SET007 Axioms: SET007+596.ax


%------------------------------------------------------------------------------
% File     : SET007+596 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Structural Induction in the Positive Propositional Language
% 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    : hilbert2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   57 (   2 unit)
%            Number of atoms       :  307 (  69 equality)
%            Maximal formula depth :   21 (   7 average)
%            Number of connectives :  281 (  31 ~  ;   1  |; 102  &)
%                                         (   9 <=>; 138 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   33 (   0 propositional; 1-5 arity)
%            Number of functors    :   43 (  13 constant; 0-4 arity)
%            Number of variables   :  154 (   0 singleton; 143 !;  11 ?)
%            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(fc1_hilbert2,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0)
    & v1_funct_1(k1_xboole_0)
    & v2_funct_1(k1_xboole_0)
    & v1_finset_1(k1_xboole_0)
    & v1_finseq_1(k1_xboole_0)
    & v1_funcop_1(k1_xboole_0)
    & v4_trees_3(k1_xboole_0) )).

fof(fc2_hilbert2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B)
        & v6_trees_3(B) )
     => ( v1_relat_1(k4_trees_4(A,B))
        & v1_funct_1(k4_trees_4(A,B))
        & v3_trees_2(k4_trees_4(A,B))
        & ~ v1_trees_9(k4_trees_4(A,B)) ) ) )).

fof(fc3_hilbert2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v3_trees_2(B) )
     => ( v1_relat_1(k5_trees_4(A,B))
        & v1_funct_1(k5_trees_4(A,B))
        & v3_trees_2(k5_trees_4(A,B))
        & ~ v1_trees_9(k5_trees_4(A,B)) ) ) )).

fof(fc4_hilbert2,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v3_trees_2(B)
        & v1_relat_1(C)
        & v1_funct_1(C)
        & v3_trees_2(C) )
     => ( v1_relat_1(k6_trees_4(A,B,C))
        & v1_funct_1(k6_trees_4(A,B,C))
        & v3_trees_2(k6_trees_4(A,B,C))
        & ~ v1_trees_9(k6_trees_4(A,B,C)) ) ) )).

fof(t1_hilbert2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ( ! [C] :
            ( r2_hidden(C,A)
           => m1_pboole(k1_funct_1(B,C),C) )
       => ! [C] :
            ( ( v1_relat_1(C)
              & v1_funct_1(C) )
           => ( C = k3_card_3(B)
             => k1_relat_1(C) = k3_tarski(A) ) ) ) ) )).

fof(t2_hilbert2,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C)
        & v1_finseq_1(C) )
     => ! [D] :
          ( ( v1_relat_1(D)
            & v1_funct_1(D)
            & v1_finseq_1(D) )
         => ( k7_finseq_1(k9_finseq_1(A),C) = k7_finseq_1(k9_finseq_1(B),D)
           => C = D ) ) ) )).

fof(t3_hilbert2,axiom,(
    ! [A,B] :
      ( m2_finseq_1(k9_finseq_1(A),B)
     => r2_hidden(A,B) ) )).

fof(t4_hilbert2,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ~ ( B != k1_xboole_0
          & ! [C] :
              ( m2_finseq_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => B != k7_finseq_1(C,k9_finseq_1(D)) ) ) ) ) )).

fof(t5_hilbert2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_trees_1(B) )
     => ! [C] :
          ( ( ~ v1_xboole_0(C)
            & v1_trees_1(C) )
         => ( r2_hidden(k9_finseq_1(A),k15_trees_3(B,C))
          <=> ( A = np__0
              | A = np__1 ) ) ) ) )).

fof(t6_hilbert2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v3_trees_2(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C)
            & v3_trees_2(C) )
         => k1_funct_1(k6_trees_4(A,B,C),k1_xboole_0) = A ) ) )).

fof(t7_hilbert2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v3_trees_2(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C)
            & v3_trees_2(C) )
         => ( k1_funct_1(k6_trees_4(A,B,C),k12_finseq_1(k5_numbers,np__0)) = k1_funct_1(B,k1_xboole_0)
            & k1_funct_1(k6_trees_4(A,B,C),k12_finseq_1(k5_numbers,np__1)) = k1_funct_1(C,k1_xboole_0) ) ) ) )).

fof(t8_hilbert2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v3_trees_2(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C)
            & v3_trees_2(C) )
         => ( k5_trees_2(k6_trees_4(A,B,C),k12_finseq_1(k5_numbers,np__0)) = B
            & k5_trees_2(k6_trees_4(A,B,C),k12_finseq_1(k5_numbers,np__1)) = C ) ) ) )).

fof(d1_hilbert2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_hilbert2(A) = k12_finseq_1(k5_numbers,k1_nat_1(np__3,A)) ) )).

fof(d2_hilbert2,axiom,(
    ! [A] :
      ( v1_hilbert1(A)
    <=> r2_hidden(k2_hilbert1,A) ) )).

fof(d3_hilbert2,axiom,(
    ! [A] :
      ( v4_hilbert1(A)
    <=> ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r2_hidden(k1_hilbert2(B),A) ) ) )).

fof(d4_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
     => ( v2_hilbert1(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_hilbert1)
           => ! [C] :
                ( m1_subset_1(C,k1_hilbert1)
               => ( ( r2_hidden(B,A)
                    & r2_hidden(C,A) )
                 => r2_hidden(k3_hilbert1(B,C),A) ) ) ) ) ) )).

fof(d5_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
     => ( v3_hilbert1(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_hilbert1)
           => ! [C] :
                ( m1_subset_1(C,k1_hilbert1)
               => ( ( r2_hidden(B,A)
                    & r2_hidden(C,A) )
                 => r2_hidden(k4_hilbert1(B,C),A) ) ) ) ) ) )).

fof(d6_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ( v1_hilbert2(A)
      <=> ? [B] :
            ( m1_subset_1(B,k1_hilbert1)
            & ? [C] :
                ( m1_subset_1(C,k1_hilbert1)
                & A = k4_hilbert1(B,C) ) ) ) ) )).

fof(d7_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ( v2_hilbert2(A)
      <=> ? [B] :
            ( m1_subset_1(B,k1_hilbert1)
            & ? [C] :
                ( m1_subset_1(C,k1_hilbert1)
                & A = k3_hilbert1(B,C) ) ) ) ) )).

fof(d8_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ( v3_hilbert2(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & A = k1_hilbert2(B) ) ) ) )).

fof(t9_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ~ ( ~ v1_hilbert2(A)
          & ~ v2_hilbert2(A)
          & ~ v3_hilbert2(A)
          & A != k2_hilbert1 ) ) )).

fof(t10_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => r1_xreal_0(np__1,k3_finseq_1(A)) ) )).

fof(t11_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ( k1_funct_1(A,np__1) = np__1
       => v2_hilbert2(A) ) ) )).

fof(t12_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ( k1_funct_1(A,np__1) = np__2
       => v1_hilbert2(A) ) ) )).

fof(t13_hilbert2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ( k1_funct_1(B,np__1) = k1_nat_1(np__3,A)
           => v3_hilbert2(B) ) ) ) )).

fof(t14_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ( k1_funct_1(A,np__1) = np__0
       => A = k2_hilbert1 ) ) )).

fof(t15_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ( ~ r1_xreal_0(k3_finseq_1(k4_hilbert1(A,B)),k3_finseq_1(A))
            & ~ r1_xreal_0(k3_finseq_1(k4_hilbert1(A,B)),k3_finseq_1(B)) ) ) ) )).

fof(t16_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ( ~ r1_xreal_0(k3_finseq_1(k3_hilbert1(A,B)),k3_finseq_1(A))
            & ~ r1_xreal_0(k3_finseq_1(k3_hilbert1(A,B)),k3_finseq_1(B)) ) ) ) )).

fof(t17_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( A = k7_finseq_1(B,C)
               => A = B ) ) ) ) )).

fof(t18_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ! [C] :
              ( m1_subset_1(C,k1_hilbert1)
             => ! [D] :
                  ( m1_subset_1(D,k1_hilbert1)
                 => ( k7_finseq_1(A,B) = k7_finseq_1(C,D)
                   => ( A = C
                      & B = D ) ) ) ) ) ) )).

fof(t19_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ! [C] :
              ( m1_subset_1(C,k1_hilbert1)
             => ! [D] :
                  ( m1_subset_1(D,k1_hilbert1)
                 => ( k4_hilbert1(A,B) = k4_hilbert1(C,D)
                   => ( A = C
                      & D = B ) ) ) ) ) ) )).

fof(t20_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ! [C] :
              ( m1_subset_1(C,k1_hilbert1)
             => ! [D] :
                  ( m1_subset_1(D,k1_hilbert1)
                 => ( k3_hilbert1(A,B) = k3_hilbert1(C,D)
                   => ( A = C
                      & D = B ) ) ) ) ) ) )).

fof(t21_hilbert2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k1_hilbert2(A) = k1_hilbert2(B)
           => A = B ) ) ) )).

fof(t22_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ! [C] :
              ( m1_subset_1(C,k1_hilbert1)
             => ! [D] :
                  ( m1_subset_1(D,k1_hilbert1)
                 => k4_hilbert1(A,B) != k3_hilbert1(C,D) ) ) ) ) )).

fof(t23_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => k4_hilbert1(A,B) != k2_hilbert1 ) ) )).

fof(t24_hilbert2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ! [C] :
              ( m1_subset_1(C,k1_hilbert1)
             => k4_hilbert1(B,C) != k1_hilbert2(A) ) ) ) )).

fof(t25_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => k3_hilbert1(A,B) != k2_hilbert1 ) ) )).

fof(t26_hilbert2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ! [C] :
              ( m1_subset_1(C,k1_hilbert1)
             => k3_hilbert1(B,C) != k1_hilbert2(A) ) ) ) )).

fof(t27_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ( k4_hilbert1(A,B) != A
            & k4_hilbert1(A,B) != B ) ) ) )).

fof(t28_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ( k3_hilbert1(A,B) != A
            & k3_hilbert1(A,B) != B ) ) ) )).

fof(t29_hilbert2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_hilbert1 != k1_hilbert2(A) ) )).

fof(d9_hilbert2,axiom,(
    ! [A] :
      ( m1_pboole(A,k1_hilbert1)
     => ( A = k2_hilbert2
      <=> ( k1_funct_1(A,k2_hilbert1) = k2_trees_4(k1_hilbert1,k2_hilbert1)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => k1_funct_1(A,k1_hilbert2(B)) = k2_trees_4(k1_hilbert1,k1_hilbert2(B)) )
          & ! [B] :
              ( m1_subset_1(B,k1_hilbert1)
             => ! [C] :
                  ( m1_subset_1(C,k1_hilbert1)
                 => ? [D] :
                      ( v1_funct_1(D)
                      & v3_trees_2(D)
                      & m3_trees_2(D,k1_hilbert1)
                      & ? [E] :
                          ( v1_funct_1(E)
                          & v3_trees_2(E)
                          & m3_trees_2(E,k1_hilbert1)
                          & D = k1_funct_1(A,B)
                          & E = k1_funct_1(A,C)
                          & k1_funct_1(A,k4_hilbert1(B,C)) = k10_trees_4(k1_hilbert1,k4_hilbert1(B,C),D,E)
                          & k1_funct_1(A,k3_hilbert1(B,C)) = k10_trees_4(k1_hilbert1,k3_hilbert1(B,C),D,E) ) ) ) ) ) ) ) )).

fof(d10_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => k3_hilbert2(A) = k1_funct_1(k2_hilbert2,A) ) )).

fof(t30_hilbert2,axiom,(
    k3_hilbert2(k2_hilbert1) = k2_trees_4(k1_hilbert1,k2_hilbert1) )).

fof(t31_hilbert2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k3_hilbert2(k1_hilbert2(A)) = k2_trees_4(k1_hilbert1,k1_hilbert2(A)) ) )).

fof(t32_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => k3_hilbert2(k4_hilbert1(A,B)) = k10_trees_4(k1_hilbert1,k4_hilbert1(A,B),k3_hilbert2(A),k3_hilbert2(B)) ) ) )).

fof(t33_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => k3_hilbert2(k3_hilbert1(A,B)) = k10_trees_4(k1_hilbert1,k3_hilbert1(A,B),k3_hilbert2(A),k3_hilbert2(B)) ) ) )).

fof(t34_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => k1_funct_1(k3_hilbert2(A),k1_xboole_0) = A ) )).

fof(t35_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_trees_1(B,k1_relat_1(k3_hilbert2(A)))
         => k7_trees_2(k1_hilbert1,k3_hilbert2(A),B) = k3_hilbert2(k3_trees_2(k1_hilbert1,k3_hilbert2(A),B)) ) ) )).

fof(t36_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ~ ( r2_hidden(A,k6_trees_2(k1_hilbert1,k3_hilbert2(B)))
              & A != k2_hilbert1
              & ~ v3_hilbert2(A) ) ) ) )).

fof(s1_hilbert2,axiom,
    ( ( p1_s1_hilbert2(k6_finseq_1(k5_numbers))
      & ! [A] :
          ( m1_trees_1(A,f1_s1_hilbert2)
         => ( p1_s1_hilbert2(A)
           => ! [B] :
                ( m2_subset_1(B,k1_numbers,k5_numbers)
               => ( r2_hidden(k8_finseq_1(k5_numbers,A,k12_finseq_1(k5_numbers,B)),f1_s1_hilbert2)
                 => p1_s1_hilbert2(k8_finseq_1(k5_numbers,A,k12_finseq_1(k5_numbers,B))) ) ) ) ) )
   => ! [A] :
        ( m1_trees_1(A,f1_s1_hilbert2)
       => p1_s1_hilbert2(A) ) )).

fof(s2_hilbert2,axiom,
    ( ( p1_s2_hilbert2(k2_hilbert1)
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => p1_s2_hilbert2(k1_hilbert2(A)) )
      & ! [A] :
          ( m1_subset_1(A,k1_hilbert1)
         => ! [B] :
              ( m1_subset_1(B,k1_hilbert1)
             => ( ( p1_s2_hilbert2(A)
                  & p1_s2_hilbert2(B) )
               => ( p1_s2_hilbert2(k4_hilbert1(A,B))
                  & p1_s2_hilbert2(k3_hilbert1(A,B)) ) ) ) ) )
   => ! [A] :
        ( m1_subset_1(A,k1_hilbert1)
       => p1_s2_hilbert2(A) ) )).

fof(s3_hilbert2,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,k1_hilbert1)
         => ! [B] :
              ( m1_subset_1(B,k1_hilbert1)
             => ! [C,D] :
                ? [E] : p1_s3_hilbert2(A,B,C,D,E) ) )
      & ! [A] :
          ( m1_subset_1(A,k1_hilbert1)
         => ! [B] :
              ( m1_subset_1(B,k1_hilbert1)
             => ! [C,D] :
                ? [E] : p2_s3_hilbert2(A,B,C,D,E) ) )
      & ! [A] :
          ( m1_subset_1(A,k1_hilbert1)
         => ! [B] :
              ( m1_subset_1(B,k1_hilbert1)
             => ! [C,D,E,F] :
                  ( ( p1_s3_hilbert2(A,B,C,D,E)
                    & p1_s3_hilbert2(A,B,C,D,F) )
                 => E = F ) ) )
      & ! [A] :
          ( m1_subset_1(A,k1_hilbert1)
         => ! [B] :
              ( m1_subset_1(B,k1_hilbert1)
             => ! [C,D,E,F] :
                  ( ( p2_s3_hilbert2(A,B,C,D,E)
                    & p2_s3_hilbert2(A,B,C,D,F) )
                 => E = F ) ) ) )
   => ? [A] :
        ( m1_pboole(A,k1_hilbert1)
        & k1_funct_1(A,k2_hilbert1) = f1_s3_hilbert2
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k1_funct_1(A,k1_hilbert2(B)) = f2_s3_hilbert2(B) )
        & ! [B] :
            ( m1_subset_1(B,k1_hilbert1)
           => ! [C] :
                ( m1_subset_1(C,k1_hilbert1)
               => ( p1_s3_hilbert2(B,C,k1_funct_1(A,B),k1_funct_1(A,C),k1_funct_1(A,k4_hilbert1(B,C)))
                  & p2_s3_hilbert2(B,C,k1_funct_1(A,B),k1_funct_1(A,C),k1_funct_1(A,k3_hilbert1(B,C))) ) ) ) ) )).

fof(s4_hilbert2,axiom,(
    ? [A] :
      ( m1_pboole(A,k1_hilbert1)
      & k1_funct_1(A,k2_hilbert1) = f1_s4_hilbert2
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_funct_1(A,k1_hilbert2(B)) = f2_s4_hilbert2(B) )
      & ! [B] :
          ( m1_subset_1(B,k1_hilbert1)
         => ! [C] :
              ( m1_subset_1(C,k1_hilbert1)
             => ( k1_funct_1(A,k4_hilbert1(B,C)) = f3_s4_hilbert2(k1_funct_1(A,B),k1_funct_1(A,C))
                & k1_funct_1(A,k3_hilbert1(B,C)) = f4_s4_hilbert2(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ) ) )).

fof(dt_k1_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m1_subset_1(k1_hilbert2(A),k1_hilbert1) ) )).

fof(dt_k2_hilbert2,axiom,(
    m1_pboole(k2_hilbert2,k1_hilbert1) )).

fof(dt_k3_hilbert2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_hilbert1)
     => ( v1_funct_1(k3_hilbert2(A))
        & v3_trees_2(k3_hilbert2(A))
        & m3_trees_2(k3_hilbert2(A),k1_hilbert1) ) ) )).
%------------------------------------------------------------------------------