## 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 ) ) ) )).
%------------------------------------------------------------------------------
```