SET007 Axioms: SET007+264.ax


%------------------------------------------------------------------------------
% File     : SET007+264 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Metrics in Cartesian Product
% 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    : metric_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   43 (  12 unit)
%            Number of atoms       :  675 (  36 equality)
%            Maximal formula depth :   34 (  13 average)
%            Number of connectives :  718 (  86 ~  ;   0  |; 478  &)
%                                         (   6 <=>; 148 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   16 (   1 propositional; 0-3 arity)
%            Number of functors    :   38 (  14 constant; 0-8 arity)
%            Number of variables   :  170 (   0 singleton; 167 !;   3 ?)
%            Maximal term depth    :    5 (   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_metric_3,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
        & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
     => ( ~ v3_struct_0(g1_metric_1(A,B))
        & v1_metric_1(g1_metric_1(A,B)) ) ) )).

fof(d1_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k1_numbers)
                & m2_relset_1(C,k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k1_numbers) )
             => ( C = k1_metric_3(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(B))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(B))
                               => ! [H] :
                                    ( m1_subset_1(H,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
                                   => ! [I] :
                                        ( m1_subset_1(I,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
                                       => ( ( H = k4_tarski(D,F)
                                            & I = k4_tarski(E,G) )
                                         => k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),C,H,I) = k3_real_1(k4_metric_1(A,D,E),k4_metric_1(B,F,G)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t1_metric_3,axiom,(
    $true )).

fof(t2_metric_3,axiom,(
    $true )).

fof(t3_metric_3,axiom,(
    $true )).

fof(t4_metric_3,axiom,(
    $true )).

fof(t5_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
             => ! [D] :
                  ( m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
                 => ( k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),C,D) = np__0
                  <=> C = D ) ) ) ) ) )).

fof(t6_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
             => ! [D] :
                  ( m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
                 => k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),C,D) = k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),D,C) ) ) ) ) )).

fof(t7_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
             => ! [D] :
                  ( m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
                 => ! [E] :
                      ( m1_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
                     => r1_xreal_0(k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),C,E),k3_real_1(k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),C,D),k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),D,E))) ) ) ) ) ) )).

fof(d2_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
             => ! [D] :
                  ( m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
                 => k2_metric_3(A,B,C,D) = k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),C,D) ) ) ) ) )).

fof(d3_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => k3_metric_3(A,B) = g1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B)) ) ) )).

fof(d4_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C))),k1_numbers)
                    & m2_relset_1(D,k2_zfmisc_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C))),k1_numbers) )
                 => ( D = k4_metric_3(A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(B))
                               => ! [H] :
                                    ( m1_subset_1(H,u1_struct_0(B))
                                   => ! [I] :
                                        ( m1_subset_1(I,u1_struct_0(C))
                                       => ! [J] :
                                            ( m1_subset_1(J,u1_struct_0(C))
                                           => ! [K] :
                                                ( m1_subset_1(K,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
                                               => ! [L] :
                                                    ( m1_subset_1(L,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
                                                   => ( ( K = k3_mcart_1(E,G,I)
                                                        & L = k3_mcart_1(F,H,J) )
                                                     => k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),D,K,L) = k3_real_1(k3_real_1(k4_metric_1(A,E,F),k4_metric_1(B,G,H)),k4_metric_1(C,I,J)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t8_metric_3,axiom,(
    $true )).

fof(t9_metric_3,axiom,(
    $true )).

fof(t10_metric_3,axiom,(
    $true )).

fof(t11_metric_3,axiom,(
    $true )).

fof(t12_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
                 => ! [E] :
                      ( m1_subset_1(E,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
                     => ( k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),D,E) = np__0
                      <=> D = E ) ) ) ) ) ) )).

fof(t13_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
                 => ! [E] :
                      ( m1_subset_1(E,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
                     => k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),D,E) = k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),E,D) ) ) ) ) ) )).

fof(t14_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
                 => ! [E] :
                      ( m1_subset_1(E,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
                     => ! [F] :
                          ( m1_subset_1(F,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
                         => r1_xreal_0(k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),D,F),k3_real_1(k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),D,E),k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),E,F))) ) ) ) ) ) ) )).

fof(d5_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => k5_metric_3(A,B,C) = g1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C)) ) ) ) )).

fof(d6_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
                 => ! [E] :
                      ( m1_subset_1(E,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
                     => k6_metric_3(A,B,C,D,E) = k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),D,E) ) ) ) ) ) )).

fof(d7_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & v6_metric_1(D)
                    & v7_metric_1(D)
                    & v8_metric_1(D)
                    & v9_metric_1(D)
                    & l1_metric_1(D) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))),k1_numbers)
                        & m2_relset_1(E,k2_zfmisc_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))),k1_numbers) )
                     => ( E = k7_metric_3(A,B,C,D)
                      <=> ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(A))
                               => ! [H] :
                                    ( m1_subset_1(H,u1_struct_0(B))
                                   => ! [I] :
                                        ( m1_subset_1(I,u1_struct_0(B))
                                       => ! [J] :
                                            ( m1_subset_1(J,u1_struct_0(C))
                                           => ! [K] :
                                                ( m1_subset_1(K,u1_struct_0(C))
                                               => ! [L] :
                                                    ( m1_subset_1(L,u1_struct_0(D))
                                                   => ! [M] :
                                                        ( m1_subset_1(M,u1_struct_0(D))
                                                       => ! [N] :
                                                            ( m1_subset_1(N,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                                                           => ! [O] :
                                                                ( m1_subset_1(O,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                                                               => ( ( N = k4_mcart_1(F,H,J,L)
                                                                    & O = k4_mcart_1(G,I,K,M) )
                                                                 => k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),E,N,O) = k3_real_1(k3_real_1(k4_metric_1(A,F,G),k4_metric_1(B,H,I)),k3_real_1(k4_metric_1(C,J,K),k4_metric_1(D,L,M))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t15_metric_3,axiom,(
    $true )).

fof(t16_metric_3,axiom,(
    $true )).

fof(t17_metric_3,axiom,(
    $true )).

fof(t18_metric_3,axiom,(
    $true )).

fof(t19_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & v6_metric_1(D)
                    & v7_metric_1(D)
                    & v8_metric_1(D)
                    & v9_metric_1(D)
                    & l1_metric_1(D) )
                 => ! [E] :
                      ( m1_subset_1(E,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                     => ! [F] :
                          ( m1_subset_1(F,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                         => ( k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),E,F) = np__0
                          <=> E = F ) ) ) ) ) ) ) )).

fof(t20_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & v6_metric_1(D)
                    & v7_metric_1(D)
                    & v8_metric_1(D)
                    & v9_metric_1(D)
                    & l1_metric_1(D) )
                 => ! [E] :
                      ( m1_subset_1(E,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                     => ! [F] :
                          ( m1_subset_1(F,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                         => k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),E,F) = k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),F,E) ) ) ) ) ) ) )).

fof(t21_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & v6_metric_1(D)
                    & v7_metric_1(D)
                    & v8_metric_1(D)
                    & v9_metric_1(D)
                    & l1_metric_1(D) )
                 => ! [E] :
                      ( m1_subset_1(E,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                     => ! [F] :
                          ( m1_subset_1(F,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                         => ! [G] :
                              ( m1_subset_1(G,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                             => r1_xreal_0(k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),E,G),k3_real_1(k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),E,F),k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),F,G))) ) ) ) ) ) ) ) )).

fof(d8_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & v6_metric_1(D)
                    & v7_metric_1(D)
                    & v8_metric_1(D)
                    & v9_metric_1(D)
                    & l1_metric_1(D) )
                 => k8_metric_3(A,B,C,D) = g1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D)) ) ) ) ) )).

fof(d9_metric_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_metric_1(C)
                & v7_metric_1(C)
                & v8_metric_1(C)
                & v9_metric_1(C)
                & l1_metric_1(C) )
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & v6_metric_1(D)
                    & v7_metric_1(D)
                    & v8_metric_1(D)
                    & v9_metric_1(D)
                    & l1_metric_1(D) )
                 => ! [E] :
                      ( m1_subset_1(E,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                     => ! [F] :
                          ( m1_subset_1(F,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
                         => k9_metric_3(A,B,C,D,E,F) = k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),E,F) ) ) ) ) ) ) )).

fof(s1_metric_3,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(k2_zfmisc_1(f1_s1_metric_3,f2_s1_metric_3),k2_zfmisc_1(f1_s1_metric_3,f2_s1_metric_3)),f3_s1_metric_3)
      & m2_relset_1(A,k2_zfmisc_1(k2_zfmisc_1(f1_s1_metric_3,f2_s1_metric_3),k2_zfmisc_1(f1_s1_metric_3,f2_s1_metric_3)),f3_s1_metric_3)
      & ! [B] :
          ( m1_subset_1(B,f1_s1_metric_3)
         => ! [C] :
              ( m1_subset_1(C,f1_s1_metric_3)
             => ! [D] :
                  ( m1_subset_1(D,f2_s1_metric_3)
                 => ! [E] :
                      ( m1_subset_1(E,f2_s1_metric_3)
                     => ! [F] :
                          ( m1_subset_1(F,k2_zfmisc_1(f1_s1_metric_3,f2_s1_metric_3))
                         => ! [G] :
                              ( m1_subset_1(G,k2_zfmisc_1(f1_s1_metric_3,f2_s1_metric_3))
                             => ( ( F = k4_tarski(B,D)
                                  & G = k4_tarski(C,E) )
                               => k1_funct_1(A,k4_tarski(F,G)) = f4_s1_metric_3(B,C,D,E) ) ) ) ) ) ) ) ) )).

fof(s2_metric_3,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(k3_zfmisc_1(f1_s2_metric_3,f2_s2_metric_3,f3_s2_metric_3),k3_zfmisc_1(f1_s2_metric_3,f2_s2_metric_3,f3_s2_metric_3)),f4_s2_metric_3)
      & m2_relset_1(A,k2_zfmisc_1(k3_zfmisc_1(f1_s2_metric_3,f2_s2_metric_3,f3_s2_metric_3),k3_zfmisc_1(f1_s2_metric_3,f2_s2_metric_3,f3_s2_metric_3)),f4_s2_metric_3)
      & ! [B] :
          ( m1_subset_1(B,f1_s2_metric_3)
         => ! [C] :
              ( m1_subset_1(C,f1_s2_metric_3)
             => ! [D] :
                  ( m1_subset_1(D,f2_s2_metric_3)
                 => ! [E] :
                      ( m1_subset_1(E,f2_s2_metric_3)
                     => ! [F] :
                          ( m1_subset_1(F,f3_s2_metric_3)
                         => ! [G] :
                              ( m1_subset_1(G,f3_s2_metric_3)
                             => ! [H] :
                                  ( m1_subset_1(H,k3_zfmisc_1(f1_s2_metric_3,f2_s2_metric_3,f3_s2_metric_3))
                                 => ! [I] :
                                      ( m1_subset_1(I,k3_zfmisc_1(f1_s2_metric_3,f2_s2_metric_3,f3_s2_metric_3))
                                     => ( ( H = k3_mcart_1(B,D,F)
                                          & I = k3_mcart_1(C,E,G) )
                                       => k1_funct_1(A,k4_tarski(H,I)) = f5_s2_metric_3(B,C,D,E,F,G) ) ) ) ) ) ) ) ) ) ) )).

fof(s3_metric_3,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(k4_zfmisc_1(f1_s3_metric_3,f2_s3_metric_3,f3_s3_metric_3,f4_s3_metric_3),k4_zfmisc_1(f1_s3_metric_3,f2_s3_metric_3,f3_s3_metric_3,f4_s3_metric_3)),f5_s3_metric_3)
      & m2_relset_1(A,k2_zfmisc_1(k4_zfmisc_1(f1_s3_metric_3,f2_s3_metric_3,f3_s3_metric_3,f4_s3_metric_3),k4_zfmisc_1(f1_s3_metric_3,f2_s3_metric_3,f3_s3_metric_3,f4_s3_metric_3)),f5_s3_metric_3)
      & ! [B] :
          ( m1_subset_1(B,f1_s3_metric_3)
         => ! [C] :
              ( m1_subset_1(C,f1_s3_metric_3)
             => ! [D] :
                  ( m1_subset_1(D,f2_s3_metric_3)
                 => ! [E] :
                      ( m1_subset_1(E,f2_s3_metric_3)
                     => ! [F] :
                          ( m1_subset_1(F,f3_s3_metric_3)
                         => ! [G] :
                              ( m1_subset_1(G,f3_s3_metric_3)
                             => ! [H] :
                                  ( m1_subset_1(H,f4_s3_metric_3)
                                 => ! [I] :
                                      ( m1_subset_1(I,f4_s3_metric_3)
                                     => ! [J] :
                                          ( m1_subset_1(J,k4_zfmisc_1(f1_s3_metric_3,f2_s3_metric_3,f3_s3_metric_3,f4_s3_metric_3))
                                         => ! [K] :
                                              ( m1_subset_1(K,k4_zfmisc_1(f1_s3_metric_3,f2_s3_metric_3,f3_s3_metric_3,f4_s3_metric_3))
                                             => ( ( J = k4_mcart_1(B,D,F,H)
                                                  & K = k4_mcart_1(C,E,G,I) )
                                               => k1_funct_1(A,k4_tarski(J,K)) = f6_s3_metric_3(B,C,D,E,F,G,H,I) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(dt_k1_metric_3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v3_struct_0(B)
        & v6_metric_1(B)
        & v7_metric_1(B)
        & v8_metric_1(B)
        & v9_metric_1(B)
        & l1_metric_1(B) )
     => ( v1_funct_1(k1_metric_3(A,B))
        & v1_funct_2(k1_metric_3(A,B),k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k1_numbers)
        & m2_relset_1(k1_metric_3(A,B),k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k1_numbers) ) ) )).

fof(dt_k2_metric_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v3_struct_0(B)
        & v6_metric_1(B)
        & v7_metric_1(B)
        & v8_metric_1(B)
        & v9_metric_1(B)
        & l1_metric_1(B)
        & m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
        & m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))) )
     => m1_subset_1(k2_metric_3(A,B,C,D),k1_numbers) ) )).

fof(dt_k3_metric_3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v3_struct_0(B)
        & v6_metric_1(B)
        & v7_metric_1(B)
        & v8_metric_1(B)
        & v9_metric_1(B)
        & l1_metric_1(B) )
     => ( ~ v3_struct_0(k3_metric_3(A,B))
        & v1_metric_1(k3_metric_3(A,B))
        & v6_metric_1(k3_metric_3(A,B))
        & v7_metric_1(k3_metric_3(A,B))
        & v8_metric_1(k3_metric_3(A,B))
        & v9_metric_1(k3_metric_3(A,B))
        & l1_metric_1(k3_metric_3(A,B)) ) ) )).

fof(dt_k4_metric_3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v3_struct_0(B)
        & v6_metric_1(B)
        & v7_metric_1(B)
        & v8_metric_1(B)
        & v9_metric_1(B)
        & l1_metric_1(B)
        & ~ v3_struct_0(C)
        & v6_metric_1(C)
        & v7_metric_1(C)
        & v8_metric_1(C)
        & v9_metric_1(C)
        & l1_metric_1(C) )
     => ( v1_funct_1(k4_metric_3(A,B,C))
        & v1_funct_2(k4_metric_3(A,B,C),k2_zfmisc_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C))),k1_numbers)
        & m2_relset_1(k4_metric_3(A,B,C),k2_zfmisc_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C))),k1_numbers) ) ) )).

fof(dt_k5_metric_3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v3_struct_0(B)
        & v6_metric_1(B)
        & v7_metric_1(B)
        & v8_metric_1(B)
        & v9_metric_1(B)
        & l1_metric_1(B)
        & ~ v3_struct_0(C)
        & v6_metric_1(C)
        & v7_metric_1(C)
        & v8_metric_1(C)
        & v9_metric_1(C)
        & l1_metric_1(C) )
     => ( ~ v3_struct_0(k5_metric_3(A,B,C))
        & v1_metric_1(k5_metric_3(A,B,C))
        & v6_metric_1(k5_metric_3(A,B,C))
        & v7_metric_1(k5_metric_3(A,B,C))
        & v8_metric_1(k5_metric_3(A,B,C))
        & v9_metric_1(k5_metric_3(A,B,C))
        & l1_metric_1(k5_metric_3(A,B,C)) ) ) )).

fof(dt_k6_metric_3,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v3_struct_0(B)
        & v6_metric_1(B)
        & v7_metric_1(B)
        & v8_metric_1(B)
        & v9_metric_1(B)
        & l1_metric_1(B)
        & ~ v3_struct_0(C)
        & v6_metric_1(C)
        & v7_metric_1(C)
        & v8_metric_1(C)
        & v9_metric_1(C)
        & l1_metric_1(C)
        & m1_subset_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
        & m1_subset_1(E,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C))) )
     => m1_subset_1(k6_metric_3(A,B,C,D,E),k1_numbers) ) )).

fof(dt_k7_metric_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v3_struct_0(B)
        & v6_metric_1(B)
        & v7_metric_1(B)
        & v8_metric_1(B)
        & v9_metric_1(B)
        & l1_metric_1(B)
        & ~ v3_struct_0(C)
        & v6_metric_1(C)
        & v7_metric_1(C)
        & v8_metric_1(C)
        & v9_metric_1(C)
        & l1_metric_1(C)
        & ~ v3_struct_0(D)
        & v6_metric_1(D)
        & v7_metric_1(D)
        & v8_metric_1(D)
        & v9_metric_1(D)
        & l1_metric_1(D) )
     => ( v1_funct_1(k7_metric_3(A,B,C,D))
        & v1_funct_2(k7_metric_3(A,B,C,D),k2_zfmisc_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))),k1_numbers)
        & m2_relset_1(k7_metric_3(A,B,C,D),k2_zfmisc_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))),k1_numbers) ) ) )).

fof(dt_k8_metric_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v3_struct_0(B)
        & v6_metric_1(B)
        & v7_metric_1(B)
        & v8_metric_1(B)
        & v9_metric_1(B)
        & l1_metric_1(B)
        & ~ v3_struct_0(C)
        & v6_metric_1(C)
        & v7_metric_1(C)
        & v8_metric_1(C)
        & v9_metric_1(C)
        & l1_metric_1(C)
        & ~ v3_struct_0(D)
        & v6_metric_1(D)
        & v7_metric_1(D)
        & v8_metric_1(D)
        & v9_metric_1(D)
        & l1_metric_1(D) )
     => ( ~ v3_struct_0(k8_metric_3(A,B,C,D))
        & v1_metric_1(k8_metric_3(A,B,C,D))
        & v6_metric_1(k8_metric_3(A,B,C,D))
        & v7_metric_1(k8_metric_3(A,B,C,D))
        & v8_metric_1(k8_metric_3(A,B,C,D))
        & v9_metric_1(k8_metric_3(A,B,C,D))
        & l1_metric_1(k8_metric_3(A,B,C,D)) ) ) )).

fof(dt_k9_metric_3,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v3_struct_0(B)
        & v6_metric_1(B)
        & v7_metric_1(B)
        & v8_metric_1(B)
        & v9_metric_1(B)
        & l1_metric_1(B)
        & ~ v3_struct_0(C)
        & v6_metric_1(C)
        & v7_metric_1(C)
        & v8_metric_1(C)
        & v9_metric_1(C)
        & l1_metric_1(C)
        & ~ v3_struct_0(D)
        & v6_metric_1(D)
        & v7_metric_1(D)
        & v8_metric_1(D)
        & v9_metric_1(D)
        & l1_metric_1(D)
        & m1_subset_1(E,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
        & m1_subset_1(F,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))) )
     => m1_subset_1(k9_metric_3(A,B,C,D,E,F),k1_numbers) ) )).
%------------------------------------------------------------------------------