SET007 Axioms: SET007+209.ax


%------------------------------------------------------------------------------
% File     : SET007+209 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Families of Subsets, Subspaces and Mappings in Topological Spaces
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : tops_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   81 (  13 unit)
%            Number of atoms       :  507 (  57 equality)
%            Maximal formula depth :   17 (   8 average)
%            Number of connectives :  454 (  28 ~  ;   0  |; 143  &)
%                                         (  16 <=>; 267 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   1 propositional; 0-3 arity)
%            Number of functors    :   27 (   1 constant; 0-5 arity)
%            Number of variables   :  227 (   0 singleton; 224 !;   3 ?)
%            Maximal term depth    :    5 (   2 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => r1_tarski(B,k1_zfmisc_1(k2_pre_topc(A))) ) ) )).

fof(t2_tops_2,axiom,(
    $true )).

fof(t3_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( r1_tarski(C,B)
             => m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ) )).

fof(t4_tops_2,axiom,(
    $true )).

fof(t5_tops_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ~ ( r1_pre_topc(A,B)
              & B = k1_xboole_0 ) ) ) )).

fof(t6_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => r1_tarski(k6_subset_1(u1_struct_0(A),k5_setfam_1(u1_struct_0(A),B),k5_setfam_1(u1_struct_0(A),C)),k5_setfam_1(u1_struct_0(A),k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C))) ) ) ) )).

fof(t7_tops_2,axiom,(
    $true )).

fof(t8_tops_2,axiom,(
    $true )).

fof(t9_tops_2,axiom,(
    $true )).

fof(t10_tops_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( ~ ( B != k1_xboole_0
            & k7_setfam_1(A,B) = k1_xboole_0 )
        & ~ ( k7_setfam_1(A,B) != k1_xboole_0
            & B = k1_xboole_0 ) ) ) )).

fof(t11_tops_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( B != k1_xboole_0
       => k6_setfam_1(A,k7_setfam_1(A,B)) = k3_subset_1(A,k5_setfam_1(A,B)) ) ) )).

fof(t12_tops_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( B != k1_xboole_0
       => k5_setfam_1(A,k7_setfam_1(A,B)) = k3_subset_1(A,k6_setfam_1(A,B)) ) ) )).

fof(t13_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_finset_1(k7_setfam_1(u1_struct_0(A),B))
          <=> v1_finset_1(B) ) ) ) )).

fof(d1_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_tops_2(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ( r2_hidden(C,B)
                 => v3_pre_topc(C,A) ) ) ) ) ) )).

fof(d2_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v2_tops_2(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ( r2_hidden(C,B)
                 => v4_pre_topc(C,A) ) ) ) ) ) )).

fof(t14_tops_2,axiom,(
    $true )).

fof(t15_tops_2,axiom,(
    $true )).

fof(t16_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v2_tops_2(B,A)
          <=> v1_tops_2(k7_setfam_1(u1_struct_0(A),B),A) ) ) ) )).

fof(t17_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_tops_2(B,A)
          <=> v2_tops_2(k7_setfam_1(u1_struct_0(A),B),A) ) ) ) )).

fof(t18_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( ( r1_tarski(B,C)
                  & v1_tops_2(C,A) )
               => v1_tops_2(B,A) ) ) ) ) )).

fof(t19_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( ( r1_tarski(B,C)
                  & v2_tops_2(C,A) )
               => v2_tops_2(B,A) ) ) ) ) )).

fof(t20_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( ( v1_tops_2(B,A)
                  & v1_tops_2(C,A) )
               => v1_tops_2(k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) )).

fof(t21_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( v1_tops_2(B,A)
               => v1_tops_2(k5_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) )).

fof(t22_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( v1_tops_2(B,A)
               => v1_tops_2(k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) )).

fof(t23_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( ( v2_tops_2(B,A)
                  & v2_tops_2(C,A) )
               => v2_tops_2(k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) )).

fof(t24_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( v2_tops_2(B,A)
               => v2_tops_2(k5_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) )).

fof(t25_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( v2_tops_2(B,A)
               => v2_tops_2(k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) )).

fof(t26_tops_2,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_tops_2(B,A)
           => v3_pre_topc(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) )).

fof(t27_tops_2,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( ( v1_tops_2(B,A)
              & v1_finset_1(B) )
           => v3_pre_topc(k6_setfam_1(u1_struct_0(A),B),A) ) ) ) )).

fof(t28_tops_2,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( ( v2_tops_2(B,A)
              & v1_finset_1(B) )
           => v4_pre_topc(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) )).

fof(t29_tops_2,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v2_tops_2(B,A)
           => v4_pre_topc(k6_setfam_1(u1_struct_0(A),B),A) ) ) ) )).

fof(t30_tops_2,axiom,(
    $true )).

fof(t31_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_pre_topc(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
             => m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ) )).

fof(t32_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_pre_topc(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ( v3_pre_topc(C,B)
              <=> ? [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & v3_pre_topc(D,A)
                    & k3_xboole_0(D,k2_pre_topc(B)) = C ) ) ) ) ) )).

fof(t33_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_pre_topc(C,A)
             => ( v3_pre_topc(B,A)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
                   => ( D = B
                     => v3_pre_topc(D,C) ) ) ) ) ) ) )).

fof(t34_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_pre_topc(C,A)
             => ( v4_pre_topc(B,A)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
                   => ( D = B
                     => v4_pre_topc(D,C) ) ) ) ) ) ) )).

fof(t35_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_pre_topc(C,A)
             => ( v1_tops_2(B,A)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(C))))
                   => ( D = B
                     => v1_tops_2(D,C) ) ) ) ) ) ) )).

fof(t36_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_pre_topc(C,A)
             => ( v2_tops_2(B,A)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(C))))
                   => ( D = B
                     => v2_tops_2(D,C) ) ) ) ) ) ) )).

fof(t37_tops_2,axiom,(
    $true )).

fof(t38_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => m1_subset_1(k5_subset_1(u1_struct_0(A),B,C),k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,C)))) ) ) ) )).

fof(d3_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B)))))
                 => ( D = k1_tops_2(A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B))))
                       => ( r2_hidden(E,D)
                        <=> ? [F] :
                              ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
                              & r2_hidden(F,C)
                              & k5_subset_1(u1_struct_0(A),F,B) = E ) ) ) ) ) ) ) ) )).

fof(t39_tops_2,axiom,(
    $true )).

fof(t40_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                 => ( r1_tarski(C,D)
                   => r1_tarski(k1_tops_2(A,B,C),k1_tops_2(A,B,D)) ) ) ) ) ) )).

fof(t41_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                 => ( r2_hidden(B,D)
                   => r2_hidden(k5_subset_1(u1_struct_0(A),B,C),k1_tops_2(A,C,D)) ) ) ) ) ) )).

fof(t42_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                 => ( r1_tarski(B,k5_setfam_1(u1_struct_0(A),D))
                   => r1_tarski(k5_subset_1(u1_struct_0(A),B,C),k5_setfam_1(u1_struct_0(k3_pre_topc(A,C)),k1_tops_2(A,C,D))) ) ) ) ) ) )).

fof(t43_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( r1_tarski(B,k5_setfam_1(u1_struct_0(A),C))
               => B = k5_setfam_1(u1_struct_0(k3_pre_topc(A,B)),k1_tops_2(A,B,C)) ) ) ) ) )).

fof(t44_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => r1_tarski(k5_setfam_1(u1_struct_0(k3_pre_topc(A,B)),k1_tops_2(A,B,C)),k5_setfam_1(u1_struct_0(A),C)) ) ) ) )).

fof(t45_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( r1_tarski(B,k5_setfam_1(u1_struct_0(k3_pre_topc(A,B)),k1_tops_2(A,B,C)))
               => r1_tarski(B,k5_setfam_1(u1_struct_0(A),C)) ) ) ) ) )).

fof(t46_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( v1_finset_1(C)
               => v1_finset_1(k1_tops_2(A,B,C)) ) ) ) ) )).

fof(t47_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( v1_tops_2(C,A)
               => v1_tops_2(k1_tops_2(A,B,C),k3_pre_topc(A,B)) ) ) ) ) )).

fof(t48_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( v2_tops_2(C,A)
               => v2_tops_2(k1_tops_2(A,B,C),k3_pre_topc(A,B)) ) ) ) ) )).

fof(t49_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_pre_topc(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
             => ~ ( v1_tops_2(C,B)
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                     => ~ ( v1_tops_2(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                             => ( E = k2_pre_topc(B)
                               => C = k1_tops_2(A,E,D) ) ) ) ) ) ) ) ) )).

fof(t50_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ? [D] :
                  ( v1_relat_1(D)
                  & v1_funct_1(D)
                  & k1_relat_1(D) = C
                  & k2_relat_1(D) = k1_tops_2(A,B,C)
                  & ! [E] :
                      ( r2_hidden(E,C)
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
                         => ( F = E
                           => k1_funct_1(D,E) = k5_subset_1(u1_struct_0(A),F,B) ) ) ) ) ) ) ) )).

fof(t51_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( k1_relat_1(C) = k2_pre_topc(A)
                & r1_tarski(k2_relat_1(C),k2_pre_topc(B)) ) ) ) ) )).

fof(t52_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => k5_pre_topc(A,B,C,k2_pre_topc(B)) = k2_pre_topc(A) ) ) ) )).

fof(t53_tops_2,axiom,(
    $true )).

fof(t54_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
                 => m1_subset_1(k9_relat_1(k3_funct_3(C),D),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ) ) )).

fof(t55_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v5_pre_topc(C,A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                   => ( v3_pre_topc(D,B)
                     => v3_pre_topc(k5_pre_topc(A,B,C,D),A) ) ) ) ) ) ) )).

fof(t56_tops_2,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v5_pre_topc(C,A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                   => r1_tarski(k6_pre_topc(A,k5_pre_topc(A,B,C,D)),k5_pre_topc(A,B,C,k6_pre_topc(B,D))) ) ) ) ) ) )).

fof(t57_tops_2,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v5_pre_topc(C,A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => r1_tarski(k4_pre_topc(A,B,C,k6_pre_topc(A,D)),k6_pre_topc(B,k4_pre_topc(A,B,C,D))) ) ) ) ) ) )).

fof(t58_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( l1_pre_topc(B)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & l1_pre_topc(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(C))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(C),u1_struct_0(B))
                        & m2_relset_1(E,u1_struct_0(C),u1_struct_0(B)) )
                     => ( ( v5_pre_topc(D,A,C)
                          & v5_pre_topc(E,C,B) )
                       => v5_pre_topc(k7_funct_2(u1_struct_0(A),u1_struct_0(C),u1_struct_0(B),D,E),A,B) ) ) ) ) ) ) )).

fof(t59_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
                 => ( ( v5_pre_topc(C,A,B)
                      & v1_tops_2(D,B) )
                   => ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                       => ( E = k9_relat_1(k3_funct_3(C),D)
                         => v1_tops_2(E,A) ) ) ) ) ) ) ) )).

fof(t60_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( l1_pre_topc(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
                 => ( ( v5_pre_topc(C,A,B)
                      & v2_tops_2(D,B) )
                   => ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                       => ( E = k9_relat_1(k3_funct_3(C),D)
                         => v2_tops_2(E,A) ) ) ) ) ) ) ) )).

fof(d4_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_struct_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( ( k2_relat_1(C) = k2_pre_topc(B)
                  & v2_funct_1(C) )
               => k2_tops_2(A,B,C) = k2_funct_1(C) ) ) ) ) )).

fof(t61_tops_2,axiom,(
    $true )).

fof(t62_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( ( k2_relat_1(C) = k2_pre_topc(B)
                  & v2_funct_1(C) )
               => ( k1_relat_1(k2_tops_2(A,B,C)) = k2_pre_topc(B)
                  & k2_relat_1(k2_tops_2(A,B,C)) = k2_pre_topc(A) ) ) ) ) ) )).

fof(t63_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_struct_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( ( k2_relat_1(C) = k2_pre_topc(B)
                  & v2_funct_1(C) )
               => v2_funct_1(k2_tops_2(A,B,C)) ) ) ) ) )).

fof(t64_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( ( k2_relat_1(C) = k2_pre_topc(B)
                  & v2_funct_1(C) )
               => k2_tops_2(B,A,k2_tops_2(A,B,C)) = C ) ) ) ) )).

fof(t65_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_struct_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( ( k2_relat_1(C) = k2_pre_topc(B)
                  & v2_funct_1(C) )
               => ( k5_relat_1(C,k2_tops_2(A,B,C)) = k6_relat_1(k1_relat_1(C))
                  & k5_relat_1(k2_tops_2(A,B,C),C) = k6_relat_1(k2_relat_1(C)) ) ) ) ) ) )).

fof(t66_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & l1_struct_0(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
                        & m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
                     => ( ( k1_relat_1(D) = k2_pre_topc(A)
                          & k2_relat_1(D) = k2_pre_topc(B)
                          & v2_funct_1(D)
                          & k1_relat_1(E) = k2_pre_topc(B)
                          & k2_relat_1(E) = k2_pre_topc(C)
                          & v2_funct_1(E) )
                       => k2_tops_2(A,C,k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E)) = k7_funct_2(u1_struct_0(C),u1_struct_0(B),u1_struct_0(A),k2_tops_2(B,C,E),k2_tops_2(A,B,D)) ) ) ) ) ) ) )).

fof(t67_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_struct_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( ( k2_relat_1(C) = k2_pre_topc(B)
                      & v2_funct_1(C) )
                   => k4_pre_topc(A,B,C,D) = k5_pre_topc(B,A,k2_tops_2(A,B,C),D) ) ) ) ) ) )).

fof(t68_tops_2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_struct_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                 => ( ( k2_relat_1(C) = k2_pre_topc(B)
                      & v2_funct_1(C) )
                   => k5_pre_topc(A,B,C,D) = k4_pre_topc(B,A,k2_tops_2(A,B,C),D) ) ) ) ) ) )).

fof(d5_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( l1_pre_topc(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v3_tops_2(C,A,B)
              <=> ( k1_relat_1(C) = k2_pre_topc(A)
                  & k2_relat_1(C) = k2_pre_topc(B)
                  & v2_funct_1(C)
                  & v5_pre_topc(C,A,B)
                  & v5_pre_topc(k2_tops_2(A,B,C),B,A) ) ) ) ) ) )).

fof(t69_tops_2,axiom,(
    $true )).

fof(t70_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v3_tops_2(C,A,B)
               => v3_tops_2(k2_tops_2(A,B,C),B,A) ) ) ) ) )).

fof(t71_tops_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & l1_pre_topc(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
                        & m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
                     => ( ( v3_tops_2(D,A,B)
                          & v3_tops_2(E,B,C) )
                       => v3_tops_2(k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E),A,C) ) ) ) ) ) ) )).

fof(t72_tops_2,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v3_tops_2(C,A,B)
              <=> ( k1_relat_1(C) = k2_pre_topc(A)
                  & k2_relat_1(C) = k2_pre_topc(B)
                  & v2_funct_1(C)
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                     => ( v4_pre_topc(D,A)
                      <=> v4_pre_topc(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ) ) )).

fof(t73_tops_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v3_tops_2(C,A,B)
              <=> ( k1_relat_1(C) = k2_pre_topc(A)
                  & k2_relat_1(C) = k2_pre_topc(B)
                  & v2_funct_1(C)
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                     => k5_pre_topc(A,B,C,k6_pre_topc(B,D)) = k6_pre_topc(A,k5_pre_topc(A,B,C,D)) ) ) ) ) ) ) )).

fof(t74_tops_2,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v3_tops_2(C,A,B)
              <=> ( k1_relat_1(C) = k2_pre_topc(A)
                  & k2_relat_1(C) = k2_pre_topc(B)
                  & v2_funct_1(C)
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                     => k4_pre_topc(A,B,C,k6_pre_topc(A,D)) = k6_pre_topc(B,k4_pre_topc(A,B,C,D)) ) ) ) ) ) ) )).

fof(dt_k1_tops_2,axiom,(
    ! [A,B,C] :
      ( ( l1_pre_topc(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
     => m1_subset_1(k1_tops_2(A,B,C),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B))))) ) )).

fof(dt_k2_tops_2,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & l1_struct_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
     => ( v1_funct_1(k2_tops_2(A,B,C))
        & v1_funct_2(k2_tops_2(A,B,C),u1_struct_0(B),u1_struct_0(A))
        & m2_relset_1(k2_tops_2(A,B,C),u1_struct_0(B),u1_struct_0(A)) ) ) )).
%------------------------------------------------------------------------------