SET007 Axioms: SET007+227.ax


%------------------------------------------------------------------------------
% File     : SET007+227 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Metric Spaces
% 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_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   67 (   6 unit)
%            Number of atoms       :  336 (  57 equality)
%            Maximal formula depth :   19 (   7 average)
%            Number of connectives :  293 (  24 ~  ;   0  |; 117  &)
%                                         (  27 <=>; 125 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   1 propositional; 0-3 arity)
%            Number of functors    :   30 (   7 constant; 0-5 arity)
%            Number of variables   :  168 (   0 singleton; 155 !;  13 ?)
%            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(rc1_metric_1,axiom,(
    ? [A] :
      ( l1_metric_1(A)
      & v1_metric_1(A) ) )).

fof(rc2_metric_1,axiom,(
    ? [A] :
      ( l1_metric_1(A)
      & ~ v3_struct_0(A)
      & v1_metric_1(A) ) )).

fof(rc3_metric_1,axiom,(
    ? [A] :
      ( l1_metric_1(A)
      & ~ v3_struct_0(A)
      & v1_metric_1(A)
      & v6_metric_1(A)
      & v7_metric_1(A)
      & v8_metric_1(A)
      & v9_metric_1(A) ) )).

fof(fc1_metric_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k6_metric_1(A))
        & v1_metric_1(k6_metric_1(A)) ) ) )).

fof(fc2_metric_1,axiom,(
    ! [A] :
      ( v1_metric_1(k6_metric_1(A))
      & v6_metric_1(k6_metric_1(A))
      & v7_metric_1(k6_metric_1(A))
      & v8_metric_1(k6_metric_1(A))
      & v9_metric_1(k6_metric_1(A)) ) )).

fof(fc3_metric_1,axiom,
    ( ~ v3_struct_0(k8_metric_1)
    & v1_metric_1(k8_metric_1) )).

fof(fc4_metric_1,axiom,
    ( ~ v3_struct_0(k8_metric_1)
    & v1_metric_1(k8_metric_1)
    & v6_metric_1(k8_metric_1)
    & v7_metric_1(k8_metric_1)
    & v8_metric_1(k8_metric_1)
    & v9_metric_1(k8_metric_1) )).

fof(d1_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k2_metric_1(A,B,C) = k1_metric_1(u1_struct_0(A),u1_struct_0(A),u1_metric_1(A),B,C) ) ) ) )).

fof(d2_metric_1,axiom,(
    k3_metric_1 = k2_funcop_1(k2_zfmisc_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0)),np__0) )).

fof(d3_metric_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
     => ( v2_metric_1(B,A)
      <=> ! [C] :
            ( m1_subset_1(C,A)
           => k1_metric_1(A,A,B,C,C) = np__0 ) ) ) )).

fof(d4_metric_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
     => ( v3_metric_1(B,A)
      <=> ! [C] :
            ( m1_subset_1(C,A)
           => ! [D] :
                ( m1_subset_1(D,A)
               => ( k1_metric_1(A,A,B,C,D) = np__0
                 => C = D ) ) ) ) ) )).

fof(d5_metric_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
     => ( v4_metric_1(B,A)
      <=> ! [C] :
            ( m1_subset_1(C,A)
           => ! [D] :
                ( m1_subset_1(D,A)
               => k1_metric_1(A,A,B,C,D) = k1_metric_1(A,A,B,D,C) ) ) ) ) )).

fof(d6_metric_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
     => ( v5_metric_1(B,A)
      <=> ! [C] :
            ( m1_subset_1(C,A)
           => ! [D] :
                ( m1_subset_1(D,A)
               => ! [E] :
                    ( m1_subset_1(E,A)
                   => r1_xreal_0(k1_metric_1(A,A,B,C,E),k3_real_1(k1_metric_1(A,A,B,C,D),k1_metric_1(A,A,B,D,E))) ) ) ) ) ) )).

fof(d7_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( v6_metric_1(A)
      <=> v2_metric_1(u1_metric_1(A),u1_struct_0(A)) ) ) )).

fof(d8_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( v7_metric_1(A)
      <=> v3_metric_1(u1_metric_1(A),u1_struct_0(A)) ) ) )).

fof(d9_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( v8_metric_1(A)
      <=> v4_metric_1(u1_metric_1(A),u1_struct_0(A)) ) ) )).

fof(d10_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( v9_metric_1(A)
      <=> v5_metric_1(u1_metric_1(A),u1_struct_0(A)) ) ) )).

fof(t1_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => k2_metric_1(A,B,B) = np__0 )
      <=> v6_metric_1(A) ) ) )).

fof(t2_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k2_metric_1(A,B,C) = np__0
                 => B = C ) ) )
      <=> v7_metric_1(A) ) ) )).

fof(t3_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => k2_metric_1(A,B,C) = k2_metric_1(A,C,B) ) )
      <=> v8_metric_1(A) ) ) )).

fof(t4_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => r1_xreal_0(k2_metric_1(A,B,D),k3_real_1(k2_metric_1(A,B,C),k2_metric_1(A,C,D))) ) ) )
      <=> v9_metric_1(A) ) ) )).

fof(t5_metric_1,axiom,(
    ! [A] :
      ( ( v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_xreal_0(np__0,k4_metric_1(A,B,C)) ) ) ) )).

fof(t6_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( ( k2_metric_1(A,B,C) = np__0
                       => B = C )
                      & ( B = C
                       => k2_metric_1(A,B,C) = np__0 )
                      & k2_metric_1(A,B,C) = k2_metric_1(A,C,B)
                      & r1_xreal_0(k2_metric_1(A,B,D),k3_real_1(k2_metric_1(A,B,C),k2_metric_1(A,C,D))) ) ) ) )
       => ( v6_metric_1(A)
          & v7_metric_1(A)
          & v8_metric_1(A)
          & v9_metric_1(A)
          & l1_metric_1(A) ) ) ) )).

fof(d11_metric_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
        & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
     => ( B = k5_metric_1(A)
      <=> ! [C] :
            ( m1_subset_1(C,A)
           => ! [D] :
                ( m1_subset_1(D,A)
               => ( k1_metric_1(A,A,B,C,C) = np__0
                  & ( C != D
                   => k1_metric_1(A,A,B,C,D) = np__1 ) ) ) ) ) ) )).

fof(d12_metric_1,axiom,(
    ! [A] : k6_metric_1(A) = g1_metric_1(A,k5_metric_1(A)) )).

fof(d13_metric_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) )
     => ( A = k7_metric_1
      <=> ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => k1_metric_1(k1_numbers,k1_numbers,A,B,C) = k18_complex1(k5_real_1(B,C)) ) ) ) ) )).

fof(t7_metric_1,axiom,(
    $true )).

fof(t8_metric_1,axiom,(
    $true )).

fof(t9_metric_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( k1_metric_1(k1_numbers,k1_numbers,k7_metric_1,A,B) = np__0
          <=> A = B ) ) ) )).

fof(t10_metric_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k1_metric_1(k1_numbers,k1_numbers,k7_metric_1,A,B) = k1_metric_1(k1_numbers,k1_numbers,k7_metric_1,B,A) ) ) )).

fof(t11_metric_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => r1_xreal_0(k1_metric_1(k1_numbers,k1_numbers,k7_metric_1,A,B),k3_real_1(k1_metric_1(k1_numbers,k1_numbers,k7_metric_1,A,C),k1_metric_1(k1_numbers,k1_numbers,k7_metric_1,C,B))) ) ) ) )).

fof(d14_metric_1,axiom,(
    k8_metric_1 = g1_metric_1(k1_numbers,k7_metric_1) )).

fof(t12_metric_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( l1_metric_1(B)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ( r2_hidden(D,k9_metric_1(B,C,A))
                  <=> ( ~ v3_struct_0(B)
                      & ~ r1_xreal_0(A,k2_metric_1(B,C,D)) ) ) ) ) ) ) )).

fof(t13_metric_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( l1_metric_1(B)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ( r2_hidden(D,k10_metric_1(B,C,A))
                  <=> ( ~ v3_struct_0(B)
                      & r1_xreal_0(k2_metric_1(B,C,D),A) ) ) ) ) ) ) )).

fof(t14_metric_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( l1_metric_1(B)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ( r2_hidden(D,k11_metric_1(B,C,A))
                  <=> ( ~ v3_struct_0(B)
                      & k2_metric_1(B,C,D) = A ) ) ) ) ) ) )).

fof(t15_metric_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( l1_metric_1(B)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => r1_tarski(k9_metric_1(B,C,A),k10_metric_1(B,C,A)) ) ) ) )).

fof(t16_metric_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( l1_metric_1(B)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => r1_tarski(k11_metric_1(B,C,A),k10_metric_1(B,C,A)) ) ) ) )).

fof(t17_metric_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( l1_metric_1(B)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => k4_subset_1(u1_struct_0(B),k11_metric_1(B,C,A),k9_metric_1(B,C,A)) = k10_metric_1(B,C,A) ) ) ) )).

fof(dt_l1_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => l1_struct_0(A) ) )).

fof(existence_l1_metric_1,axiom,(
    ? [A] : l1_metric_1(A) )).

fof(abstractness_v1_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( v1_metric_1(A)
       => A = g1_metric_1(u1_struct_0(A),u1_metric_1(A)) ) ) )).

fof(dt_k1_metric_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_funct_1(C)
        & m1_relset_1(C,k2_zfmisc_1(A,B),k1_numbers)
        & m1_subset_1(D,A)
        & m1_subset_1(E,B) )
     => m1_subset_1(k1_metric_1(A,B,C,D,E),k1_numbers) ) )).

fof(redefinition_k1_metric_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_funct_1(C)
        & m1_relset_1(C,k2_zfmisc_1(A,B),k1_numbers)
        & m1_subset_1(D,A)
        & m1_subset_1(E,B) )
     => k1_metric_1(A,B,C,D,E) = k1_binop_1(C,D,E) ) )).

fof(dt_k2_metric_1,axiom,(
    ! [A,B,C] :
      ( ( l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k2_metric_1(A,B,C),k1_numbers) ) )).

fof(dt_k3_metric_1,axiom,
    ( v1_funct_1(k3_metric_1)
    & v1_funct_2(k3_metric_1,k2_zfmisc_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0)),k1_numbers)
    & m2_relset_1(k3_metric_1,k2_zfmisc_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0)),k1_numbers) )).

fof(dt_k4_metric_1,axiom,(
    ! [A,B,C] :
      ( ( v8_metric_1(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k4_metric_1(A,B,C),k1_numbers) ) )).

fof(commutativity_k4_metric_1,axiom,(
    ! [A,B,C] :
      ( ( v8_metric_1(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k4_metric_1(A,B,C) = k4_metric_1(A,C,B) ) )).

fof(redefinition_k4_metric_1,axiom,(
    ! [A,B,C] :
      ( ( v8_metric_1(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k4_metric_1(A,B,C) = k2_metric_1(A,B,C) ) )).

fof(dt_k5_metric_1,axiom,(
    ! [A] :
      ( v1_funct_1(k5_metric_1(A))
      & v1_funct_2(k5_metric_1(A),k2_zfmisc_1(A,A),k1_numbers)
      & m2_relset_1(k5_metric_1(A),k2_zfmisc_1(A,A),k1_numbers) ) )).

fof(dt_k6_metric_1,axiom,(
    ! [A] :
      ( v1_metric_1(k6_metric_1(A))
      & l1_metric_1(k6_metric_1(A)) ) )).

fof(dt_k7_metric_1,axiom,
    ( v1_funct_1(k7_metric_1)
    & v1_funct_2(k7_metric_1,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers)
    & m2_relset_1(k7_metric_1,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) )).

fof(dt_k8_metric_1,axiom,
    ( v1_metric_1(k8_metric_1)
    & l1_metric_1(k8_metric_1) )).

fof(dt_k9_metric_1,axiom,(
    ! [A,B,C] :
      ( ( l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & v1_xreal_0(C) )
     => m1_subset_1(k9_metric_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k10_metric_1,axiom,(
    ! [A,B,C] :
      ( ( l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & v1_xreal_0(C) )
     => m1_subset_1(k10_metric_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k11_metric_1,axiom,(
    ! [A,B,C] :
      ( ( l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & v1_xreal_0(C) )
     => m1_subset_1(k11_metric_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_u1_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( v1_funct_1(u1_metric_1(A))
        & v1_funct_2(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
        & m2_relset_1(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) ) ) )).

fof(dt_g1_metric_1,axiom,(
    ! [A,B] :
      ( ( 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) )
     => ( v1_metric_1(g1_metric_1(A,B))
        & l1_metric_1(g1_metric_1(A,B)) ) ) )).

fof(free_g1_metric_1,axiom,(
    ! [A,B] :
      ( ( 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) )
     => ! [C,D] :
          ( g1_metric_1(A,B) = g1_metric_1(C,D)
         => ( A = C
            & B = D ) ) ) )).

fof(d15_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( ( ~ v3_struct_0(A)
                     => ( D = k9_metric_1(A,B,C)
                      <=> ? [E] :
                            ( ~ v3_struct_0(E)
                            & l1_metric_1(E)
                            & ? [F] :
                                ( m1_subset_1(F,u1_struct_0(E))
                                & E = A
                                & F = B
                                & D = a_3_0_metric_1(C,E,F) ) ) ) )
                    & ( v3_struct_0(A)
                     => ( D = k9_metric_1(A,B,C)
                      <=> v1_xboole_0(D) ) ) ) ) ) ) ) )).

fof(d16_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( ( ~ v3_struct_0(A)
                     => ( D = k10_metric_1(A,B,C)
                      <=> ? [E] :
                            ( ~ v3_struct_0(E)
                            & l1_metric_1(E)
                            & ? [F] :
                                ( m1_subset_1(F,u1_struct_0(E))
                                & E = A
                                & F = B
                                & D = a_3_1_metric_1(C,E,F) ) ) ) )
                    & ( v3_struct_0(A)
                     => ( D = k10_metric_1(A,B,C)
                      <=> v1_xboole_0(D) ) ) ) ) ) ) ) )).

fof(d17_metric_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( ( ~ v3_struct_0(A)
                     => ( D = k11_metric_1(A,B,C)
                      <=> ? [E] :
                            ( ~ v3_struct_0(E)
                            & l1_metric_1(E)
                            & ? [F] :
                                ( m1_subset_1(F,u1_struct_0(E))
                                & E = A
                                & F = B
                                & D = a_3_2_metric_1(C,E,F) ) ) ) )
                    & ( v3_struct_0(A)
                     => ( D = k11_metric_1(A,B,C)
                      <=> v1_xboole_0(D) ) ) ) ) ) ) ) )).

fof(t18_metric_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => k9_metric_1(B,C,A) = a_3_0_metric_1(A,B,C) ) ) ) )).

fof(t19_metric_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => k10_metric_1(B,C,A) = a_3_1_metric_1(A,B,C) ) ) ) )).

fof(t20_metric_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => k11_metric_1(B,C,A) = a_3_2_metric_1(A,B,C) ) ) ) )).

fof(fraenkel_a_3_0_metric_1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_xreal_0(B)
        & ~ v3_struct_0(C)
        & l1_metric_1(C)
        & m1_subset_1(D,u1_struct_0(C)) )
     => ( r2_hidden(A,a_3_0_metric_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(C))
            & A = E
            & ~ r1_xreal_0(B,k2_metric_1(C,D,E)) ) ) ) )).

fof(fraenkel_a_3_1_metric_1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_xreal_0(B)
        & ~ v3_struct_0(C)
        & l1_metric_1(C)
        & m1_subset_1(D,u1_struct_0(C)) )
     => ( r2_hidden(A,a_3_1_metric_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(C))
            & A = E
            & r1_xreal_0(k2_metric_1(C,D,E),B) ) ) ) )).

fof(fraenkel_a_3_2_metric_1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_xreal_0(B)
        & ~ v3_struct_0(C)
        & l1_metric_1(C)
        & m1_subset_1(D,u1_struct_0(C)) )
     => ( r2_hidden(A,a_3_2_metric_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(C))
            & A = E
            & k2_metric_1(C,D,E) = B ) ) ) )).
%------------------------------------------------------------------------------