SET007 Axioms: SET007+812.ax


%------------------------------------------------------------------------------
% File     : SET007+812 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Some Set Series in Finite Topological Spaces.
% Version  : [Urb08] axioms.
% English  : Fundamental Concepts for Image Processing

% 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    : fintopo3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   71 (   0 unit)
%            Number of atoms       :  401 (  57 equality)
%            Maximal formula depth :   14 (   8 average)
%            Number of connectives :  405 (  75 ~  ;   0  |; 116  &)
%                                         (   8 <=>; 206 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   12 (   0 propositional; 1-3 arity)
%            Number of functors    :   29 (   5 constant; 0-4 arity)
%            Number of variables   :  209 (   0 singleton; 208 !;   1 ?)
%            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(t1_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_fin_topo(A)
           => r1_tarski(B,k11_fin_topo(A,B)) ) ) ) )).

fof(t2_fintopo3,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,u1_struct_0(A))
             => ( r2_hidden(C,k1_fintopo3(A,B))
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ~ ( r2_hidden(D,k3_subset_1(u1_struct_0(A),B))
                        & r2_hidden(C,k1_fin_topo(A,D)) ) ) ) ) ) ) )).

fof(t3_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_fin_topo(A)
           => r1_tarski(k1_fintopo3(A,B),B) ) ) ) )).

fof(t4_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k1_fintopo3(A,B) = k3_subset_1(u1_struct_0(A),k11_fin_topo(A,k3_subset_1(u1_struct_0(A),B))) ) ) )).

fof(t5_fintopo3,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(k11_fin_topo(A,B),k11_fin_topo(A,C)) ) ) ) ) )).

fof(t6_fintopo3,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(k1_fintopo3(A,B),k1_fintopo3(A,C)) ) ) ) ) )).

fof(t7_fintopo3,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(k8_fin_topo(A,k5_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),k8_fin_topo(A,B),k8_fin_topo(A,C))) ) ) ) )).

fof(t8_fintopo3,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)))
             => k8_fin_topo(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k8_fin_topo(A,B),k8_fin_topo(A,C)) ) ) ) )).

fof(t9_fintopo3,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(k4_subset_1(u1_struct_0(A),k7_fin_topo(A,B),k7_fin_topo(A,C)),k7_fin_topo(A,k4_subset_1(u1_struct_0(A),B,C))) ) ) ) )).

fof(t10_fintopo3,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)))
             => k5_subset_1(u1_struct_0(A),k7_fin_topo(A,B),k7_fin_topo(A,C)) = k7_fin_topo(A,k5_subset_1(u1_struct_0(A),B,C)) ) ) ) )).

fof(t11_fintopo3,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)))
             => k4_subset_1(u1_struct_0(A),k11_fin_topo(A,B),k11_fin_topo(A,C)) = k11_fin_topo(A,k4_subset_1(u1_struct_0(A),B,C)) ) ) ) )).

fof(t12_fintopo3,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)))
             => k5_subset_1(u1_struct_0(A),k1_fintopo3(A,B),k1_fintopo3(A,C)) = k1_fintopo3(A,k5_subset_1(u1_struct_0(A),B,C)) ) ) ) )).

fof(d2_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
                & m2_relset_1(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A))) )
             => ( C = k2_fintopo3(A,B)
              <=> ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                         => ( E = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,D)
                           => k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,k1_nat_1(D,np__1)) = k8_fin_topo(A,E) ) ) )
                  & k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,np__0) = B ) ) ) ) ) )).

fof(d3_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k3_fintopo3(A,B,C) = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),k2_fintopo3(A,B),C) ) ) ) )).

fof(d4_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
                & m2_relset_1(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A))) )
             => ( C = k4_fintopo3(A,B)
              <=> ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                         => ( E = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,D)
                           => k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,k1_nat_1(D,np__1)) = k7_fin_topo(A,E) ) ) )
                  & k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,np__0) = B ) ) ) ) ) )).

fof(d5_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k5_fintopo3(A,B,C) = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),k4_fintopo3(A,B),C) ) ) ) )).

fof(t13_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k3_fintopo3(A,B,k1_nat_1(C,np__1)) = k8_fin_topo(A,k3_fintopo3(A,B,C)) ) ) ) )).

fof(t14_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k3_fintopo3(A,B,np__0) = B ) ) )).

fof(t15_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k3_fintopo3(A,B,np__1) = k8_fin_topo(A,B) ) ) )).

fof(t16_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k3_fintopo3(A,B,np__2) = k8_fin_topo(A,k8_fin_topo(A,B)) ) ) )).

fof(t17_fintopo3,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)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k3_fintopo3(A,k4_subset_1(u1_struct_0(A),B,C),D) = k4_subset_1(u1_struct_0(A),k3_fintopo3(A,B,D),k3_fintopo3(A,C,D)) ) ) ) ) )).

fof(t18_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k5_fintopo3(A,B,k1_nat_1(C,np__1)) = k7_fin_topo(A,k5_fintopo3(A,B,C)) ) ) ) )).

fof(t19_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k5_fintopo3(A,B,np__0) = B ) ) )).

fof(t20_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k5_fintopo3(A,B,np__1) = k7_fin_topo(A,B) ) ) )).

fof(t21_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k5_fintopo3(A,B,np__2) = k7_fin_topo(A,k7_fin_topo(A,B)) ) ) )).

fof(t22_fintopo3,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)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k5_fintopo3(A,k5_subset_1(u1_struct_0(A),B,C),D) = k5_subset_1(u1_struct_0(A),k5_fintopo3(A,B,D),k5_fintopo3(A,C,D)) ) ) ) ) )).

fof(t23_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_fin_topo(A)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_tarski(B,k3_fintopo3(A,B,C)) ) ) ) ) )).

fof(t24_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_fin_topo(A)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_tarski(k5_fintopo3(A,B,C),B) ) ) ) ) )).

fof(t25_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_fin_topo(A)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_tarski(k3_fintopo3(A,B,C),k3_fintopo3(A,B,k1_nat_1(C,np__1))) ) ) ) ) )).

fof(t26_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_fin_topo(A)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_tarski(k5_fintopo3(A,B,k1_nat_1(C,np__1)),k5_fintopo3(A,B,C)) ) ) ) ) )).

fof(t27_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k3_subset_1(u1_struct_0(A),k5_fintopo3(A,k3_subset_1(u1_struct_0(A),B),C)) = k3_fintopo3(A,B,C) ) ) ) )).

fof(t28_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k3_subset_1(u1_struct_0(A),k3_fintopo3(A,k3_subset_1(u1_struct_0(A),B),C)) = k5_fintopo3(A,B,C) ) ) ) )).

fof(t29_fintopo3,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)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k4_subset_1(u1_struct_0(A),k3_fintopo3(A,B,D),k3_fintopo3(A,C,D)) = k3_subset_1(u1_struct_0(A),k5_fintopo3(A,k3_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),B,C)),D)) ) ) ) ) )).

fof(t30_fintopo3,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)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k5_subset_1(u1_struct_0(A),k5_fintopo3(A,B,D),k5_fintopo3(A,C,D)) = k3_subset_1(u1_struct_0(A),k3_fintopo3(A,k3_subset_1(u1_struct_0(A),k5_subset_1(u1_struct_0(A),B,C)),D)) ) ) ) ) )).

fof(d6_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
                & m2_relset_1(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A))) )
             => ( C = k6_fintopo3(A,B)
              <=> ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                         => ( E = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,D)
                           => k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,k1_nat_1(D,np__1)) = k11_fin_topo(A,E) ) ) )
                  & k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,np__0) = B ) ) ) ) ) )).

fof(d7_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k7_fintopo3(A,B,C) = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),k6_fintopo3(A,B),C) ) ) ) )).

fof(d8_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
                & m2_relset_1(C,k5_numbers,k1_zfmisc_1(u1_struct_0(A))) )
             => ( C = k8_fintopo3(A,B)
              <=> ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                         => ( E = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,D)
                           => k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,k1_nat_1(D,np__1)) = k1_fintopo3(A,E) ) ) )
                  & k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),C,np__0) = B ) ) ) ) ) )).

fof(d9_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k9_fintopo3(A,B,C) = k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),k8_fintopo3(A,B),C) ) ) ) )).

fof(t31_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k7_fintopo3(A,B,k1_nat_1(C,np__1)) = k11_fin_topo(A,k7_fintopo3(A,B,C)) ) ) ) )).

fof(t32_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k7_fintopo3(A,B,np__0) = B ) ) )).

fof(t33_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k7_fintopo3(A,B,np__1) = k11_fin_topo(A,B) ) ) )).

fof(t34_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k7_fintopo3(A,B,np__2) = k11_fin_topo(A,k11_fin_topo(A,B)) ) ) )).

fof(t35_fintopo3,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)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k7_fintopo3(A,k4_subset_1(u1_struct_0(A),B,C),D) = k4_subset_1(u1_struct_0(A),k7_fintopo3(A,B,D),k7_fintopo3(A,C,D)) ) ) ) ) )).

fof(t36_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_fin_topo(A)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_tarski(B,k7_fintopo3(A,B,C)) ) ) ) ) )).

fof(t37_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_fin_topo(A)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_tarski(k7_fintopo3(A,B,C),k7_fintopo3(A,B,k1_nat_1(C,np__1))) ) ) ) ) )).

fof(t38_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k9_fintopo3(A,B,k1_nat_1(C,np__1)) = k1_fintopo3(A,k9_fintopo3(A,B,C)) ) ) ) )).

fof(t39_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k9_fintopo3(A,B,np__0) = B ) ) )).

fof(t40_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k9_fintopo3(A,B,np__1) = k1_fintopo3(A,B) ) ) )).

fof(t41_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k9_fintopo3(A,B,np__2) = k1_fintopo3(A,k1_fintopo3(A,B)) ) ) )).

fof(t42_fintopo3,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)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k9_fintopo3(A,k5_subset_1(u1_struct_0(A),B,C),D) = k5_subset_1(u1_struct_0(A),k9_fintopo3(A,B,D),k9_fintopo3(A,C,D)) ) ) ) ) )).

fof(t43_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_fin_topo(A)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_tarski(k9_fintopo3(A,B,C),B) ) ) ) ) )).

fof(t44_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_fin_topo(A)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_tarski(k9_fintopo3(A,B,k1_nat_1(C,np__1)),k9_fintopo3(A,B,C)) ) ) ) ) )).

fof(t45_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k9_fintopo3(A,B,C) = k3_subset_1(u1_struct_0(A),k7_fintopo3(A,k3_subset_1(u1_struct_0(A),B),C)) ) ) ) )).

fof(t46_fintopo3,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)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k5_subset_1(u1_struct_0(A),k9_fintopo3(A,B,D),k9_fintopo3(A,C,D)) = k3_subset_1(u1_struct_0(A),k7_fintopo3(A,k3_subset_1(u1_struct_0(A),k5_subset_1(u1_struct_0(A),B,C)),D)) ) ) ) ) )).

fof(d10_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k10_fintopo3(A,B,C) = k7_fintopo3(A,k1_fin_topo(A,C),B) ) ) ) )).

fof(t47_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k10_fintopo3(A,np__0,B) = k1_fin_topo(A,B) ) ) )).

fof(t48_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k10_fintopo3(A,k1_nat_1(C,np__1),B) = k11_fin_topo(A,k10_fintopo3(A,C,B)) ) ) ) )).

fof(d11_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_fin_topo(B) )
         => ( r1_fintopo3(A,B)
          <=> ( u1_struct_0(A) = u1_struct_0(B)
              & ! [C,D] :
                  ( ( r2_hidden(C,u1_struct_0(A))
                    & r2_hidden(D,u1_struct_0(B)) )
                 => ( r2_hidden(D,k1_funct_1(u1_fin_topo(A),C))
                  <=> r2_hidden(C,k1_funct_1(u1_fin_topo(B),D)) ) ) ) ) ) ) )).

fof(symmetry_r1_fintopo3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & ~ v3_struct_0(B)
        & l1_fin_topo(B) )
     => ( r1_fintopo3(A,B)
       => r1_fintopo3(B,A) ) ) )).

fof(dt_k1_fintopo3,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(k1_fintopo3(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k2_fintopo3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_funct_1(k2_fintopo3(A,B))
        & v1_funct_2(k2_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
        & m2_relset_1(k2_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(dt_k3_fintopo3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k3_fintopo3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k4_fintopo3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_funct_1(k4_fintopo3(A,B))
        & v1_funct_2(k4_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
        & m2_relset_1(k4_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(dt_k5_fintopo3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k5_fintopo3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k6_fintopo3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_funct_1(k6_fintopo3(A,B))
        & v1_funct_2(k6_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
        & m2_relset_1(k6_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(dt_k7_fintopo3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k7_fintopo3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k8_fintopo3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_funct_1(k8_fintopo3(A,B))
        & v1_funct_2(k8_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
        & m2_relset_1(k8_fintopo3(A,B),k5_numbers,k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(dt_k9_fintopo3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k9_fintopo3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k10_fintopo3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k10_fintopo3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(d1_fintopo3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k1_fintopo3(A,B) = a_2_0_fintopo3(A,B) ) ) )).

fof(fraenkel_a_2_0_fintopo3,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_fintopo3(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,k3_subset_1(u1_struct_0(B),C))
                    & r2_hidden(D,k1_fin_topo(B,E)) ) ) ) ) ) )).
%------------------------------------------------------------------------------