SET007 Axioms: SET007+665.ax


%------------------------------------------------------------------------------
% File     : SET007+665 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Incompleteness of the Lattice of Substitutions
% 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    : heyting3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   64 (   5 unit)
%            Number of atoms       :  305 (  39 equality)
%            Maximal formula depth :   18 (   8 average)
%            Number of connectives :  294 (  53 ~  ;   2  |;  99  &)
%                                         (  12 <=>; 128 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   42 (   1 propositional; 0-3 arity)
%            Number of functors    :   43 (   8 constant; 0-4 arity)
%            Number of variables   :  165 (   0 singleton; 159 !;   6 ?)
%            Maximal term depth    :    6 (   2 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_heyting3,axiom,(
    ! [A,B] : v1_funct_1(k2_zfmisc_1(A,k1_tarski(B))) )).

fof(fc2_heyting3,axiom,(
    ! [A,B] :
      ( ~ v3_struct_0(k1_heyting3(A,B))
      & v1_lattice3(k1_heyting3(A,B))
      & v2_lattice3(k1_heyting3(A,B)) ) )).

fof(fc3_heyting3,axiom,(
    ! [A,B] :
      ( ~ v3_struct_0(k1_heyting3(A,B))
      & v2_orders_2(k1_heyting3(A,B))
      & v3_orders_2(k1_heyting3(A,B))
      & v4_orders_2(k1_heyting3(A,B))
      & v1_lattice3(k1_heyting3(A,B))
      & v2_lattice3(k1_heyting3(A,B)) ) )).

fof(fc4_heyting3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => v1_finset_1(k2_heyting3(A,B)) ) )).

fof(fc5_heyting3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => v1_finset_1(k3_heyting3(A,B)) ) )).

fof(fc6_heyting3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( ~ v1_xboole_0(k2_heyting3(A,B))
        & v1_finset_1(k2_heyting3(A,B)) ) ) )).

fof(fc7_heyting3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v1_xboole_0(k4_heyting3(np__1,A))
        & v1_finset_1(k4_heyting3(np__1,A)) ) ) )).

fof(rc1_heyting3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ? [B] :
          ( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
          & ~ v1_xboole_0(B) ) ) )).

fof(fc8_heyting3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v3_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v2_orders_2(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v3_orders_2(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v4_orders_2(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v1_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v2_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))
        & ~ v3_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))) ) ) )).

fof(fc9_heyting3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v3_struct_0(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v3_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v4_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v5_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v6_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v7_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v8_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v9_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v10_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v11_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v12_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v13_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v14_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & v15_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
        & ~ v4_lattice3(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A))) ) ) )).

fof(t1_heyting3,axiom,(
    ! [A] :
      ( ( ~ v1_abian(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => r1_xreal_0(np__1,A) ) )).

fof(t2_heyting3,axiom,(
    $true )).

fof(t3_heyting3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
     => ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & r1_tarski(A,k1_finsub_1(k1_zfmisc_1(k5_numbers),k2_finseq_1(B),k6_domain_1(k5_numbers,np__0))) ) ) )).

fof(t4_heyting3,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
     => ~ ! [B] :
            ( ( ~ v1_abian(B)
              & m2_subset_1(B,k1_numbers,k5_numbers) )
           => r2_hidden(B,A) ) ) )).

fof(t5_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k12_mcart_1(k1_numbers,k5_numbers,k5_numbers,k6_domain_1(k5_numbers,A)))) )
         => ? [C] :
              ( ~ v1_xboole_0(C)
              & m2_subset_1(C,k1_numbers,k5_numbers)
              & r1_tarski(B,k12_mcart_1(k5_numbers,k5_numbers,k1_finsub_1(k1_zfmisc_1(k5_numbers),k2_finseq_1(C),k6_domain_1(k5_numbers,np__0)),k6_domain_1(k5_numbers,A))) ) ) ) )).

fof(t6_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k12_mcart_1(k1_numbers,k5_numbers,k5_numbers,k6_domain_1(k5_numbers,A)))) )
         => ~ ! [C] :
                ( ( ~ v1_xboole_0(C)
                  & m2_subset_1(C,k1_numbers,k5_numbers) )
               => r2_hidden(k1_domain_1(k5_numbers,k5_numbers,k1_nat_1(k2_nat_1(np__2,C),np__1),A),B) ) ) ) )).

fof(t7_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_finset_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k12_mcart_1(k1_numbers,k5_numbers,k5_numbers,k6_domain_1(k5_numbers,A)))) )
         => ? [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
              & ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ~ ( r1_xreal_0(C,D)
                      & r2_hidden(k1_domain_1(k5_numbers,k5_numbers,D,A),B) ) ) ) ) ) )).

fof(t8_heyting3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v14_lattices(A)
        & l3_lattices(A) )
     => k6_lattices(A) = k4_yellow_0(k3_lattice3(A)) ) )).

fof(t9_heyting3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v13_lattices(A)
        & l3_lattices(A) )
     => k5_lattices(A) = k3_yellow_0(k3_lattice3(A)) ) )).

fof(t10_heyting3,axiom,(
    $true )).

fof(t11_heyting3,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
             => ( C = k1_xboole_0
               => ( D = k1_xboole_0
                  | k3_heyting2(A,B,D,C) = k1_xboole_0 ) ) ) ) ) )).

fof(t12_heyting3,axiom,(
    ! [A,B,C,D] :
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => r1_tarski(k1_substlat(A,C),k1_substlat(B,D)) ) )).

fof(t13_heyting3,axiom,(
    ! [A,B,C,D,E] :
      ( m1_subset_1(E,k5_finsub_1(k4_partfun1(A,C)))
     => ! [F] :
          ( m1_subset_1(F,k5_finsub_1(k4_partfun1(B,D)))
         => ( ( r1_tarski(A,B)
              & r1_tarski(C,D)
              & E = F )
           => k3_substlat(A,C,E) = k3_substlat(B,D,F) ) ) ) )).

fof(t14_heyting3,axiom,(
    ! [A,B,C,D] :
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => u2_lattices(k5_substlat(A,C)) = k1_realset1(u2_lattices(k5_substlat(B,D)),u1_struct_0(k5_substlat(A,C))) ) )).

fof(d1_heyting3,axiom,(
    ! [A,B] : k1_heyting3(A,B) = k3_lattice3(k5_substlat(A,B)) )).

fof(t15_heyting3,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,u1_struct_0(k1_heyting3(A,B)))
     => ! [D] :
          ( m1_subset_1(D,u1_struct_0(k1_heyting3(A,B)))
         => ( r3_orders_2(k1_heyting3(A,B),C,D)
          <=> ! [E] :
                ~ ( r2_hidden(E,C)
                  & ! [F] :
                      ~ ( r2_hidden(F,D)
                        & r1_tarski(F,E) ) ) ) ) ) )).

fof(t16_heyting3,axiom,(
    ! [A,B,C,D] :
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => ( v4_yellow_0(k1_heyting3(A,C),k1_heyting3(B,D))
        & m1_yellow_0(k1_heyting3(A,C),k1_heyting3(B,D)) ) ) )).

fof(d2_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B)))
             => ( C = k2_heyting3(A,B)
              <=> ! [D] :
                    ( r2_hidden(D,C)
                  <=> ~ ( ! [E] :
                            ( ( ~ v1_abian(E)
                              & m2_subset_1(E,k1_numbers,k5_numbers) )
                           => ~ ( r1_xreal_0(E,k2_nat_1(np__2,A))
                                & k1_domain_1(k5_numbers,k5_numbers,E,B) = D ) )
                        & k1_domain_1(k5_numbers,k5_numbers,k2_nat_1(np__2,A),B) != D ) ) ) ) ) ) )).

fof(d3_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B)))
             => ( C = k3_heyting3(A,B)
              <=> ! [D] :
                    ( r2_hidden(D,C)
                  <=> ? [E] :
                        ( ~ v1_abian(E)
                        & m2_subset_1(E,k1_numbers,k5_numbers)
                        & r1_xreal_0(E,k1_nat_1(k2_nat_1(np__2,A),np__1))
                        & k1_domain_1(k5_numbers,k5_numbers,E,B) = D ) ) ) ) ) ) )).

fof(t17_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r2_hidden(k1_domain_1(k5_numbers,k5_numbers,k1_nat_1(k2_nat_1(np__2,A),np__1),B),k3_heyting3(A,B)) ) ) )).

fof(t18_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r1_xboole_0(k3_heyting3(A,B),k6_domain_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,k1_nat_1(k2_nat_1(np__2,A),np__3),B))) ) ) )).

fof(t19_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k3_heyting3(k1_nat_1(A,np__1),B) = k2_xboole_0(k3_heyting3(A,B),k6_domain_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,k1_nat_1(k2_nat_1(np__2,A),np__3),B))) ) ) )).

fof(t20_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r2_xboole_0(k3_heyting3(A,B),k3_heyting3(k1_nat_1(A,np__1),B)) ) ) )).

fof(t21_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ r1_tarski(k2_heyting3(A,B),k3_heyting3(C,B)) ) ) ) )).

fof(t22_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(A,C)
               => r1_tarski(k3_heyting3(A,B),k3_heyting3(C,B)) ) ) ) ) )).

fof(t23_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_heyting3(np__1,A) = k7_domain_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__1,A),k1_domain_1(k5_numbers,k5_numbers,np__2,A)) ) )).

fof(d4_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,k5_finsub_1(k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B))))
             => ( C = k4_heyting3(A,B)
              <=> ! [D] :
                    ( r2_hidden(D,C)
                  <=> ~ ( ! [E] :
                            ( ( ~ v1_xboole_0(E)
                              & m2_subset_1(E,k1_numbers,k5_numbers) )
                           => ~ ( r1_xreal_0(E,A)
                                & D = k2_heyting3(E,B) ) )
                        & D != k3_heyting3(A,B) ) ) ) ) ) ) )).

fof(t24_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ~ ( r2_hidden(C,k4_heyting3(k1_nat_1(A,np__1),B))
                & ! [D] :
                    ~ ( r2_hidden(D,k4_heyting3(A,B))
                      & r1_tarski(D,C) ) ) ) ) )).

fof(t25_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ r2_hidden(k3_heyting3(A,B),k4_heyting3(k1_nat_1(A,np__1),B)) ) ) )).

fof(t26_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_tarski(k2_heyting3(A,B),k2_heyting3(C,B))
               => A = C ) ) ) ) )).

fof(t27_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_tarski(k3_heyting3(A,B),k2_heyting3(C,B))
              <=> ~ r1_xreal_0(C,A) ) ) ) ) )).

fof(t28_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => m1_subset_1(k4_heyting3(A,B),u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,B)))) ) ) )).

fof(d5_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))))
         => ( B = k5_heyting3(A)
          <=> ! [C] :
                ( r2_hidden(C,B)
              <=> ? [D] :
                    ( ~ v1_xboole_0(D)
                    & m2_subset_1(D,k1_numbers,k5_numbers)
                    & C = k4_heyting3(D,A) ) ) ) ) ) )).

fof(t29_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k4_heyting3(np__1,A) = k7_domain_1(k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,A)),k2_heyting3(np__1,A),k3_heyting3(np__1,A)) ) )).

fof(t30_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k4_heyting3(np__1,A) != k1_tarski(k1_xboole_0) ) )).

fof(t31_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => m1_subset_1(k6_domain_1(k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B)),k2_heyting3(A,B)),u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,B)))) ) ) )).

fof(t32_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B,C,D] :
          ( m1_subset_1(D,u1_struct_0(k1_heyting3(B,k6_domain_1(k5_numbers,A))))
         => ( r2_hidden(C,D)
           => ( v1_finset_1(C)
              & m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(B,k6_domain_1(k5_numbers,A)))) ) ) ) ) )).

fof(t33_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
         => ( r1_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),k5_heyting3(A),B)
           => ! [C] :
                ( ~ v1_xboole_0(C)
               => ~ ( r2_hidden(C,B)
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ~ ( r2_hidden(k1_domain_1(k5_numbers,k5_numbers,D,A),C)
                            & v1_abian(D) ) ) ) ) ) ) ) )).

fof(t34_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))))
                 => ( ( r1_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),D,B)
                      & r1_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),D,C) )
                   => r1_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),D,k13_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),B,C)) ) ) ) ) ) )).

fof(t35_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
         => ( r2_hidden(k1_xboole_0,B)
           => B = k1_tarski(k1_xboole_0) ) ) ) )).

fof(t36_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))) )
         => ~ ( B != k1_tarski(k1_xboole_0)
              & ! [C] :
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v1_finset_1(C) )
                 => ~ ( r2_hidden(C,B)
                      & C != k1_xboole_0 ) ) ) ) ) )).

fof(t37_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))) )
         => ! [C] :
              ( m1_subset_1(C,k5_finsub_1(k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,A))))
             => ( B = C
               => ( B = k1_tarski(k1_xboole_0)
                  | ( ~ v1_xboole_0(k1_heyting2(k5_numbers,k6_domain_1(k5_numbers,A),C))
                    & v1_finset_1(k1_heyting2(k5_numbers,k6_domain_1(k5_numbers,A),C))
                    & m1_subset_1(k1_heyting2(k5_numbers,k6_domain_1(k5_numbers,A),C),k1_zfmisc_1(k5_numbers)) ) ) ) ) ) ) )).

fof(t38_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
         => ! [C] :
              ( m1_subset_1(C,k5_finsub_1(k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,A))))
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & v1_finset_1(D)
                    & m1_subset_1(D,k1_zfmisc_1(k5_numbers)) )
                 => ( ( D = k1_heyting2(k5_numbers,k6_domain_1(k5_numbers,A),C)
                      & C = B )
                   => ! [E] :
                        ( r2_hidden(E,B)
                       => ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => ~ ( ~ r1_xreal_0(F,k2_xcmplx_0(k1_pre_circ(D),np__1))
                                & r2_hidden(k1_domain_1(k5_numbers,k5_numbers,F,A),E) ) ) ) ) ) ) ) ) )).

fof(t39_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k4_yellow_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))) = k1_tarski(k1_xboole_0) ) )).

fof(t40_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k3_yellow_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))) = k1_xboole_0 ) )).

fof(t41_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
             => ( ( r3_orders_2(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),B,C)
                  & B = k1_tarski(k1_xboole_0) )
               => C = k1_tarski(k1_xboole_0) ) ) ) ) )).

fof(t42_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
             => ( ( r3_orders_2(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),B,C)
                  & C = k1_xboole_0 )
               => B = k1_xboole_0 ) ) ) ) )).

fof(t43_heyting3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
         => ~ ( r1_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),k5_heyting3(A),B)
              & B = k1_tarski(k1_xboole_0) ) ) ) )).

fof(s1_heyting3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(f1_s1_heyting3)))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(f1_s1_heyting3)))
         => ( ( ! [C] :
                  ( r2_hidden(C,A)
                <=> p1_s1_heyting3(C) )
              & ! [C] :
                  ( r2_hidden(C,B)
                <=> p1_s1_heyting3(C) ) )
           => A = B ) ) ) )).

fof(dt_k1_heyting3,axiom,(
    ! [A,B] : l1_orders_2(k1_heyting3(A,B)) )).

fof(dt_k2_heyting3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m1_subset_1(k2_heyting3(A,B),k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B))) ) )).

fof(dt_k3_heyting3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m1_subset_1(k3_heyting3(A,B),k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B))) ) )).

fof(dt_k4_heyting3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m1_subset_1(k4_heyting3(A,B),k5_finsub_1(k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B)))) ) )).

fof(dt_k5_heyting3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m1_subset_1(k5_heyting3(A),k1_zfmisc_1(u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))) ) )).
%------------------------------------------------------------------------------