SET007 Axioms: SET007+548.ax
%------------------------------------------------------------------------------
% File : SET007+548 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : First-countable, Sequential, and Frechet 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 : frechet [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 56 ( 10 unt; 0 def)
% Number of atoms : 339 ( 32 equ)
% Maximal formula atoms : 19 ( 6 avg)
% Number of connectives : 336 ( 53 ~; 5 |; 149 &)
% ( 14 <=>; 115 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 33 ( 31 usr; 1 prp; 0-3 aty)
% Number of functors : 41 ( 41 usr; 9 con; 0-4 aty)
% Number of variables : 122 ( 114 !; 8 ?)
% 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(cc1_frechet,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ! [C] :
( m1_yellow_8(C,A,B)
=> ~ v1_xboole_0(C) ) ) ).
fof(fc1_frechet,axiom,
( ~ v3_struct_0(k3_topmetr)
& v1_pre_topc(k3_topmetr)
& v2_pre_topc(k3_topmetr)
& v1_frechet(k3_topmetr) ) ).
fof(cc2_frechet,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_frechet(A) )
=> ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_frechet(A) ) ) ) ).
fof(cc3_frechet,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_frechet(A) )
=> ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v4_frechet(A) ) ) ) ).
fof(t1_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> m1_subset_1(k2_relat_1(B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(t2_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( l1_struct_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( r1_tarski(k1_frechet(A,C),u1_struct_0(B))
=> ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(B))
& m2_relset_1(C,k5_numbers,u1_struct_0(B)) ) ) ) ) ) ).
fof(t3_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_yellow_8(C,A,B)
=> C != k1_xboole_0 ) ) ) ).
fof(t4_frechet,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_pre_topc(B,A)
& v4_pre_topc(C,A) )
=> v3_pre_topc(k6_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t5_frechet,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( v4_pre_topc(k1_pre_topc(A),A)
& v4_pre_topc(k2_pre_topc(A),A)
& ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_pre_topc(B,A)
& v4_pre_topc(C,A) )
=> v4_pre_topc(k4_subset_1(u1_struct_0(A),B,C),A) ) ) )
& ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v2_tops_2(B,A)
=> v4_pre_topc(k6_setfam_1(u1_struct_0(A),B),A) ) ) )
=> ( v2_pre_topc(A)
& l1_pre_topc(A) ) ) ) ).
fof(t6_frechet,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( v4_pre_topc(D,B)
<=> v4_pre_topc(k5_pre_topc(A,B,C,D),A) ) )
=> ( v2_pre_topc(B)
& l1_pre_topc(B) ) ) ) ) ) ).
fof(t7_frechet,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k8_metric_1))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( B = A
=> ( r1_xreal_0(C,np__0)
| k9_metric_1(k8_metric_1,A,C) = k2_rcomp_1(k5_real_1(B,C),k3_real_1(B,C)) ) ) ) ) ) ).
fof(t8_frechet,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( v3_pre_topc(A,k3_topmetr)
<=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r2_hidden(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& r1_tarski(k2_rcomp_1(k5_real_1(B,C),k3_real_1(B,C)),A) ) ) ) ) ) ) ).
fof(t9_frechet,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(k3_topmetr))
& m2_relset_1(A,k5_numbers,u1_struct_0(k3_topmetr)) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r2_hidden(k8_funct_2(k5_numbers,u1_struct_0(k3_topmetr),A,B),k2_rcomp_1(k5_real_1(B,k6_real_1(np__1,np__4)),k3_real_1(B,k6_real_1(np__1,np__4)))) )
=> v4_pre_topc(k1_frechet(k3_topmetr,A),k3_topmetr) ) ) ).
fof(t10_frechet,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = k5_numbers
=> v4_pre_topc(A,k3_topmetr) ) ) ).
fof(t12_frechet,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k2_relat_1(k1_funct_4(A,B)) = k2_xboole_0(k9_relat_1(A,k4_xboole_0(k1_relat_1(A),k1_relat_1(B))),k2_relat_1(B)) ) ) ).
fof(t13_frechet,axiom,
! [A,B] :
( r1_tarski(B,A)
=> k9_relat_1(k6_relat_1(A),B) = B ) ).
fof(t14_frechet,axiom,
$true ).
fof(t15_frechet,axiom,
! [A,B,C] : k1_relat_1(k1_funct_4(k6_relat_1(A),k2_funcop_1(B,C))) = k2_xboole_0(A,B) ).
fof(t16_frechet,axiom,
! [A,B,C] :
( B != k1_xboole_0
=> k2_relat_1(k1_funct_4(k6_relat_1(A),k2_funcop_1(B,C))) = k2_xboole_0(k4_xboole_0(A,B),k1_tarski(C)) ) ).
fof(t17_frechet,axiom,
! [A,B,C,D] :
( r1_tarski(C,A)
=> k10_relat_1(k1_funct_4(k6_relat_1(A),k2_funcop_1(B,D)),k4_xboole_0(C,k1_tarski(D))) = k4_xboole_0(k4_xboole_0(C,B),k1_tarski(D)) ) ).
fof(t18_frechet,axiom,
! [A,B,C] :
( ~ r2_hidden(C,A)
=> k10_relat_1(k1_funct_4(k6_relat_1(A),k2_funcop_1(B,C)),k1_tarski(C)) = B ) ).
fof(t19_frechet,axiom,
! [A,B,C,D] :
( r1_tarski(C,A)
=> ( r2_hidden(D,A)
| k10_relat_1(k1_funct_4(k6_relat_1(A),k2_funcop_1(B,D)),k2_xboole_0(C,k1_tarski(D))) = k2_xboole_0(C,B) ) ) ).
fof(t20_frechet,axiom,
! [A,B,C,D] :
( r1_tarski(C,A)
=> ( r2_hidden(D,A)
| k10_relat_1(k1_funct_4(k6_relat_1(A),k2_funcop_1(B,D)),k4_xboole_0(C,k1_tarski(D))) = k4_xboole_0(C,B) ) ) ).
fof(d1_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( v1_frechet(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_yellow_8(C,A,B)
& v1_card_4(C) ) ) ) ) ).
fof(t21_frechet,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) )
=> v1_frechet(k5_pcomps_1(A)) ) ).
fof(t22_frechet,axiom,
v1_frechet(k3_topmetr) ).
fof(d2_frechet,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_frechet(A,B,C)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& r2_hidden(C,D)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(E,F)
& ~ r2_hidden(k8_funct_2(k5_numbers,u1_struct_0(A),B,F),D) ) ) ) ) ) ) ) ) ).
fof(t23_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( C = k2_funcop_1(k5_numbers,B)
=> r1_frechet(A,C,B) ) ) ) ) ).
fof(d3_frechet,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v2_frechet(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& r1_frechet(A,B,C) ) ) ) ) ).
fof(d4_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = k2_frechet(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,C)
<=> r1_frechet(A,B,D) ) ) ) ) ) ) ).
fof(d5_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( v3_frechet(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r2_hidden(C,k6_pre_topc(A,B))
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ~ ( r1_tarski(k1_frechet(A,D),B)
& r2_hidden(C,k2_frechet(A,D)) ) ) ) ) ) ) ) ).
fof(d6_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( v4_frechet(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_pre_topc(B,A)
<=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( v2_frechet(C,A)
& r1_tarski(k1_frechet(A,C),B) )
=> r1_tarski(k2_frechet(A,C),B) ) ) ) ) ) ) ).
fof(t24_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_frechet(A)
=> v3_frechet(A) ) ) ).
fof(t25_frechet,axiom,
$true ).
fof(t26_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_pre_topc(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( v2_frechet(C,A)
& r1_tarski(k1_frechet(A,C),B) )
=> r1_tarski(k2_frechet(A,C),B) ) ) ) ) ) ).
fof(t27_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( v2_frechet(C,A)
& r1_tarski(k1_frechet(A,C),B) )
=> r1_tarski(k2_frechet(A,C),B) ) )
=> v4_pre_topc(B,A) ) )
=> v4_frechet(A) ) ) ).
fof(t28_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_frechet(A)
=> v4_frechet(A) ) ) ).
fof(d7_frechet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( A = k3_frechet
<=> ( u1_struct_0(A) = k2_xboole_0(k4_xboole_0(k1_numbers,k5_numbers),k1_tarski(k1_numbers))
& ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k3_topmetr),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(k3_topmetr),u1_struct_0(A))
& B = k1_funct_4(k6_relat_1(k1_numbers),k2_funcop_1(k5_numbers,k1_numbers))
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_pre_topc(C,A)
<=> v4_pre_topc(k5_pre_topc(k3_topmetr,A,B,C),k3_topmetr) ) ) ) ) ) ) ).
fof(t29_frechet,axiom,
$true ).
fof(t30_frechet,axiom,
m1_subset_1(k1_numbers,u1_struct_0(k3_frechet)) ).
fof(t31_frechet,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_frechet)))
=> ( ( v3_pre_topc(A,k3_frechet)
& r2_hidden(k1_numbers,A) )
<=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
& v3_pre_topc(B,k3_topmetr)
& r1_tarski(k5_numbers,B)
& A = k2_xboole_0(k4_xboole_0(B,k5_numbers),k1_tarski(k1_numbers)) ) ) ) ).
fof(t32_frechet,axiom,
! [A] :
( ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_frechet)))
=> ( r2_hidden(k1_numbers,A)
| ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
& k3_xboole_0(k5_numbers,A) = k1_xboole_0 ) ) )
& ( ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
& k3_xboole_0(k5_numbers,A) = k1_xboole_0 )
=> ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_frechet)))
& ~ r2_hidden(k1_numbers,A) ) ) ) ).
fof(t33_frechet,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_frechet)))
=> ( A = B
=> ( ( ( k3_xboole_0(k5_numbers,A) = k1_xboole_0
& v3_pre_topc(A,k3_topmetr) )
=> ( ~ r2_hidden(k1_numbers,B)
& v3_pre_topc(B,k3_frechet) ) )
& ( v3_pre_topc(B,k3_frechet)
=> ( r2_hidden(k1_numbers,B)
| ( k3_xboole_0(k5_numbers,A) = k1_xboole_0
& v3_pre_topc(A,k3_topmetr) ) ) ) ) ) ) ) ).
fof(t34_frechet,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_frechet)))
=> ( A = k1_tarski(k1_numbers)
=> v4_pre_topc(A,k3_frechet) ) ) ).
fof(t35_frechet,axiom,
~ v1_frechet(k3_frechet) ).
fof(t36_frechet,axiom,
v3_frechet(k3_frechet) ).
fof(t37_frechet,axiom,
? [A] :
( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v3_frechet(A)
& ~ v1_frechet(A) ) ).
fof(t38_frechet,axiom,
$true ).
fof(t39_frechet,axiom,
$true ).
fof(t40_frechet,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,k6_real_1(np__1,B))
& ~ r1_xreal_0(B,np__0) ) ) ) ) ).
fof(dt_k1_frechet,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> m1_subset_1(k1_frechet(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(redefinition_k1_frechet,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k1_frechet(A,B) = k2_relat_1(B) ) ).
fof(dt_k2_frechet,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> m1_subset_1(k2_frechet(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k3_frechet,axiom,
( ~ v3_struct_0(k3_frechet)
& v1_pre_topc(k3_frechet)
& v2_pre_topc(k3_frechet)
& l1_pre_topc(k3_frechet) ) ).
fof(t11_frechet,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] :
( m1_subset_1(B,u1_struct_0(k5_pcomps_1(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B = C
& ! [D] :
( m1_yellow_8(D,k5_pcomps_1(A),B)
=> ~ ( D = a_2_0_frechet(A,C)
& v1_card_4(D)
& ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,D)
& m2_relset_1(E,k5_numbers,D)
& ! [F] :
~ ( r2_hidden(F,k5_numbers)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ~ ( F = G
& k1_funct_1(E,F) = k9_metric_1(A,C,k6_real_1(np__1,k1_nat_1(G,np__1))) ) ) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_2_0_frechet,axiom,
! [A,B,C] :
( ( ~ 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,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_0_frechet(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k9_metric_1(B,C,k6_real_1(np__1,D))
& D != np__0 ) ) ) ).
%------------------------------------------------------------------------------