Y2 )
          | ? [Z3: v] :
              ( ( member_v @ Z3 @ ( Successors @ X ) )
              & ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E4 )
              & ( sCC_Bl4291963740693775144ding_v @ Successors @ Z3 @ Y2 @ E4 ) ) ) ) ) ).

% graph.ra_cases
thf(fact_795_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,X: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
       => ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y2 ) @ E4 )
         => ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y2 @ E4 ) ) ) ) ).

% graph.edge_ra
thf(fact_796_graph_Oedge__ra,axiom,
    ! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v,E4: set_Product_prod_v_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( member_v @ Y2 @ ( Successors @ X ) )
       => ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ E4 )
         => ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 ) ) ) ) ).

% graph.edge_ra
thf(fact_797_select__convs_I6_J,axiom,
    ! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
      ( ( sCC_Bl2536197123907397897t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
      = Sccs ) ).

% select_convs(6)
thf(fact_798_select__convs_I9_J,axiom,
    ! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
      ( ( sCC_Bl3567736435408124606t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
      = More ) ).

% select_convs(9)
thf(fact_799_subset__insert__iff,axiom,
    ! [A3: set_set_v,X: set_v,B: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X @ B ) )
      = ( ( ( member_set_v @ X @ A3 )
         => ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) @ B ) )
        & ( ~ ( member_set_v @ X @ A3 )
         => ( ord_le5216385588623774835_set_v @ A3 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_800_subset__insert__iff,axiom,
    ! [A3: set_v,X: v,B: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ ( insert_v @ X @ B ) )
      = ( ( ( member_v @ X @ A3 )
         => ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ X @ bot_bot_set_v ) ) @ B ) )
        & ( ~ ( member_v @ X @ A3 )
         => ( ord_less_eq_set_v @ A3 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_801_subset__insert__iff,axiom,
    ! [A3: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) )
      = ( ( ( member7453568604450474000od_v_v @ X @ A3 )
         => ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B ) )
        & ( ~ ( member7453568604450474000od_v_v @ X @ A3 )
         => ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_802_Diff__single__insert,axiom,
    ! [A3: set_set_v,X: set_v,B: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) @ B )
     => ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_803_Diff__single__insert,axiom,
    ! [A3: set_v,X: v,B: set_v] :
      ( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ X @ bot_bot_set_v ) ) @ B )
     => ( ord_less_eq_set_v @ A3 @ ( insert_v @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_804_Diff__single__insert,axiom,
    ! [A3: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B )
     => ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_805_graph_Ora__add__edge,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v,V3: v,W: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
       => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
          | ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ V3 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
            & ( sCC_Bl4291963740693775144ding_v @ Successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).

% graph.ra_add_edge
thf(fact_806_graph_Ois__scc__def,axiom,
    ! [Vertices: set_set_v,Successors: set_v > set_set_v,S4: set_set_v] :
      ( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
     => ( ( sCC_Bl1515522642333523865_set_v @ Successors @ S4 )
        = ( ( S4 != bot_bot_set_set_v )
          & ( sCC_Bl7907073126578335045_set_v @ Successors @ S4 )
          & ! [S5: set_set_v] :
              ( ( ( ord_le5216385588623774835_set_v @ S4 @ S5 )
                & ( sCC_Bl7907073126578335045_set_v @ Successors @ S5 ) )
             => ( S5 = S4 ) ) ) ) ) ).

% graph.is_scc_def
thf(fact_807_graph_Ois__scc__def,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S4: set_Product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S4 )
        = ( ( S4 != bot_bo723834152578015283od_v_v )
          & ( sCC_Bl2301996248249672505od_v_v @ Successors @ S4 )
          & ! [S5: set_Product_prod_v_v] :
              ( ( ( ord_le7336532860387713383od_v_v @ S4 @ S5 )
                & ( sCC_Bl2301996248249672505od_v_v @ Successors @ S5 ) )
             => ( S5 = S4 ) ) ) ) ) ).

% graph.is_scc_def
thf(fact_808_graph_Ois__scc__def,axiom,
    ! [Vertices: set_v,Successors: v > set_v,S4: set_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bloemen_is_scc_v @ Successors @ S4 )
        = ( ( S4 != bot_bot_set_v )
          & ( sCC_Bl5398416737448265317bscc_v @ Successors @ S4 )
          & ! [S5: set_v] :
              ( ( ( ord_less_eq_set_v @ S4 @ S5 )
                & ( sCC_Bl5398416737448265317bscc_v @ Successors @ S5 ) )
             => ( S5 = S4 ) ) ) ) ) ).

% graph.is_scc_def
thf(fact_809_graph_Oavoiding__explored,axiom,
    ! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,X: v,Y2: v,E4: set_Product_prod_v_v,W: v,V3: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
       => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
         => ( ~ ( member_v @ Y2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
           => ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
             => ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ) ).

% graph.avoiding_explored
thf(fact_810_diff__shunt__var,axiom,
    ! [X: set_set_v,Y2: set_set_v] :
      ( ( ( minus_7228012346218142266_set_v @ X @ Y2 )
        = bot_bot_set_set_v )
      = ( ord_le5216385588623774835_set_v @ X @ Y2 ) ) ).

% diff_shunt_var
thf(fact_811_diff__shunt__var,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( ( minus_minus_set_v @ X @ Y2 )
        = bot_bot_set_v )
      = ( ord_less_eq_set_v @ X @ Y2 ) ) ).

% diff_shunt_var
thf(fact_812_diff__shunt__var,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( ( minus_4183494784930505774od_v_v @ X @ Y2 )
        = bot_bo723834152578015283od_v_v )
      = ( ord_le7336532860387713383od_v_v @ X @ Y2 ) ) ).

% diff_shunt_var
thf(fact_813_scc__partition,axiom,
    ! [S4: set_v,S6: set_v,X: v] :
      ( ( sCC_Bloemen_is_scc_v @ successors @ S4 )
     => ( ( sCC_Bloemen_is_scc_v @ successors @ S6 )
       => ( ( member_v @ X @ ( inf_inf_set_v @ S4 @ S6 ) )
         => ( S4 = S6 ) ) ) ) ).

% scc_partition
thf(fact_814_ra__mono,axiom,
    ! [X: v,Y2: v,E4: set_Product_prod_v_v,E6: set_Product_prod_v_v] :
      ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
     => ( ( ord_le7336532860387713383od_v_v @ E6 @ E4 )
       => ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E6 ) ) ) ).

% ra_mono
thf(fact_815_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_816_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_817_Collect__empty__eq__bot,axiom,
    ! [P: set_v > $o] :
      ( ( ( collect_set_v @ P )
        = bot_bot_set_set_v )
      = ( P = bot_bot_set_v_o ) ) ).

% Collect_empty_eq_bot
thf(fact_818_bot__empty__eq,axiom,
    ( bot_bot_v_o
    = ( ^ [X2: v] : ( member_v @ X2 @ bot_bot_set_v ) ) ) ).

% bot_empty_eq
thf(fact_819_bot__empty__eq,axiom,
    ( bot_bo8461541820394803818_v_v_o
    = ( ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ).

% bot_empty_eq
thf(fact_820_bot__empty__eq,axiom,
    ( bot_bot_set_v_o
    = ( ^ [X2: set_v] : ( member_set_v @ X2 @ bot_bot_set_set_v ) ) ) ).

% bot_empty_eq
thf(fact_821_vfin,axiom,
    finite_finite_v @ vertices ).

% vfin
thf(fact_822_inf__right__idem,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Y2 )
      = ( inf_inf_set_v @ X @ Y2 ) ) ).

% inf_right_idem
thf(fact_823_inf_Oright__idem,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B2 ) @ B2 )
      = ( inf_inf_set_v @ A @ B2 ) ) ).

% inf.right_idem
thf(fact_824_inf__left__idem,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y2 ) )
      = ( inf_inf_set_v @ X @ Y2 ) ) ).

% inf_left_idem
thf(fact_825_inf_Oleft__idem,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B2 ) )
      = ( inf_inf_set_v @ A @ B2 ) ) ).

% inf.left_idem
thf(fact_826_inf__idem,axiom,
    ! [X: set_v] :
      ( ( inf_inf_set_v @ X @ X )
      = X ) ).

% inf_idem
thf(fact_827_inf_Oidem,axiom,
    ! [A: set_v] :
      ( ( inf_inf_set_v @ A @ A )
      = A ) ).

% inf.idem
thf(fact_828_Int__iff,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
      = ( ( member7453568604450474000od_v_v @ C @ A3 )
        & ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% Int_iff
thf(fact_829_Int__iff,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
      = ( ( member_v @ C @ A3 )
        & ( member_v @ C @ B ) ) ) ).

% Int_iff
thf(fact_830_IntI,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ A3 )
     => ( ( member7453568604450474000od_v_v @ C @ B )
       => ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).

% IntI
thf(fact_831_IntI,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ A3 )
     => ( ( member_v @ C @ B )
       => ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).

% IntI
thf(fact_832_inf_Obounded__iff,axiom,
    ! [A: set_v,B2: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) )
      = ( ( ord_less_eq_set_v @ A @ B2 )
        & ( ord_less_eq_set_v @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_833_inf_Obounded__iff,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
      = ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
        & ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_834_le__inf__iff,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
      = ( ( ord_less_eq_set_v @ X @ Y2 )
        & ( ord_less_eq_set_v @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_835_le__inf__iff,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
      = ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
        & ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_836_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_v] :
      ( ( inf_inf_set_v @ X @ bot_bot_set_v )
      = bot_bot_set_v ) ).

% boolean_algebra.conj_zero_right
thf(fact_837_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
      = bot_bo723834152578015283od_v_v ) ).

% boolean_algebra.conj_zero_right
thf(fact_838_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_set_v] :
      ( ( inf_inf_set_set_v @ X @ bot_bot_set_set_v )
      = bot_bot_set_set_v ) ).

% boolean_algebra.conj_zero_right
thf(fact_839_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_v] :
      ( ( inf_inf_set_v @ bot_bot_set_v @ X )
      = bot_bot_set_v ) ).

% boolean_algebra.conj_zero_left
thf(fact_840_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
      = bot_bo723834152578015283od_v_v ) ).

% boolean_algebra.conj_zero_left
thf(fact_841_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_set_v] :
      ( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X )
      = bot_bot_set_set_v ) ).

% boolean_algebra.conj_zero_left
thf(fact_842_inf__bot__left,axiom,
    ! [X: set_v] :
      ( ( inf_inf_set_v @ bot_bot_set_v @ X )
      = bot_bot_set_v ) ).

% inf_bot_left
thf(fact_843_inf__bot__left,axiom,
    ! [X: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
      = bot_bo723834152578015283od_v_v ) ).

% inf_bot_left
thf(fact_844_inf__bot__left,axiom,
    ! [X: set_set_v] :
      ( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X )
      = bot_bot_set_set_v ) ).

% inf_bot_left
thf(fact_845_inf__bot__right,axiom,
    ! [X: set_v] :
      ( ( inf_inf_set_v @ X @ bot_bot_set_v )
      = bot_bot_set_v ) ).

% inf_bot_right
thf(fact_846_inf__bot__right,axiom,
    ! [X: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
      = bot_bo723834152578015283od_v_v ) ).

% inf_bot_right
thf(fact_847_inf__bot__right,axiom,
    ! [X: set_set_v] :
      ( ( inf_inf_set_set_v @ X @ bot_bot_set_set_v )
      = bot_bot_set_set_v ) ).

% inf_bot_right
thf(fact_848_List_Ofinite__set,axiom,
    ! [Xs: list_v] : ( finite_finite_v @ ( set_v2 @ Xs ) ) ).

% List.finite_set
thf(fact_849_sup__inf__absorb,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) )
      = X ) ).

% sup_inf_absorb
thf(fact_850_sup__inf__absorb,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ X @ Y2 ) )
      = X ) ).

% sup_inf_absorb
thf(fact_851_sup__inf__absorb,axiom,
    ! [X: set_set_v,Y2: set_set_v] :
      ( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ X @ Y2 ) )
      = X ) ).

% sup_inf_absorb
thf(fact_852_inf__sup__absorb,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
      = X ) ).

% inf_sup_absorb
thf(fact_853_inf__sup__absorb,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) )
      = X ) ).

% inf_sup_absorb
thf(fact_854_inf__sup__absorb,axiom,
    ! [X: set_set_v,Y2: set_set_v] :
      ( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ X @ Y2 ) )
      = X ) ).

% inf_sup_absorb
thf(fact_855_Int__subset__iff,axiom,
    ! [C3: set_v,A3: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ C3 @ ( inf_inf_set_v @ A3 @ B ) )
      = ( ( ord_less_eq_set_v @ C3 @ A3 )
        & ( ord_less_eq_set_v @ C3 @ B ) ) ) ).

% Int_subset_iff
thf(fact_856_Int__subset__iff,axiom,
    ! [C3: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C3 @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
      = ( ( ord_le7336532860387713383od_v_v @ C3 @ A3 )
        & ( ord_le7336532860387713383od_v_v @ C3 @ B ) ) ) ).

% Int_subset_iff
thf(fact_857_Int__insert__left__if0,axiom,
    ! [A: set_v,C3: set_set_v,B: set_set_v] :
      ( ~ ( member_set_v @ A @ C3 )
     => ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C3 )
        = ( inf_inf_set_set_v @ B @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_858_Int__insert__left__if0,axiom,
    ! [A: product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ A @ C3 )
     => ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
        = ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_859_Int__insert__left__if0,axiom,
    ! [A: v,C3: set_v,B: set_v] :
      ( ~ ( member_v @ A @ C3 )
     => ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C3 )
        = ( inf_inf_set_v @ B @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_860_Int__insert__left__if1,axiom,
    ! [A: set_v,C3: set_set_v,B: set_set_v] :
      ( ( member_set_v @ A @ C3 )
     => ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C3 )
        = ( insert_set_v @ A @ ( inf_inf_set_set_v @ B @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_861_Int__insert__left__if1,axiom,
    ! [A: product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A @ C3 )
     => ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
        = ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_862_Int__insert__left__if1,axiom,
    ! [A: v,C3: set_v,B: set_v] :
      ( ( member_v @ A @ C3 )
     => ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C3 )
        = ( insert_v @ A @ ( inf_inf_set_v @ B @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_863_insert__inter__insert,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ ( insert1338601472111419319od_v_v @ A @ B ) )
      = ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ).

% insert_inter_insert
thf(fact_864_insert__inter__insert,axiom,
    ! [A: set_v,A3: set_set_v,B: set_set_v] :
      ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A3 ) @ ( insert_set_v @ A @ B ) )
      = ( insert_set_v @ A @ ( inf_inf_set_set_v @ A3 @ B ) ) ) ).

% insert_inter_insert
thf(fact_865_insert__inter__insert,axiom,
    ! [A: v,A3: set_v,B: set_v] :
      ( ( inf_inf_set_v @ ( insert_v @ A @ A3 ) @ ( insert_v @ A @ B ) )
      = ( insert_v @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) ).

% insert_inter_insert
thf(fact_866_Int__insert__right__if0,axiom,
    ! [A: set_v,A3: set_set_v,B: set_set_v] :
      ( ~ ( member_set_v @ A @ A3 )
     => ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
        = ( inf_inf_set_set_v @ A3 @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_867_Int__insert__right__if0,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ A @ A3 )
     => ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
        = ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_868_Int__insert__right__if0,axiom,
    ! [A: v,A3: set_v,B: set_v] :
      ( ~ ( member_v @ A @ A3 )
     => ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B ) )
        = ( inf_inf_set_v @ A3 @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_869_Int__insert__right__if1,axiom,
    ! [A: set_v,A3: set_set_v,B: set_set_v] :
      ( ( member_set_v @ A @ A3 )
     => ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
        = ( insert_set_v @ A @ ( inf_inf_set_set_v @ A3 @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_870_Int__insert__right__if1,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ A @ A3 )
     => ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
        = ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_871_Int__insert__right__if1,axiom,
    ! [A: v,A3: set_v,B: set_v] :
      ( ( member_v @ A @ A3 )
     => ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B ) )
        = ( insert_v @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_872_Un__Int__eq_I1_J,axiom,
    ! [S4: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S4 @ T2 ) @ S4 )
      = S4 ) ).

% Un_Int_eq(1)
thf(fact_873_Un__Int__eq_I1_J,axiom,
    ! [S4: set_v,T2: set_v] :
      ( ( inf_inf_set_v @ ( sup_sup_set_v @ S4 @ T2 ) @ S4 )
      = S4 ) ).

% Un_Int_eq(1)
thf(fact_874_Un__Int__eq_I1_J,axiom,
    ! [S4: set_set_v,T2: set_set_v] :
      ( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S4 @ T2 ) @ S4 )
      = S4 ) ).

% Un_Int_eq(1)
thf(fact_875_Un__Int__eq_I2_J,axiom,
    ! [S4: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S4 @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_876_Un__Int__eq_I2_J,axiom,
    ! [S4: set_v,T2: set_v] :
      ( ( inf_inf_set_v @ ( sup_sup_set_v @ S4 @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_877_Un__Int__eq_I2_J,axiom,
    ! [S4: set_set_v,T2: set_set_v] :
      ( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S4 @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_878_Un__Int__eq_I3_J,axiom,
    ! [S4: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ S4 @ ( sup_su414716646722978715od_v_v @ S4 @ T2 ) )
      = S4 ) ).

% Un_Int_eq(3)
thf(fact_879_Un__Int__eq_I3_J,axiom,
    ! [S4: set_v,T2: set_v] :
      ( ( inf_inf_set_v @ S4 @ ( sup_sup_set_v @ S4 @ T2 ) )
      = S4 ) ).

% Un_Int_eq(3)
thf(fact_880_Un__Int__eq_I3_J,axiom,
    ! [S4: set_set_v,T2: set_set_v] :
      ( ( inf_inf_set_set_v @ S4 @ ( sup_sup_set_set_v @ S4 @ T2 ) )
      = S4 ) ).

% Un_Int_eq(3)
thf(fact_881_Un__Int__eq_I4_J,axiom,
    ! [T2: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ T2 @ ( sup_su414716646722978715od_v_v @ S4 @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_882_Un__Int__eq_I4_J,axiom,
    ! [T2: set_v,S4: set_v] :
      ( ( inf_inf_set_v @ T2 @ ( sup_sup_set_v @ S4 @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_883_Un__Int__eq_I4_J,axiom,
    ! [T2: set_set_v,S4: set_set_v] :
      ( ( inf_inf_set_set_v @ T2 @ ( sup_sup_set_set_v @ S4 @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_884_Int__Un__eq_I1_J,axiom,
    ! [S4: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S4 @ T2 ) @ S4 )
      = S4 ) ).

% Int_Un_eq(1)
thf(fact_885_Int__Un__eq_I1_J,axiom,
    ! [S4: set_v,T2: set_v] :
      ( ( sup_sup_set_v @ ( inf_inf_set_v @ S4 @ T2 ) @ S4 )
      = S4 ) ).

% Int_Un_eq(1)
thf(fact_886_Int__Un__eq_I1_J,axiom,
    ! [S4: set_set_v,T2: set_set_v] :
      ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S4 @ T2 ) @ S4 )
      = S4 ) ).

% Int_Un_eq(1)
thf(fact_887_Int__Un__eq_I2_J,axiom,
    ! [S4: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S4 @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_888_Int__Un__eq_I2_J,axiom,
    ! [S4: set_v,T2: set_v] :
      ( ( sup_sup_set_v @ ( inf_inf_set_v @ S4 @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_889_Int__Un__eq_I2_J,axiom,
    ! [S4: set_set_v,T2: set_set_v] :
      ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S4 @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_890_Int__Un__eq_I3_J,axiom,
    ! [S4: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ S4 @ ( inf_in6271465464967711157od_v_v @ S4 @ T2 ) )
      = S4 ) ).

% Int_Un_eq(3)
thf(fact_891_Int__Un__eq_I3_J,axiom,
    ! [S4: set_v,T2: set_v] :
      ( ( sup_sup_set_v @ S4 @ ( inf_inf_set_v @ S4 @ T2 ) )
      = S4 ) ).

% Int_Un_eq(3)
thf(fact_892_Int__Un__eq_I3_J,axiom,
    ! [S4: set_set_v,T2: set_set_v] :
      ( ( sup_sup_set_set_v @ S4 @ ( inf_inf_set_set_v @ S4 @ T2 ) )
      = S4 ) ).

% Int_Un_eq(3)
thf(fact_893_Int__Un__eq_I4_J,axiom,
    ! [T2: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ T2 @ ( inf_in6271465464967711157od_v_v @ S4 @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_894_Int__Un__eq_I4_J,axiom,
    ! [T2: set_v,S4: set_v] :
      ( ( sup_sup_set_v @ T2 @ ( inf_inf_set_v @ S4 @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_895_Int__Un__eq_I4_J,axiom,
    ! [T2: set_set_v,S4: set_set_v] :
      ( ( sup_sup_set_set_v @ T2 @ ( inf_inf_set_set_v @ S4 @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_896_disjoint__insert_I2_J,axiom,
    ! [A3: set_v,B2: v,B: set_v] :
      ( ( bot_bot_set_v
        = ( inf_inf_set_v @ A3 @ ( insert_v @ B2 @ B ) ) )
      = ( ~ ( member_v @ B2 @ A3 )
        & ( bot_bot_set_v
          = ( inf_inf_set_v @ A3 @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_897_disjoint__insert_I2_J,axiom,
    ! [A3: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) )
      = ( ~ ( member7453568604450474000od_v_v @ B2 @ A3 )
        & ( bot_bo723834152578015283od_v_v
          = ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_898_disjoint__insert_I2_J,axiom,
    ! [A3: set_set_v,B2: set_v,B: set_set_v] :
      ( ( bot_bot_set_set_v
        = ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ B2 @ B ) ) )
      = ( ~ ( member_set_v @ B2 @ A3 )
        & ( bot_bot_set_set_v
          = ( inf_inf_set_set_v @ A3 @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_899_disjoint__insert_I1_J,axiom,
    ! [B: set_v,A: v,A3: set_v] :
      ( ( ( inf_inf_set_v @ B @ ( insert_v @ A @ A3 ) )
        = bot_bot_set_v )
      = ( ~ ( member_v @ A @ B )
        & ( ( inf_inf_set_v @ B @ A3 )
          = bot_bot_set_v ) ) ) ).

% disjoint_insert(1)
thf(fact_900_disjoint__insert_I1_J,axiom,
    ! [B: set_Product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ( inf_in6271465464967711157od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
        = bot_bo723834152578015283od_v_v )
      = ( ~ ( member7453568604450474000od_v_v @ A @ B )
        & ( ( inf_in6271465464967711157od_v_v @ B @ A3 )
          = bot_bo723834152578015283od_v_v ) ) ) ).

% disjoint_insert(1)
thf(fact_901_disjoint__insert_I1_J,axiom,
    ! [B: set_set_v,A: set_v,A3: set_set_v] :
      ( ( ( inf_inf_set_set_v @ B @ ( insert_set_v @ A @ A3 ) )
        = bot_bot_set_set_v )
      = ( ~ ( member_set_v @ A @ B )
        & ( ( inf_inf_set_set_v @ B @ A3 )
          = bot_bot_set_set_v ) ) ) ).

% disjoint_insert(1)
thf(fact_902_insert__disjoint_I2_J,axiom,
    ! [A: v,A3: set_v,B: set_v] :
      ( ( bot_bot_set_v
        = ( inf_inf_set_v @ ( insert_v @ A @ A3 ) @ B ) )
      = ( ~ ( member_v @ A @ B )
        & ( bot_bot_set_v
          = ( inf_inf_set_v @ A3 @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_903_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( bot_bo723834152578015283od_v_v
        = ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B ) )
      = ( ~ ( member7453568604450474000od_v_v @ A @ B )
        & ( bot_bo723834152578015283od_v_v
          = ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_904_insert__disjoint_I2_J,axiom,
    ! [A: set_v,A3: set_set_v,B: set_set_v] :
      ( ( bot_bot_set_set_v
        = ( inf_inf_set_set_v @ ( insert_set_v @ A @ A3 ) @ B ) )
      = ( ~ ( member_set_v @ A @ B )
        & ( bot_bot_set_set_v
          = ( inf_inf_set_set_v @ A3 @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_905_insert__disjoint_I1_J,axiom,
    ! [A: v,A3: set_v,B: set_v] :
      ( ( ( inf_inf_set_v @ ( insert_v @ A @ A3 ) @ B )
        = bot_bot_set_v )
      = ( ~ ( member_v @ A @ B )
        & ( ( inf_inf_set_v @ A3 @ B )
          = bot_bot_set_v ) ) ) ).

% insert_disjoint(1)
thf(fact_906_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B )
        = bot_bo723834152578015283od_v_v )
      = ( ~ ( member7453568604450474000od_v_v @ A @ B )
        & ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
          = bot_bo723834152578015283od_v_v ) ) ) ).

% insert_disjoint(1)
thf(fact_907_insert__disjoint_I1_J,axiom,
    ! [A: set_v,A3: set_set_v,B: set_set_v] :
      ( ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A3 ) @ B )
        = bot_bot_set_set_v )
      = ( ~ ( member_set_v @ A @ B )
        & ( ( inf_inf_set_set_v @ A3 @ B )
          = bot_bot_set_set_v ) ) ) ).

% insert_disjoint(1)
thf(fact_908_Diff__disjoint,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
      = bot_bo723834152578015283od_v_v ) ).

% Diff_disjoint
thf(fact_909_Diff__disjoint,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( inf_inf_set_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B @ A3 ) )
      = bot_bot_set_set_v ) ).

% Diff_disjoint
thf(fact_910_Diff__disjoint,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B @ A3 ) )
      = bot_bot_set_v ) ).

% Diff_disjoint
thf(fact_911_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) @ X )
      = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_912_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y2: set_v,Z: set_v,X: set_v] :
      ( ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ Z ) @ X )
      = ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_913_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y2: set_set_v,Z: set_set_v,X: set_set_v] :
      ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y2 @ Z ) @ X )
      = ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y2 @ X ) @ ( sup_sup_set_set_v @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_914_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X )
      = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_915_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y2: set_v,Z: set_v,X: set_v] :
      ( ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X )
      = ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_916_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y2: set_set_v,Z: set_set_v,X: set_set_v] :
      ( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y2 @ Z ) @ X )
      = ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y2 @ X ) @ ( inf_inf_set_set_v @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_917_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
      = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_918_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
      = ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_919_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] :
      ( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y2 @ Z ) )
      = ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y2 ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_920_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
      = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_921_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
      = ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_922_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] :
      ( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y2 @ Z ) )
      = ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y2 ) @ ( inf_inf_set_set_v @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_923_Int__left__commute,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B @ C3 ) )
      = ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A3 @ C3 ) ) ) ).

% Int_left_commute
thf(fact_924_Int__left__absorb,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ A3 @ B ) )
      = ( inf_inf_set_v @ A3 @ B ) ) ).

% Int_left_absorb
thf(fact_925_Int__commute,axiom,
    ( inf_inf_set_v
    = ( ^ [A5: set_v,B5: set_v] : ( inf_inf_set_v @ B5 @ A5 ) ) ) ).

% Int_commute
thf(fact_926_Int__absorb,axiom,
    ! [A3: set_v] :
      ( ( inf_inf_set_v @ A3 @ A3 )
      = A3 ) ).

% Int_absorb
thf(fact_927_Int__assoc,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C3 )
      = ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B @ C3 ) ) ) ).

% Int_assoc
thf(fact_928_IntD2,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
     => ( member7453568604450474000od_v_v @ C @ B ) ) ).

% IntD2
thf(fact_929_IntD2,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
     => ( member_v @ C @ B ) ) ).

% IntD2
thf(fact_930_IntD1,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
     => ( member7453568604450474000od_v_v @ C @ A3 ) ) ).

% IntD1
thf(fact_931_IntD1,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
     => ( member_v @ C @ A3 ) ) ).

% IntD1
thf(fact_932_IntE,axiom,
    ! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
     => ~ ( ( member7453568604450474000od_v_v @ C @ A3 )
         => ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).

% IntE
thf(fact_933_IntE,axiom,
    ! [C: v,A3: set_v,B: set_v] :
      ( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
     => ~ ( ( member_v @ C @ A3 )
         => ~ ( member_v @ C @ B ) ) ) ).

% IntE
thf(fact_934_inf__left__commute,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
      = ( inf_inf_set_v @ Y2 @ ( inf_inf_set_v @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_935_inf_Oleft__commute,axiom,
    ! [B2: set_v,A: set_v,C: set_v] :
      ( ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A @ C ) )
      = ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ).

% inf.left_commute
thf(fact_936_inf__commute,axiom,
    ( inf_inf_set_v
    = ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).

% inf_commute
thf(fact_937_inf_Ocommute,axiom,
    ( inf_inf_set_v
    = ( ^ [A4: set_v,B4: set_v] : ( inf_inf_set_v @ B4 @ A4 ) ) ) ).

% inf.commute
thf(fact_938_inf__assoc,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Z )
      = ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ).

% inf_assoc
thf(fact_939_inf_Oassoc,axiom,
    ! [A: set_v,B2: set_v,C: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C )
      = ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ).

% inf.assoc
thf(fact_940_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_v
    = ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_941_inf__sup__aci_I2_J,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Z )
      = ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_942_inf__sup__aci_I3_J,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
      = ( inf_inf_set_v @ Y2 @ ( inf_inf_set_v @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_943_inf__sup__aci_I4_J,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y2 ) )
      = ( inf_inf_set_v @ X @ Y2 ) ) ).

% inf_sup_aci(4)
thf(fact_944_inf_OcoboundedI2,axiom,
    ! [B2: set_v,C: set_v,A: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ C )
     => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_945_inf_OcoboundedI2,axiom,
    ! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
     => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_946_inf_OcoboundedI1,axiom,
    ! [A: set_v,C: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A @ C )
     => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_947_inf_OcoboundedI1,axiom,
    ! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ C )
     => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_948_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_v
    = ( ^ [B4: set_v,A4: set_v] :
          ( ( inf_inf_set_v @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_949_inf_Oabsorb__iff2,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
          ( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_950_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_v
    = ( ^ [A4: set_v,B4: set_v] :
          ( ( inf_inf_set_v @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_951_inf_Oabsorb__iff1,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
          ( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_952_inf_Ocobounded2,axiom,
    ! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_953_inf_Ocobounded2,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_954_inf_Ocobounded1,axiom,
    ! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ A ) ).

% inf.cobounded1
thf(fact_955_inf_Ocobounded1,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ A ) ).

% inf.cobounded1
thf(fact_956_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_v
    = ( ^ [A4: set_v,B4: set_v] :
          ( A4
          = ( inf_inf_set_v @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_957_inf_Oorder__iff,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
          ( A4
          = ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_958_inf__greatest,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( ord_less_eq_set_v @ X @ Y2 )
     => ( ( ord_less_eq_set_v @ X @ Z )
       => ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_959_inf__greatest,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
     => ( ( ord_le7336532860387713383od_v_v @ X @ Z )
       => ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_960_inf_OboundedI,axiom,
    ! [A: set_v,B2: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A @ B2 )
     => ( ( ord_less_eq_set_v @ A @ C )
       => ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_961_inf_OboundedI,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
     => ( ( ord_le7336532860387713383od_v_v @ A @ C )
       => ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_962_inf_OboundedE,axiom,
    ! [A: set_v,B2: set_v,C: set_v] :
      ( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) )
     => ~ ( ( ord_less_eq_set_v @ A @ B2 )
         => ~ ( ord_less_eq_set_v @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_963_inf_OboundedE,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
     => ~ ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
         => ~ ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_964_inf__absorb2,axiom,
    ! [Y2: set_v,X: set_v] :
      ( ( ord_less_eq_set_v @ Y2 @ X )
     => ( ( inf_inf_set_v @ X @ Y2 )
        = Y2 ) ) ).

% inf_absorb2
thf(fact_965_inf__absorb2,axiom,
    ! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
     => ( ( inf_in6271465464967711157od_v_v @ X @ Y2 )
        = Y2 ) ) ).

% inf_absorb2
thf(fact_966_inf__absorb1,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( ord_less_eq_set_v @ X @ Y2 )
     => ( ( inf_inf_set_v @ X @ Y2 )
        = X ) ) ).

% inf_absorb1
thf(fact_967_inf__absorb1,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
     => ( ( inf_in6271465464967711157od_v_v @ X @ Y2 )
        = X ) ) ).

% inf_absorb1
thf(fact_968_inf_Oabsorb2,axiom,
    ! [B2: set_v,A: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ A )
     => ( ( inf_inf_set_v @ A @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_969_inf_Oabsorb2,axiom,
    ! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
     => ( ( inf_in6271465464967711157od_v_v @ A @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_970_inf_Oabsorb1,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A @ B2 )
     => ( ( inf_inf_set_v @ A @ B2 )
        = A ) ) ).

% inf.absorb1
thf(fact_971_inf_Oabsorb1,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
     => ( ( inf_in6271465464967711157od_v_v @ A @ B2 )
        = A ) ) ).

% inf.absorb1
thf(fact_972_le__iff__inf,axiom,
    ( ord_less_eq_set_v
    = ( ^ [X2: set_v,Y3: set_v] :
          ( ( inf_inf_set_v @ X2 @ Y3 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_973_le__iff__inf,axiom,
    ( ord_le7336532860387713383od_v_v
    = ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
          ( ( inf_in6271465464967711157od_v_v @ X2 @ Y3 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_974_inf__unique,axiom,
    ! [F: set_v > set_v > set_v,X: set_v,Y2: set_v] :
      ( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y ) @ X3 )
     => ( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y ) @ Y )
       => ( ! [X3: set_v,Y: set_v,Z3: set_v] :
              ( ( ord_less_eq_set_v @ X3 @ Y )
             => ( ( ord_less_eq_set_v @ X3 @ Z3 )
               => ( ord_less_eq_set_v @ X3 @ ( F @ Y @ Z3 ) ) ) )
         => ( ( inf_inf_set_v @ X @ Y2 )
            = ( F @ X @ Y2 ) ) ) ) ) ).

% inf_unique
thf(fact_975_inf__unique,axiom,
    ! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
      ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y ) @ X3 )
     => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y ) @ Y )
       => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
              ( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
             => ( ( ord_le7336532860387713383od_v_v @ X3 @ Z3 )
               => ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ Y @ Z3 ) ) ) )
         => ( ( inf_in6271465464967711157od_v_v @ X @ Y2 )
            = ( F @ X @ Y2 ) ) ) ) ) ).

% inf_unique
thf(fact_976_inf_OorderI,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( A
        = ( inf_inf_set_v @ A @ B2 ) )
     => ( ord_less_eq_set_v @ A @ B2 ) ) ).

% inf.orderI
thf(fact_977_inf_OorderI,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( A
        = ( inf_in6271465464967711157od_v_v @ A @ B2 ) )
     => ( ord_le7336532860387713383od_v_v @ A @ B2 ) ) ).

% inf.orderI
thf(fact_978_inf_OorderE,axiom,
    ! [A: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A @ B2 )
     => ( A
        = ( inf_inf_set_v @ A @ B2 ) ) ) ).

% inf.orderE
thf(fact_979_inf_OorderE,axiom,
    ! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
     => ( A
        = ( inf_in6271465464967711157od_v_v @ A @ B2 ) ) ) ).

% inf.orderE
thf(fact_980_le__infI2,axiom,
    ! [B2: set_v,X: set_v,A: set_v] :
      ( ( ord_less_eq_set_v @ B2 @ X )
     => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_981_le__infI2,axiom,
    ! [B2: set_Product_prod_v_v,X: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B2 @ X )
     => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_982_le__infI1,axiom,
    ! [A: set_v,X: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ A @ X )
     => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_983_le__infI1,axiom,
    ! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A @ X )
     => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_984_inf__mono,axiom,
    ! [A: set_v,C: set_v,B2: set_v,D: set_v] :
      ( ( ord_less_eq_set_v @ A @ C )
     => ( ( ord_less_eq_set_v @ B2 @ D )
       => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ ( inf_inf_set_v @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_985_inf__mono,axiom,
    ! [A: 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 @ A @ C )
     => ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
       => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_986_le__infI,axiom,
    ! [X: set_v,A: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ X @ A )
     => ( ( ord_less_eq_set_v @ X @ B2 )
       => ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).

% le_infI
thf(fact_987_le__infI,axiom,
    ! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X @ A )
     => ( ( ord_le7336532860387713383od_v_v @ X @ B2 )
       => ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) ) ) ) ).

% le_infI
thf(fact_988_le__infE,axiom,
    ! [X: set_v,A: set_v,B2: set_v] :
      ( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B2 ) )
     => ~ ( ( ord_less_eq_set_v @ X @ A )
         => ~ ( ord_less_eq_set_v @ X @ B2 ) ) ) ).

% le_infE
thf(fact_989_le__infE,axiom,
    ! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) )
     => ~ ( ( ord_le7336532860387713383od_v_v @ X @ A )
         => ~ ( ord_le7336532860387713383od_v_v @ X @ B2 ) ) ) ).

% le_infE
thf(fact_990_inf__le2,axiom,
    ! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Y2 ) ).

% inf_le2
thf(fact_991_inf__le2,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ Y2 ) ).

% inf_le2
thf(fact_992_inf__le1,axiom,
    ! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ X ) ).

% inf_le1
thf(fact_993_inf__le1,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ X ) ).

% inf_le1
thf(fact_994_inf__sup__ord_I1_J,axiom,
    ! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ X ) ).

% inf_sup_ord(1)
thf(fact_995_inf__sup__ord_I1_J,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ X ) ).

% inf_sup_ord(1)
thf(fact_996_inf__sup__ord_I2_J,axiom,
    ! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Y2 ) ).

% inf_sup_ord(2)
thf(fact_997_inf__sup__ord_I2_J,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ Y2 ) ).

% inf_sup_ord(2)
thf(fact_998_sup__inf__distrib2,axiom,
    ! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) @ X )
      = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_999_sup__inf__distrib2,axiom,
    ! [Y2: set_v,Z: set_v,X: set_v] :
      ( ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ Z ) @ X )
      = ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_1000_sup__inf__distrib2,axiom,
    ! [Y2: set_set_v,Z: set_set_v,X: set_set_v] :
      ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y2 @ Z ) @ X )
      = ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y2 @ X ) @ ( sup_sup_set_set_v @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_1001_sup__inf__distrib1,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
      = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_1002_sup__inf__distrib1,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
      = ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_1003_sup__inf__distrib1,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] :
      ( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y2 @ Z ) )
      = ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y2 ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_1004_inf__sup__distrib2,axiom,
    ! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X )
      = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_1005_inf__sup__distrib2,axiom,
    ! [Y2: set_v,Z: set_v,X: set_v] :
      ( ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X )
      = ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_1006_inf__sup__distrib2,axiom,
    ! [Y2: set_set_v,Z: set_set_v,X: set_set_v] :
      ( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y2 @ Z ) @ X )
      = ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y2 @ X ) @ ( inf_inf_set_set_v @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_1007_inf__sup__distrib1,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
      = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_1008_inf__sup__distrib1,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
      = ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_1009_inf__sup__distrib1,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] :
      ( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y2 @ Z ) )
      = ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y2 ) @ ( inf_inf_set_set_v @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_1010_distrib__imp2,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
          ( ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y @ Z3 ) )
          = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y ) @ ( sup_su414716646722978715od_v_v @ X3 @ Z3 ) ) )
     => ( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
        = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_1011_distrib__imp2,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ! [X3: set_v,Y: set_v,Z3: set_v] :
          ( ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ Y @ Z3 ) )
          = ( inf_inf_set_v @ ( sup_sup_set_v @ X3 @ Y ) @ ( sup_sup_set_v @ X3 @ Z3 ) ) )
     => ( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
        = ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_1012_distrib__imp2,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] :
      ( ! [X3: set_set_v,Y: set_set_v,Z3: set_set_v] :
          ( ( sup_sup_set_set_v @ X3 @ ( inf_inf_set_set_v @ Y @ Z3 ) )
          = ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X3 @ Y ) @ ( sup_sup_set_set_v @ X3 @ Z3 ) ) )
     => ( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y2 @ Z ) )
        = ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y2 ) @ ( inf_inf_set_set_v @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_1013_distrib__imp1,axiom,
    ! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
      ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
          ( ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y @ Z3 ) )
          = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y ) @ ( inf_in6271465464967711157od_v_v @ X3 @ Z3 ) ) )
     => ( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
        = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_1014_distrib__imp1,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] :
      ( ! [X3: set_v,Y: set_v,Z3: set_v] :
          ( ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ Y @ Z3 ) )
          = ( sup_sup_set_v @ ( inf_inf_set_v @ X3 @ Y ) @ ( inf_inf_set_v @ X3 @ Z3 ) ) )
     => ( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
        = ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_1015_distrib__imp1,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] :
      ( ! [X3: set_set_v,Y: set_set_v,Z3: set_set_v] :
          ( ( inf_inf_set_set_v @ X3 @ ( sup_sup_set_set_v @ Y @ Z3 ) )
          = ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X3 @ Y ) @ ( inf_inf_set_set_v @ X3 @ Z3 ) ) )
     => ( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y2 @ Z ) )
        = ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y2 ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_1016_Int__emptyI,axiom,
    ! [A3: set_v,B: set_v] :
      ( ! [X3: v] :
          ( ( member_v @ X3 @ A3 )
         => ~ ( member_v @ X3 @ B ) )
     => ( ( inf_inf_set_v @ A3 @ B )
        = bot_bot_set_v ) ) ).

% Int_emptyI
thf(fact_1017_Int__emptyI,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ! [X3: product_prod_v_v] :
          ( ( member7453568604450474000od_v_v @ X3 @ A3 )
         => ~ ( member7453568604450474000od_v_v @ X3 @ B ) )
     => ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
        = bot_bo723834152578015283od_v_v ) ) ).

% Int_emptyI
thf(fact_1018_Int__emptyI,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ! [X3: set_v] :
          ( ( member_set_v @ X3 @ A3 )
         => ~ ( member_set_v @ X3 @ B ) )
     => ( ( inf_inf_set_set_v @ A3 @ B )
        = bot_bot_set_set_v ) ) ).

% Int_emptyI
thf(fact_1019_disjoint__iff,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( ( inf_inf_set_v @ A3 @ B )
        = bot_bot_set_v )
      = ( ! [X2: v] :
            ( ( member_v @ X2 @ A3 )
           => ~ ( member_v @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_1020_disjoint__iff,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
        = bot_bo723834152578015283od_v_v )
      = ( ! [X2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X2 @ A3 )
           => ~ ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_1021_disjoint__iff,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( ( inf_inf_set_set_v @ A3 @ B )
        = bot_bot_set_set_v )
      = ( ! [X2: set_v] :
            ( ( member_set_v @ X2 @ A3 )
           => ~ ( member_set_v @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_1022_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_1023_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_1024_Int__empty__left,axiom,
    ! [B: set_set_v] :
      ( ( inf_inf_set_set_v @ bot_bot_set_set_v @ B )
      = bot_bot_set_set_v ) ).

% Int_empty_left
thf(fact_1025_Int__empty__right,axiom,
    ! [A3: set_v] :
      ( ( inf_inf_set_v @ A3 @ bot_bot_set_v )
      = bot_bot_set_v ) ).

% Int_empty_right
thf(fact_1026_Int__empty__right,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
      = bot_bo723834152578015283od_v_v ) ).

% Int_empty_right
thf(fact_1027_Int__empty__right,axiom,
    ! [A3: set_set_v] :
      ( ( inf_inf_set_set_v @ A3 @ bot_bot_set_set_v )
      = bot_bot_set_set_v ) ).

% Int_empty_right
thf(fact_1028_disjoint__iff__not__equal,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( ( inf_inf_set_v @ A3 @ B )
        = bot_bot_set_v )
      = ( ! [X2: v] :
            ( ( member_v @ X2 @ A3 )
           => ! [Y3: v] :
                ( ( member_v @ Y3 @ B )
               => ( X2 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1029_disjoint__iff__not__equal,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
        = bot_bo723834152578015283od_v_v )
      = ( ! [X2: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X2 @ A3 )
           => ! [Y3: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ Y3 @ B )
               => ( X2 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1030_disjoint__iff__not__equal,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( ( inf_inf_set_set_v @ A3 @ B )
        = bot_bot_set_set_v )
      = ( ! [X2: set_v] :
            ( ( member_set_v @ X2 @ A3 )
           => ! [Y3: set_v] :
                ( ( member_set_v @ Y3 @ B )
               => ( X2 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1031_Int__Collect__mono,axiom,
    ! [A3: set_set_v,B: set_set_v,P: set_v > $o,Q: set_v > $o] :
      ( ( ord_le5216385588623774835_set_v @ A3 @ B )
     => ( ! [X3: set_v] :
            ( ( member_set_v @ X3 @ A3 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le5216385588623774835_set_v @ ( inf_inf_set_set_v @ A3 @ ( collect_set_v @ P ) ) @ ( inf_inf_set_set_v @ B @ ( collect_set_v @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1032_Int__Collect__mono,axiom,
    ! [A3: set_v,B: set_v,P: v > $o,Q: v > $o] :
      ( ( ord_less_eq_set_v @ A3 @ B )
     => ( ! [X3: v] :
            ( ( member_v @ X3 @ A3 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B @ ( collect_v @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1033_Int__Collect__mono,axiom,
    ! [A3: 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 @ A3 @ B )
     => ( ! [X3: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ A3 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1034_Int__greatest,axiom,
    ! [C3: set_v,A3: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ C3 @ A3 )
     => ( ( ord_less_eq_set_v @ C3 @ B )
       => ( ord_less_eq_set_v @ C3 @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).

% Int_greatest
thf(fact_1035_Int__greatest,axiom,
    ! [C3: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ C3 @ A3 )
     => ( ( ord_le7336532860387713383od_v_v @ C3 @ B )
       => ( ord_le7336532860387713383od_v_v @ C3 @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).

% Int_greatest
thf(fact_1036_Int__absorb2,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ B )
     => ( ( inf_inf_set_v @ A3 @ B )
        = A3 ) ) ).

% Int_absorb2
thf(fact_1037_Int__absorb2,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
     => ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
        = A3 ) ) ).

% Int_absorb2
thf(fact_1038_Int__absorb1,axiom,
    ! [B: set_v,A3: set_v] :
      ( ( ord_less_eq_set_v @ B @ A3 )
     => ( ( inf_inf_set_v @ A3 @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_1039_Int__absorb1,axiom,
    ! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ B @ A3 )
     => ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_1040_Int__lower2,axiom,
    ! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ B ) ).

% Int_lower2
thf(fact_1041_Int__lower2,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ B ) ).

% Int_lower2
thf(fact_1042_Int__lower1,axiom,
    ! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ A3 ) ).

% Int_lower1
thf(fact_1043_Int__lower1,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ A3 ) ).

% Int_lower1
thf(fact_1044_Int__mono,axiom,
    ! [A3: set_v,C3: set_v,B: set_v,D2: set_v] :
      ( ( ord_less_eq_set_v @ A3 @ C3 )
     => ( ( ord_less_eq_set_v @ B @ D2 )
       => ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ C3 @ D2 ) ) ) ) ).

% Int_mono
thf(fact_1045_Int__mono,axiom,
    ! [A3: set_Product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
      ( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
     => ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
       => ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ C3 @ D2 ) ) ) ) ).

% Int_mono
thf(fact_1046_Int__insert__left,axiom,
    ! [A: set_v,C3: set_set_v,B: set_set_v] :
      ( ( ( member_set_v @ A @ C3 )
       => ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C3 )
          = ( insert_set_v @ A @ ( inf_inf_set_set_v @ B @ C3 ) ) ) )
      & ( ~ ( member_set_v @ A @ C3 )
       => ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C3 )
          = ( inf_inf_set_set_v @ B @ C3 ) ) ) ) ).

% Int_insert_left
thf(fact_1047_Int__insert__left,axiom,
    ! [A: product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( member7453568604450474000od_v_v @ A @ C3 )
       => ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
          = ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) ) )
      & ( ~ ( member7453568604450474000od_v_v @ A @ C3 )
       => ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
          = ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) ) ) ).

% Int_insert_left
thf(fact_1048_Int__insert__left,axiom,
    ! [A: v,C3: set_v,B: set_v] :
      ( ( ( member_v @ A @ C3 )
       => ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C3 )
          = ( insert_v @ A @ ( inf_inf_set_v @ B @ C3 ) ) ) )
      & ( ~ ( member_v @ A @ C3 )
       => ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C3 )
          = ( inf_inf_set_v @ B @ C3 ) ) ) ) ).

% Int_insert_left
thf(fact_1049_Int__insert__right,axiom,
    ! [A: set_v,A3: set_set_v,B: set_set_v] :
      ( ( ( member_set_v @ A @ A3 )
       => ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
          = ( insert_set_v @ A @ ( inf_inf_set_set_v @ A3 @ B ) ) ) )
      & ( ~ ( member_set_v @ A @ A3 )
       => ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B ) )
          = ( inf_inf_set_set_v @ A3 @ B ) ) ) ) ).

% Int_insert_right
thf(fact_1050_Int__insert__right,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( member7453568604450474000od_v_v @ A @ A3 )
       => ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
          = ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) )
      & ( ~ ( member7453568604450474000od_v_v @ A @ A3 )
       => ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
          = ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).

% Int_insert_right
thf(fact_1051_Int__insert__right,axiom,
    ! [A: v,A3: set_v,B: set_v] :
      ( ( ( member_v @ A @ A3 )
       => ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B ) )
          = ( insert_v @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) )
      & ( ~ ( member_v @ A @ A3 )
       => ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B ) )
          = ( inf_inf_set_v @ A3 @ B ) ) ) ) ).

% Int_insert_right
thf(fact_1052_Un__Int__distrib2,axiom,
    ! [B: set_Product_prod_v_v,C3: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) @ A3 )
      = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ A3 ) @ ( sup_su414716646722978715od_v_v @ C3 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_1053_Un__Int__distrib2,axiom,
    ! [B: set_v,C3: set_v,A3: set_v] :
      ( ( sup_sup_set_v @ ( inf_inf_set_v @ B @ C3 ) @ A3 )
      = ( inf_inf_set_v @ ( sup_sup_set_v @ B @ A3 ) @ ( sup_sup_set_v @ C3 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_1054_Un__Int__distrib2,axiom,
    ! [B: set_set_v,C3: set_set_v,A3: set_set_v] :
      ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B @ C3 ) @ A3 )
      = ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B @ A3 ) @ ( sup_sup_set_set_v @ C3 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_1055_Int__Un__distrib2,axiom,
    ! [B: set_Product_prod_v_v,C3: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C3 ) @ A3 )
      = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ A3 ) @ ( inf_in6271465464967711157od_v_v @ C3 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_1056_Int__Un__distrib2,axiom,
    ! [B: set_v,C3: set_v,A3: set_v] :
      ( ( inf_inf_set_v @ ( sup_sup_set_v @ B @ C3 ) @ A3 )
      = ( sup_sup_set_v @ ( inf_inf_set_v @ B @ A3 ) @ ( inf_inf_set_v @ C3 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_1057_Int__Un__distrib2,axiom,
    ! [B: set_set_v,C3: set_set_v,A3: set_set_v] :
      ( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B @ C3 ) @ A3 )
      = ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B @ A3 ) @ ( inf_inf_set_set_v @ C3 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_1058_Un__Int__distrib,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) )
      = ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ A3 @ C3 ) ) ) ).

% Un_Int_distrib
thf(fact_1059_Un__Int__distrib,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( sup_sup_set_v @ A3 @ ( inf_inf_set_v @ B @ C3 ) )
      = ( inf_inf_set_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_sup_set_v @ A3 @ C3 ) ) ) ).

% Un_Int_distrib
thf(fact_1060_Un__Int__distrib,axiom,
    ! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
      ( ( sup_sup_set_set_v @ A3 @ ( inf_inf_set_set_v @ B @ C3 ) )
      = ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ ( sup_sup_set_set_v @ A3 @ C3 ) ) ) ).

% Un_Int_distrib
thf(fact_1061_Int__Un__distrib,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) )
      = ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ A3 @ C3 ) ) ) ).

% Int_Un_distrib
thf(fact_1062_Int__Un__distrib,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( inf_inf_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) )
      = ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ A3 @ C3 ) ) ) ).

% Int_Un_distrib
thf(fact_1063_Int__Un__distrib,axiom,
    ! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
      ( ( inf_inf_set_set_v @ A3 @ ( sup_sup_set_set_v @ B @ C3 ) )
      = ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A3 @ B ) @ ( inf_inf_set_set_v @ A3 @ C3 ) ) ) ).

% Int_Un_distrib
thf(fact_1064_Un__Int__crazy,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) @ ( inf_in6271465464967711157od_v_v @ C3 @ A3 ) )
      = ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) @ ( sup_su414716646722978715od_v_v @ C3 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_1065_Un__Int__crazy,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ B @ C3 ) ) @ ( inf_inf_set_v @ C3 @ A3 ) )
      = ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_sup_set_v @ B @ C3 ) ) @ ( sup_sup_set_v @ C3 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_1066_Un__Int__crazy,axiom,
    ! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
      ( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A3 @ B ) @ ( inf_inf_set_set_v @ B @ C3 ) ) @ ( inf_inf_set_set_v @ C3 @ A3 ) )
      = ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A3 @ B ) @ ( sup_sup_set_set_v @ B @ C3 ) ) @ ( sup_sup_set_set_v @ C3 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_1067_finite__list,axiom,
    ! [A3: set_v] :
      ( ( finite_finite_v @ A3 )
     => ? [Xs2: list_v] :
          ( ( set_v2 @ Xs2 )
          = A3 ) ) ).

% finite_list
thf(fact_1068_Int__Diff,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C3 )
      = ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B @ C3 ) ) ) ).

% Int_Diff
thf(fact_1069_Diff__Int2,axiom,
    ! [A3: set_v,C3: set_v,B: set_v] :
      ( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C3 ) @ ( inf_inf_set_v @ B @ C3 ) )
      = ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C3 ) @ B ) ) ).

% Diff_Int2
thf(fact_1070_Diff__Diff__Int,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( minus_minus_set_v @ A3 @ ( minus_minus_set_v @ A3 @ B ) )
      = ( inf_inf_set_v @ A3 @ B ) ) ).

% Diff_Diff_Int
thf(fact_1071_Diff__Int__distrib,axiom,
    ! [C3: set_v,A3: set_v,B: set_v] :
      ( ( inf_inf_set_v @ C3 @ ( minus_minus_set_v @ A3 @ B ) )
      = ( minus_minus_set_v @ ( inf_inf_set_v @ C3 @ A3 ) @ ( inf_inf_set_v @ C3 @ B ) ) ) ).

% Diff_Int_distrib
thf(fact_1072_Diff__Int__distrib2,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B ) @ C3 )
      = ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C3 ) @ ( inf_inf_set_v @ B @ C3 ) ) ) ).

% Diff_Int_distrib2
thf(fact_1073_graph_Odfss_Ocases,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: produc5741669702376414499t_unit] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ~ ! [V2: v,E7: sCC_Bl1394983891496994913t_unit] :
            ( X
           != ( produc3862955338007567901t_unit @ V2 @ E7 ) ) ) ).

% graph.dfss.cases
thf(fact_1074_graph_Ovfin,axiom,
    ! [Vertices: set_v,Successors: v > set_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( finite_finite_v @ Vertices ) ) ).

% graph.vfin
thf(fact_1075_distrib__sup__le,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y2 @ Z ) ) @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y2 ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1076_distrib__sup__le,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1077_distrib__sup__le,axiom,
    ! [X: 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 @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1078_distrib__inf__le,axiom,
    ! [X: set_set_v,Y2: set_set_v,Z: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y2 ) @ ( inf_inf_set_set_v @ X @ Z ) ) @ ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y2 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1079_distrib__inf__le,axiom,
    ! [X: set_v,Y2: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) @ ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1080_distrib__inf__le,axiom,
    ! [X: 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 @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1081_successively_Ocases,axiom,
    ! [X: produc8237170675765753490list_v] :
      ( ! [P2: v > v > $o] :
          ( X
         != ( produc601102195597853570list_v @ P2 @ nil_v ) )
     => ( ! [P2: v > v > $o,X3: v] :
            ( X
           != ( produc601102195597853570list_v @ P2 @ ( cons_v @ X3 @ nil_v ) ) )
       => ~ ! [P2: v > v > $o,X3: v,Y: v,Xs2: list_v] :
              ( X
             != ( produc601102195597853570list_v @ P2 @ ( cons_v @ X3 @ ( cons_v @ Y @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_1082_sorted__wrt_Ocases,axiom,
    ! [X: produc8237170675765753490list_v] :
      ( ! [P2: v > v > $o] :
          ( X
         != ( produc601102195597853570list_v @ P2 @ nil_v ) )
     => ~ ! [P2: v > v > $o,X3: v,Ys2: list_v] :
            ( X
           != ( produc601102195597853570list_v @ P2 @ ( cons_v @ X3 @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_1083_Int__Diff__disjoint,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
      = bot_bo723834152578015283od_v_v ) ).

% Int_Diff_disjoint
thf(fact_1084_Int__Diff__disjoint,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ A3 @ B ) @ ( minus_7228012346218142266_set_v @ A3 @ B ) )
      = bot_bot_set_set_v ) ).

% Int_Diff_disjoint
thf(fact_1085_Int__Diff__disjoint,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ B ) )
      = bot_bot_set_v ) ).

% Int_Diff_disjoint
thf(fact_1086_Diff__triv,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
        = bot_bo723834152578015283od_v_v )
     => ( ( minus_4183494784930505774od_v_v @ A3 @ B )
        = A3 ) ) ).

% Diff_triv
thf(fact_1087_Diff__triv,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( ( inf_inf_set_set_v @ A3 @ B )
        = bot_bot_set_set_v )
     => ( ( minus_7228012346218142266_set_v @ A3 @ B )
        = A3 ) ) ).

% Diff_triv
thf(fact_1088_Diff__triv,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( ( inf_inf_set_v @ A3 @ B )
        = bot_bot_set_v )
     => ( ( minus_minus_set_v @ A3 @ B )
        = A3 ) ) ).

% Diff_triv
thf(fact_1089_Un__Int__assoc__eq,axiom,
    ! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
      ( ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A3 @ B ) @ C3 )
        = ( inf_inf_set_set_v @ A3 @ ( sup_sup_set_set_v @ B @ C3 ) ) )
      = ( ord_le5216385588623774835_set_v @ C3 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_1090_Un__Int__assoc__eq,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C3 )
        = ( inf_inf_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) ) )
      = ( ord_less_eq_set_v @ C3 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_1091_Un__Int__assoc__eq,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ C3 )
        = ( inf_in6271465464967711157od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) )
      = ( ord_le7336532860387713383od_v_v @ C3 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_1092_shuffles_Ocases,axiom,
    ! [X: produc1391462591744249447list_v] :
      ( ! [Ys2: list_v] :
          ( X
         != ( produc6795410681906604247list_v @ nil_v @ Ys2 ) )
     => ( ! [Xs2: list_v] :
            ( X
           != ( produc6795410681906604247list_v @ Xs2 @ nil_v ) )
       => ~ ! [X3: v,Xs2: list_v,Y: v,Ys2: list_v] :
              ( X
             != ( produc6795410681906604247list_v @ ( cons_v @ X3 @ Xs2 ) @ ( cons_v @ Y @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_1093_splice_Ocases,axiom,
    ! [X: produc1391462591744249447list_v] :
      ( ! [Ys2: list_v] :
          ( X
         != ( produc6795410681906604247list_v @ nil_v @ Ys2 ) )
     => ~ ! [X3: v,Xs2: list_v,Ys2: list_v] :
            ( X
           != ( produc6795410681906604247list_v @ ( cons_v @ X3 @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_1094_Un__Diff__Int,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_1095_Un__Diff__Int,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) @ ( inf_inf_set_set_v @ A3 @ B ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_1096_Un__Diff__Int,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( inf_inf_set_v @ A3 @ B ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_1097_Int__Diff__Un,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_1098_Int__Diff__Un,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A3 @ B ) @ ( minus_7228012346218142266_set_v @ A3 @ B ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_1099_Int__Diff__Un,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ B ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_1100_Diff__Int,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) )
      = ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ C3 ) ) ) ).

% Diff_Int
thf(fact_1101_Diff__Int,axiom,
    ! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
      ( ( minus_7228012346218142266_set_v @ A3 @ ( inf_inf_set_set_v @ B @ C3 ) )
      = ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) @ ( minus_7228012346218142266_set_v @ A3 @ C3 ) ) ) ).

% Diff_Int
thf(fact_1102_Diff__Int,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( minus_minus_set_v @ A3 @ ( inf_inf_set_v @ B @ C3 ) )
      = ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ C3 ) ) ) ).

% Diff_Int
thf(fact_1103_Diff__Un,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
      ( ( minus_4183494784930505774od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) )
      = ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ C3 ) ) ) ).

% Diff_Un
thf(fact_1104_Diff__Un,axiom,
    ! [A3: set_set_v,B: set_set_v,C3: set_set_v] :
      ( ( minus_7228012346218142266_set_v @ A3 @ ( sup_sup_set_set_v @ B @ C3 ) )
      = ( inf_inf_set_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) @ ( minus_7228012346218142266_set_v @ A3 @ C3 ) ) ) ).

% Diff_Un
thf(fact_1105_Diff__Un,axiom,
    ! [A3: set_v,B: set_v,C3: set_v] :
      ( ( minus_minus_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) )
      = ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ C3 ) ) ) ).

% Diff_Un
thf(fact_1106_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 )
          & ! [X2: product_prod_v_v] :
              ( ( member7453568604450474000od_v_v @ X2 @ Vertices2 )
             => ( ord_le7336532860387713383od_v_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).

% SCC_Bloemen_Sequential.graph_def
thf(fact_1107_SCC__Bloemen__Sequential_Ograph__def,axiom,
    ( sCC_Bloemen_graph_v
    = ( ^ [Vertices2: set_v,Successors2: v > set_v] :
          ( ( finite_finite_v @ Vertices2 )
          & ! [X2: v] :
              ( ( member_v @ X2 @ Vertices2 )
             => ( ord_less_eq_set_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).

% SCC_Bloemen_Sequential.graph_def
thf(fact_1108_graph_Ointro,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ Vertices )
     => ( ! [X3: product_prod_v_v] :
            ( ( member7453568604450474000od_v_v @ X3 @ Vertices )
           => ( ord_le7336532860387713383od_v_v @ ( Successors @ X3 ) @ Vertices ) )
       => ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors ) ) ) ).

% graph.intro
thf(fact_1109_graph_Ointro,axiom,
    ! [Vertices: set_v,Successors: v > set_v] :
      ( ( finite_finite_v @ Vertices )
     => ( ! [X3: v] :
            ( ( member_v @ X3 @ Vertices )
           => ( ord_less_eq_set_v @ ( Successors @ X3 ) @ Vertices ) )
       => ( sCC_Bloemen_graph_v @ Vertices @ Successors ) ) ) ).

% graph.intro
thf(fact_1110_graph_Ora__mono,axiom,
    ! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v,E6: set_Product_prod_v_v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
       => ( ( ord_le7336532860387713383od_v_v @ E6 @ E4 )
         => ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E6 ) ) ) ) ).

% graph.ra_mono
thf(fact_1111_graph_Oscc__partition,axiom,
    ! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S4: set_Product_prod_v_v,S6: set_Product_prod_v_v,X: product_prod_v_v] :
      ( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
     => ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S4 )
       => ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S6 )
         => ( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ S4 @ S6 ) )
           => ( S4 = S6 ) ) ) ) ) ).

% graph.scc_partition
thf(fact_1112_graph_Oscc__partition,axiom,
    ! [Vertices: set_v,Successors: v > set_v,S4: set_v,S6: set_v,X: v] :
      ( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
     => ( ( sCC_Bloemen_is_scc_v @ Successors @ S4 )
       => ( ( sCC_Bloemen_is_scc_v @ Successors @ S6 )
         => ( ( member_v @ X @ ( inf_inf_set_v @ S4 @ S6 ) )
           => ( S4 = S6 ) ) ) ) ) ).

% graph.scc_partition
thf(fact_1113_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_Product_prod_v_v,K: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
      ( ( A3
        = ( sup_su414716646722978715od_v_v @ K @ A ) )
     => ( ( sup_su414716646722978715od_v_v @ A3 @ B2 )
        = ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1114_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_v,K: set_v,A: set_v,B2: set_v] :
      ( ( A3
        = ( sup_sup_set_v @ K @ A ) )
     => ( ( sup_sup_set_v @ A3 @ B2 )
        = ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1115_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_set_v,K: set_set_v,A: set_set_v,B2: set_set_v] :
      ( ( A3
        = ( sup_sup_set_set_v @ K @ A ) )
     => ( ( sup_sup_set_set_v @ A3 @ B2 )
        = ( sup_sup_set_set_v @ K @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1116_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_Product_prod_v_v,K: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
      ( ( B
        = ( sup_su414716646722978715od_v_v @ K @ B2 ) )
     => ( ( sup_su414716646722978715od_v_v @ A @ B )
        = ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1117_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_v,K: set_v,B2: set_v,A: set_v] :
      ( ( B
        = ( sup_sup_set_v @ K @ B2 ) )
     => ( ( sup_sup_set_v @ A @ B )
        = ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1118_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_set_v,K: set_set_v,B2: set_set_v,A: set_set_v] :
      ( ( B
        = ( sup_sup_set_set_v @ K @ B2 ) )
     => ( ( sup_sup_set_set_v @ A @ B )
        = ( sup_sup_set_set_v @ K @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1119_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_v] :
      ( ( sup_sup_set_v @ X @ bot_bot_set_v )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_1120_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Product_prod_v_v] :
      ( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_1121_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_set_v] :
      ( ( sup_sup_set_set_v @ X @ bot_bot_set_set_v )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_1122_finite__Diff__insert,axiom,
    ! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) ) )
      = ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).

% finite_Diff_insert
thf(fact_1123_finite__Diff__insert,axiom,
    ! [A3: set_set_v,A: set_v,B: set_set_v] :
      ( ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ B ) ) )
      = ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B ) ) ) ).

% finite_Diff_insert
thf(fact_1124_finite__Diff__insert,axiom,
    ! [A3: set_v,A: v,B: set_v] :
      ( ( finite_finite_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ B ) ) )
      = ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B ) ) ) ).

% finite_Diff_insert
thf(fact_1125_finite__Diff2,axiom,
    ! [B: set_v,A3: set_v] :
      ( ( finite_finite_v @ B )
     => ( ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B ) )
        = ( finite_finite_v @ A3 ) ) ) ).

% finite_Diff2
thf(fact_1126_finite__Diff,axiom,
    ! [A3: set_v,B: set_v] :
      ( ( finite_finite_v @ A3 )
     => ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B ) ) ) ).

% finite_Diff
thf(fact_1127_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_1128_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_1129_finite__Un,axiom,
    ! [F3: set_set_v,G: set_set_v] :
      ( ( finite_finite_set_v @ ( sup_sup_set_set_v @ F3 @ G ) )
      = ( ( finite_finite_set_v @ F3 )
        & ( finite_finite_set_v @ G ) ) ) ).

% finite_Un
thf(fact_1130_finite__insert,axiom,
    ! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
      = ( finite3348123685078250256od_v_v @ A3 ) ) ).

% finite_insert
thf(fact_1131_finite__insert,axiom,
    ! [A: set_v,A3: set_set_v] :
      ( ( finite_finite_set_v @ ( insert_set_v @ A @ A3 ) )
      = ( finite_finite_set_v @ A3 ) ) ).

% finite_insert
thf(fact_1132_finite__insert,axiom,
    ! [A: v,A3: set_v] :
      ( ( finite_finite_v @ ( insert_v @ A @ A3 ) )
      = ( finite_finite_v @ A3 ) ) ).

% finite_insert
thf(fact_1133_infinite__imp__nonempty,axiom,
    ! [S4: set_v] :
      ( ~ ( finite_finite_v @ S4 )
     => ( S4 != bot_bot_set_v ) ) ).

% infinite_imp_nonempty
thf(fact_1134_infinite__imp__nonempty,axiom,
    ! [S4: set_Product_prod_v_v] :
      ( ~ ( finite3348123685078250256od_v_v @ S4 )
     => ( S4 != bot_bo723834152578015283od_v_v ) ) ).

% infinite_imp_nonempty
thf(fact_1135_infinite__imp__nonempty,axiom,
    ! [S4: set_set_v] :
      ( ~ ( finite_finite_set_v @ S4 )
     => ( S4 != bot_bot_set_set_v ) ) ).

% infinite_imp_nonempty
thf(fact_1136_finite_OemptyI,axiom,
    finite_finite_v @ bot_bot_set_v ).

% finite.emptyI
thf(fact_1137_finite_OemptyI,axiom,
    finite3348123685078250256od_v_v @ bot_bo723834152578015283od_v_v ).

% finite.emptyI
thf(fact_1138_finite_OemptyI,axiom,
    finite_finite_set_v @ bot_bot_set_set_v ).

% finite.emptyI
thf(fact_1139_finite_OinsertI,axiom,
    ! [A3: set_Product_prod_v_v,A: product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ A3 )
     => ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) ) ) ).

% finite.insertI
thf(fact_1140_finite_OinsertI,axiom,
    ! [A3: set_set_v,A: set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( finite_finite_set_v @ ( insert_set_v @ A @ A3 ) ) ) ).

% finite.insertI
thf(fact_1141_finite_OinsertI,axiom,
    ! [A3: set_v,A: v] :
      ( ( finite_finite_v @ A3 )
     => ( finite_finite_v @ ( insert_v @ A @ A3 ) ) ) ).

% finite.insertI
thf(fact_1142_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_1143_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_1144_finite__UnI,axiom,
    ! [F3: set_set_v,G: set_set_v] :
      ( ( finite_finite_set_v @ F3 )
     => ( ( finite_finite_set_v @ G )
       => ( finite_finite_set_v @ ( sup_sup_set_set_v @ F3 @ G ) ) ) ) ).

% finite_UnI
thf(fact_1145_Un__infinite,axiom,
    ! [S4: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ~ ( finite3348123685078250256od_v_v @ S4 )
     => ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S4 @ T2 ) ) ) ).

% Un_infinite
thf(fact_1146_Un__infinite,axiom,
    ! [S4: set_v,T2: set_v] :
      ( ~ ( finite_finite_v @ S4 )
     => ~ ( finite_finite_v @ ( sup_sup_set_v @ S4 @ T2 ) ) ) ).

% Un_infinite
thf(fact_1147_Un__infinite,axiom,
    ! [S4: set_set_v,T2: set_set_v] :
      ( ~ ( finite_finite_set_v @ S4 )
     => ~ ( finite_finite_set_v @ ( sup_sup_set_set_v @ S4 @ T2 ) ) ) ).

% Un_infinite
thf(fact_1148_infinite__Un,axiom,
    ! [S4: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
      ( ( ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S4 @ T2 ) ) )
      = ( ~ ( finite3348123685078250256od_v_v @ S4 )
        | ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).

% infinite_Un
thf(fact_1149_infinite__Un,axiom,
    ! [S4: set_v,T2: set_v] :
      ( ( ~ ( finite_finite_v @ ( sup_sup_set_v @ S4 @ T2 ) ) )
      = ( ~ ( finite_finite_v @ S4 )
        | ~ ( finite_finite_v @ T2 ) ) ) ).

% infinite_Un
thf(fact_1150_infinite__Un,axiom,
    ! [S4: set_set_v,T2: set_set_v] :
      ( ( ~ ( finite_finite_set_v @ ( sup_sup_set_set_v @ S4 @ T2 ) ) )
      = ( ~ ( finite_finite_set_v @ S4 )
        | ~ ( finite_finite_set_v @ T2 ) ) ) ).

% infinite_Un
thf(fact_1151_Diff__infinite__finite,axiom,
    ! [T2: set_v,S4: set_v] :
      ( ( finite_finite_v @ T2 )
     => ( ~ ( finite_finite_v @ S4 )
       => ~ ( finite_finite_v @ ( minus_minus_set_v @ S4 @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1152_finite__has__maximal,axiom,
    ! [A3: set_set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( A3 != bot_bot_set_set_v )
       => ? [X3: set_v] :
            ( ( member_set_v @ X3 @ A3 )
            & ! [Xa2: set_v] :
                ( ( member_set_v @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_v @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1153_finite__has__maximal,axiom,
    ! [A3: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ( A3 != bot_bo3497076220358800403od_v_v )
       => ? [X3: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X3 @ A3 )
            & ! [Xa2: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ Xa2 @ A3 )
               => ( ( ord_le7336532860387713383od_v_v @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1154_finite__has__minimal,axiom,
    ! [A3: set_set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( A3 != bot_bot_set_set_v )
       => ? [X3: set_v] :
            ( ( member_set_v @ X3 @ A3 )
            & ! [Xa2: set_v] :
                ( ( member_set_v @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_v @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1155_finite__has__minimal,axiom,
    ! [A3: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ( A3 != bot_bo3497076220358800403od_v_v )
       => ? [X3: set_Product_prod_v_v] :
            ( ( member8406446414694345712od_v_v @ X3 @ A3 )
            & ! [Xa2: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ Xa2 @ A3 )
               => ( ( ord_le7336532860387713383od_v_v @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1156_infinite__finite__induct,axiom,
    ! [P: set_v > $o,A3: set_v] :
      ( ! [A7: set_v] :
          ( ~ ( finite_finite_v @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_v )
       => ( ! [X3: v,F4: set_v] :
              ( ( finite_finite_v @ F4 )
             => ( ~ ( member_v @ X3 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_v @ X3 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1157_infinite__finite__induct,axiom,
    ! [P: set_Product_prod_v_v > $o,A3: set_Product_prod_v_v] :
      ( ! [A7: set_Product_prod_v_v] :
          ( ~ ( finite3348123685078250256od_v_v @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo723834152578015283od_v_v )
       => ( ! [X3: product_prod_v_v,F4: set_Product_prod_v_v] :
              ( ( finite3348123685078250256od_v_v @ F4 )
             => ( ~ ( member7453568604450474000od_v_v @ X3 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert1338601472111419319od_v_v @ X3 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1158_infinite__finite__induct,axiom,
    ! [P: set_set_v > $o,A3: set_set_v] :
      ( ! [A7: set_set_v] :
          ( ~ ( finite_finite_set_v @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_set_v )
       => ( ! [X3: set_v,F4: set_set_v] :
              ( ( finite_finite_set_v @ F4 )
             => ( ~ ( member_set_v @ X3 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_set_v @ X3 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1159_finite__ne__induct,axiom,
    ! [F3: set_v,P: set_v > $o] :
      ( ( finite_finite_v @ F3 )
     => ( ( F3 != bot_bot_set_v )
       => ( ! [X3: v] : ( P @ ( insert_v @ X3 @ bot_bot_set_v ) )
         => ( ! [X3: v,F4: set_v] :
                ( ( finite_finite_v @ F4 )
               => ( ( F4 != bot_bot_set_v )
                 => ( ~ ( member_v @ X3 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_v @ X3 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1160_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 )
       => ( ! [X3: product_prod_v_v] : ( P @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
         => ( ! [X3: product_prod_v_v,F4: set_Product_prod_v_v] :
                ( ( finite3348123685078250256od_v_v @ F4 )
               => ( ( F4 != bot_bo723834152578015283od_v_v )
                 => ( ~ ( member7453568604450474000od_v_v @ X3 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert1338601472111419319od_v_v @ X3 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1161_finite__ne__induct,axiom,
    ! [F3: set_set_v,P: set_set_v > $o] :
      ( ( finite_finite_set_v @ F3 )
     => ( ( F3 != bot_bot_set_set_v )
       => ( ! [X3: set_v] : ( P @ ( insert_set_v @ X3 @ bot_bot_set_set_v ) )
         => ( ! [X3: set_v,F4: set_set_v] :
                ( ( finite_finite_set_v @ F4 )
               => ( ( F4 != bot_bot_set_set_v )
                 => ( ~ ( member_set_v @ X3 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_set_v @ X3 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1162_finite__induct,axiom,
    ! [F3: set_v,P: set_v > $o] :
      ( ( finite_finite_v @ F3 )
     => ( ( P @ bot_bot_set_v )
       => ( ! [X3: v,F4: set_v] :
              ( ( finite_finite_v @ F4 )
             => ( ~ ( member_v @ X3 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_v @ X3 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_1163_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 )
       => ( ! [X3: product_prod_v_v,F4: set_Product_prod_v_v] :
              ( ( finite3348123685078250256od_v_v @ F4 )
             => ( ~ ( member7453568604450474000od_v_v @ X3 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert1338601472111419319od_v_v @ X3 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_1164_finite__induct,axiom,
    ! [F3: set_set_v,P: set_set_v > $o] :
      ( ( finite_finite_set_v @ F3 )
     => ( ( P @ bot_bot_set_set_v )
       => ( ! [X3: set_v,F4: set_set_v] :
              ( ( finite_finite_set_v @ F4 )
             => ( ~ ( member_set_v @ X3 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_set_v @ X3 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_1165_finite_Osimps,axiom,
    ( finite_finite_v
    = ( ^ [A4: set_v] :
          ( ( A4 = bot_bot_set_v )
          | ? [A5: set_v,B4: v] :
              ( ( A4
                = ( insert_v @ B4 @ A5 ) )
              & ( finite_finite_v @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_1166_finite_Osimps,axiom,
    ( finite3348123685078250256od_v_v
    = ( ^ [A4: set_Product_prod_v_v] :
          ( ( A4 = bot_bo723834152578015283od_v_v )
          | ? [A5: set_Product_prod_v_v,B4: product_prod_v_v] :
              ( ( A4
                = ( insert1338601472111419319od_v_v @ B4 @ A5 ) )
              & ( finite3348123685078250256od_v_v @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_1167_finite_Osimps,axiom,
    ( finite_finite_set_v
    = ( ^ [A4: set_set_v] :
          ( ( A4 = bot_bot_set_set_v )
          | ? [A5: set_set_v,B4: set_v] :
              ( ( A4
                = ( insert_set_v @ B4 @ A5 ) )
              & ( finite_finite_set_v @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_1168_finite_Ocases,axiom,
    ! [A: set_v] :
      ( ( finite_finite_v @ A )
     => ( ( A != bot_bot_set_v )
       => ~ ! [A7: set_v] :
              ( ? [A8: v] :
                  ( A
                  = ( insert_v @ A8 @ A7 ) )
             => ~ ( finite_finite_v @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1169_finite_Ocases,axiom,
    ! [A: set_Product_prod_v_v] :
      ( ( finite3348123685078250256od_v_v @ A )
     => ( ( A != bot_bo723834152578015283od_v_v )
       => ~ ! [A7: set_Product_prod_v_v] :
              ( ? [A8: product_prod_v_v] :
                  ( A
                  = ( insert1338601472111419319od_v_v @ A8 @ A7 ) )
             => ~ ( finite3348123685078250256od_v_v @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1170_finite_Ocases,axiom,
    ! [A: set_set_v] :
      ( ( finite_finite_set_v @ A )
     => ( ( A != bot_bot_set_set_v )
       => ~ ! [A7: set_set_v] :
              ( ? [A8: set_v] :
                  ( A
                  = ( insert_set_v @ A8 @ A7 ) )
             => ~ ( finite_finite_set_v @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1171_finite__subset__induct,axiom,
    ! [F3: set_set_v,A3: set_set_v,P: set_set_v > $o] :
      ( ( finite_finite_set_v @ F3 )
     => ( ( ord_le5216385588623774835_set_v @ F3 @ A3 )
       => ( ( P @ bot_bot_set_set_v )
         => ( ! [A8: set_v,F4: set_set_v] :
                ( ( finite_finite_set_v @ F4 )
               => ( ( member_set_v @ A8 @ A3 )
                 => ( ~ ( member_set_v @ A8 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_set_v @ A8 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1172_finite__subset__induct,axiom,
    ! [F3: set_v,A3: set_v,P: set_v > $o] :
      ( ( finite_finite_v @ F3 )
     => ( ( ord_less_eq_set_v @ F3 @ A3 )
       => ( ( P @ bot_bot_set_v )
         => ( ! [A8: v,F4: set_v] :
                ( ( finite_finite_v @ F4 )
               => ( ( member_v @ A8 @ A3 )
                 => ( ~ ( member_v @ A8 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_v @ A8 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1173_finite__subset__induct,axiom,
    ! [F3: set_Product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
      ( ( finite3348123685078250256od_v_v @ F3 )
     => ( ( ord_le7336532860387713383od_v_v @ F3 @ A3 )
       => ( ( P @ bot_bo723834152578015283od_v_v )
         => ( ! [A8: product_prod_v_v,F4: set_Product_prod_v_v] :
                ( ( finite3348123685078250256od_v_v @ F4 )
               => ( ( member7453568604450474000od_v_v @ A8 @ A3 )
                 => ( ~ ( member7453568604450474000od_v_v @ A8 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert1338601472111419319od_v_v @ A8 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1174_finite__subset__induct_H,axiom,
    ! [F3: set_set_v,A3: set_set_v,P: set_set_v > $o] :
      ( ( finite_finite_set_v @ F3 )
     => ( ( ord_le5216385588623774835_set_v @ F3 @ A3 )
       => ( ( P @ bot_bot_set_set_v )
         => ( ! [A8: set_v,F4: set_set_v] :
                ( ( finite_finite_set_v @ F4 )
               => ( ( member_set_v @ A8 @ A3 )
                 => ( ( ord_le5216385588623774835_set_v @ F4 @ A3 )
                   => ( ~ ( member_set_v @ A8 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_set_v @ A8 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1175_finite__subset__induct_H,axiom,
    ! [F3: set_v,A3: set_v,P: set_v > $o] :
      ( ( finite_finite_v @ F3 )
     => ( ( ord_less_eq_set_v @ F3 @ A3 )
       => ( ( P @ bot_bot_set_v )
         => ( ! [A8: v,F4: set_v] :
                ( ( finite_finite_v @ F4 )
               => ( ( member_v @ A8 @ A3 )
                 => ( ( ord_less_eq_set_v @ F4 @ A3 )
                   => ( ~ ( member_v @ A8 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_v @ A8 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1176_finite__subset__induct_H,axiom,
    ! [F3: set_Product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
      ( ( finite3348123685078250256od_v_v @ F3 )
     => ( ( ord_le7336532860387713383od_v_v @ F3 @ A3 )
       => ( ( P @ bot_bo723834152578015283od_v_v )
         => ( ! [A8: product_prod_v_v,F4: set_Product_prod_v_v] :
                ( ( finite3348123685078250256od_v_v @ F4 )
               => ( ( member7453568604450474000od_v_v @ A8 @ A3 )
                 => ( ( ord_le7336532860387713383od_v_v @ F4 @ A3 )
                   => ( ~ ( member7453568604450474000od_v_v @ A8 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert1338601472111419319od_v_v @ A8 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1177_infinite__remove,axiom,
    ! [S4: set_Product_prod_v_v,A: product_prod_v_v] :
      ( ~ ( finite3348123685078250256od_v_v @ S4 )
     => ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ S4 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% infinite_remove
thf(fact_1178_infinite__remove,axiom,
    ! [S4: set_set_v,A: set_v] :
      ( ~ ( finite_finite_set_v @ S4 )
     => ~ ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ S4 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ) ).

% infinite_remove
thf(fact_1179_infinite__remove,axiom,
    ! [S4: set_v,A: v] :
      ( ~ ( finite_finite_v @ S4 )
     => ~ ( finite_finite_v @ ( minus_minus_set_v @ S4 @ ( insert_v @ A @ bot_bot_set_v ) ) ) ) ).

% infinite_remove
thf(fact_1180_infinite__coinduct,axiom,
    ! [X5: set_Product_prod_v_v > $o,A3: set_Product_prod_v_v] :
      ( ( X5 @ A3 )
     => ( ! [A7: set_Product_prod_v_v] :
            ( ( X5 @ A7 )
           => ? [X4: product_prod_v_v] :
                ( ( member7453568604450474000od_v_v @ X4 @ A7 )
                & ( ( X5 @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) )
                  | ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
       => ~ ( finite3348123685078250256od_v_v @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_1181_infinite__coinduct,axiom,
    ! [X5: set_set_v > $o,A3: set_set_v] :
      ( ( X5 @ A3 )
     => ( ! [A7: set_set_v] :
            ( ( X5 @ A7 )
           => ? [X4: set_v] :
                ( ( member_set_v @ X4 @ A7 )
                & ( ( X5 @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ X4 @ bot_bot_set_set_v ) ) )
                  | ~ ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ X4 @ bot_bot_set_set_v ) ) ) ) ) )
       => ~ ( finite_finite_set_v @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_1182_infinite__coinduct,axiom,
    ! [X5: set_v > $o,A3: set_v] :
      ( ( X5 @ A3 )
     => ( ! [A7: set_v] :
            ( ( X5 @ A7 )
           => ? [X4: v] :
                ( ( member_v @ X4 @ A7 )
                & ( ( X5 @ ( minus_minus_set_v @ A7 @ ( insert_v @ X4 @ bot_bot_set_v ) ) )
                  | ~ ( finite_finite_v @ ( minus_minus_set_v @ A7 @ ( insert_v @ X4 @ bot_bot_set_v ) ) ) ) ) )
       => ~ ( finite_finite_v @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_1183_finite__empty__induct,axiom,
    ! [A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
      ( ( finite3348123685078250256od_v_v @ A3 )
     => ( ( P @ A3 )
       => ( ! [A8: product_prod_v_v,A7: set_Product_prod_v_v] :
              ( ( finite3348123685078250256od_v_v @ A7 )
             => ( ( member7453568604450474000od_v_v @ A8 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ A8 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
         => ( P @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% finite_empty_induct
thf(fact_1184_finite__empty__induct,axiom,
    ! [A3: set_set_v,P: set_set_v > $o] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( P @ A3 )
       => ( ! [A8: set_v,A7: set_set_v] :
              ( ( finite_finite_set_v @ A7 )
             => ( ( member_set_v @ A8 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ A8 @ bot_bot_set_set_v ) ) ) ) ) )
         => ( P @ bot_bot_set_set_v ) ) ) ) ).

% finite_empty_induct
thf(fact_1185_finite__empty__induct,axiom,
    ! [A3: set_v,P: set_v > $o] :
      ( ( finite_finite_v @ A3 )
     => ( ( P @ A3 )
       => ( ! [A8: v,A7: set_v] :
              ( ( finite_finite_v @ A7 )
             => ( ( member_v @ A8 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_v @ A7 @ ( insert_v @ A8 @ bot_bot_set_v ) ) ) ) ) )
         => ( P @ bot_bot_set_v ) ) ) ) ).

% finite_empty_induct
thf(fact_1186_remove__induct,axiom,
    ! [P: set_set_v > $o,B: set_set_v] :
      ( ( P @ bot_bot_set_set_v )
     => ( ( ~ ( finite_finite_set_v @ B )
         => ( P @ B ) )
       => ( ! [A7: set_set_v] :
              ( ( finite_finite_set_v @ A7 )
             => ( ( A7 != bot_bot_set_set_v )
               => ( ( ord_le5216385588623774835_set_v @ A7 @ B )
                 => ( ! [X4: set_v] :
                        ( ( member_set_v @ X4 @ A7 )
                       => ( P @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ X4 @ bot_bot_set_set_v ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_1187_remove__induct,axiom,
    ! [P: set_v > $o,B: set_v] :
      ( ( P @ bot_bot_set_v )
     => ( ( ~ ( finite_finite_v @ B )
         => ( P @ B ) )
       => ( ! [A7: set_v] :
              ( ( finite_finite_v @ A7 )
             => ( ( A7 != bot_bot_set_v )
               => ( ( ord_less_eq_set_v @ A7 @ B )
                 => ( ! [X4: v] :
                        ( ( member_v @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_v @ A7 @ ( insert_v @ X4 @ bot_bot_set_v ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_1188_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 ) )
       => ( ! [A7: set_Product_prod_v_v] :
              ( ( finite3348123685078250256od_v_v @ A7 )
             => ( ( A7 != bot_bo723834152578015283od_v_v )
               => ( ( ord_le7336532860387713383od_v_v @ A7 @ B )
                 => ( ! [X4: product_prod_v_v] :
                        ( ( member7453568604450474000od_v_v @ X4 @ A7 )
                       => ( P @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_1189_finite__remove__induct,axiom,
    ! [B: set_set_v,P: set_set_v > $o] :
      ( ( finite_finite_set_v @ B )
     => ( ( P @ bot_bot_set_set_v )
       => ( ! [A7: set_set_v] :
              ( ( finite_finite_set_v @ A7 )
             => ( ( A7 != bot_bot_set_set_v )
               => ( ( ord_le5216385588623774835_set_v @ A7 @ B )
                 => ( ! [X4: set_v] :
                        ( ( member_set_v @ X4 @ A7 )
                       => ( P @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ X4 @ bot_bot_set_set_v ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_1190_finite__remove__induct,axiom,
    ! [B: set_v,P: set_v > $o] :
      ( ( finite_finite_v @ B )
     => ( ( P @ bot_bot_set_v )
       => ( ! [A7: set_v] :
              ( ( finite_finite_v @ A7 )
             => ( ( A7 != bot_bot_set_v )
               => ( ( ord_less_eq_set_v @ A7 @ B )
                 => ( ! [X4: v] :
                        ( ( member_v @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_v @ A7 @ ( insert_v @ X4 @ bot_bot_set_v ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_1191_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 )
       => ( ! [A7: set_Product_prod_v_v] :
              ( ( finite3348123685078250256od_v_v @ A7 )
             => ( ( A7 != bot_bo723834152578015283od_v_v )
               => ( ( ord_le7336532860387713383od_v_v @ A7 @ B )
                 => ( ! [X4: product_prod_v_v] :
                        ( ( member7453568604450474000od_v_v @ X4 @ A7 )
                       => ( P @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_1192_less__by__empty,axiom,
    ! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
      ( ( A3 = bot_bo723834152578015283od_v_v )
     => ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).

% less_by_empty
thf(fact_1193_rotate1__hd__tl,axiom,
    ! [Xs: list_v] :
      ( ( Xs != nil_v )
     => ( ( rotate1_v @ Xs )
        = ( append_v @ ( tl_v @ Xs ) @ ( cons_v @ ( hd_v @ Xs ) @ nil_v ) ) ) ) ).

% rotate1_hd_tl
thf(fact_1194_set__rotate1,axiom,
    ! [Xs: list_v] :
      ( ( set_v2 @ ( rotate1_v @ Xs ) )
      = ( set_v2 @ Xs ) ) ).

% set_rotate1
thf(fact_1195_rotate1_Osimps_I2_J,axiom,
    ! [X: v,Xs: list_v] :
      ( ( rotate1_v @ ( cons_v @ X @ Xs ) )
      = ( append_v @ Xs @ ( cons_v @ X @ nil_v ) ) ) ).

% rotate1.simps(2)
thf(fact_1196_set__removeAll,axiom,
    ! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
      ( ( set_Product_prod_v_v2 @ ( remove481895986417801203od_v_v @ X @ Xs ) )
      = ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ).

% set_removeAll
thf(fact_1197_set__removeAll,axiom,
    ! [X: set_v,Xs: list_set_v] :
      ( ( set_set_v2 @ ( removeAll_set_v @ X @ Xs ) )
      = ( minus_7228012346218142266_set_v @ ( set_set_v2 @ Xs ) @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ).

% set_removeAll
thf(fact_1198_set__removeAll,axiom,
    ! [X: v,Xs: list_v] :
      ( ( set_v2 @ ( removeAll_v @ X @ Xs ) )
      = ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v @ X @ bot_bot_set_v ) ) ) ).

% set_removeAll
thf(fact_1199_insert__subsetI,axiom,
    ! [X: set_v,A3: set_set_v,X5: set_set_v] :
      ( ( member_set_v @ X @ A3 )
     => ( ( ord_le5216385588623774835_set_v @ X5 @ A3 )
       => ( ord_le5216385588623774835_set_v @ ( insert_set_v @ X @ X5 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_1200_insert__subsetI,axiom,
    ! [X: v,A3: set_v,X5: set_v] :
      ( ( member_v @ X @ A3 )
     => ( ( ord_less_eq_set_v @ X5 @ A3 )
       => ( ord_less_eq_set_v @ ( insert_v @ X @ X5 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_1201_insert__subsetI,axiom,
    ! [X: product_prod_v_v,A3: set_Product_prod_v_v,X5: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ X @ A3 )
     => ( ( ord_le7336532860387713383od_v_v @ X5 @ A3 )
       => ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ X5 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_1202_removeAll__id,axiom,
    ! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
      ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
     => ( ( remove481895986417801203od_v_v @ X @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_1203_removeAll__id,axiom,
    ! [X: v,Xs: list_v] :
      ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
     => ( ( removeAll_v @ X @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_1204_removeAll_Osimps_I2_J,axiom,
    ! [X: v,Y2: v,Xs: list_v] :
      ( ( ( X = Y2 )
       => ( ( removeAll_v @ X @ ( cons_v @ Y2 @ Xs ) )
          = ( removeAll_v @ X @ Xs ) ) )
      & ( ( X != Y2 )
       => ( ( removeAll_v @ X @ ( cons_v @ Y2 @ Xs ) )
          = ( cons_v @ Y2 @ ( removeAll_v @ X @ Xs ) ) ) ) ) ).

% removeAll.simps(2)
thf(fact_1205_subset__emptyI,axiom,
    ! [A3: set_set_v] :
      ( ! [X3: set_v] :
          ~ ( member_set_v @ X3 @ A3 )
     => ( ord_le5216385588623774835_set_v @ A3 @ bot_bot_set_set_v ) ) ).

% subset_emptyI
thf(fact_1206_subset__emptyI,axiom,
    ! [A3: set_v] :
      ( ! [X3: v] :
          ~ ( member_v @ X3 @ A3 )
     => ( ord_less_eq_set_v @ A3 @ bot_bot_set_v ) ) ).

% subset_emptyI
thf(fact_1207_subset__emptyI,axiom,
    ! [A3: set_Product_prod_v_v] :
      ( ! [X3: product_prod_v_v] :
          ~ ( member7453568604450474000od_v_v @ X3 @ A3 )
     => ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) ).

% subset_emptyI
thf(fact_1208_Field__insert,axiom,
    ! [A: v,B2: v,R: set_Product_prod_v_v] :
      ( ( field_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R ) )
      = ( sup_sup_set_v @ ( insert_v @ A @ ( insert_v @ B2 @ bot_bot_set_v ) ) @ ( field_v @ R ) ) ) ).

% Field_insert
thf(fact_1209_Field__insert,axiom,
    ! [A: product_prod_v_v,B2: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
      ( ( field_7153129647634986036od_v_v @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R ) )
      = ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R ) ) ) ).

% Field_insert
thf(fact_1210_Field__insert,axiom,
    ! [A: set_v,B2: set_v,R: set_Pr8199228935972127175_set_v] :
      ( ( field_set_v @ ( insert1457770702614273975_set_v @ ( produc3441907479644599895_set_v @ A @ B2 ) @ R ) )
      = ( sup_sup_set_set_v @ ( insert_set_v @ A @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) ) @ ( field_set_v @ R ) ) ) ).

% Field_insert
thf(fact_1211_dom,axiom,
    accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ va @ ea ) ) ).

% dom
thf(fact_1212_dfss_Ocases,axiom,
    ! [X: produc5741669702376414499t_unit] :
      ~ ! [V2: v,E7: sCC_Bl1394983891496994913t_unit] :
          ( X
         != ( produc3862955338007567901t_unit @ V2 @ E7 ) ) ).

% dfss.cases
thf(fact_1213_Field__empty,axiom,
    ( ( field_7153129647634986036od_v_v @ bot_bo3282589961317712691od_v_v )
    = bot_bo723834152578015283od_v_v ) ).

% Field_empty
thf(fact_1214_Field__empty,axiom,
    ( ( field_set_v @ bot_bo8153096493302634547_set_v )
    = bot_bot_set_set_v ) ).

% Field_empty
thf(fact_1215_Field__empty,axiom,
    ( ( field_v @ bot_bo723834152578015283od_v_v )
    = bot_bot_set_v ) ).

% Field_empty
thf(fact_1216_Field__Un,axiom,
    ! [R: set_Pr2149350503807050951od_v_v,S7: set_Pr2149350503807050951od_v_v] :
      ( ( field_7153129647634986036od_v_v @ ( sup_su1742609618068805275od_v_v @ R @ S7 ) )
      = ( sup_su414716646722978715od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ ( field_7153129647634986036od_v_v @ S7 ) ) ) ).

% Field_Un
thf(fact_1217_Field__Un,axiom,
    ! [R: set_Pr8199228935972127175_set_v,S7: set_Pr8199228935972127175_set_v] :
      ( ( field_set_v @ ( sup_su7977902838240902043_set_v @ R @ S7 ) )
      = ( sup_sup_set_set_v @ ( field_set_v @ R ) @ ( field_set_v @ S7 ) ) ) ).

% Field_Un
thf(fact_1218_Field__Un,axiom,
    ! [R: set_Product_prod_v_v,S7: set_Product_prod_v_v] :
      ( ( field_v @ ( sup_su414716646722978715od_v_v @ R @ S7 ) )
      = ( sup_sup_set_v @ ( field_v @ R ) @ ( field_v @ S7 ) ) ) ).

% Field_Un
thf(fact_1219_remove__def,axiom,
    ( remove5001965847480235980od_v_v
    = ( ^ [X2: product_prod_v_v,A5: set_Product_prod_v_v] : ( minus_4183494784930505774od_v_v @ A5 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).

% remove_def
thf(fact_1220_remove__def,axiom,
    ( remove_set_v
    = ( ^ [X2: set_v,A5: set_set_v] : ( minus_7228012346218142266_set_v @ A5 @ ( insert_set_v @ X2 @ bot_bot_set_set_v ) ) ) ) ).

% remove_def
thf(fact_1221_remove__def,axiom,
    ( remove_v
    = ( ^ [X2: v,A5: set_v] : ( minus_minus_set_v @ A5 @ ( insert_v @ X2 @ bot_bot_set_v ) ) ) ) ).

% remove_def
thf(fact_1222_Sup__fin_Oinsert__remove,axiom,
    ! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
            = bot_bo3497076220358800403od_v_v )
         => ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
           != bot_bo3497076220358800403od_v_v )
         => ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
            = ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_1223_Sup__fin_Oinsert__remove,axiom,
    ! [A3: set_set_set_v,X: set_set_v] :
      ( ( finite8701002811114149628_set_v @ A3 )
     => ( ( ( ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X @ bot_bo5775917114081396255_set_v ) )
            = bot_bo5775917114081396255_set_v )
         => ( ( lattic1829858174534819978_set_v @ ( insert_set_set_v @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X @ bot_bo5775917114081396255_set_v ) )
           != bot_bo5775917114081396255_set_v )
         => ( ( lattic1829858174534819978_set_v @ ( insert_set_set_v @ X @ A3 ) )
            = ( sup_sup_set_set_v @ X @ ( lattic1829858174534819978_set_v @ ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X @ bot_bo5775917114081396255_set_v ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_1224_Sup__fin_Oinsert__remove,axiom,
    ! [A3: set_set_v,X: set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
            = bot_bot_set_set_v )
         => ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
           != bot_bot_set_set_v )
         => ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A3 ) )
            = ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_1225_Sup__fin_Oremove,axiom,
    ! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ( member8406446414694345712od_v_v @ X @ A3 )
       => ( ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
              = bot_bo3497076220358800403od_v_v )
           => ( ( lattic5151207300795964030od_v_v @ A3 )
              = X ) )
          & ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
             != bot_bo3497076220358800403od_v_v )
           => ( ( lattic5151207300795964030od_v_v @ A3 )
              = ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_1226_Sup__fin_Oremove,axiom,
    ! [A3: set_set_set_v,X: set_set_v] :
      ( ( finite8701002811114149628_set_v @ A3 )
     => ( ( member_set_set_v @ X @ A3 )
       => ( ( ( ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X @ bot_bo5775917114081396255_set_v ) )
              = bot_bo5775917114081396255_set_v )
           => ( ( lattic1829858174534819978_set_v @ A3 )
              = X ) )
          & ( ( ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X @ bot_bo5775917114081396255_set_v ) )
             != bot_bo5775917114081396255_set_v )
           => ( ( lattic1829858174534819978_set_v @ A3 )
              = ( sup_sup_set_set_v @ X @ ( lattic1829858174534819978_set_v @ ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X @ bot_bo5775917114081396255_set_v ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_1227_Sup__fin_Oremove,axiom,
    ! [A3: set_set_v,X: set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( member_set_v @ X @ A3 )
       => ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
              = bot_bot_set_set_v )
           => ( ( lattic2918178447194608042_set_v @ A3 )
              = X ) )
          & ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
             != bot_bot_set_set_v )
           => ( ( lattic2918178447194608042_set_v @ A3 )
              = ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_1228_member__remove,axiom,
    ! [X: v,Y2: v,A3: set_v] :
      ( ( member_v @ X @ ( remove_v @ Y2 @ A3 ) )
      = ( ( member_v @ X @ A3 )
        & ( X != Y2 ) ) ) ).

% member_remove
thf(fact_1229_member__remove,axiom,
    ! [X: product_prod_v_v,Y2: product_prod_v_v,A3: set_Product_prod_v_v] :
      ( ( member7453568604450474000od_v_v @ X @ ( remove5001965847480235980od_v_v @ Y2 @ A3 ) )
      = ( ( member7453568604450474000od_v_v @ X @ A3 )
        & ( X != Y2 ) ) ) ).

% member_remove
thf(fact_1230_Sup__fin_Osingleton,axiom,
    ! [X: set_v] :
      ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
      = X ) ).

% Sup_fin.singleton
thf(fact_1231_Sup__fin_Oinsert,axiom,
    ! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ( A3 != bot_bo3497076220358800403od_v_v )
       => ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
          = ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_1232_Sup__fin_Oinsert,axiom,
    ! [A3: set_set_set_v,X: set_set_v] :
      ( ( finite8701002811114149628_set_v @ A3 )
     => ( ( A3 != bot_bo5775917114081396255_set_v )
       => ( ( lattic1829858174534819978_set_v @ ( insert_set_set_v @ X @ A3 ) )
          = ( sup_sup_set_set_v @ X @ ( lattic1829858174534819978_set_v @ A3 ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_1233_Sup__fin_Oinsert,axiom,
    ! [A3: set_set_v,X: set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( A3 != bot_bot_set_set_v )
       => ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A3 ) )
          = ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_1234_Sup__fin_Oin__idem,axiom,
    ! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ( member8406446414694345712od_v_v @ X @ A3 )
       => ( ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A3 ) )
          = ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_1235_Sup__fin_Oin__idem,axiom,
    ! [A3: set_set_v,X: set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( member_set_v @ X @ A3 )
       => ( ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ A3 ) )
          = ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_1236_Sup__fin_Oin__idem,axiom,
    ! [A3: set_set_set_v,X: set_set_v] :
      ( ( finite8701002811114149628_set_v @ A3 )
     => ( ( member_set_set_v @ X @ A3 )
       => ( ( sup_sup_set_set_v @ X @ ( lattic1829858174534819978_set_v @ A3 ) )
          = ( lattic1829858174534819978_set_v @ A3 ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_1237_Sup__fin_Obounded__iff,axiom,
    ! [A3: set_set_v,X: set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( A3 != bot_bot_set_set_v )
       => ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X )
          = ( ! [X2: set_v] :
                ( ( member_set_v @ X2 @ A3 )
               => ( ord_less_eq_set_v @ X2 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1238_Sup__fin_Obounded__iff,axiom,
    ! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ( A3 != bot_bo3497076220358800403od_v_v )
       => ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X )
          = ( ! [X2: set_Product_prod_v_v] :
                ( ( member8406446414694345712od_v_v @ X2 @ A3 )
               => ( ord_le7336532860387713383od_v_v @ X2 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1239_Sup__fin_OboundedI,axiom,
    ! [A3: set_set_v,X: set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( A3 != bot_bot_set_set_v )
       => ( ! [A8: set_v] :
              ( ( member_set_v @ A8 @ A3 )
             => ( ord_less_eq_set_v @ A8 @ X ) )
         => ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1240_Sup__fin_OboundedI,axiom,
    ! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ( A3 != bot_bo3497076220358800403od_v_v )
       => ( ! [A8: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ A8 @ A3 )
             => ( ord_le7336532860387713383od_v_v @ A8 @ X ) )
         => ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1241_Sup__fin_OboundedE,axiom,
    ! [A3: set_set_v,X: set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( A3 != bot_bot_set_set_v )
       => ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X )
         => ! [A9: set_v] :
              ( ( member_set_v @ A9 @ A3 )
             => ( ord_less_eq_set_v @ A9 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1242_Sup__fin_OboundedE,axiom,
    ! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ( A3 != bot_bo3497076220358800403od_v_v )
       => ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X )
         => ! [A9: set_Product_prod_v_v] :
              ( ( member8406446414694345712od_v_v @ A9 @ A3 )
             => ( ord_le7336532860387713383od_v_v @ A9 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1243_Sup__fin_Osubset__imp,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( ord_le5216385588623774835_set_v @ A3 @ B )
     => ( ( A3 != bot_bot_set_set_v )
       => ( ( finite_finite_set_v @ B )
         => ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ ( lattic2918178447194608042_set_v @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_1244_Sup__fin_Osubset__imp,axiom,
    ! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( ord_le4714265922333009223od_v_v @ A3 @ B )
     => ( ( A3 != bot_bo3497076220358800403od_v_v )
       => ( ( finite6084192165098772208od_v_v @ B )
         => ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_1245_Sup__fin_Osubset,axiom,
    ! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ( B != bot_bo3497076220358800403od_v_v )
       => ( ( ord_le4714265922333009223od_v_v @ B @ A3 )
         => ( ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ B ) @ ( lattic5151207300795964030od_v_v @ A3 ) )
            = ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_1246_Sup__fin_Osubset,axiom,
    ! [A3: set_set_set_v,B: set_set_set_v] :
      ( ( finite8701002811114149628_set_v @ A3 )
     => ( ( B != bot_bo5775917114081396255_set_v )
       => ( ( ord_le8117609702905084755_set_v @ B @ A3 )
         => ( ( sup_sup_set_set_v @ ( lattic1829858174534819978_set_v @ B ) @ ( lattic1829858174534819978_set_v @ A3 ) )
            = ( lattic1829858174534819978_set_v @ A3 ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_1247_Sup__fin_Osubset,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( B != bot_bot_set_set_v )
       => ( ( ord_le5216385588623774835_set_v @ B @ A3 )
         => ( ( sup_sup_set_v @ ( lattic2918178447194608042_set_v @ B ) @ ( lattic2918178447194608042_set_v @ A3 ) )
            = ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_1248_Sup__fin_Oclosed,axiom,
    ! [A3: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ( A3 != bot_bo3497076220358800403od_v_v )
       => ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y ) @ ( insert7504383016908236695od_v_v @ X3 @ ( insert7504383016908236695od_v_v @ Y @ bot_bo3497076220358800403od_v_v ) ) )
         => ( member8406446414694345712od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ A3 ) ) ) ) ).

% Sup_fin.closed
thf(fact_1249_Sup__fin_Oclosed,axiom,
    ! [A3: set_set_set_v] :
      ( ( finite8701002811114149628_set_v @ A3 )
     => ( ( A3 != bot_bo5775917114081396255_set_v )
       => ( ! [X3: set_set_v,Y: set_set_v] : ( member_set_set_v @ ( sup_sup_set_set_v @ X3 @ Y ) @ ( insert_set_set_v @ X3 @ ( insert_set_set_v @ Y @ bot_bo5775917114081396255_set_v ) ) )
         => ( member_set_set_v @ ( lattic1829858174534819978_set_v @ A3 ) @ A3 ) ) ) ) ).

% Sup_fin.closed
thf(fact_1250_Sup__fin_Oclosed,axiom,
    ! [A3: set_set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( A3 != bot_bot_set_set_v )
       => ( ! [X3: set_v,Y: set_v] : ( member_set_v @ ( sup_sup_set_v @ X3 @ Y ) @ ( insert_set_v @ X3 @ ( insert_set_v @ Y @ bot_bot_set_set_v ) ) )
         => ( member_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ A3 ) ) ) ) ).

% Sup_fin.closed
thf(fact_1251_Sup__fin_Oinsert__not__elem,axiom,
    ! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ~ ( member8406446414694345712od_v_v @ X @ A3 )
       => ( ( A3 != bot_bo3497076220358800403od_v_v )
         => ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
            = ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_1252_Sup__fin_Oinsert__not__elem,axiom,
    ! [A3: set_set_set_v,X: set_set_v] :
      ( ( finite8701002811114149628_set_v @ A3 )
     => ( ~ ( member_set_set_v @ X @ A3 )
       => ( ( A3 != bot_bo5775917114081396255_set_v )
         => ( ( lattic1829858174534819978_set_v @ ( insert_set_set_v @ X @ A3 ) )
            = ( sup_sup_set_set_v @ X @ ( lattic1829858174534819978_set_v @ A3 ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_1253_Sup__fin_Oinsert__not__elem,axiom,
    ! [A3: set_set_v,X: set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ~ ( member_set_v @ X @ A3 )
       => ( ( A3 != bot_bot_set_set_v )
         => ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A3 ) )
            = ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_1254_Sup__fin_Ounion,axiom,
    ! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
      ( ( finite6084192165098772208od_v_v @ A3 )
     => ( ( A3 != bot_bo3497076220358800403od_v_v )
       => ( ( finite6084192165098772208od_v_v @ B )
         => ( ( B != bot_bo3497076220358800403od_v_v )
           => ( ( lattic5151207300795964030od_v_v @ ( sup_su335656005089752955od_v_v @ A3 @ B ) )
              = ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ B ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_1255_Sup__fin_Ounion,axiom,
    ! [A3: set_set_set_v,B: set_set_set_v] :
      ( ( finite8701002811114149628_set_v @ A3 )
     => ( ( A3 != bot_bo5775917114081396255_set_v )
       => ( ( finite8701002811114149628_set_v @ B )
         => ( ( B != bot_bo5775917114081396255_set_v )
           => ( ( lattic1829858174534819978_set_v @ ( sup_su4471370308589719943_set_v @ A3 @ B ) )
              = ( sup_sup_set_set_v @ ( lattic1829858174534819978_set_v @ A3 ) @ ( lattic1829858174534819978_set_v @ B ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_1256_Sup__fin_Ounion,axiom,
    ! [A3: set_set_v,B: set_set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( A3 != bot_bot_set_set_v )
       => ( ( finite_finite_set_v @ B )
         => ( ( B != bot_bot_set_set_v )
           => ( ( lattic2918178447194608042_set_v @ ( sup_sup_set_set_v @ A3 @ B ) )
              = ( sup_sup_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ ( lattic2918178447194608042_set_v @ B ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_1257_remove__code_I1_J,axiom,
    ! [X: v,Xs: list_v] :
      ( ( remove_v @ X @ ( set_v2 @ Xs ) )
      = ( set_v2 @ ( removeAll_v @ X @ Xs ) ) ) ).

% remove_code(1)
thf(fact_1258_Inf__fin_Oremove,axiom,
    ! [A3: set_set_v,X: set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( member_set_v @ X @ A3 )
       => ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
              = bot_bot_set_set_v )
           => ( ( lattic8209813555532694032_set_v @ A3 )
              = X ) )
          & ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
             != bot_bot_set_set_v )
           => ( ( lattic8209813555532694032_set_v @ A3 )
              = ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_1259_Inf__fin_Oinsert__remove,axiom,
    ! [A3: set_set_v,X: set_v] :
      ( ( finite_finite_set_v @ A3 )
     => ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
            = bot_bot_set_set_v )
         => ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
           != bot_bot_set_set_v )
         => ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
            = ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_1260_prepostdfs,axiom,
    ! [Vs: set_v,W: v] :
      ( ( Vs
        = ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) )
     => ( ( Vs != bot_bot_set_v )
       => ( ( W
            = ( fChoice_v
              @ ^ [X2: v] : ( member_v @ X2 @ Vs ) ) )
         => ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ ea ) )
           => ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
             => ( ( sCC_Bl36166008131615352t_unit @ successors @ W @ ea )
               => ( sCC_Bl8953792750115413617t_unit @ successors @ W @ ea @ ( sCC_Bloemen_dfs_v @ successors @ W @ ea ) ) ) ) ) ) ) ) ).

% prepostdfs
thf(fact_1261_singleton__conv,axiom,
    ! [A: v] :
      ( ( collect_v
        @ ^ [X2: v] : ( X2 = A ) )
      = ( insert_v @ A @ bot_bot_set_v ) ) ).

% singleton_conv
thf(fact_1262_singleton__conv,axiom,
    ! [A: product_prod_v_v] :
      ( ( collec140062887454715474od_v_v
        @ ^ [X2: product_prod_v_v] : ( X2 = A ) )
      = ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).

% singleton_conv
thf(fact_1263_singleton__conv,axiom,
    ! [A: set_v] :
      ( ( collect_set_v
        @ ^ [X2: set_v] : ( X2 = A ) )
      = ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).

% singleton_conv
thf(fact_1264_singleton__conv2,axiom,
    ! [A: v] :
      ( ( collect_v
        @ ( ^ [Y4: v,Z4: v] : ( Y4 = Z4 )
          @ A ) )
      = ( insert_v @ A @ bot_bot_set_v ) ) ).

% singleton_conv2
thf(fact_1265_singleton__conv2,axiom,
    ! [A: product_prod_v_v] :
      ( ( collec140062887454715474od_v_v
        @ ( ^ [Y4: product_prod_v_v,Z4: product_prod_v_v] : ( Y4 = Z4 )
          @ A ) )
      = ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).

% singleton_conv2
thf(fact_1266_singleton__conv2,axiom,
    ! [A: set_v] :
      ( ( collect_set_v
        @ ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 )
          @ A ) )
      = ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).

% singleton_conv2
thf(fact_1267_w__def,axiom,
    ( w
    = ( fChoice_v
      @ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) ) ) ) ).

% w_def
thf(fact_1268_prepostdfss,axiom,
    ! [Vs: set_v,W: v,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
      ( ( Vs
        = ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) )
     => ( ( Vs != bot_bot_set_v )
       => ( ( W
            = ( fChoice_v
              @ ^ [X2: v] : ( member_v @ X2 @ Vs ) ) )
         => ( ( ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ ea ) )
               => ( E2 = ea ) )
              & ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ ea ) )
               => ( ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
                   => ( E2
                      = ( sCC_Bloemen_dfs_v @ successors @ W @ ea ) ) )
                  & ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
                   => ( E2
                      = ( sCC_Bloemen_unite_v @ va @ W @ ea ) ) ) ) ) )
           => ( ( E3
                = ( sCC_Bl48393358579903213t_unit
                  @ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = va ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ E2 @ va ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 ) )
                  @ E2 ) )
             => ( ( sCC_Bl1748261141445803503t_unit @ successors @ va @ E3 )
               => ( sCC_Bl6082031138996704384t_unit @ successors @ va @ E3 @ ( sCC_Bloemen_dfss_v @ successors @ va @ E3 ) ) ) ) ) ) ) ) ).

% prepostdfss
thf(fact_1269_dfss_Opsimps,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
      ( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) )
     => ( ( sCC_Bloemen_dfss_v @ successors @ V3 @ E )
        = ( if_SCC4926449794303880475t_unit
          @ ( ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
            = bot_bot_set_v )
          @ E
          @ ( sCC_Bloemen_dfss_v @ successors @ V3
            @ ( sCC_Bl48393358579903213t_unit
              @ ^ [Uu: v > set_v,X2: v] :
                  ( if_set_v @ ( X2 = V3 )
                  @ ( sup_sup_set_v
                    @ ( sCC_Bl3795065053823578884t_unit
                      @ ( if_SCC4926449794303880475t_unit
                        @ ( member_v
                          @ ( fChoice_v
                            @ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                          @ ( sCC_Bl157864678168468314t_unit @ E ) )
                        @ E
                        @ ( if_SCC4926449794303880475t_unit
                          @ ~ ( member_v
                              @ ( fChoice_v
                                @ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                              @ ( sCC_Bl4645233313691564917t_unit @ E ) )
                          @ ( sCC_Bloemen_dfs_v @ successors
                            @ ( fChoice_v
                              @ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                            @ E )
                          @ ( sCC_Bloemen_unite_v @ V3
                            @ ( fChoice_v
                              @ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                            @ E ) ) )
                      @ V3 )
                    @ ( insert_v
                      @ ( fChoice_v
                        @ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                      @ bot_bot_set_v ) )
                  @ ( sCC_Bl3795065053823578884t_unit
                    @ ( if_SCC4926449794303880475t_unit
                      @ ( member_v
                        @ ( fChoice_v
                          @ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                        @ ( sCC_Bl157864678168468314t_unit @ E ) )
                      @ E
                      @ ( if_SCC4926449794303880475t_unit
                        @ ~ ( member_v
                            @ ( fChoice_v
                              @ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                            @ ( sCC_Bl4645233313691564917t_unit @ E ) )
                        @ ( sCC_Bloemen_dfs_v @ successors
                          @ ( fChoice_v
                            @ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                          @ E )
                        @ ( sCC_Bloemen_unite_v @ V3
                          @ ( fChoice_v
                            @ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                          @ E ) ) )
                    @ X2 ) )
              @ ( if_SCC4926449794303880475t_unit
                @ ( member_v
                  @ ( fChoice_v
                    @ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                  @ ( sCC_Bl157864678168468314t_unit @ E ) )
                @ E
                @ ( if_SCC4926449794303880475t_unit
                  @ ~ ( member_v
                      @ ( fChoice_v
                        @ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                      @ ( sCC_Bl4645233313691564917t_unit @ E ) )
                  @ ( sCC_Bloemen_dfs_v @ successors
                    @ ( fChoice_v
                      @ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                    @ E )
                  @ ( sCC_Bloemen_unite_v @ V3
                    @ ( fChoice_v
                      @ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
                    @ E ) ) ) ) ) ) ) ) ).

% dfss.psimps
thf(fact_1270_e_H_H__def,axiom,
    ( e
    = ( sCC_Bl48393358579903213t_unit
      @ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = va ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ e2 @ va ) @ ( insert_v @ w @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ e2 @ X2 ) )
      @ e2 ) ) ).

% e''_def
thf(fact_1271_pre__dfss__unite__pre__dfss,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
      ( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
     => ( ( member_v @ W @ ( successors @ V3 ) )
       => ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
         => ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
           => ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
             => ( sCC_Bl1748261141445803503t_unit @ successors @ V3
                @ ( sCC_Bl48393358579903213t_unit
                  @ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = V3 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ V3 ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ X2 ) )
                  @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).

% pre_dfss_unite_pre_dfss
thf(fact_1272_pre__dfss__post__dfs__pre__dfss,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
      ( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
     => ( ( member_v @ W @ ( successors @ V3 ) )
       => ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
         => ( ( sCC_Bl8953792750115413617t_unit @ successors @ W @ E @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) )
           => ( sCC_Bl1748261141445803503t_unit @ successors @ V3
              @ ( sCC_Bl48393358579903213t_unit
                @ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = V3 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ V3 ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ X2 ) )
                @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) ) ) ) ) ) ) ).

% pre_dfss_post_dfs_pre_dfss
thf(fact_1273_pre__dfs__implies__post__dfs,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
      ( ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ E )
     => ( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) )
       => ( ( sCC_Bl6082031138996704384t_unit @ successors @ V3
            @ ( sCC_Bl7876664385711583351t_unit
              @ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
              @ ( sCC_Bl349061681862590396t_unit
                @ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                @ ( sCC_Bl7870604408699998558t_unit
                  @ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
                  @ E ) ) )
            @ ( sCC_Bloemen_dfss_v @ successors @ V3
              @ ( sCC_Bl7876664385711583351t_unit
                @ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
                @ ( sCC_Bl349061681862590396t_unit
                  @ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                  @ ( sCC_Bl7870604408699998558t_unit
                    @ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
                    @ E ) ) ) ) )
         => ( sCC_Bl8953792750115413617t_unit @ successors @ V3 @ E @ ( sCC_Bloemen_dfs_v @ successors @ V3 @ E ) ) ) ) ) ).

% pre_dfs_implies_post_dfs
thf(fact_1274_dfs__dfss_Odomintros_I1_J,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
      ( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors )
        @ ( sum_In5289330923152326972t_unit
          @ ( produc3862955338007567901t_unit @ V3
            @ ( sCC_Bl7876664385711583351t_unit
              @ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
              @ ( sCC_Bl349061681862590396t_unit
                @ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                @ ( sCC_Bl7870604408699998558t_unit
                  @ ^ [Uu: set_v] : ( insert_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
                  @ E ) ) ) ) ) )
     => ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) ) ) ).

% dfs_dfss.domintros(1)
thf(fact_1275_dfs_Opsimps,axiom,
    ! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
      ( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) )
     => ( ( sCC_Bloemen_dfs_v @ successors @ V3 @ E )
        = ( if_SCC4926449794303880475t_unit
          @ ( V3
            = ( hd_v
              @ ( sCC_Bl8828226123343373779t_unit
                @ ( sCC_Bloemen_dfss_v @ successors @ V3
                  @ ( sCC_Bl7876664385711583351t_unit
                    @ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
                    @ ( sCC_Bl349061681862590396t_unit
                      @ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                      @ ( sCC_Bl7870604408699998558t_unit
                        @ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
                        @ E ) ) ) ) ) ) )
          @ ( sCC_Bl7876664385711583351t_unit
            @ ^ [Uu: list_v] :
                ( tl_v
                @ ( sCC_Bl9201514103433284750t_unit
                  @ ( sCC_Bloemen_dfss_v @ successors @ V3
                    @ ( sCC_Bl7876664385711583351t_unit
                      @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
                      @ ( sCC_Bl349061681862590396t_unit
                        @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                        @ ( sCC_Bl7870604408699998558t_unit
                          @ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
                          @ E ) ) ) ) ) )
            @ ( sCC_Bl349061681862590396t_unit
              @ ^ [Uu: list_v] :
                  ( tl_v
                  @ ( sCC_Bl8828226123343373779t_unit
                    @ ( sCC_Bloemen_dfss_v @ successors @ V3
                      @ ( sCC_Bl7876664385711583351t_unit
                        @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
                        @ ( sCC_Bl349061681862590396t_unit
                          @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                          @ ( sCC_Bl7870604408699998558t_unit
                            @ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
                            @ E ) ) ) ) ) )
              @ ( sCC_Bl2708505634401380163t_unit
                @ ^ [Uu: set_v] :
                    ( sup_sup_set_v
                    @ ( sCC_Bl157864678168468314t_unit
                      @ ( sCC_Bloemen_dfss_v @ successors @ V3
                        @ ( sCC_Bl7876664385711583351t_unit
                          @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
                          @ ( sCC_Bl349061681862590396t_unit
                            @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                            @ ( sCC_Bl7870604408699998558t_unit
                              @ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
                              @ E ) ) ) ) )
                    @ ( sCC_Bl1280885523602775798t_unit
                      @ ( sCC_Bloemen_dfss_v @ successors @ V3
                        @ ( sCC_Bl7876664385711583351t_unit
                          @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
                          @ ( sCC_Bl349061681862590396t_unit
                            @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                            @ ( sCC_Bl7870604408699998558t_unit
                              @ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
                              @ E ) ) ) )
                      @ V3 ) )
                @ ( sCC_Bl6816368539212994290t_unit
                  @ ^ [Uu: set_set_v] :
                      ( sup_sup_set_set_v
                      @ ( sCC_Bl2536197123907397897t_unit
                        @ ( sCC_Bloemen_dfss_v @ successors @ V3
                          @ ( sCC_Bl7876664385711583351t_unit
                            @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
                            @ ( sCC_Bl349061681862590396t_unit
                              @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                              @ ( sCC_Bl7870604408699998558t_unit
                                @ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
                                @ E ) ) ) ) )
                      @ ( insert_set_v
                        @ ( sCC_Bl1280885523602775798t_unit
                          @ ( sCC_Bloemen_dfss_v @ successors @ V3
                            @ ( sCC_Bl7876664385711583351t_unit
                              @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
                              @ ( sCC_Bl349061681862590396t_unit
                                @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                                @ ( sCC_Bl7870604408699998558t_unit
                                  @ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
                                  @ E ) ) ) )
                          @ V3 )
                        @ bot_bot_set_set_v ) )
                  @ ( sCC_Bloemen_dfss_v @ successors @ V3
                    @ ( sCC_Bl7876664385711583351t_unit
                      @ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
                      @ ( sCC_Bl349061681862590396t_unit
                        @ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                        @ ( sCC_Bl7870604408699998558t_unit
                          @ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
                          @ E ) ) ) ) ) ) ) )
          @ ( sCC_Bl7876664385711583351t_unit
            @ ^ [Uu: list_v] :
                ( tl_v
                @ ( sCC_Bl9201514103433284750t_unit
                  @ ( sCC_Bloemen_dfss_v @ successors @ V3
                    @ ( sCC_Bl7876664385711583351t_unit
                      @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
                      @ ( sCC_Bl349061681862590396t_unit
                        @ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                        @ ( sCC_Bl7870604408699998558t_unit
                          @ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
                          @ E ) ) ) ) ) )
            @ ( sCC_Bloemen_dfss_v @ successors @ V3
              @ ( sCC_Bl7876664385711583351t_unit
                @ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
                @ ( sCC_Bl349061681862590396t_unit
                  @ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
                  @ ( sCC_Bl7870604408699998558t_unit
                    @ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
                    @ E ) ) ) ) ) ) ) ) ).

% dfs.psimps
thf(fact_1276_unite__S__equal,axiom,
    ! [E: sCC_Bl1394983891496994913t_unit,W: v,V3: v] :
      ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
     => ( ( member_v @ W @ ( successors @ V3 ) )
       => ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
         => ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
           => ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
             => ( ( comple2307003700295860064_set_v
                  @ ( collect_set_v
                    @ ^ [Uu: set_v] :
                      ? [N3: v] :
                        ( ( Uu
                          = ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N3 ) )
                        & ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) )
                = ( comple2307003700295860064_set_v
                  @ ( collect_set_v
                    @ ^ [Uu: set_v] :
                      ? [N3: v] :
                        ( ( Uu
                          = ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) )
                        & ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ).

% unite_S_equal
thf(fact_1277_unite__def,axiom,
    ( sCC_Bloemen_unite_v
    = ( ^ [V4: v,W2: v,E8: sCC_Bl1394983891496994913t_unit] :
          ( sCC_Bl349061681862590396t_unit
          @ ^ [Uu: list_v] :
              ( dropWhile_v
              @ ^ [X2: v] :
                  ~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ X2 ) )
              @ ( sCC_Bl8828226123343373779t_unit @ E8 ) )
          @ ( sCC_Bl3155122997657187039t_unit
            @ ^ [Uu: v > set_v,X2: v] :
                ( if_set_v
                @ ( member_v @ X2
                  @ ( comple2307003700295860064_set_v
                    @ ( collect_set_v
                      @ ^ [Uv: set_v] :
                        ? [Y3: v] :
                          ( ( Uv
                            = ( sCC_Bl1280885523602775798t_unit @ E8 @ Y3 ) )
                          & ( member_v @ Y3
                            @ ( sup_sup_set_v
                              @ ( set_v2
                                @ ( takeWhile_v
                                  @ ^ [Z2: v] :
                                      ~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
                                  @ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
                              @ ( insert_v
                                @ ( hd_v
                                  @ ( dropWhile_v
                                    @ ^ [Z2: v] :
                                        ~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
                                    @ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
                                @ bot_bot_set_v ) ) ) ) ) ) )
                @ ( comple2307003700295860064_set_v
                  @ ( collect_set_v
                    @ ^ [Uv: set_v] :
                      ? [Y3: v] :
                        ( ( Uv
                          = ( sCC_Bl1280885523602775798t_unit @ E8 @ Y3 ) )
                        & ( member_v @ Y3
                          @ ( sup_sup_set_v
                            @ ( set_v2
                              @ ( takeWhile_v
                                @ ^ [Z2: v] :
                                    ~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
                                @ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
                            @ ( insert_v
                              @ ( hd_v
                                @ ( dropWhile_v
                                  @ ^ [Z2: v] :
                                      ~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
                                  @ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
                              @ bot_bot_set_v ) ) ) ) ) )
                @ ( sCC_Bl1280885523602775798t_unit @ E8 @ X2 ) )
            @ E8 ) ) ) ) ).

% unite_def

% Helper facts (6)
thf(help_fChoice_1_1_fChoice_001tf__v_T,axiom,
    ! [P: v > $o] :
      ( ( P @ ( fChoice_v @ P ) )
      = ( ? [X6: v] : ( P @ X6 ) ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( if_set_v @ $false @ X @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
    ! [X: set_v,Y2: set_v] :
      ( ( if_set_v @ $true @ X @ Y2 )
      = X ) ).

thf(help_If_3_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
    ! [X: sCC_Bl1394983891496994913t_unit,Y2: sCC_Bl1394983891496994913t_unit] :
      ( ( if_SCC4926449794303880475t_unit @ $false @ X @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
    ! [X: sCC_Bl1394983891496994913t_unit,Y2: sCC_Bl1394983891496994913t_unit] :
      ( ( if_SCC4926449794303880475t_unit @ $true @ X @ Y2 )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( sCC_Bl8828226123343373779t_unit @ e )
    = ( sCC_Bl8828226123343373779t_unit @ ea ) ) ).

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