SET007 Axioms: SET007+455.ax


%------------------------------------------------------------------------------
% File     : SET007+455 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Closure Operator and the Closure System of Many Sorted Sets
% 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    : closure2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  140 (   4 unit)
%            Number of atoms       :  846 (  52 equality)
%            Maximal formula depth :   15 (   8 average)
%            Number of connectives :  731 (  25 ~  ;   0  |; 360  &)
%                                         (  30 <=>; 316 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   46 (   1 propositional; 0-4 arity)
%            Number of functors    :   48 (   1 constant; 0-4 arity)
%            Number of variables   :  435 (   0 singleton; 418 !;  17 ?)
%            Maximal term depth    :    4 (   1 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(cc1_closure2,axiom,(
    ! [A] :
      ( v1_xboole_0(A)
     => v1_fraenkel(A) ) )).

fof(rc1_closure2,axiom,(
    ? [A] :
      ( v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_finset_1(A)
      & v1_fraenkel(A) ) )).

fof(fc1_closure2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( ~ v1_xboole_0(k2_tarski(A,B))
        & v1_finset_1(k2_tarski(A,B))
        & v1_fraenkel(k2_tarski(A,B)) ) ) )).

fof(fc2_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ( ~ v1_xboole_0(k1_closure2(A,B))
        & v1_fraenkel(k1_closure2(A,B))
        & v1_pralg_2(k1_closure2(A,B)) ) ) )).

fof(rc2_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ? [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
          & ~ v1_xboole_0(C)
          & v1_fraenkel(C)
          & v1_pralg_2(C) ) ) )).

fof(rc3_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ? [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
          & v1_xboole_0(C)
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v2_funct_1(C)
          & v1_finset_1(C)
          & v1_fraenkel(C) ) ) )).

fof(fc3_closure2,axiom,(
    ! [A] :
      ( ( v1_xboole_0(A)
        & v1_fraenkel(A) )
     => ( v1_xboole_0(k3_closure2(A))
        & v1_relat_1(k3_closure2(A))
        & v1_funct_1(k3_closure2(A))
        & v2_funct_1(k3_closure2(A))
        & v1_finset_1(k3_closure2(A))
        & v1_fraenkel(k3_closure2(A)) ) ) )).

fof(fc4_closure2,axiom,(
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => ( v1_relat_1(k4_closure2(A,B,C))
        & v3_relat_1(k4_closure2(A,B,C))
        & v1_funct_1(k4_closure2(A,B,C))
        & v1_pre_circ(k4_closure2(A,B,C),A) ) ) )).

fof(fc5_closure2,axiom,(
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => ( v1_relat_1(k4_closure2(A,B,C))
        & v2_relat_1(k4_closure2(A,B,C))
        & v1_funct_1(k4_closure2(A,B,C)) ) ) )).

fof(rc4_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ? [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
          & ~ v1_xboole_0(C)
          & v1_fraenkel(C)
          & v1_pralg_2(C)
          & v1_closure2(C,A,B)
          & v2_closure2(C,A,B)
          & v3_closure2(C,A,B)
          & v4_closure2(C,A,B)
          & v5_closure2(C,A,B)
          & v6_closure2(C,A,B) ) ) )).

fof(cc2_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( v2_closure2(C,A,B)
           => ( v1_fraenkel(C)
              & v1_closure2(C,A,B) ) ) ) ) )).

fof(cc3_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( v4_closure2(C,A,B)
           => ( v1_fraenkel(C)
              & v3_closure2(C,A,B) ) ) ) ) )).

fof(cc4_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( v4_closure2(C,A,B)
           => ( v1_fraenkel(C)
              & v5_closure2(C,A,B) ) ) ) ) )).

fof(cc5_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( v5_closure2(C,A,B)
           => ( ~ v1_xboole_0(C)
              & v1_fraenkel(C) ) ) ) ) )).

fof(cc6_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( v2_closure2(C,A,B)
           => ( v1_fraenkel(C)
              & v6_closure2(C,A,B) ) ) ) ) )).

fof(cc7_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( v6_closure2(C,A,B)
           => ( ~ v1_xboole_0(C)
              & v1_fraenkel(C) ) ) ) ) )).

fof(rc5_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ? [C] :
          ( m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
          & ~ v1_xboole_0(C)
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
          & v1_partfun1(C,k6_closure2(A,B),k6_closure2(A,B))
          & v7_closure2(C,A,B)
          & v8_closure2(C,A,B)
          & v9_closure2(C,A,B)
          & v10_closure2(C,A,B) ) ) )).

fof(cc8_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
         => ( ( v1_funct_1(C)
              & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
              & v10_closure2(C,A,B) )
           => ( ~ v1_xboole_0(C)
              & v1_funct_1(C)
              & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
              & v1_partfun1(C,k6_closure2(A,B),k6_closure2(A,B))
              & v8_closure2(C,A,B) ) ) ) ) )).

fof(rc6_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ? [B] :
          ( l1_closure2(B,A)
          & v11_closure2(B,A) ) ) )).

fof(fc6_closure2,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & l2_msualg_1(B,A) )
     => ( v11_closure2(k9_closure2(A,B),A)
        & v12_closure2(k9_closure2(A,B),A)
        & v13_closure2(k9_closure2(A,B),A)
        & v14_closure2(k9_closure2(A,B),A)
        & v15_closure2(k9_closure2(A,B),A)
        & v16_closure2(k9_closure2(A,B),A)
        & v17_closure2(k9_closure2(A,B),A) ) ) )).

fof(fc7_closure2,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & v5_msualg_1(B,A)
        & l2_msualg_1(B,A) )
     => ( v5_msualg_1(k9_closure2(A,B),A)
        & v11_closure2(k9_closure2(A,B),A)
        & v12_closure2(k9_closure2(A,B),A)
        & v13_closure2(k9_closure2(A,B),A)
        & v14_closure2(k9_closure2(A,B),A)
        & v15_closure2(k9_closure2(A,B),A)
        & v16_closure2(k9_closure2(A,B),A)
        & v17_closure2(k9_closure2(A,B),A) ) ) )).

fof(rc7_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ? [B] :
          ( l1_closure2(B,A)
          & v5_msualg_1(B,A)
          & v11_closure2(B,A)
          & v12_closure2(B,A)
          & v13_closure2(B,A)
          & v14_closure2(B,A)
          & v15_closure2(B,A)
          & v16_closure2(B,A)
          & v17_closure2(B,A) ) ) )).

fof(fc8_closure2,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & v12_closure2(B,A)
        & l1_closure2(B,A) )
     => ( v1_fraenkel(u1_closure2(A,B))
        & v1_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) )).

fof(fc9_closure2,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & v13_closure2(B,A)
        & l1_closure2(B,A) )
     => ( ~ v1_xboole_0(u1_closure2(A,B))
        & v1_fraenkel(u1_closure2(A,B))
        & v1_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
        & v2_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
        & v6_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) )).

fof(fc10_closure2,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & v14_closure2(B,A)
        & l1_closure2(B,A) )
     => ( v1_fraenkel(u1_closure2(A,B))
        & v3_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) )).

fof(fc11_closure2,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & v15_closure2(B,A)
        & l1_closure2(B,A) )
     => ( ~ v1_xboole_0(u1_closure2(A,B))
        & v1_fraenkel(u1_closure2(A,B))
        & v3_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
        & v4_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
        & v5_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) )).

fof(fc12_closure2,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & v16_closure2(B,A)
        & l1_closure2(B,A) )
     => ( ~ v1_xboole_0(u1_closure2(A,B))
        & v1_fraenkel(u1_closure2(A,B))
        & v5_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) )).

fof(fc13_closure2,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & v17_closure2(B,A)
        & l1_closure2(B,A) )
     => ( ~ v1_xboole_0(u1_closure2(A,B))
        & v1_fraenkel(u1_closure2(A,B))
        & v6_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) )).

fof(fc14_closure2,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),B))) )
     => ( v5_msualg_1(g1_closure2(A,B,C),A)
        & v11_closure2(g1_closure2(A,B,C),A) ) ) )).

fof(fc15_closure2,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & l2_msualg_1(B,A)
        & v1_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) )
     => ( v11_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
        & v12_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A) ) ) )).

fof(fc16_closure2,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & l2_msualg_1(B,A)
        & v2_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) )
     => ( v11_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
        & v12_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
        & v13_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A) ) ) )).

fof(fc17_closure2,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & l2_msualg_1(B,A)
        & v3_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) )
     => ( v11_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
        & v14_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A) ) ) )).

fof(fc18_closure2,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & l2_msualg_1(B,A)
        & v4_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) )
     => ( v11_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
        & v14_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
        & v15_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A) ) ) )).

fof(fc19_closure2,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & l2_msualg_1(B,A)
        & v5_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) )
     => ( v11_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
        & v16_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A) ) ) )).

fof(fc20_closure2,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & l2_msualg_1(B,A)
        & v6_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) )
     => ( v11_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
        & v17_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A) ) ) )).

fof(cc9_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_closure2(B,A)
         => ( v13_closure2(B,A)
           => v12_closure2(B,A) ) ) ) )).

fof(cc10_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_closure2(B,A)
         => ( v15_closure2(B,A)
           => v14_closure2(B,A) ) ) ) )).

fof(cc11_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_closure2(B,A)
         => ( v15_closure2(B,A)
           => v16_closure2(B,A) ) ) ) )).

fof(cc12_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_closure2(B,A)
         => ( v13_closure2(B,A)
           => v17_closure2(B,A) ) ) ) )).

fof(t1_closure2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( r1_tarski(B,C)
               => r1_tarski(k8_funct_2(A,A,k6_partfun1(A),B),k8_funct_2(A,A,k6_partfun1(A),C)) ) ) ) ) )).

fof(t2_closure2,axiom,(
    $true )).

fof(t3_closure2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m4_pboole(C,A,B)
             => r1_tarski(k2_relat_1(C),k3_tarski(k2_relat_1(k7_mssubfam(A,B)))) ) ) ) )).

fof(d1_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( C = k1_closure2(A,B)
        <=> ! [D] :
              ( r2_hidden(D,C)
            <=> m4_pboole(D,A,B) ) ) ) )).

fof(t4_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => m1_subset_1(k4_subset_1(k1_closure2(A,B),C,D),k1_zfmisc_1(k1_closure2(A,B))) ) ) ) )).

fof(t5_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => m1_subset_1(k5_subset_1(k1_closure2(A,B),C,D),k1_zfmisc_1(k1_closure2(A,B))) ) ) ) )).

fof(t6_closure2,axiom,(
    ! [A,B,C] :
      ( m1_pboole(C,A)
     => ! [D] :
          ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,C)))
         => m1_subset_1(k4_xboole_0(D,B),k1_zfmisc_1(k1_closure2(A,C))) ) ) )).

fof(t7_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => m1_subset_1(k7_subset_1(k1_closure2(A,B),C,D),k1_zfmisc_1(k1_closure2(A,B))) ) ) ) )).

fof(t8_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( r2_pboole(A,B,C)
           => m1_subset_1(k1_tarski(B),k1_zfmisc_1(k1_closure2(A,C))) ) ) ) )).

fof(t9_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_pboole(D,A)
             => ( ( r2_pboole(A,B,C)
                  & r2_pboole(A,D,C) )
               => m1_subset_1(k2_tarski(B,D),k1_zfmisc_1(k1_closure2(A,C))) ) ) ) ) )).

fof(t10_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_closure2(C,A,B,k2_closure2(A,B))
         => ! [D] :
              ( m1_closure2(D,A,B,k2_closure2(A,B))
             => r2_hidden(k3_pboole(A,C,D),k2_closure2(A,B)) ) ) ) )).

fof(t11_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_closure2(C,A,B,k2_closure2(A,B))
         => ! [D] :
              ( m1_closure2(D,A,B,k2_closure2(A,B))
             => r2_hidden(k2_pboole(A,C,D),k2_closure2(A,B)) ) ) ) )).

fof(t12_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_closure2(D,A,C,k2_closure2(A,C))
             => r2_hidden(k4_pboole(A,D,B),k2_closure2(A,C)) ) ) ) )).

fof(t13_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_closure2(C,A,B,k2_closure2(A,B))
         => ! [D] :
              ( m1_closure2(D,A,B,k2_closure2(A,B))
             => r2_hidden(k5_pboole(A,C,D),k2_closure2(A,B)) ) ) ) )).

fof(d2_closure2,axiom,(
    $true )).

fof(t14_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( ~ v1_xboole_0(C)
            & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
         => k1_relat_1(k3_closure2(C)) = A ) ) )).

fof(d4_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( ( C != k1_xboole_0
             => k4_closure2(A,B,C) = k3_closure2(C) )
            & ( C = k1_xboole_0
             => k4_closure2(A,B,C) = k1_pboole(A) ) ) ) ) )).

fof(t16_closure2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k1_relat_1(k3_closure2(k1_tarski(A))) = k1_relat_1(A) ) )).

fof(t17_closure2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => k1_relat_1(k3_closure2(k2_tarski(A,B))) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B)) ) ) )).

fof(t18_closure2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k1_relat_1(B))
       => k1_funct_1(k3_closure2(k1_tarski(B)),A) = k1_tarski(k1_funct_1(B,A)) ) ) )).

fof(t19_closure2,axiom,(
    ! [A,B,C] :
      ( m1_pboole(C,B)
     => ! [D] :
          ( ( v1_relat_1(D)
            & v1_funct_1(D) )
         => ! [E] :
              ( m1_subset_1(E,k1_zfmisc_1(k1_closure2(B,C)))
             => ( ( r2_hidden(A,B)
                  & E = k1_tarski(D) )
               => k1_funct_1(k4_closure2(B,C,E),A) = k1_tarski(k1_funct_1(D,A)) ) ) ) ) )).

fof(t20_closure2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(k3_closure2(k2_tarski(B,C))))
           => k1_funct_1(k3_closure2(k2_tarski(B,C)),A) = k2_tarski(k1_funct_1(B,A),k1_funct_1(C,A)) ) ) ) )).

fof(t21_closure2,axiom,(
    ! [A,B,C] :
      ( m1_pboole(C,B)
     => ! [D] :
          ( ( v1_relat_1(D)
            & v1_funct_1(D) )
         => ! [E] :
              ( ( v1_relat_1(E)
                & v1_funct_1(E) )
             => ! [F] :
                  ( m1_subset_1(F,k1_zfmisc_1(k1_closure2(B,C)))
                 => ( ( r2_hidden(A,B)
                      & F = k2_tarski(D,E) )
                   => k1_funct_1(k4_closure2(B,C,F),A) = k2_tarski(k1_funct_1(D,A),k1_funct_1(E,A)) ) ) ) ) ) )).

fof(t22_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => ( r2_hidden(C,D)
               => r1_pboole(A,C,k5_closure2(A,B,D)) ) ) ) ) )).

fof(t23_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_pboole(D,A)
             => ! [E] :
                  ( m1_subset_1(E,k1_zfmisc_1(k1_closure2(A,B)))
                 => ( E = k2_tarski(C,D)
                   => r6_pboole(A,k2_mboolean(A,k5_closure2(A,B,E)),k2_pboole(A,C,D)) ) ) ) ) ) )).

fof(t24_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m1_closure2(D,A,B,k2_closure2(A,B))
             => ! [E] :
                  ( m1_closure2(E,A,B,k2_closure2(A,B))
                 => ( C = k2_tarski(D,E)
                   => r6_pboole(A,k6_mssubfam(A,B,k5_closure2(A,B,C)),k3_pboole(A,D,E)) ) ) ) ) ) )).

fof(t25_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m4_pboole(D,A,B)
             => ( ! [E] :
                    ( m1_pboole(E,A)
                   => ( r2_hidden(E,C)
                     => r2_pboole(A,D,E) ) )
               => r2_pboole(A,D,k6_mssubfam(A,B,k5_closure2(A,B,C))) ) ) ) ) )).

fof(t26_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => r6_pboole(A,k5_closure2(A,B,k2_closure2(A,B)),k7_mssubfam(A,B)) ) )).

fof(d5_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( v1_closure2(C,A,B)
          <=> ! [D] :
                ( m1_pboole(D,A)
               => ! [E] :
                    ( m1_pboole(E,A)
                   => ( ( r2_hidden(D,C)
                        & r2_hidden(E,C) )
                     => r2_hidden(k2_pboole(A,D,E),C) ) ) ) ) ) ) )).

fof(d6_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( v2_closure2(C,A,B)
          <=> ! [D] :
                ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
               => ( r1_tarski(D,C)
                 => r2_hidden(k2_mboolean(A,k5_closure2(A,B,D)),C) ) ) ) ) ) )).

fof(d7_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( v3_closure2(C,A,B)
          <=> ! [D] :
                ( m1_pboole(D,A)
               => ! [E] :
                    ( m1_pboole(E,A)
                   => ( ( r2_hidden(D,C)
                        & r2_hidden(E,C) )
                     => r2_hidden(k3_pboole(A,D,E),C) ) ) ) ) ) ) )).

fof(d8_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( v4_closure2(C,A,B)
          <=> ! [D] :
                ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
               => ( r1_tarski(D,C)
                 => r2_hidden(k6_mssubfam(A,B,k5_closure2(A,B,D)),C) ) ) ) ) ) )).

fof(d9_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( v5_closure2(C,A,B)
          <=> r2_hidden(B,C) ) ) ) )).

fof(d10_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( v6_closure2(C,A,B)
          <=> r2_hidden(k1_pboole(A),C) ) ) ) )).

fof(d11_closure2,axiom,(
    $true )).

fof(d12_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
            & m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
         => ( v7_closure2(C,A,B)
          <=> ! [D] :
                ( m1_closure2(D,A,B,k6_closure2(A,B))
               => r2_pboole(A,D,k7_closure2(A,B,C,D)) ) ) ) ) )).

fof(d13_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
            & m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
         => ( v8_closure2(C,A,B)
          <=> ! [D] :
                ( m1_closure2(D,A,B,k6_closure2(A,B))
               => ! [E] :
                    ( m1_closure2(E,A,B,k6_closure2(A,B))
                   => ( r2_pboole(A,D,E)
                     => r2_pboole(A,k7_closure2(A,B,C,D),k7_closure2(A,B,C,E)) ) ) ) ) ) ) )).

fof(d14_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
            & m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
         => ( v9_closure2(C,A,B)
          <=> ! [D] :
                ( m1_closure2(D,A,B,k6_closure2(A,B))
               => r6_pboole(A,k7_closure2(A,B,C,D),k7_closure2(A,B,C,k7_closure2(A,B,C,D))) ) ) ) ) )).

fof(d15_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
            & m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
         => ( v10_closure2(C,A,B)
          <=> ! [D] :
                ( m1_closure2(D,A,B,k6_closure2(A,B))
               => ! [E] :
                    ( m1_closure2(E,A,B,k6_closure2(A,B))
                   => k1_funct_1(C,k2_pboole(A,D,E)) = k2_pboole(A,k7_closure2(A,B,C,D),k7_closure2(A,B,C,E)) ) ) ) ) ) )).

fof(t27_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ( v1_funct_1(k6_partfun1(k6_closure2(A,B)))
        & v1_funct_2(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B))
        & v7_closure2(k6_partfun1(k6_closure2(A,B)),A,B)
        & m2_relset_1(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B)) ) ) )).

fof(t28_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ( v1_funct_1(k6_partfun1(k6_closure2(A,B)))
        & v1_funct_2(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B))
        & v8_closure2(k6_partfun1(k6_closure2(A,B)),A,B)
        & m2_relset_1(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B)) ) ) )).

fof(t29_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ( v1_funct_1(k6_partfun1(k6_closure2(A,B)))
        & v1_funct_2(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B))
        & v9_closure2(k6_partfun1(k6_closure2(A,B)),A,B)
        & m2_relset_1(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B)) ) ) )).

fof(t30_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ( v1_funct_1(k6_partfun1(k6_closure2(A,B)))
        & v1_funct_2(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B))
        & v10_closure2(k6_partfun1(k6_closure2(A,B)),A,B)
        & m2_relset_1(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B)) ) ) )).

fof(t31_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_closure2(C,A,B,k2_closure2(A,B))
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
                & m2_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
             => ( ( r6_pboole(A,C,B)
                  & v7_closure2(D,A,B) )
               => r6_pboole(A,C,k7_closure2(A,B,D,C)) ) ) ) ) )).

fof(t32_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
            & m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
         => ( ( v7_closure2(C,A,B)
              & ! [D] :
                  ( m1_closure2(D,A,B,k6_closure2(A,B))
                 => r2_pboole(A,k7_closure2(A,B,C,D),D) ) )
           => v9_closure2(C,A,B) ) ) ) )).

fof(t33_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_closure2(C,A,B,k2_closure2(A,B))
         => ! [D] :
              ( m1_closure2(D,A,B,k2_closure2(A,B))
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,k6_closure2(A,B),k6_closure2(A,B))
                    & m2_relset_1(E,k6_closure2(A,B),k6_closure2(A,B)) )
                 => ! [F] :
                      ( m1_closure2(F,A,B,k6_closure2(A,B))
                     => ( ( r6_pboole(A,F,k3_pboole(A,C,D))
                          & v8_closure2(E,A,B) )
                       => r2_pboole(A,k7_closure2(A,B,E,F),k3_pboole(A,k7_closure2(A,B,E,C),k7_closure2(A,B,E,D))) ) ) ) ) ) ) )).

fof(t34_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_closure2(C,A,B,k2_closure2(A,B))
         => ! [D] :
              ( m1_closure2(D,A,B,k2_closure2(A,B))
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,k6_closure2(A,B),k6_closure2(A,B))
                    & m2_relset_1(E,k6_closure2(A,B),k6_closure2(A,B)) )
                 => ! [F] :
                      ( m1_closure2(F,A,B,k6_closure2(A,B))
                     => ( ( r6_pboole(A,F,k4_pboole(A,C,D))
                          & v10_closure2(E,A,B) )
                       => r2_pboole(A,k4_pboole(A,k7_closure2(A,B,E,C),k7_closure2(A,B,E,D)),k7_closure2(A,B,E,F)) ) ) ) ) ) ) )).

fof(t35_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
            & m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
                & m2_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
             => ( ( v7_closure2(C,A,B)
                  & v7_closure2(D,A,B) )
               => v7_closure2(k8_closure2(A,B,D,C),A,B) ) ) ) ) )).

fof(t36_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
            & m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
                & m2_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
             => ( ( v8_closure2(C,A,B)
                  & v8_closure2(D,A,B) )
               => v8_closure2(k8_closure2(A,B,D,C),A,B) ) ) ) ) )).

fof(t37_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
            & m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
                & m2_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
             => ( ( v9_closure2(C,A,B)
                  & v9_closure2(D,A,B)
                  & k8_closure2(A,B,D,C) = k8_closure2(A,B,C,D) )
               => v9_closure2(k8_closure2(A,B,D,C),A,B) ) ) ) ) )).

fof(t38_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
            & m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
                & m2_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
             => ( ( v10_closure2(C,A,B)
                  & v10_closure2(D,A,B) )
               => v10_closure2(k8_closure2(A,B,D,C),A,B) ) ) ) ) )).

fof(d16_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_closure2(B,A)
         => ( v12_closure2(B,A)
          <=> v1_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) )).

fof(d17_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_closure2(B,A)
         => ( v13_closure2(B,A)
          <=> v2_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) )).

fof(d18_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_closure2(B,A)
         => ( v14_closure2(B,A)
          <=> v3_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) )).

fof(d19_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_closure2(B,A)
         => ( v15_closure2(B,A)
          <=> v4_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) )).

fof(d20_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_closure2(B,A)
         => ( v16_closure2(B,A)
          <=> v5_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) )).

fof(d21_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_closure2(B,A)
         => ( v17_closure2(B,A)
          <=> v6_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) )).

fof(d22_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l2_msualg_1(B,A)
         => k9_closure2(A,B) = g1_closure2(A,u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) ) ) )).

fof(t40_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( v15_closure2(B,A)
            & l1_closure2(B,A) )
         => ! [C] :
              ( m1_closure2(C,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
                    & m2_relset_1(D,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) )
                 => ( ( r2_hidden(C,u1_closure2(A,B))
                      & ! [E] :
                          ( m1_closure2(E,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
                         => r6_pboole(u1_struct_0(A),k7_closure2(u1_struct_0(A),u4_msualg_1(A,B),D,E),k11_closure2(A,B,E)) ) )
                   => r6_pboole(u1_struct_0(A),k7_closure2(u1_struct_0(A),u4_msualg_1(A,B),D,C),C) ) ) ) ) ) )).

fof(t41_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( v15_closure2(B,A)
            & l1_closure2(B,A) )
         => ! [C] :
              ( m1_closure2(C,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
                    & m2_relset_1(D,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) )
                 => ( ( r6_pboole(u1_struct_0(A),k7_closure2(u1_struct_0(A),u4_msualg_1(A,B),D,C),C)
                      & ! [E] :
                          ( m1_closure2(E,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
                         => r6_pboole(u1_struct_0(A),k7_closure2(u1_struct_0(A),u4_msualg_1(A,B),D,E),k11_closure2(A,B,E)) ) )
                   => r2_hidden(C,u1_closure2(A,B)) ) ) ) ) ) )).

fof(t42_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( v15_closure2(B,A)
            & l1_closure2(B,A) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
                & m2_relset_1(C,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) )
             => ( ! [D] :
                    ( m1_closure2(D,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
                   => r6_pboole(u1_struct_0(A),k7_closure2(u1_struct_0(A),u4_msualg_1(A,B),C,D),k11_closure2(A,B,D)) )
               => ( v7_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
                  & v8_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
                  & v9_closure2(C,u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) ) ) )).

fof(d25_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( v15_closure2(B,A)
            & l1_closure2(B,A) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
                & v7_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
                & v8_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
                & v9_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
                & m2_relset_1(C,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) )
             => ( C = k12_closure2(A,B)
              <=> ! [D] :
                    ( m1_closure2(D,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
                   => r6_pboole(u1_struct_0(A),k7_closure2(u1_struct_0(A),u4_msualg_1(A,B),C,D),k11_closure2(A,B,D)) ) ) ) ) ) )).

fof(t43_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_pboole(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B))
                & v7_closure2(C,u1_struct_0(A),B)
                & v8_closure2(C,u1_struct_0(A),B)
                & v9_closure2(C,u1_struct_0(A),B)
                & m2_relset_1(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B)) )
             => k12_closure2(A,k10_closure2(A,B,C)) = C ) ) ) )).

fof(t44_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( v15_closure2(B,A)
            & l1_closure2(B,A) )
         => k10_closure2(A,u4_msualg_1(A,B),k12_closure2(A,B)) = g1_closure2(A,u4_msualg_1(A,B),u1_closure2(A,B)) ) ) )).

fof(dt_m1_closure2,axiom,(
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => ! [D] :
          ( m1_closure2(D,A,B,C)
         => m4_pboole(D,A,B) ) ) )).

fof(existence_m1_closure2,axiom,(
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => ? [D] : m1_closure2(D,A,B,C) ) )).

fof(redefinition_m1_closure2,axiom,(
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => ! [D] :
          ( m1_closure2(D,A,B,C)
        <=> m1_subset_1(D,C) ) ) )).

fof(dt_l1_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_closure2(B,A)
         => l2_msualg_1(B,A) ) ) )).

fof(existence_l1_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ? [B] : l1_closure2(B,A) ) )).

fof(abstractness_v11_closure2,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & l1_closure2(B,A) )
     => ( v11_closure2(B,A)
       => B = g1_closure2(A,u4_msualg_1(A,B),u1_closure2(A,B)) ) ) )).

fof(dt_k1_closure2,axiom,(
    $true )).

fof(dt_k2_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => m1_subset_1(k2_closure2(A,B),k1_zfmisc_1(k1_closure2(A,B))) ) )).

fof(redefinition_k2_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => k2_closure2(A,B) = k1_closure2(A,B) ) )).

fof(dt_k3_closure2,axiom,(
    ! [A] :
      ( v1_fraenkel(A)
     => ( v1_relat_1(k3_closure2(A))
        & v1_funct_1(k3_closure2(A)) ) ) )).

fof(dt_k4_closure2,axiom,(
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => m1_pboole(k4_closure2(A,B,C),A) ) )).

fof(dt_k5_closure2,axiom,(
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => m4_pboole(k5_closure2(A,B,C),A,k1_mboolean(A,B)) ) )).

fof(redefinition_k5_closure2,axiom,(
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => k5_closure2(A,B,C) = k4_closure2(A,B,C) ) )).

fof(dt_k6_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ( v1_closure2(k6_closure2(A,B),A,B)
        & v2_closure2(k6_closure2(A,B),A,B)
        & v3_closure2(k6_closure2(A,B),A,B)
        & v4_closure2(k6_closure2(A,B),A,B)
        & v5_closure2(k6_closure2(A,B),A,B)
        & v6_closure2(k6_closure2(A,B),A,B)
        & m1_subset_1(k6_closure2(A,B),k1_zfmisc_1(k1_closure2(A,B))) ) ) )).

fof(redefinition_k6_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => k6_closure2(A,B) = k1_closure2(A,B) ) )).

fof(dt_k7_closure2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
        & m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
        & m1_subset_1(D,k6_closure2(A,B)) )
     => m1_closure2(k7_closure2(A,B,C,D),A,B,k6_closure2(A,B)) ) )).

fof(redefinition_k7_closure2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
        & m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
        & m1_subset_1(D,k6_closure2(A,B)) )
     => k7_closure2(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k8_closure2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
        & m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
        & v1_funct_1(D)
        & v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
        & m1_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
     => ( v1_funct_1(k8_closure2(A,B,C,D))
        & v1_funct_2(k8_closure2(A,B,C,D),k6_closure2(A,B),k6_closure2(A,B))
        & m2_relset_1(k8_closure2(A,B,C,D),k6_closure2(A,B),k6_closure2(A,B)) ) ) )).

fof(redefinition_k8_closure2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
        & m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
        & v1_funct_1(D)
        & v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
        & m1_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
     => k8_closure2(A,B,C,D) = k5_relat_1(C,D) ) )).

fof(dt_k9_closure2,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & l2_msualg_1(B,A) )
     => l1_closure2(k9_closure2(A,B),A) ) )).

fof(dt_k10_closure2,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & m1_pboole(B,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B))
        & v7_closure2(C,u1_struct_0(A),B)
        & v8_closure2(C,u1_struct_0(A),B)
        & v9_closure2(C,u1_struct_0(A),B)
        & m1_relset_1(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B)) )
     => ( v11_closure2(k10_closure2(A,B,C),A)
        & v15_closure2(k10_closure2(A,B,C),A)
        & l1_closure2(k10_closure2(A,B,C),A) ) ) )).

fof(dt_k11_closure2,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & v15_closure2(B,A)
        & l1_closure2(B,A)
        & m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B)) )
     => m1_closure2(k11_closure2(A,B,C),u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) ) )).

fof(dt_k12_closure2,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & v15_closure2(B,A)
        & l1_closure2(B,A) )
     => ( v1_funct_1(k12_closure2(A,B))
        & v1_funct_2(k12_closure2(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
        & v7_closure2(k12_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
        & v8_closure2(k12_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
        & v9_closure2(k12_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
        & m2_relset_1(k12_closure2(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) ) ) )).

fof(dt_u1_closure2,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & l1_closure2(B,A) )
     => m1_subset_1(u1_closure2(A,B),k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) ) )).

fof(dt_g1_closure2,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),B))) )
     => ( v11_closure2(g1_closure2(A,B,C),A)
        & l1_closure2(g1_closure2(A,B,C),A) ) ) )).

fof(free_g1_closure2,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),B))) )
     => ! [D,E,F] :
          ( g1_closure2(A,B,C) = g1_closure2(D,E,F)
         => ( A = D
            & B = E
            & C = F ) ) ) )).

fof(d3_closure2,axiom,(
    ! [A] :
      ( v1_fraenkel(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( A != k1_xboole_0
             => ( B = k3_closure2(A)
              <=> ? [C] :
                    ( ~ v1_xboole_0(C)
                    & v1_fraenkel(C)
                    & C = A
                    & k1_relat_1(B) = k1_setfam_1(a_1_0_closure2(C))
                    & ! [D] :
                        ( r2_hidden(D,k1_relat_1(B))
                       => k1_funct_1(B,D) = a_2_0_closure2(C,D) ) ) ) )
            & ( A = k1_xboole_0
             => ( B = k3_closure2(A)
              <=> B = k1_xboole_0 ) ) ) ) ) )).

fof(t15_closure2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ( ~ v1_xboole_0(C)
           => ! [D] :
                ( r2_hidden(D,A)
               => k1_funct_1(k4_closure2(A,B,C),D) = a_4_0_closure2(A,B,C,D) ) ) ) ) )).

fof(t39_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_pboole(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B))
                & v7_closure2(C,u1_struct_0(A),B)
                & v8_closure2(C,u1_struct_0(A),B)
                & m2_relset_1(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B)) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(u1_struct_0(A),B)))
                 => ( D = a_3_0_closure2(A,B,C)
                   => ( v15_closure2(g1_closure2(A,B,D),A)
                      & l1_closure2(g1_closure2(A,B,D),A) ) ) ) ) ) ) )).

fof(d23_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_pboole(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B))
                & v7_closure2(C,u1_struct_0(A),B)
                & v8_closure2(C,u1_struct_0(A),B)
                & v9_closure2(C,u1_struct_0(A),B)
                & m2_relset_1(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B)) )
             => ! [D] :
                  ( ( v11_closure2(D,A)
                    & v15_closure2(D,A)
                    & l1_closure2(D,A) )
                 => ( D = k10_closure2(A,B,C)
                  <=> ( r6_pboole(u1_struct_0(A),u4_msualg_1(A,D),B)
                      & u1_closure2(A,D) = a_3_1_closure2(A,B,C) ) ) ) ) ) ) )).

fof(d24_closure2,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( ( v15_closure2(B,A)
            & l1_closure2(B,A) )
         => ! [C] :
              ( m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B))
             => ! [D] :
                  ( m1_closure2(D,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
                 => ( D = k11_closure2(A,B,C)
                  <=> ? [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B))))
                        & r6_pboole(u1_struct_0(A),D,k6_mssubfam(u1_struct_0(A),u4_msualg_1(A,B),k5_closure2(u1_struct_0(A),u4_msualg_1(A,B),E)))
                        & E = a_3_2_closure2(A,B,C) ) ) ) ) ) ) )).

fof(fraenkel_a_1_0_closure2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_fraenkel(B) )
     => ( r2_hidden(A,a_1_0_closure2(B))
      <=> ? [C] :
            ( m1_subset_1(C,B)
            & A = k1_relat_1(C) ) ) ) )).

fof(fraenkel_a_2_0_closure2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_fraenkel(B) )
     => ( r2_hidden(A,a_2_0_closure2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,B)
            & A = k1_funct_1(D,C) ) ) ) )).

fof(fraenkel_a_4_0_closure2,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_pboole(C,B)
        & m1_subset_1(D,k1_zfmisc_1(k1_closure2(B,C))) )
     => ( r2_hidden(A,a_4_0_closure2(B,C,D,E))
      <=> ? [F] :
            ( m1_closure2(F,B,C,k2_closure2(B,C))
            & A = k1_funct_1(F,E)
            & r2_hidden(F,D) ) ) ) )).

fof(fraenkel_a_3_0_closure2,axiom,(
    ! [A,B,C,D] :
      ( ( l1_struct_0(B)
        & m1_pboole(C,u1_struct_0(B))
        & v1_funct_1(D)
        & v1_funct_2(D,k6_closure2(u1_struct_0(B),C),k6_closure2(u1_struct_0(B),C))
        & v7_closure2(D,u1_struct_0(B),C)
        & v8_closure2(D,u1_struct_0(B),C)
        & m2_relset_1(D,k6_closure2(u1_struct_0(B),C),k6_closure2(u1_struct_0(B),C)) )
     => ( r2_hidden(A,a_3_0_closure2(B,C,D))
      <=> ? [E] :
            ( m1_closure2(E,u1_struct_0(B),C,k6_closure2(u1_struct_0(B),C))
            & A = E
            & r6_pboole(u1_struct_0(B),k7_closure2(u1_struct_0(B),C,D,E),E) ) ) ) )).

fof(fraenkel_a_3_1_closure2,axiom,(
    ! [A,B,C,D] :
      ( ( l1_struct_0(B)
        & m1_pboole(C,u1_struct_0(B))
        & v1_funct_1(D)
        & v1_funct_2(D,k6_closure2(u1_struct_0(B),C),k6_closure2(u1_struct_0(B),C))
        & v7_closure2(D,u1_struct_0(B),C)
        & v8_closure2(D,u1_struct_0(B),C)
        & v9_closure2(D,u1_struct_0(B),C)
        & m2_relset_1(D,k6_closure2(u1_struct_0(B),C),k6_closure2(u1_struct_0(B),C)) )
     => ( r2_hidden(A,a_3_1_closure2(B,C,D))
      <=> ? [E] :
            ( m1_closure2(E,u1_struct_0(B),C,k6_closure2(u1_struct_0(B),C))
            & A = E
            & r6_pboole(u1_struct_0(B),k7_closure2(u1_struct_0(B),C,D,E),E) ) ) ) )).

fof(fraenkel_a_3_2_closure2,axiom,(
    ! [A,B,C,D] :
      ( ( l1_struct_0(B)
        & v15_closure2(C,B)
        & l1_closure2(C,B)
        & m4_pboole(D,u1_struct_0(B),u4_msualg_1(B,C)) )
     => ( r2_hidden(A,a_3_2_closure2(B,C,D))
      <=> ? [E] :
            ( m1_closure2(E,u1_struct_0(B),u4_msualg_1(B,C),k6_closure2(u1_struct_0(B),u4_msualg_1(B,C)))
            & A = E
            & r2_pboole(u1_struct_0(B),D,E)
            & r2_hidden(E,u1_closure2(B,C)) ) ) ) )).
%------------------------------------------------------------------------------