SET007 Axioms: SET007+360.ax


%------------------------------------------------------------------------------
% File     : SET007+360 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Finite 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    : fin_topo [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   83 (   9 unit)
%            Number of atoms       :  414 (  46 equality)
%            Maximal formula depth :   22 (   6 average)
%            Number of connectives :  418 (  87 ~  ;   5  |; 154  &)
%                                         (  24 <=>; 148 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   27 (   1 propositional; 0-3 arity)
%            Number of functors    :   49 (   9 constant; 0-4 arity)
%            Number of variables   :  183 (   0 singleton; 168 !;  15 ?)
%            Maximal term depth    :    6 (   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(rc1_fin_topo,axiom,(
    ? [A] :
      ( l1_fin_topo(A)
      & v1_fin_topo(A) ) )).

fof(rc2_fin_topo,axiom,(
    ? [A] :
      ( l1_fin_topo(A)
      & ~ v3_struct_0(A)
      & v1_fin_topo(A) ) )).

fof(fc1_fin_topo,axiom,
    ( ~ v3_struct_0(k3_fin_topo)
    & v1_fin_topo(k3_fin_topo) )).

fof(rc3_fin_topo,axiom,(
    ? [A] :
      ( l1_fin_topo(A)
      & ~ v3_struct_0(A)
      & v6_group_1(A)
      & v1_fin_topo(A)
      & v2_fin_topo(A) ) )).

fof(t1_fin_topo,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,k1_zfmisc_1(A))
     => ( ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,C)
             => ( r1_xreal_0(k3_finseq_1(B),C)
                | r1_tarski(k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,C),k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,k1_nat_1(C,np__1))) ) ) )
       => ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => ! [D] :
                ( m2_subset_1(D,k1_numbers,k5_numbers)
               => ( ( r1_xreal_0(C,D)
                    & r1_xreal_0(np__1,C)
                    & r1_xreal_0(D,k3_finseq_1(B)) )
                 => r1_tarski(k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,C),k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,D)) ) ) ) ) ) )).

fof(t2_fin_topo,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,k1_zfmisc_1(A))
     => ( ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,C)
             => ( r1_xreal_0(k3_finseq_1(B),C)
                | r1_tarski(k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,C),k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,k1_nat_1(C,np__1))) ) ) )
       => ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => ! [D] :
                ( m2_subset_1(D,k1_numbers,k5_numbers)
               => ( ( r1_xreal_0(np__1,C)
                    & r1_xreal_0(D,k3_finseq_1(B))
                    & r1_tarski(k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,D),k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,C)) )
                 => ( r1_xreal_0(D,C)
                    | ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( ( r1_xreal_0(C,E)
                            & r1_xreal_0(E,D) )
                         => k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,D) = k4_finseq_4(k5_numbers,k1_zfmisc_1(A),B,E) ) ) ) ) ) ) ) ) )).

fof(d1_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_fin_topo(A,B) = k8_funct_2(u1_struct_0(A),k1_zfmisc_1(u1_struct_0(A)),u1_fin_topo(A),B) ) ) )).

fof(d2_fin_topo,axiom,(
    k3_fin_topo = g1_fin_topo(k6_domain_1(k5_numbers,np__0),k2_fin_topo(np__0,k2_subset_1(k6_domain_1(k5_numbers,np__0)))) )).

fof(d3_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ( v2_fin_topo(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => r2_hidden(B,k1_fin_topo(A,B)) ) ) ) )).

fof(t3_fin_topo,axiom,(
    $true )).

fof(t4_fin_topo,axiom,(
    $true )).

fof(t5_fin_topo,axiom,(
    $true )).

fof(t6_fin_topo,axiom,(
    $true )).

fof(t7_fin_topo,axiom,(
    v2_fin_topo(k3_fin_topo) )).

fof(t8_fin_topo,axiom,(
    v6_group_1(k3_fin_topo) )).

fof(d4_fin_topo,axiom,(
    $true )).

fof(d5_fin_topo,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( r1_fin_topo(A,B)
        <=> r1_tarski(u1_struct_0(A),k3_tarski(B)) ) ) )).

fof(t10_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r2_hidden(B,k4_fin_topo(A,C))
              <=> ( ~ r1_xboole_0(k1_fin_topo(A,B),C)
                  & ~ r1_xboole_0(k1_fin_topo(A,B),k3_subset_1(u1_struct_0(A),C)) ) ) ) ) ) )).

fof(d7_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k5_fin_topo(A,B) = k5_subset_1(u1_struct_0(A),B,k4_fin_topo(A,B)) ) ) )).

fof(d8_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k6_fin_topo(A,B) = k5_subset_1(u1_struct_0(A),k3_subset_1(u1_struct_0(A),B),k4_fin_topo(A,B)) ) ) )).

fof(t11_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k4_fin_topo(A,B) = k4_subset_1(u1_struct_0(A),k5_fin_topo(A,B),k6_fin_topo(A,B)) ) ) )).

fof(d12_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k10_fin_topo(A,B) = k6_subset_1(u1_struct_0(A),B,k9_fin_topo(A,B)) ) ) )).

fof(d14_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ( v3_fin_topo(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( r2_hidden(C,k1_fin_topo(A,B))
                 => r2_hidden(B,k1_fin_topo(A,C)) ) ) ) ) ) )).

fof(t12_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r2_hidden(B,k7_fin_topo(A,C))
              <=> r1_tarski(k1_fin_topo(A,B),C) ) ) ) ) )).

fof(t13_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r2_hidden(B,k8_fin_topo(A,C))
              <=> ~ r1_xboole_0(k1_fin_topo(A,B),C) ) ) ) ) )).

fof(t14_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r2_hidden(B,k9_fin_topo(A,C))
              <=> ( r2_hidden(B,C)
                  & r1_xboole_0(k6_subset_1(u1_struct_0(A),k1_fin_topo(A,B),k6_domain_1(u1_struct_0(A),B)),C) ) ) ) ) ) )).

fof(t15_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r2_hidden(B,k10_fin_topo(A,C))
              <=> ( r2_hidden(B,C)
                  & ~ r1_xboole_0(k6_subset_1(u1_struct_0(A),k1_fin_topo(A,B),k6_domain_1(u1_struct_0(A),B)),C) ) ) ) ) ) )).

fof(t16_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r2_hidden(B,k11_fin_topo(A,C))
              <=> ? [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                    & r2_hidden(D,C)
                    & r2_hidden(B,k1_fin_topo(A,D)) ) ) ) ) ) )).

fof(t17_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ( v3_fin_topo(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => k8_fin_topo(A,B) = k11_fin_topo(A,B) ) ) ) )).

fof(d15_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v4_fin_topo(B,A)
          <=> B = k7_fin_topo(A,B) ) ) ) )).

fof(d16_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v5_fin_topo(B,A)
          <=> B = k8_fin_topo(A,B) ) ) ) )).

fof(d17_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v6_fin_topo(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ~ ( B = k4_subset_1(u1_struct_0(A),C,D)
                        & C != k1_xboole_0
                        & D != k1_xboole_0
                        & r1_xboole_0(C,D)
                        & r1_xboole_0(k8_fin_topo(A,C),D) ) ) ) ) ) ) )).

fof(t18_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => r1_tarski(B,k8_fin_topo(A,B)) ) ) )).

fof(t19_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(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)))
             => ( r1_tarski(B,C)
               => r1_tarski(k8_fin_topo(A,B),k8_fin_topo(A,C)) ) ) ) ) )).

fof(t20_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_group_1(A)
        & v2_fin_topo(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v6_fin_topo(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( r2_hidden(C,B)
                    & ! [D] :
                        ( m2_finseq_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ~ ( ~ r1_xreal_0(k3_finseq_1(D),np__0)
                            & k4_finseq_4(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),D,np__1) = k6_domain_1(u1_struct_0(A),C)
                            & ! [E] :
                                ( m2_subset_1(E,k1_numbers,k5_numbers)
                               => ~ ( ~ r1_xreal_0(E,np__0)
                                    & ~ r1_xreal_0(k3_finseq_1(D),E)
                                    & k4_finseq_4(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),D,k1_nat_1(E,np__1)) != k5_subset_1(u1_struct_0(A),k8_fin_topo(A,k4_finseq_4(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),D,E)),B) ) )
                            & r1_tarski(B,k4_finseq_4(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),D,k3_finseq_1(D))) ) ) ) ) ) ) ) )).

fof(t21_fin_topo,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( r2_hidden(C,k3_subset_1(A,B))
              <=> ~ r2_hidden(C,B) ) ) ) ) )).

fof(t22_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k3_subset_1(u1_struct_0(A),k7_fin_topo(A,k3_subset_1(u1_struct_0(A),B))) = k8_fin_topo(A,B) ) ) )).

fof(t23_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k3_subset_1(u1_struct_0(A),k8_fin_topo(A,k3_subset_1(u1_struct_0(A),B))) = k7_fin_topo(A,B) ) ) )).

fof(t24_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k4_fin_topo(A,B) = k5_subset_1(u1_struct_0(A),k8_fin_topo(A,B),k8_fin_topo(A,k3_subset_1(u1_struct_0(A),B))) ) ) )).

fof(t25_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k4_fin_topo(A,k3_subset_1(u1_struct_0(A),B)) = k4_fin_topo(A,B) ) ) )).

fof(t26_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ~ ( r2_hidden(B,k9_fin_topo(A,C))
                  & r2_hidden(B,k8_fin_topo(A,k6_subset_1(u1_struct_0(A),C,k6_domain_1(u1_struct_0(A),B)))) ) ) ) ) )).

fof(t27_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ~ ( k9_fin_topo(A,B) != k1_xboole_0
              & k1_card_1(B) != np__1
              & v6_fin_topo(B,A) ) ) ) )).

fof(t28_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => r1_tarski(k7_fin_topo(A,B),B) ) ) )).

fof(t29_fin_topo,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( B = C
          <=> k3_subset_1(A,B) = k3_subset_1(A,C) ) ) ) )).

fof(t30_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v4_fin_topo(B,A)
           => v5_fin_topo(k3_subset_1(u1_struct_0(A),B),A) ) ) ) )).

fof(t31_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v5_fin_topo(B,A)
           => v4_fin_topo(k3_subset_1(u1_struct_0(A),B),A) ) ) ) )).

fof(s1_fin_topo,axiom,
    ( ( v1_finset_1(f2_s1_fin_topo)
      & r1_tarski(f3_s1_fin_topo,f2_s1_fin_topo)
      & ! [A] :
          ( m1_subset_1(A,k1_zfmisc_1(f1_s1_fin_topo))
         => ( r1_tarski(A,f2_s1_fin_topo)
           => ( r1_tarski(A,f4_s1_fin_topo(A))
              & r1_tarski(f4_s1_fin_topo(A),f2_s1_fin_topo) ) ) ) )
   => ? [A] :
        ( m2_finseq_1(A,k1_zfmisc_1(f1_s1_fin_topo))
        & ~ r1_xreal_0(k3_finseq_1(A),np__0)
        & k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,np__1) = f3_s1_fin_topo
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ ( ~ r1_xreal_0(B,np__0)
                & ~ r1_xreal_0(k3_finseq_1(A),B)
                & k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,k1_nat_1(B,np__1)) != f4_s1_fin_topo(k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,B)) ) )
        & f4_s1_fin_topo(k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,k3_finseq_1(A))) = k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,k3_finseq_1(A))
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r1_xreal_0(C,k3_finseq_1(A))
                 => ( r1_xreal_0(B,np__0)
                    | r1_xreal_0(C,B)
                    | ( r1_tarski(k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,B),f2_s1_fin_topo)
                      & r2_xboole_0(k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,B),k4_finseq_4(k5_numbers,k1_zfmisc_1(f1_s1_fin_topo),A,C)) ) ) ) ) ) ) )).

fof(dt_l1_fin_topo,axiom,(
    ! [A] :
      ( l1_fin_topo(A)
     => l1_struct_0(A) ) )).

fof(existence_l1_fin_topo,axiom,(
    ? [A] : l1_fin_topo(A) )).

fof(abstractness_v1_fin_topo,axiom,(
    ! [A] :
      ( l1_fin_topo(A)
     => ( v1_fin_topo(A)
       => A = g1_fin_topo(u1_struct_0(A),u1_fin_topo(A)) ) ) )).

fof(dt_k1_fin_topo,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m1_subset_1(k1_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k2_fin_topo,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_tarski(A)))
     => ( v1_funct_1(k2_fin_topo(A,B))
        & v1_funct_2(k2_fin_topo(A,B),k1_tarski(A),k1_zfmisc_1(k1_tarski(A)))
        & m2_relset_1(k2_fin_topo(A,B),k1_tarski(A),k1_zfmisc_1(k1_tarski(A))) ) ) )).

fof(redefinition_k2_fin_topo,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_tarski(A)))
     => k2_fin_topo(A,B) = k3_cqc_lang(A,B) ) )).

fof(dt_k3_fin_topo,axiom,
    ( v1_fin_topo(k3_fin_topo)
    & l1_fin_topo(k3_fin_topo) )).

fof(dt_k4_fin_topo,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k4_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k5_fin_topo,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k5_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k6_fin_topo,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k6_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k7_fin_topo,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k7_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k8_fin_topo,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k8_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k9_fin_topo,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k9_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k10_fin_topo,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k10_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k11_fin_topo,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k11_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k12_fin_topo,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k12_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k13_fin_topo,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k13_fin_topo(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_u1_fin_topo,axiom,(
    ! [A] :
      ( l1_fin_topo(A)
     => ( v1_funct_1(u1_fin_topo(A))
        & v1_funct_2(u1_fin_topo(A),u1_struct_0(A),k1_zfmisc_1(u1_struct_0(A)))
        & m2_relset_1(u1_fin_topo(A),u1_struct_0(A),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(dt_g1_fin_topo,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_zfmisc_1(A))
        & m1_relset_1(B,A,k1_zfmisc_1(A)) )
     => ( v1_fin_topo(g1_fin_topo(A,B))
        & l1_fin_topo(g1_fin_topo(A,B)) ) ) )).

fof(free_g1_fin_topo,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_zfmisc_1(A))
        & m1_relset_1(B,A,k1_zfmisc_1(A)) )
     => ! [C,D] :
          ( g1_fin_topo(A,B) = g1_fin_topo(C,D)
         => ( A = C
            & B = D ) ) ) )).

fof(t9_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(A) )
     => r1_fin_topo(A,a_1_0_fin_topo(A)) ) )).

fof(d6_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k4_fin_topo(A,B) = a_2_0_fin_topo(A,B) ) ) )).

fof(d9_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k7_fin_topo(A,B) = a_2_1_fin_topo(A,B) ) ) )).

fof(d10_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k8_fin_topo(A,B) = a_2_2_fin_topo(A,B) ) ) )).

fof(d11_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k9_fin_topo(A,B) = a_2_3_fin_topo(A,B) ) ) )).

fof(d13_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k11_fin_topo(A,B) = a_2_4_fin_topo(A,B) ) ) )).

fof(d18_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k12_fin_topo(A,B) = k1_setfam_1(a_2_5_fin_topo(A,B)) ) ) )).

fof(d19_fin_topo,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k13_fin_topo(A,B) = k3_tarski(a_2_6_fin_topo(A,B)) ) ) )).

fof(fraenkel_a_1_0_fin_topo,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_fin_topo(B)
        & l1_fin_topo(B) )
     => ( r2_hidden(A,a_1_0_fin_topo(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(B))
            & A = k1_fin_topo(B,C) ) ) ) )).

fof(fraenkel_a_2_0_fin_topo,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_fin_topo(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_2_0_fin_topo(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(B))
            & A = D
            & ~ r1_xboole_0(k1_fin_topo(B,D),C)
            & ~ r1_xboole_0(k1_fin_topo(B,D),k3_subset_1(u1_struct_0(B),C)) ) ) ) )).

fof(fraenkel_a_2_1_fin_topo,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_fin_topo(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_2_1_fin_topo(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(B))
            & A = D
            & r1_tarski(k1_fin_topo(B,D),C) ) ) ) )).

fof(fraenkel_a_2_2_fin_topo,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_fin_topo(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_2_2_fin_topo(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(B))
            & A = D
            & ~ r1_xboole_0(k1_fin_topo(B,D),C) ) ) ) )).

fof(fraenkel_a_2_3_fin_topo,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_fin_topo(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_2_3_fin_topo(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(B))
            & A = D
            & r2_hidden(D,C)
            & r1_xboole_0(k6_subset_1(u1_struct_0(B),k1_fin_topo(B,D),k6_domain_1(u1_struct_0(B),D)),C) ) ) ) )).

fof(fraenkel_a_2_4_fin_topo,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_fin_topo(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_2_4_fin_topo(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(B))
            & A = D
            & ? [E] :
                ( m1_subset_1(E,u1_struct_0(B))
                & r2_hidden(E,C)
                & r2_hidden(D,k1_fin_topo(B,E)) ) ) ) ) )).

fof(fraenkel_a_2_5_fin_topo,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_fin_topo(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_2_5_fin_topo(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
            & A = D
            & r1_tarski(C,D)
            & v5_fin_topo(D,B) ) ) ) )).

fof(fraenkel_a_2_6_fin_topo,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_fin_topo(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_2_6_fin_topo(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
            & A = D
            & r1_tarski(C,D)
            & v4_fin_topo(D,B) ) ) ) )).
%------------------------------------------------------------------------------