SET007 Axioms: SET007+297.ax


%------------------------------------------------------------------------------
% File     : SET007+297 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Lattice of Natural Numbers and The Sublattice of it.
% 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    : nat_lat [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  116 (  69 unit)
%            Number of atoms       :  290 (  46 equality)
%            Maximal formula depth :   13 (   3 average)
%            Number of connectives :  199 (  25 ~  ;   0  |;  96  &)
%                                         (   8 <=>;  70 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   33 (   1 propositional; 0-4 arity)
%            Number of functors    :   30 (  13 constant; 0-6 arity)
%            Number of variables   :   78 (   0 singleton;  75 !;   3 ?)
%            Maximal term depth    :    3 (   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(fc1_nat_lat,axiom,
    ( ~ v3_struct_0(k6_nat_lat)
    & v3_lattices(k6_nat_lat)
    & v4_lattices(k6_nat_lat)
    & v5_lattices(k6_nat_lat)
    & v6_lattices(k6_nat_lat)
    & v7_lattices(k6_nat_lat)
    & v8_lattices(k6_nat_lat)
    & v9_lattices(k6_nat_lat)
    & v10_lattices(k6_nat_lat) )).

fof(fc2_nat_lat,axiom,
    ( ~ v1_xboole_0(k7_nat_lat)
    & v1_membered(k7_nat_lat)
    & v2_membered(k7_nat_lat)
    & v3_membered(k7_nat_lat)
    & v4_membered(k7_nat_lat)
    & v5_membered(k7_nat_lat) )).

fof(cc1_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B) ) ) ) )).

fof(cc2_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k5_numbers))
     => ! [B] :
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B) ) ) ) )).

fof(fc3_nat_lat,axiom,
    ( ~ v3_struct_0(k13_nat_lat)
    & v3_lattices(k13_nat_lat)
    & v4_lattices(k13_nat_lat)
    & v5_lattices(k13_nat_lat)
    & v6_lattices(k13_nat_lat)
    & v7_lattices(k13_nat_lat)
    & v8_lattices(k13_nat_lat)
    & v9_lattices(k13_nat_lat)
    & v10_lattices(k13_nat_lat) )).

fof(rc1_nat_lat,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & l3_lattices(A) )
     => ? [B] :
          ( m2_nat_lat(B,A)
          & ~ v3_struct_0(B)
          & v3_lattices(B)
          & v4_lattices(B)
          & v5_lattices(B)
          & v6_lattices(B)
          & v7_lattices(B)
          & v8_lattices(B)
          & v9_lattices(B)
          & v10_lattices(B) ) ) )).

fof(d1_nat_lat,axiom,(
    $true )).

fof(d2_nat_lat,axiom,(
    $true )).

fof(d3_nat_lat,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) )
     => ( A = k1_nat_lat
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_binop_1(k5_numbers,k5_numbers,k5_numbers,A,B,C) = k6_nat_1(B,C) ) ) ) ) )).

fof(d4_nat_lat,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) )
     => ( A = k2_nat_lat
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_binop_1(k5_numbers,k5_numbers,k5_numbers,A,B,C) = k5_nat_1(B,C) ) ) ) ) )).

fof(d5_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat)))
     => k3_nat_lat(A) = A ) )).

fof(t1_nat_lat,axiom,(
    $true )).

fof(t2_nat_lat,axiom,(
    $true )).

fof(t3_nat_lat,axiom,(
    $true )).

fof(t4_nat_lat,axiom,(
    $true )).

fof(t5_nat_lat,axiom,(
    $true )).

fof(t6_nat_lat,axiom,(
    $true )).

fof(t7_nat_lat,axiom,(
    $true )).

fof(t8_nat_lat,axiom,(
    $true )).

fof(t9_nat_lat,axiom,(
    $true )).

fof(t10_nat_lat,axiom,(
    $true )).

fof(t11_nat_lat,axiom,(
    $true )).

fof(t12_nat_lat,axiom,(
    $true )).

fof(t13_nat_lat,axiom,(
    $true )).

fof(t14_nat_lat,axiom,(
    $true )).

fof(t15_nat_lat,axiom,(
    $true )).

fof(t16_nat_lat,axiom,(
    $true )).

fof(t17_nat_lat,axiom,(
    $true )).

fof(t18_nat_lat,axiom,(
    $true )).

fof(t19_nat_lat,axiom,(
    $true )).

fof(t20_nat_lat,axiom,(
    $true )).

fof(t21_nat_lat,axiom,(
    $true )).

fof(t22_nat_lat,axiom,(
    $true )).

fof(t23_nat_lat,axiom,(
    $true )).

fof(t24_nat_lat,axiom,(
    $true )).

fof(t25_nat_lat,axiom,(
    $true )).

fof(t26_nat_lat,axiom,(
    $true )).

fof(t27_nat_lat,axiom,(
    $true )).

fof(t28_nat_lat,axiom,(
    $true )).

fof(t29_nat_lat,axiom,(
    $true )).

fof(t30_nat_lat,axiom,(
    $true )).

fof(t31_nat_lat,axiom,(
    $true )).

fof(t32_nat_lat,axiom,(
    $true )).

fof(t33_nat_lat,axiom,(
    $true )).

fof(t34_nat_lat,axiom,(
    $true )).

fof(t35_nat_lat,axiom,(
    $true )).

fof(t36_nat_lat,axiom,(
    $true )).

fof(t37_nat_lat,axiom,(
    $true )).

fof(t38_nat_lat,axiom,(
    $true )).

fof(t39_nat_lat,axiom,(
    $true )).

fof(t40_nat_lat,axiom,(
    $true )).

fof(t41_nat_lat,axiom,(
    $true )).

fof(t42_nat_lat,axiom,(
    $true )).

fof(t43_nat_lat,axiom,(
    $true )).

fof(t44_nat_lat,axiom,(
    $true )).

fof(t45_nat_lat,axiom,(
    $true )).

fof(t46_nat_lat,axiom,(
    $true )).

fof(t47_nat_lat,axiom,(
    $true )).

fof(t48_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat)))
         => k1_lattices(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat),A,B) = k5_nat_1(k3_nat_lat(A),k3_nat_lat(B)) ) ) )).

fof(t49_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat)))
         => k2_lattices(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat),A,B) = k6_nat_1(k3_nat_lat(A),k3_nat_lat(B)) ) ) )).

fof(t50_nat_lat,axiom,(
    $true )).

fof(t51_nat_lat,axiom,(
    $true )).

fof(t52_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat)))
         => ( r1_lattices(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat),A,B)
           => r1_nat_1(k3_nat_lat(A),k3_nat_lat(B)) ) ) ) )).

fof(d6_nat_lat,axiom,(
    k4_nat_lat = np__1 )).

fof(d7_nat_lat,axiom,(
    k5_nat_lat = np__0 )).

fof(t53_nat_lat,axiom,(
    $true )).

fof(t54_nat_lat,axiom,(
    $true )).

fof(t55_nat_lat,axiom,(
    k3_nat_lat(k4_nat_lat) = np__1 )).

fof(t56_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat)))
     => ( k2_lattices(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat),k4_nat_lat,A) = k4_nat_lat
        & k2_lattices(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat),A,k4_nat_lat) = k4_nat_lat ) ) )).

fof(d8_nat_lat,axiom,(
    k6_nat_lat = g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat) )).

fof(t57_nat_lat,axiom,(
    $true )).

fof(t58_nat_lat,axiom,(
    $true )).

fof(t59_nat_lat,axiom,(
    $true )).

fof(t60_nat_lat,axiom,
    ( ~ v3_struct_0(k6_nat_lat)
    & v10_lattices(k6_nat_lat)
    & v13_lattices(k6_nat_lat)
    & l3_lattices(k6_nat_lat) )).

fof(t61_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k6_nat_lat))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_nat_lat))
         => k1_binop_1(k2_nat_lat,A,B) = k1_binop_1(k2_nat_lat,B,A) ) ) )).

fof(t62_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k6_nat_lat))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_nat_lat))
         => k1_binop_1(k1_nat_lat,A,B) = k1_binop_1(k1_nat_lat,B,A) ) ) )).

fof(t63_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k6_nat_lat))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_nat_lat))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k6_nat_lat))
             => k1_binop_1(k2_nat_lat,A,k1_binop_1(k2_nat_lat,B,C)) = k1_binop_1(k2_nat_lat,k1_binop_1(k2_nat_lat,A,B),C) ) ) ) )).

fof(t64_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k6_nat_lat))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_nat_lat))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k6_nat_lat))
             => ( k1_binop_1(k2_nat_lat,A,k1_binop_1(k2_nat_lat,B,C)) = k1_binop_1(k2_nat_lat,k1_binop_1(k2_nat_lat,B,A),C)
                & k1_binop_1(k2_nat_lat,A,k1_binop_1(k2_nat_lat,B,C)) = k1_binop_1(k2_nat_lat,k1_binop_1(k2_nat_lat,A,C),B)
                & k1_binop_1(k2_nat_lat,A,k1_binop_1(k2_nat_lat,B,C)) = k1_binop_1(k2_nat_lat,k1_binop_1(k2_nat_lat,C,B),A)
                & k1_binop_1(k2_nat_lat,A,k1_binop_1(k2_nat_lat,B,C)) = k1_binop_1(k2_nat_lat,k1_binop_1(k2_nat_lat,C,A),B) ) ) ) ) )).

fof(t65_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k6_nat_lat))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_nat_lat))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k6_nat_lat))
             => k1_binop_1(k1_nat_lat,A,k1_binop_1(k1_nat_lat,B,C)) = k1_binop_1(k1_nat_lat,k1_binop_1(k1_nat_lat,A,B),C) ) ) ) )).

fof(t66_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k6_nat_lat))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_nat_lat))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k6_nat_lat))
             => ( k1_binop_1(k1_nat_lat,A,k1_binop_1(k1_nat_lat,B,C)) = k1_binop_1(k1_nat_lat,k1_binop_1(k1_nat_lat,B,A),C)
                & k1_binop_1(k1_nat_lat,A,k1_binop_1(k1_nat_lat,B,C)) = k1_binop_1(k1_nat_lat,k1_binop_1(k1_nat_lat,A,C),B)
                & k1_binop_1(k1_nat_lat,A,k1_binop_1(k1_nat_lat,B,C)) = k1_binop_1(k1_nat_lat,k1_binop_1(k1_nat_lat,C,B),A)
                & k1_binop_1(k1_nat_lat,A,k1_binop_1(k1_nat_lat,B,C)) = k1_binop_1(k1_nat_lat,k1_binop_1(k1_nat_lat,C,A),B) ) ) ) ) )).

fof(t67_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k6_nat_lat))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_nat_lat))
         => ( k1_binop_1(k1_nat_lat,A,k1_binop_1(k2_nat_lat,A,B)) = A
            & k1_binop_1(k1_nat_lat,k1_binop_1(k2_nat_lat,B,A),A) = A
            & k1_binop_1(k1_nat_lat,A,k1_binop_1(k2_nat_lat,B,A)) = A
            & k1_binop_1(k1_nat_lat,k1_binop_1(k2_nat_lat,A,B),A) = A ) ) ) )).

fof(t68_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k6_nat_lat))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k6_nat_lat))
         => ( k1_binop_1(k2_nat_lat,A,k1_binop_1(k1_nat_lat,A,B)) = A
            & k1_binop_1(k2_nat_lat,k1_binop_1(k1_nat_lat,B,A),A) = A
            & k1_binop_1(k2_nat_lat,A,k1_binop_1(k1_nat_lat,B,A)) = A
            & k1_binop_1(k2_nat_lat,k1_binop_1(k1_nat_lat,A,B),A) = A ) ) ) )).

fof(d9_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k5_numbers))
     => ( A = k7_nat_lat
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r2_hidden(B,A)
            <=> ~ r1_xreal_0(B,np__0) ) ) ) ) )).

fof(d10_nat_lat,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => k8_nat_lat(A) = A ) ) )).

fof(d11_nat_lat,axiom,(
    ! [A] :
      ( m1_nat_lat(A,k1_numbers,k5_numbers,k7_nat_lat)
     => k9_nat_lat(A) = A ) )).

fof(d12_nat_lat,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k7_nat_lat,k7_nat_lat),k7_nat_lat)
        & m2_relset_1(A,k2_zfmisc_1(k7_nat_lat,k7_nat_lat),k7_nat_lat) )
     => ( A = k10_nat_lat
      <=> ! [B] :
            ( m1_nat_lat(B,k1_numbers,k5_numbers,k7_nat_lat)
           => ! [C] :
                ( m1_nat_lat(C,k1_numbers,k5_numbers,k7_nat_lat)
               => k2_binop_1(k7_nat_lat,k7_nat_lat,k7_nat_lat,A,B,C) = k6_nat_1(B,C) ) ) ) ) )).

fof(d13_nat_lat,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k7_nat_lat,k7_nat_lat),k7_nat_lat)
        & m2_relset_1(A,k2_zfmisc_1(k7_nat_lat,k7_nat_lat),k7_nat_lat) )
     => ( A = k11_nat_lat
      <=> ! [B] :
            ( m1_nat_lat(B,k1_numbers,k5_numbers,k7_nat_lat)
           => ! [C] :
                ( m1_nat_lat(C,k1_numbers,k5_numbers,k7_nat_lat)
               => k2_binop_1(k7_nat_lat,k7_nat_lat,k7_nat_lat,A,B,C) = k5_nat_1(B,C) ) ) ) ) )).

fof(d14_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g3_lattices(k7_nat_lat,k11_nat_lat,k10_nat_lat)))
     => k12_nat_lat(A) = A ) )).

fof(t69_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g3_lattices(k7_nat_lat,k11_nat_lat,k10_nat_lat)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(g3_lattices(k7_nat_lat,k11_nat_lat,k10_nat_lat)))
         => k1_lattices(g3_lattices(k7_nat_lat,k11_nat_lat,k10_nat_lat),A,B) = k5_nat_1(k12_nat_lat(A),k12_nat_lat(B)) ) ) )).

fof(t70_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g3_lattices(k7_nat_lat,k11_nat_lat,k10_nat_lat)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(g3_lattices(k7_nat_lat,k11_nat_lat,k10_nat_lat)))
         => k2_lattices(g3_lattices(k7_nat_lat,k11_nat_lat,k10_nat_lat),A,B) = k6_nat_1(k12_nat_lat(A),k12_nat_lat(B)) ) ) )).

fof(d15_nat_lat,axiom,(
    k13_nat_lat = g3_lattices(k7_nat_lat,k11_nat_lat,k10_nat_lat) )).

fof(d16_nat_lat,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v10_lattices(B)
            & l3_lattices(B) )
         => ( m2_nat_lat(B,A)
          <=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
              & u2_lattices(B) = k1_realset1(u2_lattices(A),u1_struct_0(B))
              & u1_lattices(B) = k1_realset1(u1_lattices(A),u1_struct_0(B)) ) ) ) ) )).

fof(t71_nat_lat,axiom,(
    $true )).

fof(t72_nat_lat,axiom,(
    $true )).

fof(t73_nat_lat,axiom,(
    $true )).

fof(t74_nat_lat,axiom,(
    $true )).

fof(t75_nat_lat,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & l3_lattices(A) )
     => m2_nat_lat(A,A) ) )).

fof(t76_nat_lat,axiom,(
    m2_nat_lat(k13_nat_lat,k6_nat_lat) )).

fof(dt_m1_nat_lat,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(B)) )
     => ! [D] :
          ( m1_nat_lat(D,A,B,C)
         => m2_subset_1(D,A,B) ) ) )).

fof(existence_m1_nat_lat,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(B)) )
     => ? [D] : m1_nat_lat(D,A,B,C) ) )).

fof(redefinition_m1_nat_lat,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(B)) )
     => ! [D] :
          ( m1_nat_lat(D,A,B,C)
        <=> m1_subset_1(D,C) ) ) )).

fof(dt_m2_nat_lat,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ( ~ v3_struct_0(B)
            & v10_lattices(B)
            & l3_lattices(B) ) ) ) )).

fof(existence_m2_nat_lat,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & l3_lattices(A) )
     => ? [B] : m2_nat_lat(B,A) ) )).

fof(dt_k1_nat_lat,axiom,
    ( v1_funct_1(k1_nat_lat)
    & v1_funct_2(k1_nat_lat,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers)
    & m2_relset_1(k1_nat_lat,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) )).

fof(dt_k2_nat_lat,axiom,
    ( v1_funct_1(k2_nat_lat)
    & v1_funct_2(k2_nat_lat,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers)
    & m2_relset_1(k2_nat_lat,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) )).

fof(dt_k3_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat)))
     => m2_subset_1(k3_nat_lat(A),k1_numbers,k5_numbers) ) )).

fof(dt_k4_nat_lat,axiom,(
    m1_subset_1(k4_nat_lat,u1_struct_0(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat))) )).

fof(dt_k5_nat_lat,axiom,(
    m1_subset_1(k5_nat_lat,u1_struct_0(g3_lattices(k5_numbers,k2_nat_lat,k1_nat_lat))) )).

fof(dt_k6_nat_lat,axiom,
    ( ~ v3_struct_0(k6_nat_lat)
    & v10_lattices(k6_nat_lat)
    & l3_lattices(k6_nat_lat) )).

fof(dt_k7_nat_lat,axiom,(
    m1_subset_1(k7_nat_lat,k1_zfmisc_1(k5_numbers)) )).

fof(dt_k8_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m1_nat_lat(k8_nat_lat(A),k1_numbers,k5_numbers,k7_nat_lat) ) )).

fof(dt_k9_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,k7_nat_lat)
     => m1_nat_lat(k9_nat_lat(A),k1_numbers,k5_numbers,k7_nat_lat) ) )).

fof(dt_k10_nat_lat,axiom,
    ( v1_funct_1(k10_nat_lat)
    & v1_funct_2(k10_nat_lat,k2_zfmisc_1(k7_nat_lat,k7_nat_lat),k7_nat_lat)
    & m2_relset_1(k10_nat_lat,k2_zfmisc_1(k7_nat_lat,k7_nat_lat),k7_nat_lat) )).

fof(dt_k11_nat_lat,axiom,
    ( v1_funct_1(k11_nat_lat)
    & v1_funct_2(k11_nat_lat,k2_zfmisc_1(k7_nat_lat,k7_nat_lat),k7_nat_lat)
    & m2_relset_1(k11_nat_lat,k2_zfmisc_1(k7_nat_lat,k7_nat_lat),k7_nat_lat) )).

fof(dt_k12_nat_lat,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g3_lattices(k7_nat_lat,k11_nat_lat,k10_nat_lat)))
     => m1_nat_lat(k12_nat_lat(A),k1_numbers,k5_numbers,k7_nat_lat) ) )).

fof(dt_k13_nat_lat,axiom,
    ( ~ v3_struct_0(k13_nat_lat)
    & v10_lattices(k13_nat_lat)
    & l3_lattices(k13_nat_lat) )).
%------------------------------------------------------------------------------