SET007 Axioms: SET007+325.ax
%------------------------------------------------------------------------------
% File : SET007+325 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Sequences in 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_6 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 65 ( 24 unt; 0 def)
% Number of atoms : 543 ( 13 equ)
% Maximal formula atoms : 18 ( 8 avg)
% Number of connectives : 531 ( 53 ~; 0 |; 312 &)
% ( 15 <=>; 151 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 34 ( 32 usr; 1 prp; 0-3 aty)
% Number of functors : 26 ( 26 usr; 4 con; 0-5 aty)
% Number of variables : 140 ( 130 !; 10 ?)
% 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_metric_6,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_relset_1(B,k5_numbers,u1_struct_0(A))
=> ( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v5_seqm_3(B) )
=> ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v2_tbsp_1(B,A) ) ) ) ) ).
fof(cc2_metric_6,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_relset_1(B,k5_numbers,u1_struct_0(A))
=> ( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v3_tbsp_1(B,A) )
=> ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_metric_6(B,A) ) ) ) ) ).
fof(rc1_metric_6,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_relset_1(B,k5_numbers,u1_struct_0(A))
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v2_tbsp_1(B,A)
& v3_tbsp_1(B,A)
& v5_seqm_3(B)
& v1_metric_6(B,A) ) ) ).
fof(t1_metric_6,axiom,
! [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(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r1_xreal_0(k18_complex1(k5_real_1(k4_metric_1(A,B,C),k4_metric_1(A,D,C))),k4_metric_1(A,B,D)) ) ) ) ) ).
fof(t2_metric_6,axiom,
! [A] :
( ~ v1_xboole_0(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) )
=> ( r1_pcomps_1(A,B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> r1_xreal_0(np__0,k1_metric_1(A,A,B,C,D)) ) ) ) ) ) ).
fof(t3_metric_6,axiom,
! [A] :
( ~ v1_xboole_0(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) )
=> ( r1_pcomps_1(A,B)
<=> ( v2_metric_1(B,A)
& v3_metric_1(B,A)
& v4_metric_1(B,A)
& v5_metric_1(B,A) ) ) ) ) ).
fof(t4_metric_6,axiom,
! [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)
& l1_metric_1(A) )
=> ( v2_metric_1(u1_metric_1(A),u1_struct_0(A))
& v3_metric_1(u1_metric_1(A),u1_struct_0(A))
& v4_metric_1(u1_metric_1(A),u1_struct_0(A))
& v5_metric_1(u1_metric_1(A),u1_struct_0(A)) ) ) ).
fof(t5_metric_6,axiom,
! [A] :
( ~ v1_xboole_0(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) )
=> ( r1_pcomps_1(A,B)
<=> ( v2_metric_1(B,A)
& v3_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,D,E),k3_real_1(k1_metric_1(A,A,B,C,D),k1_metric_1(A,A,B,C,E))) ) ) ) ) ) ) ) ).
fof(d1_metric_6,axiom,
$true ).
fof(d2_metric_6,axiom,
$true ).
fof(d3_metric_6,axiom,
$true ).
fof(d4_metric_6,axiom,
! [A] :
( ~ v1_xboole_0(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) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),k1_numbers)
& m2_relset_1(C,k2_zfmisc_1(A,A),k1_numbers) )
=> ( C = k1_metric_6(A,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> k1_metric_1(A,A,C,D,E) = k6_real_1(k1_metric_1(A,A,B,D,E),k3_real_1(np__1,k1_metric_1(A,A,B,D,E))) ) ) ) ) ) ) ).
fof(t6_metric_6,axiom,
! [A] :
( ~ v1_xboole_0(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) )
=> ( r1_pcomps_1(A,B)
=> r1_pcomps_1(A,k1_metric_6(A,B)) ) ) ) ).
fof(t7_metric_6,axiom,
$true ).
fof(t8_metric_6,axiom,
$true ).
fof(t9_metric_6,axiom,
$true ).
fof(t10_metric_6,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(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))
& k3_tbsp_1(A,C) = k1_struct_0(A,B) ) ) ) ).
fof(d5_metric_6,axiom,
$true ).
fof(d6_metric_6,axiom,
$true ).
fof(d7_metric_6,axiom,
$true ).
fof(d8_metric_6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(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_metric_6(A,B,C)
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(E,F)
& r1_xreal_0(D,k2_metric_1(A,k2_normsp_1(A,B,F),C)) ) ) ) ) ) ) ) ) ).
fof(d9_metric_6,axiom,
$true ).
fof(d10_metric_6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v6_tbsp_1(B,A)
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ~ r1_xreal_0(C,np__0)
& r1_tarski(B,k9_metric_1(A,D,C)) ) ) ) ) ) ).
fof(d11_metric_6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(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)) )
=> ( v1_metric_6(B,A)
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ~ r1_xreal_0(C,np__0)
& r1_tarski(k3_tbsp_1(A,B),k9_metric_1(A,D,C)) ) ) ) ) ) ).
fof(d12_metric_6,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,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)) )
=> ( r2_metric_6(A,B,C)
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(D,E)
=> r2_hidden(k2_normsp_1(A,C,E),B) ) ) ) ) ) ) ) ).
fof(t11_metric_6,axiom,
$true ).
fof(t12_metric_6,axiom,
$true ).
fof(t13_metric_6,axiom,
$true ).
fof(t14_metric_6,axiom,
$true ).
fof(t15_metric_6,axiom,
$true ).
fof(t16_metric_6,axiom,
$true ).
fof(t17_metric_6,axiom,
$true ).
fof(t18_metric_6,axiom,
$true ).
fof(t19_metric_6,axiom,
$true ).
fof(t20_metric_6,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v1_metric_6(B,A)
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ~ r1_xreal_0(C,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> r2_hidden(k2_normsp_1(A,B,E),k9_metric_1(A,D,C)) ) ) ) ) ) ) ).
fof(t21_metric_6,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(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)) )
=> ( r1_metric_6(A,C,B)
=> v2_tbsp_1(C,A) ) ) ) ) ).
fof(t22_metric_6,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] :
( ( 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_tbsp_1(B,A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ r1_metric_6(A,B,C) ) ) ) ) ).
fof(d13_metric_6,axiom,
$true ).
fof(d14_metric_6,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] :
( ( 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))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( D = k2_metric_6(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = k4_metric_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) ).
fof(d15_metric_6,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] :
( ( 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] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( D = k3_metric_6(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = k4_metric_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,C,E)) ) ) ) ) ) ) ).
fof(t23_metric_6,axiom,
$true ).
fof(t24_metric_6,axiom,
$true ).
fof(t25_metric_6,axiom,
$true ).
fof(t26_metric_6,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(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)) )
=> ( r1_metric_6(A,C,B)
=> k1_tbsp_1(A,C) = B ) ) ) ) ).
fof(t27_metric_6,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(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)) )
=> ( r1_metric_6(A,C,B)
<=> ( v2_tbsp_1(C,A)
& k1_tbsp_1(A,C) = B ) ) ) ) ) ).
fof(t28_metric_6,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] :
( ( 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_tbsp_1(B,A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r1_metric_6(A,B,C)
& k1_tbsp_1(A,B) = C ) ) ) ) ) ).
fof(t29_metric_6,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(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)) )
=> ( r1_metric_6(A,C,B)
<=> ( v4_seq_2(k2_metric_6(A,C,B))
& k2_seq_2(k2_metric_6(A,C,B)) = np__0 ) ) ) ) ) ).
fof(t30_metric_6,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(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)) )
=> ( r1_metric_6(A,C,B)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ~ r1_xreal_0(D,np__0)
=> r2_metric_6(A,k9_metric_1(A,B,D),C) ) ) ) ) ) ) ).
fof(t31_metric_6,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(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)) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ~ r1_xreal_0(D,np__0)
=> r2_metric_6(A,k9_metric_1(A,B,D),C) ) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(B,D)
& r2_hidden(D,k4_pcomps_1(A)) )
=> r2_metric_6(A,D,C) ) ) ) ) ) ) ).
fof(t32_metric_6,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(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)) )
=> ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(B,D)
& r2_hidden(D,k4_pcomps_1(A)) )
=> r2_metric_6(A,D,C) ) )
=> r1_metric_6(A,C,B) ) ) ) ) ).
fof(t33_metric_6,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(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)) )
=> ( r1_metric_6(A,C,B)
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ~ r1_xreal_0(D,np__0)
=> r2_metric_6(A,k9_metric_1(A,B,D),C) ) ) ) ) ) ) ).
fof(t34_metric_6,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(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)) )
=> ( r1_metric_6(A,C,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(B,D)
& r2_hidden(D,k4_pcomps_1(A)) )
=> r2_metric_6(A,D,C) ) ) ) ) ) ) ).
fof(t35_metric_6,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(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)) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ~ r1_xreal_0(D,np__0)
=> r2_metric_6(A,k9_metric_1(A,B,D),C) ) )
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(B,D)
& r2_hidden(D,k4_pcomps_1(A)) )
=> r2_metric_6(A,D,C) ) ) ) ) ) ) ).
fof(t36_metric_6,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] :
( ( 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] :
( ( 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_tbsp_1(B,A)
& v2_tbsp_1(C,A) )
=> k4_metric_1(A,k1_tbsp_1(A,B),k1_tbsp_1(A,C)) = k2_seq_2(k3_metric_6(A,B,C)) ) ) ) ) ).
fof(t37_metric_6,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(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [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_metric_6(A,D,B)
& r1_metric_6(A,D,C) )
=> B = C ) ) ) ) ) ).
fof(t38_metric_6,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v5_seqm_3(B)
=> v2_tbsp_1(B,A) ) ) ) ).
fof(t39_metric_6,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(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)) )
=> ! [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_metric_6(A,C,B)
& m1_bhsp_3(D,A,C) )
=> r1_metric_6(A,D,B) ) ) ) ) ) ).
fof(t40_metric_6,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] :
( ( 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] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( v3_tbsp_1(B,A)
& m1_bhsp_3(C,A,B) )
=> v3_tbsp_1(C,A) ) ) ) ) ).
fof(t41_metric_6,axiom,
$true ).
fof(t42_metric_6,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v5_seqm_3(B)
=> v3_tbsp_1(B,A) ) ) ) ).
fof(t43_metric_6,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] :
( ( 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_tbsp_1(B,A)
=> v1_metric_6(B,A) ) ) ) ).
fof(t44_metric_6,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v3_tbsp_1(B,A)
=> v1_metric_6(B,A) ) ) ) ).
fof(dt_k1_metric_6,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) )
=> ( v1_funct_1(k1_metric_6(A,B))
& v1_funct_2(k1_metric_6(A,B),k2_zfmisc_1(A,A),k1_numbers)
& m2_relset_1(k1_metric_6(A,B),k2_zfmisc_1(A,A),k1_numbers) ) ) ).
fof(dt_k2_metric_6,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)
& 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(C,u1_struct_0(A)) )
=> ( v1_funct_1(k2_metric_6(A,B,C))
& v1_funct_2(k2_metric_6(A,B,C),k5_numbers,k1_numbers)
& m2_relset_1(k2_metric_6(A,B,C),k5_numbers,k1_numbers) ) ) ).
fof(dt_k3_metric_6,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)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k3_metric_6(A,B,C))
& v1_funct_2(k3_metric_6(A,B,C),k5_numbers,k1_numbers)
& m2_relset_1(k3_metric_6(A,B,C),k5_numbers,k1_numbers) ) ) ).
%------------------------------------------------------------------------------