SET007 Axioms: SET007+310.ax
%------------------------------------------------------------------------------
% File : SET007+310 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Totally Bounded 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 : tbsp_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 56 ( 8 unt; 0 def)
% Number of atoms : 452 ( 21 equ)
% Maximal formula atoms : 23 ( 8 avg)
% Number of connectives : 485 ( 89 ~; 3 |; 227 &)
% ( 12 <=>; 154 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 36 ( 34 usr; 1 prp; 0-3 aty)
% Number of functors : 31 ( 31 usr; 6 con; 0-4 aty)
% Number of variables : 140 ( 125 !; 15 ?)
% 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_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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))
& v2_tbsp_1(B,A) )
=> ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v3_tbsp_1(B,A) ) ) ) ) ).
fof(fc1_tbsp_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k6_metric_1(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))
& v5_tbsp_1(k6_metric_1(A)) ) ) ).
fof(rc1_tbsp_1,axiom,
? [A] :
( l1_metric_1(A)
& ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& v5_tbsp_1(A) ) ).
fof(fc2_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( v1_xboole_0(k1_pre_topc(A))
& v1_membered(k1_pre_topc(A))
& v2_membered(k1_pre_topc(A))
& v3_membered(k1_pre_topc(A))
& v4_membered(k1_pre_topc(A))
& v5_membered(k1_pre_topc(A))
& v1_finset_1(k1_pre_topc(A))
& v6_tbsp_1(k1_pre_topc(A),A) ) ) ).
fof(rc2_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v6_tbsp_1(B,A) ) ) ).
fof(cc2_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_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)))
=> ( v1_finset_1(B)
=> v6_tbsp_1(B,A) ) ) ) ).
fof(rc3_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_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)))
& ~ v1_xboole_0(B)
& v1_finset_1(B)
& v6_tbsp_1(B,A) ) ) ).
fof(fc3_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_tbsp_1(A)
& l1_metric_1(A) )
=> ( ~ v1_xboole_0(k2_pre_topc(A))
& v6_tbsp_1(k2_pre_topc(A),A) ) ) ).
fof(t1_tbsp_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,A)
& ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& r1_xreal_0(B,C)
& ~ r1_xreal_0(k4_power(A,C),k4_power(A,B)) ) ) ) ) ).
fof(t2_tbsp_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,A)
& ~ ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k4_power(A,B),np__1)
& ~ r1_xreal_0(k4_power(A,B),np__0) ) ) ) ) ).
fof(t3_tbsp_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,A)
& ? [B] :
( m1_subset_1(B,k1_numbers)
& ~ r1_xreal_0(B,np__0)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(B,k4_power(A,C)) ) ) ) ) ).
fof(d1_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( v1_tbsp_1(A)
<=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( v1_finset_1(C)
& u1_struct_0(A) = k5_setfam_1(u1_struct_0(A),C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(D,C)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> D != k9_metric_1(A,E,B) ) ) ) ) ) ) ) ) ) ).
fof(t4_tbsp_1,axiom,
$true ).
fof(t5_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(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)) )
<=> ( k1_relat_1(B) = k5_numbers
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> m1_subset_1(k1_funct_1(B,C),u1_struct_0(A)) ) ) ) ) ) ).
fof(d2_tbsp_1,axiom,
$true ).
fof(d3_tbsp_1,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)) )
=> ( v2_tbsp_1(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& ! [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(d4_tbsp_1,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))
=> ( C = k1_tbsp_1(A,B)
<=> ! [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,k4_metric_1(A,k2_normsp_1(A,B,F),C)) ) ) ) ) ) ) ) ) ) ).
fof(d5_tbsp_1,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)) )
=> ( v3_tbsp_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(D,E)
& r1_xreal_0(D,F)
& r1_xreal_0(C,k2_metric_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F))) ) ) ) ) ) ) ) ) ).
fof(d6_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( v4_tbsp_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)
=> v2_tbsp_1(B,A) ) ) ) ) ).
fof(t6_tbsp_1,axiom,
$true ).
fof(t7_tbsp_1,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)) )
=> ( ( v9_metric_1(A)
& v8_metric_1(A)
& v2_tbsp_1(B,A) )
=> v3_tbsp_1(B,A) ) ) ) ).
fof(t8_tbsp_1,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)) )
=> ( v8_metric_1(A)
=> ( v3_tbsp_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(D,E)
& r1_xreal_0(C,k2_metric_1(A,k2_normsp_1(A,B,k1_nat_1(E,F)),k2_normsp_1(A,B,E))) ) ) ) ) ) ) ) ) ) ).
fof(t9_tbsp_1,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_ali2(B,A)
=> ~ ( v4_tbsp_1(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C) = C
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D) = D
=> D = C ) ) ) ) ) ) ) ).
fof(t10_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ( v2_compts_1(k5_pcomps_1(A))
=> v4_tbsp_1(A) ) ) ).
fof(t11_tbsp_1,axiom,
$true ).
fof(t12_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( ( v6_metric_1(A)
& v9_metric_1(A)
& v2_compts_1(k5_pcomps_1(A)) )
=> v1_tbsp_1(A) ) ) ).
fof(d7_tbsp_1,axiom,
$true ).
fof(d8_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( v5_tbsp_1(A)
<=> ? [B] :
( m1_subset_1(B,k1_numbers)
& ~ r1_xreal_0(B,np__0)
& ! [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,C,D),B) ) ) ) ) ) ).
fof(d9_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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)
& ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r2_hidden(D,B)
& r2_hidden(E,B) )
=> r1_xreal_0(k2_metric_1(A,D,E),C) ) ) ) ) ) ) ) ).
fof(t13_tbsp_1,axiom,
$true ).
fof(t14_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> v6_tbsp_1(k1_pre_topc(A),A) ) ).
fof(t15_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ~ ( B != k1_xboole_0
& 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)
& r2_hidden(D,B)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,B)
=> r1_xreal_0(k2_metric_1(A,D,E),C) ) ) ) ) ) )
& ( ( v8_metric_1(A)
& v9_metric_1(A) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( ~ r1_xreal_0(C,np__0)
& r2_hidden(D,B)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,B)
=> r1_xreal_0(k2_metric_1(A,D,E),C) ) ) ) ) )
| v6_tbsp_1(B,A) ) ) ) ) ) ).
fof(t16_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( v1_xreal_0(C)
=> ( v6_metric_1(A)
=> ( r1_xreal_0(C,np__0)
| r2_hidden(B,k9_metric_1(A,B,C)) ) ) ) ) ) ).
fof(t17_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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] :
( v1_xreal_0(C)
=> ( r1_xreal_0(C,np__0)
=> k9_metric_1(A,B,C) = k1_xboole_0 ) ) ) ) ).
fof(t18_tbsp_1,axiom,
$true ).
fof(t19_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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] :
( v1_xreal_0(C)
=> v6_tbsp_1(k9_metric_1(A,B,C),A) ) ) ) ).
fof(t20_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v6_tbsp_1(B,A)
& v6_tbsp_1(C,A) )
=> v6_tbsp_1(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t21_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(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)))
=> ( ( v6_tbsp_1(B,A)
& r1_tarski(C,B) )
=> v6_tbsp_1(C,A) ) ) ) ) ).
fof(t22_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = k1_struct_0(A,B)
=> v6_tbsp_1(C,A) ) ) ) ) ).
fof(t23_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_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)))
=> ( v1_finset_1(B)
=> v6_tbsp_1(B,A) ) ) ) ).
fof(t24_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( v1_finset_1(B)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v6_tbsp_1(C,A) ) ) )
=> v6_tbsp_1(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t25_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( v5_tbsp_1(A)
<=> v6_tbsp_1(k2_pre_topc(A),A) ) ) ).
fof(t26_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ( v1_tbsp_1(A)
=> v5_tbsp_1(A) ) ) ).
fof(d10_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_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)
=> ( ( B != k1_xboole_0
=> ( C = k2_tbsp_1(A,B)
<=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r2_hidden(D,B)
& r2_hidden(E,B) )
=> r1_xreal_0(k2_metric_1(A,D,E),C) ) ) )
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( r2_hidden(E,B)
& r2_hidden(F,B) )
=> r1_xreal_0(k2_metric_1(A,E,F),D) ) ) )
=> r1_xreal_0(C,D) ) ) ) ) )
& ( B = k1_xboole_0
=> ( C = k2_tbsp_1(A,B)
<=> C = np__0 ) ) ) ) ) ) ) ).
fof(t27_tbsp_1,axiom,
$true ).
fof(t28_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B,C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = k1_tarski(B)
=> k2_tbsp_1(A,C) = np__0 ) ) ) ).
fof(t29_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v6_tbsp_1(B,A)
=> r1_xreal_0(np__0,k2_tbsp_1(A,B)) ) ) ) ).
fof(t30_tbsp_1,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)))
=> ~ ( B != k1_xboole_0
& v6_tbsp_1(B,A)
& k2_tbsp_1(A,B) = np__0
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> B != k1_struct_0(A,C) ) ) ) ) ).
fof(t31_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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,k1_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> r1_xreal_0(k2_tbsp_1(A,k9_metric_1(A,B,C)),k4_real_1(np__2,C)) ) ) ) ) ).
fof(t32_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& l1_metric_1(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)))
=> ( ( v6_tbsp_1(B,A)
& r1_tarski(C,B) )
=> r1_xreal_0(k2_tbsp_1(A,C),k2_tbsp_1(A,B)) ) ) ) ) ).
fof(t33_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v6_tbsp_1(B,A)
& v6_tbsp_1(C,A) )
=> ( r1_xboole_0(B,C)
| r1_xreal_0(k2_tbsp_1(A,k4_subset_1(u1_struct_0(A),B,C)),k3_real_1(k2_tbsp_1(A,B),k2_tbsp_1(A,C))) ) ) ) ) ) ).
fof(t34_tbsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_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)
=> v6_tbsp_1(k3_tbsp_1(A,B),A) ) ) ) ).
fof(dt_k1_tbsp_1,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)
& 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_tbsp_1(A,B),u1_struct_0(A)) ) ).
fof(dt_k2_tbsp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& l1_metric_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k2_tbsp_1(A,B),k1_numbers) ) ).
fof(dt_k3_tbsp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(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(k3_tbsp_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(redefinition_k3_tbsp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(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)) )
=> k3_tbsp_1(A,B) = k2_relat_1(B) ) ).
%------------------------------------------------------------------------------