SET007 Axioms: SET007+376.ax


%------------------------------------------------------------------------------
% File     : SET007+376 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Representation Theorem for Heyting Lattices
% 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    : openlatt [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   99 (   2 unit)
%            Number of atoms       :  698 (  63 equality)
%            Maximal formula depth :   19 (   8 average)
%            Number of connectives :  737 ( 138 ~  ;   0  |; 418  &)
%                                         (  19 <=>; 162 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   44 (   1 propositional; 0-3 arity)
%            Number of functors    :   47 (   1 constant; 0-6 arity)
%            Number of variables   :  209 (   0 singleton; 202 !;   7 ?)
%            Maximal term depth    :    6 (   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_openlatt,axiom,(
    ! [A] :
      ( l3_lattices(A)
     => ( ( ~ v3_struct_0(A)
          & v10_lattices(A)
          & v13_lattices(A)
          & v1_lattice2(A) )
       => ( ~ v3_struct_0(A)
          & v4_lattices(A)
          & v5_lattices(A)
          & v6_lattices(A)
          & v7_lattices(A)
          & v8_lattices(A)
          & v9_lattices(A)
          & v10_lattices(A)
          & v11_lattices(A)
          & v12_lattices(A)
          & v13_lattices(A)
          & v1_lattice2(A)
          & v3_filter_0(A) ) ) ) )).

fof(cc2_openlatt,axiom,(
    ! [A] :
      ( l3_lattices(A)
     => ( ( ~ v3_struct_0(A)
          & v10_lattices(A)
          & v3_filter_0(A) )
       => ( ~ v3_struct_0(A)
          & v4_lattices(A)
          & v5_lattices(A)
          & v6_lattices(A)
          & v7_lattices(A)
          & v8_lattices(A)
          & v9_lattices(A)
          & v10_lattices(A)
          & v14_lattices(A) ) ) ) )).

fof(fc1_openlatt,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ~ v1_xboole_0(k1_openlatt(A)) ) )).

fof(fc2_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ~ v3_struct_0(k6_openlatt(A))
        & v4_lattices(k6_openlatt(A))
        & v5_lattices(k6_openlatt(A))
        & v6_lattices(k6_openlatt(A))
        & v7_lattices(k6_openlatt(A))
        & v8_lattices(k6_openlatt(A))
        & v9_lattices(k6_openlatt(A))
        & v10_lattices(k6_openlatt(A))
        & v11_lattices(k6_openlatt(A))
        & v12_lattices(k6_openlatt(A))
        & v13_lattices(k6_openlatt(A))
        & v14_lattices(k6_openlatt(A))
        & v15_lattices(k6_openlatt(A))
        & v1_lattice2(k6_openlatt(A))
        & v3_filter_0(k6_openlatt(A)) ) ) )).

fof(fc3_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ~ v1_xboole_0(k9_openlatt(A)) ) )).

fof(fc4_openlatt,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => ~ v1_xboole_0(k10_openlatt(A,B)) ) )).

fof(rc1_openlatt,axiom,(
    ? [A] :
      ( l3_lattices(A)
      & ~ v3_struct_0(A)
      & v4_lattices(A)
      & v5_lattices(A)
      & v6_lattices(A)
      & v7_lattices(A)
      & v8_lattices(A)
      & v9_lattices(A)
      & v10_lattices(A)
      & v11_lattices(A)
      & v12_lattices(A)
      & v13_lattices(A)
      & v14_lattices(A)
      & v15_lattices(A)
      & v1_lattice2(A)
      & v3_filter_0(A)
      & ~ v3_realset2(A) ) )).

fof(fc5_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ~ v1_xboole_0(k7_openlatt(A)) ) )).

fof(fc6_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ( ~ v3_struct_0(k17_openlatt(A))
        & v1_pre_topc(k17_openlatt(A))
        & v2_pre_topc(k17_openlatt(A)) ) ) )).

fof(t1_openlatt,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => r1_tarski(k5_subset_1(u1_struct_0(A),B,k1_tops_1(A,k4_subset_1(u1_struct_0(A),k3_subset_1(u1_struct_0(A),B),C))),C) ) ) ) )).

fof(t2_openlatt,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( ( v3_pre_topc(D,A)
                      & r1_tarski(k5_subset_1(u1_struct_0(A),B,D),C) )
                   => r1_tarski(D,k1_tops_1(A,k4_subset_1(u1_struct_0(A),k3_subset_1(u1_struct_0(A),B),C))) ) ) ) ) ) )).

fof(d1_openlatt,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => k1_openlatt(A) = u1_pre_topc(A) ) )).

fof(t3_openlatt,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v3_pre_topc(B,A)
          <=> r2_hidden(B,k1_openlatt(A)) ) ) ) )).

fof(d2_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A))
            & m2_relset_1(B,k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A)) )
         => ( B = k4_openlatt(A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A))
               => ! [D] :
                    ( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A))
                   => k2_binop_1(k1_openlatt(A),k1_openlatt(A),k1_openlatt(A),B,C,D) = k2_openlatt(A,C,D) ) ) ) ) ) )).

fof(d3_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A))
            & m2_relset_1(B,k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A)) )
         => ( B = k5_openlatt(A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A))
               => ! [D] :
                    ( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A))
                   => k2_binop_1(k1_openlatt(A),k1_openlatt(A),k1_openlatt(A),B,C,D) = k3_openlatt(A,C,D) ) ) ) ) ) )).

fof(t4_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ~ v3_struct_0(g3_lattices(k1_openlatt(A),k4_openlatt(A),k5_openlatt(A)))
        & v10_lattices(g3_lattices(k1_openlatt(A),k4_openlatt(A),k5_openlatt(A)))
        & l3_lattices(g3_lattices(k1_openlatt(A),k4_openlatt(A),k5_openlatt(A))) ) ) )).

fof(d4_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => k6_openlatt(A) = g3_lattices(k1_openlatt(A),k4_openlatt(A),k5_openlatt(A)) ) )).

fof(t5_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => u1_struct_0(k6_openlatt(A)) = k1_openlatt(A) ) )).

fof(t6_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_openlatt(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k6_openlatt(A)))
             => ( k3_lattices(k6_openlatt(A),B,C) = k2_xboole_0(B,C)
                & k4_lattices(k6_openlatt(A),B,C) = k3_xboole_0(B,C) ) ) ) ) )).

fof(t7_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_openlatt(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k6_openlatt(A)))
             => ( r3_lattices(k6_openlatt(A),B,C)
              <=> r1_tarski(B,C) ) ) ) ) )).

fof(t8_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_openlatt(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k6_openlatt(A)))
             => ! [D] :
                  ( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A))
                 => ! [E] :
                      ( m2_subset_1(E,k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A))
                     => ( ( B = D
                          & C = E )
                       => ( r3_lattices(k6_openlatt(A),B,C)
                        <=> r1_tarski(D,E) ) ) ) ) ) ) ) )).

fof(t9_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => v3_filter_0(k6_openlatt(A)) ) )).

fof(t10_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v13_lattices(k6_openlatt(A))
        & k5_lattices(k6_openlatt(A)) = k1_xboole_0 ) ) )).

fof(t11_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => k6_lattices(k6_openlatt(A)) = u1_struct_0(A) ) )).

fof(t12_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_filter_0(B,A)
         => ( r2_hidden(B,k7_openlatt(A))
          <=> ( B != u1_struct_0(A)
              & v2_filter_0(B,A) ) ) ) ) )).

fof(t13_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_filter_0(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(B,k1_funct_1(k8_openlatt(A),C))
              <=> ( r2_hidden(B,k7_openlatt(A))
                  & r2_hidden(C,B) ) ) ) ) ) )).

fof(t14_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( r2_hidden(C,k1_funct_1(k8_openlatt(A),B))
            <=> ? [D] :
                  ( m1_filter_0(D,A)
                  & D = C
                  & D != u1_struct_0(A)
                  & v2_filter_0(D,A)
                  & r2_hidden(B,D) ) ) ) ) )).

fof(d7_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => k9_openlatt(A) = k2_relat_1(k8_openlatt(A)) ) )).

fof(t15_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( r2_hidden(B,k9_openlatt(A))
        <=> ? [C] :
              ( m1_subset_1(C,u1_struct_0(A))
              & B = k1_funct_1(k8_openlatt(A),C) ) ) ) )).

fof(t16_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_funct_1(k8_openlatt(A),k3_lattices(A,B,C)) = k2_xboole_0(k1_funct_1(k8_openlatt(A),B),k1_funct_1(k8_openlatt(A),C)) ) ) ) )).

fof(t17_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_funct_1(k8_openlatt(A),k4_lattices(A,B,C)) = k3_xboole_0(k1_funct_1(k8_openlatt(A),B),k1_funct_1(k8_openlatt(A),C)) ) ) ) )).

fof(t18_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( r2_hidden(C,k10_openlatt(A,B))
            <=> ( m1_filter_0(C,A)
                & r2_hidden(B,C) ) ) ) ) )).

fof(t19_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( r2_hidden(D,k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),k10_openlatt(A,B),k10_openlatt(A,C)))
                 => ( m1_filter_0(D,A)
                    & r2_hidden(B,D)
                    & ~ r2_hidden(C,D) ) ) ) ) ) )).

fof(t20_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ~ ( D != k1_xboole_0
                    & r1_tarski(D,k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),k10_openlatt(A,B),k10_openlatt(A,C)))
                    & v6_ordinal1(D)
                    & ! [E] :
                        ~ ( r2_hidden(E,k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),k10_openlatt(A,B),k10_openlatt(A,C)))
                          & ! [F] :
                              ( r2_hidden(F,D)
                             => r1_tarski(F,E) ) ) ) ) ) ) )).

fof(t21_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ~ r3_lattices(A,B,C)
               => r2_hidden(k2_filter_0(A,B),k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),k10_openlatt(A,B),k10_openlatt(A,C))) ) ) ) ) )).

fof(t22_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( ~ r3_lattices(A,B,C)
                  & ! [D] :
                      ( m1_filter_0(D,A)
                     => ~ ( r2_hidden(D,k7_openlatt(A))
                          & ~ r2_hidden(C,D)
                          & r2_hidden(B,D) ) ) ) ) ) ) )).

fof(t23_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( B != C
                  & ! [D] :
                      ( m1_filter_0(D,A)
                     => ~ r2_hidden(D,k7_openlatt(A)) ) ) ) ) ) )).

fof(t24_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( B != C
                  & k1_funct_1(k8_openlatt(A),B) = k1_funct_1(k8_openlatt(A),C) ) ) ) ) )).

fof(t25_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => v2_funct_1(k8_openlatt(A)) ) )).

fof(d9_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A))
            & m2_relset_1(B,k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A)) )
         => ( B = k13_openlatt(A)
          <=> ! [C] :
                ( m1_subset_1(C,k9_openlatt(A))
               => ! [D] :
                    ( m1_subset_1(D,k9_openlatt(A))
                   => k2_binop_1(k9_openlatt(A),k9_openlatt(A),k9_openlatt(A),B,C,D) = k11_openlatt(A,C,D) ) ) ) ) ) )).

fof(d10_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A))
            & m2_relset_1(B,k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A)) )
         => ( B = k14_openlatt(A)
          <=> ! [C] :
                ( m1_subset_1(C,k9_openlatt(A))
               => ! [D] :
                    ( m1_subset_1(D,k9_openlatt(A))
                   => k2_binop_1(k9_openlatt(A),k9_openlatt(A),k9_openlatt(A),B,C,D) = k12_openlatt(A,C,D) ) ) ) ) ) )).

fof(t26_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ( ~ v3_struct_0(g3_lattices(k9_openlatt(A),k13_openlatt(A),k14_openlatt(A)))
        & v10_lattices(g3_lattices(k9_openlatt(A),k13_openlatt(A),k14_openlatt(A)))
        & l3_lattices(g3_lattices(k9_openlatt(A),k13_openlatt(A),k14_openlatt(A))) ) ) )).

fof(d11_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => k15_openlatt(A) = g3_lattices(k9_openlatt(A),k13_openlatt(A),k14_openlatt(A)) ) )).

fof(t27_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => u1_struct_0(k15_openlatt(A)) = k9_openlatt(A) ) )).

fof(t28_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_openlatt(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_openlatt(A)))
             => ( k3_lattices(k15_openlatt(A),B,C) = k2_xboole_0(B,C)
                & k4_lattices(k15_openlatt(A),B,C) = k3_xboole_0(B,C) ) ) ) ) )).

fof(t29_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_openlatt(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_openlatt(A)))
             => ( r3_lattices(k15_openlatt(A),B,C)
              <=> r1_tarski(B,C) ) ) ) ) )).

fof(t30_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => v3_lattice4(k16_openlatt(A),A,k15_openlatt(A)) ) )).

fof(t31_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => v11_lattices(k15_openlatt(A)) ) )).

fof(t32_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => r1_filter_1(A,k15_openlatt(A)) ) )).

fof(t33_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k8_funct_2(u1_struct_0(A),u1_struct_0(k15_openlatt(A)),k16_openlatt(A),k6_lattices(A)) = k7_openlatt(A) ) )).

fof(t34_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k8_funct_2(u1_struct_0(A),u1_struct_0(k15_openlatt(A)),k16_openlatt(A),k5_lattices(A)) = k1_xboole_0 ) )).

fof(t35_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => r1_tarski(k9_openlatt(A),k1_zfmisc_1(k7_openlatt(A))) ) )).

fof(t37_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => r1_tarski(k9_openlatt(A),u1_struct_0(k6_openlatt(k17_openlatt(A)))) ) )).

fof(t38_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => v1_lattice4(k18_openlatt(A),A,k6_openlatt(k17_openlatt(A))) ) )).

fof(t39_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k8_funct_2(u1_struct_0(A),u1_struct_0(k6_openlatt(k17_openlatt(A))),k18_openlatt(A),k4_filter_0(A,B,C)) = k4_filter_0(k6_openlatt(k17_openlatt(A)),k8_funct_2(u1_struct_0(A),u1_struct_0(k6_openlatt(k17_openlatt(A))),k18_openlatt(A),B),k8_funct_2(u1_struct_0(A),u1_struct_0(k6_openlatt(k17_openlatt(A))),k18_openlatt(A),C)) ) ) ) )).

fof(t40_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => r1_lattice4(A,k6_openlatt(k17_openlatt(A)),k18_openlatt(A)) ) )).

fof(t41_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => r2_lattice4(A,k6_openlatt(k17_openlatt(A)),k18_openlatt(A)) ) )).

fof(t42_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => r3_lattice4(A,k6_openlatt(k17_openlatt(A)),k18_openlatt(A)) ) )).

fof(dt_k1_openlatt,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => m1_subset_1(k1_openlatt(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) )).

fof(dt_k2_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_openlatt(A))
        & m1_subset_1(C,k1_openlatt(A)) )
     => m2_subset_1(k2_openlatt(A,B,C),k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A)) ) )).

fof(commutativity_k2_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_openlatt(A))
        & m1_subset_1(C,k1_openlatt(A)) )
     => k2_openlatt(A,B,C) = k2_openlatt(A,C,B) ) )).

fof(idempotence_k2_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_openlatt(A))
        & m1_subset_1(C,k1_openlatt(A)) )
     => k2_openlatt(A,B,B) = B ) )).

fof(redefinition_k2_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_openlatt(A))
        & m1_subset_1(C,k1_openlatt(A)) )
     => k2_openlatt(A,B,C) = k2_xboole_0(B,C) ) )).

fof(dt_k3_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_openlatt(A))
        & m1_subset_1(C,k1_openlatt(A)) )
     => m2_subset_1(k3_openlatt(A,B,C),k1_zfmisc_1(u1_struct_0(A)),k1_openlatt(A)) ) )).

fof(commutativity_k3_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_openlatt(A))
        & m1_subset_1(C,k1_openlatt(A)) )
     => k3_openlatt(A,B,C) = k3_openlatt(A,C,B) ) )).

fof(idempotence_k3_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_openlatt(A))
        & m1_subset_1(C,k1_openlatt(A)) )
     => k3_openlatt(A,B,B) = B ) )).

fof(redefinition_k3_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_openlatt(A))
        & m1_subset_1(C,k1_openlatt(A)) )
     => k3_openlatt(A,B,C) = k3_xboole_0(B,C) ) )).

fof(dt_k4_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_funct_1(k4_openlatt(A))
        & v1_funct_2(k4_openlatt(A),k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A))
        & m2_relset_1(k4_openlatt(A),k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A)) ) ) )).

fof(dt_k5_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_funct_1(k5_openlatt(A))
        & v1_funct_2(k5_openlatt(A),k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A))
        & m2_relset_1(k5_openlatt(A),k2_zfmisc_1(k1_openlatt(A),k1_openlatt(A)),k1_openlatt(A)) ) ) )).

fof(dt_k6_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ~ v3_struct_0(k6_openlatt(A))
        & v10_lattices(k6_openlatt(A))
        & l3_lattices(k6_openlatt(A)) ) ) )).

fof(dt_k7_openlatt,axiom,(
    $true )).

fof(dt_k8_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ( v1_relat_1(k8_openlatt(A))
        & v1_funct_1(k8_openlatt(A)) ) ) )).

fof(dt_k9_openlatt,axiom,(
    $true )).

fof(dt_k10_openlatt,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m1_subset_1(k10_openlatt(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) )).

fof(dt_k11_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A)
        & m1_subset_1(B,k9_openlatt(A))
        & m1_subset_1(C,k9_openlatt(A)) )
     => m1_subset_1(k11_openlatt(A,B,C),k9_openlatt(A)) ) )).

fof(commutativity_k11_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A)
        & m1_subset_1(B,k9_openlatt(A))
        & m1_subset_1(C,k9_openlatt(A)) )
     => k11_openlatt(A,B,C) = k11_openlatt(A,C,B) ) )).

fof(idempotence_k11_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A)
        & m1_subset_1(B,k9_openlatt(A))
        & m1_subset_1(C,k9_openlatt(A)) )
     => k11_openlatt(A,B,B) = B ) )).

fof(redefinition_k11_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A)
        & m1_subset_1(B,k9_openlatt(A))
        & m1_subset_1(C,k9_openlatt(A)) )
     => k11_openlatt(A,B,C) = k2_xboole_0(B,C) ) )).

fof(dt_k12_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A)
        & m1_subset_1(B,k9_openlatt(A))
        & m1_subset_1(C,k9_openlatt(A)) )
     => m1_subset_1(k12_openlatt(A,B,C),k9_openlatt(A)) ) )).

fof(commutativity_k12_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A)
        & m1_subset_1(B,k9_openlatt(A))
        & m1_subset_1(C,k9_openlatt(A)) )
     => k12_openlatt(A,B,C) = k12_openlatt(A,C,B) ) )).

fof(idempotence_k12_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A)
        & m1_subset_1(B,k9_openlatt(A))
        & m1_subset_1(C,k9_openlatt(A)) )
     => k12_openlatt(A,B,B) = B ) )).

fof(redefinition_k12_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A)
        & m1_subset_1(B,k9_openlatt(A))
        & m1_subset_1(C,k9_openlatt(A)) )
     => k12_openlatt(A,B,C) = k3_xboole_0(B,C) ) )).

fof(dt_k13_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ( v1_funct_1(k13_openlatt(A))
        & v1_funct_2(k13_openlatt(A),k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A))
        & m2_relset_1(k13_openlatt(A),k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A)) ) ) )).

fof(dt_k14_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ( v1_funct_1(k14_openlatt(A))
        & v1_funct_2(k14_openlatt(A),k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A))
        & m2_relset_1(k14_openlatt(A),k2_zfmisc_1(k9_openlatt(A),k9_openlatt(A)),k9_openlatt(A)) ) ) )).

fof(dt_k15_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ( ~ v3_struct_0(k15_openlatt(A))
        & v10_lattices(k15_openlatt(A))
        & l3_lattices(k15_openlatt(A)) ) ) )).

fof(dt_k16_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => m1_lattice4(k16_openlatt(A),A,k15_openlatt(A)) ) )).

fof(redefinition_k16_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => k16_openlatt(A) = k8_openlatt(A) ) )).

fof(dt_k17_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ( v1_pre_topc(k17_openlatt(A))
        & l1_pre_topc(k17_openlatt(A)) ) ) )).

fof(dt_k18_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => m1_lattice4(k18_openlatt(A),A,k6_openlatt(k17_openlatt(A))) ) )).

fof(redefinition_k18_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => k18_openlatt(A) = k8_openlatt(A) ) )).

fof(d5_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => k7_openlatt(A) = a_1_0_openlatt(A) ) )).

fof(d6_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k8_openlatt(A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k1_relat_1(B) = u1_struct_0(A)
                  & k1_funct_1(B,C) = a_2_0_openlatt(A,C) ) ) ) ) ) )).

fof(d8_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v11_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k10_openlatt(A,B) = a_2_1_openlatt(A,B) ) ) )).

fof(d12_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( v1_pre_topc(B)
            & l1_pre_topc(B) )
         => ( B = k17_openlatt(A)
          <=> ( u1_struct_0(B) = k7_openlatt(A)
              & u1_pre_topc(B) = a_1_1_openlatt(A) ) ) ) ) )).

fof(t36_openlatt,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v1_lattice2(A)
        & ~ v3_realset2(A)
        & l3_lattices(A) )
     => u1_struct_0(k6_openlatt(k17_openlatt(A))) = a_1_1_openlatt(A) ) )).

fof(fraenkel_a_1_0_openlatt,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v11_lattices(B)
        & l3_lattices(B) )
     => ( r2_hidden(A,a_1_0_openlatt(B))
      <=> ? [C] :
            ( m1_filter_0(C,B)
            & A = C
            & C != u1_struct_0(B)
            & v2_filter_0(C,B) ) ) ) )).

fof(fraenkel_a_2_0_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v11_lattices(B)
        & l3_lattices(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_2_0_openlatt(B,C))
      <=> ? [D] :
            ( m1_filter_0(D,B)
            & A = D
            & r2_hidden(D,k7_openlatt(B))
            & r2_hidden(C,D) ) ) ) )).

fof(fraenkel_a_2_1_openlatt,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v11_lattices(B)
        & l3_lattices(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_2_1_openlatt(B,C))
      <=> ? [D] :
            ( m1_filter_0(D,B)
            & A = D
            & r2_hidden(C,D) ) ) ) )).

fof(fraenkel_a_1_1_openlatt,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v1_lattice2(B)
        & ~ v3_realset2(B)
        & l3_lattices(B) )
     => ( r2_hidden(A,a_1_1_openlatt(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_zfmisc_1(k9_openlatt(B)))
            & A = k3_tarski(C) ) ) ) )).
%------------------------------------------------------------------------------