SET007 Axioms: SET007+266.ax
%------------------------------------------------------------------------------
% File : SET007+266 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On Pseudometric 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_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 99 ( 20 unt; 0 def)
% Number of atoms : 617 ( 47 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 602 ( 84 ~; 0 |; 308 &)
% ( 35 <=>; 175 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-4 aty)
% Number of functors : 35 ( 35 usr; 2 con; 0-6 aty)
% Number of variables : 239 ( 193 !; 46 ?)
% 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_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_metric_2(B,A)
=> ~ v1_xboole_0(B) ) ) ).
fof(fc1_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ~ v1_xboole_0(k2_metric_2(A)) ) ).
fof(fc2_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ( ~ v3_struct_0(k11_metric_2(A))
& v1_metric_1(k11_metric_2(A))
& v6_metric_1(k11_metric_2(A))
& v7_metric_1(k11_metric_2(A))
& v8_metric_1(k11_metric_2(A))
& v9_metric_1(k11_metric_2(A)) ) ) ).
fof(d1_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_metric_2(A,B,C)
<=> k2_metric_1(A,B,C) = np__0 ) ) ) ) ).
fof(d3_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( m1_metric_2(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& B = k1_metric_2(A,C) ) ) ) ) ).
fof(t1_metric_2,axiom,
$true ).
fof(t2_metric_2,axiom,
$true ).
fof(t3_metric_2,axiom,
$true ).
fof(t4_metric_2,axiom,
$true ).
fof(t5_metric_2,axiom,
$true ).
fof(t6_metric_2,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,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r3_metric_2(A,B,C)
& r3_metric_2(A,C,D) )
=> r3_metric_2(A,B,D) ) ) ) ) ) ).
fof(t7_metric_2,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,u1_struct_0(A))
=> ( r2_hidden(C,k1_metric_2(A,B))
<=> r3_metric_2(A,C,B) ) ) ) ) ).
fof(t8_metric_2,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,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,k1_metric_2(A,B))
& r2_hidden(D,k1_metric_2(A,B)) )
=> r3_metric_2(A,C,D) ) ) ) ) ) ).
fof(t9_metric_2,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))
=> r2_hidden(B,k1_metric_2(A,B)) ) ) ).
fof(t10_metric_2,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,u1_struct_0(A))
=> ( r2_hidden(B,k1_metric_2(A,C))
<=> r2_hidden(C,k1_metric_2(A,B)) ) ) ) ) ).
fof(t11_metric_2,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,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(B,k1_metric_2(A,C))
& r3_metric_2(A,C,D) )
=> r2_hidden(B,k1_metric_2(A,D)) ) ) ) ) ) ).
fof(t12_metric_2,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,u1_struct_0(A))
=> ( r2_hidden(C,k1_metric_2(A,B))
=> k1_metric_2(A,B) = k1_metric_2(A,C) ) ) ) ) ).
fof(t13_metric_2,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,u1_struct_0(A))
=> ( k1_metric_2(A,B) = k1_metric_2(A,C)
<=> r3_metric_2(A,B,C) ) ) ) ) ).
fof(t14_metric_2,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,u1_struct_0(A))
=> ( ~ r1_xboole_0(k1_metric_2(A,B),k1_metric_2(A,C))
<=> r3_metric_2(A,B,C) ) ) ) ) ).
fof(t15_metric_2,axiom,
$true ).
fof(t16_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_metric_2(B,A)
=> ~ v1_xboole_0(B) ) ) ).
fof(t17_metric_2,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,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,k1_metric_2(A,B))
& r2_hidden(D,k1_metric_2(A,B)) )
=> k4_metric_1(A,C,D) = np__0 ) ) ) ) ) ).
fof(t18_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_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))
=> ( r2_metric_2(A,B,C)
<=> B = C ) ) ) ) ).
fof(t19_metric_2,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))
=> ( r2_hidden(C,k1_metric_2(A,B))
<=> C = B ) ) ) ) ).
fof(t20_metric_2,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))
=> k1_metric_2(A,B) = k1_struct_0(A,B) ) ) ).
fof(t21_metric_2,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)))
=> ( m1_metric_2(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& B = k1_struct_0(A,C) ) ) ) ) ).
fof(t22_metric_2,axiom,
$true ).
fof(t23_metric_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ( r2_hidden(A,k2_metric_2(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(B))
& A = k1_metric_2(B,C) ) ) ) ).
fof(t24_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r2_hidden(k1_metric_2(A,B),k2_metric_2(A)) ) ) ).
fof(t25_metric_2,axiom,
$true ).
fof(t26_metric_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ( r2_hidden(A,k2_metric_2(B))
<=> m1_metric_2(A,B) ) ) ).
fof(t27_metric_2,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))
=> r2_hidden(k1_struct_0(A,B),k2_metric_2(A)) ) ) ).
fof(t28_metric_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ( r2_hidden(A,k2_metric_2(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(B))
& A = k1_struct_0(B,C) ) ) ) ).
fof(t29_metric_2,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,k2_metric_2(A))
=> ! [C] :
( m1_subset_1(C,k2_metric_2(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( ( r2_hidden(D,B)
& r2_hidden(F,C)
& r2_hidden(E,B)
& r2_hidden(G,C) )
=> k4_metric_1(A,D,F) = k4_metric_1(A,E,G) ) ) ) ) ) ) ) ) ).
fof(d5_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_metric_2(A))
=> ! [C] :
( m1_subset_1(C,k2_metric_2(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r4_metric_2(A,B,C,D)
<=> ! [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,C) )
=> k2_metric_1(A,E,F) = D ) ) ) ) ) ) ) ) ).
fof(t30_metric_2,axiom,
$true ).
fof(t31_metric_2,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,k2_metric_2(A))
=> ! [C] :
( m1_subset_1(C,k2_metric_2(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r4_metric_2(A,B,C,D)
<=> ? [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,C)
& k4_metric_1(A,E,F) = D ) ) ) ) ) ) ) ).
fof(t32_metric_2,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,k2_metric_2(A))
=> ! [C] :
( m1_subset_1(C,k2_metric_2(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r4_metric_2(A,B,C,D)
<=> r4_metric_2(A,C,B,D) ) ) ) ) ) ).
fof(t33_metric_2,axiom,
$true ).
fof(t34_metric_2,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,k2_metric_2(A))
=> ! [C] :
( m1_subset_1(C,k2_metric_2(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k3_metric_2(A,B,C))
<=> r4_metric_2(A,B,C,D) ) ) ) ) ) ).
fof(t35_metric_2,axiom,
$true ).
fof(t36_metric_2,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_numbers)
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)))
=> ( r2_hidden(C,k4_metric_2(A,B))
<=> ? [D] :
( m1_subset_1(D,k2_metric_2(A))
& ? [E] :
( m1_subset_1(E,k2_metric_2(A))
& C = k1_domain_1(k2_metric_2(A),k2_metric_2(A),D,E)
& r4_metric_2(A,D,E,B) ) ) ) ) ) ) ).
fof(t37_metric_2,axiom,
$true ).
fof(t38_metric_2,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_numbers)
=> ( r2_hidden(B,k5_metric_2(A))
<=> ? [C] :
( m1_subset_1(C,k2_metric_2(A))
& ? [D] :
( m1_subset_1(D,k2_metric_2(A))
& r4_metric_2(A,C,D,B) ) ) ) ) ) ).
fof(t39_metric_2,axiom,
$true ).
fof(t40_metric_2,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,k2_metric_2(A))
=> ( r2_hidden(B,k6_metric_2(A))
<=> ? [C] :
( m1_subset_1(C,k2_metric_2(A))
& ? [D] :
( m1_subset_1(D,k1_numbers)
& r4_metric_2(A,B,C,D) ) ) ) ) ) ).
fof(t41_metric_2,axiom,
$true ).
fof(t42_metric_2,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,k2_metric_2(A))
=> ( r2_hidden(B,k7_metric_2(A))
<=> ? [C] :
( m1_subset_1(C,k2_metric_2(A))
& ? [D] :
( m1_subset_1(D,k1_numbers)
& r4_metric_2(A,C,B,D) ) ) ) ) ) ).
fof(t43_metric_2,axiom,
$true ).
fof(t44_metric_2,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,k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)))
=> ( r2_hidden(B,k8_metric_2(A))
<=> ? [C] :
( m1_subset_1(C,k2_metric_2(A))
& ? [D] :
( m1_subset_1(D,k2_metric_2(A))
& ? [E] :
( m1_subset_1(E,k1_numbers)
& B = k1_domain_1(k2_metric_2(A),k2_metric_2(A),C,D)
& r4_metric_2(A,C,D,E) ) ) ) ) ) ) ).
fof(t45_metric_2,axiom,
$true ).
fof(t46_metric_2,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,k3_zfmisc_1(k2_metric_2(A),k2_metric_2(A),k1_numbers))
=> ( r2_hidden(B,k9_metric_2(A))
<=> ? [C] :
( m1_subset_1(C,k2_metric_2(A))
& ? [D] :
( m1_subset_1(D,k2_metric_2(A))
& ? [E] :
( m1_subset_1(E,k1_numbers)
& B = k3_mcart_1(C,D,E)
& r4_metric_2(A,C,D,E) ) ) ) ) ) ) ).
fof(t47_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> k6_metric_2(A) = k7_metric_2(A) ) ).
fof(t48_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> r1_tarski(k9_metric_2(A),k13_mcart_1(k2_metric_2(A),k2_metric_2(A),k1_numbers,k6_metric_2(A),k7_metric_2(A),k5_metric_2(A))) ) ).
fof(t49_metric_2,axiom,
$true ).
fof(t50_metric_2,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,k2_metric_2(A))
=> ! [C] :
( m1_subset_1(C,k2_metric_2(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r4_metric_2(A,B,C,D)
& r4_metric_2(A,B,C,E) )
=> D = E ) ) ) ) ) ) ).
fof(t51_metric_2,axiom,
$true ).
fof(t52_metric_2,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,k2_metric_2(A))
=> ! [C] :
( m1_subset_1(C,k2_metric_2(A))
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& r4_metric_2(A,B,C,D) ) ) ) ) ).
fof(d13_metric_2,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,k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)),k1_numbers) )
=> ( B = k10_metric_2(A)
<=> ! [C] :
( m1_subset_1(C,k2_metric_2(A))
=> ! [D] :
( m1_subset_1(D,k2_metric_2(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( r2_hidden(E,C)
& r2_hidden(F,D) )
=> k1_metric_1(k2_metric_2(A),k2_metric_2(A),B,C,D) = k4_metric_1(A,E,F) ) ) ) ) ) ) ) ) ).
fof(t53_metric_2,axiom,
$true ).
fof(t54_metric_2,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,k2_metric_2(A))
=> ! [C] :
( m1_subset_1(C,k2_metric_2(A))
=> ( k1_metric_1(k2_metric_2(A),k2_metric_2(A),k10_metric_2(A),B,C) = np__0
<=> B = C ) ) ) ) ).
fof(t55_metric_2,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,k2_metric_2(A))
=> ! [C] :
( m1_subset_1(C,k2_metric_2(A))
=> k1_metric_1(k2_metric_2(A),k2_metric_2(A),k10_metric_2(A),B,C) = k1_metric_1(k2_metric_2(A),k2_metric_2(A),k10_metric_2(A),C,B) ) ) ) ).
fof(t56_metric_2,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,k2_metric_2(A))
=> ! [C] :
( m1_subset_1(C,k2_metric_2(A))
=> ! [D] :
( m1_subset_1(D,k2_metric_2(A))
=> r1_xreal_0(k1_metric_1(k2_metric_2(A),k2_metric_2(A),k10_metric_2(A),B,D),k2_xcmplx_0(k1_metric_1(k2_metric_2(A),k2_metric_2(A),k10_metric_2(A),B,C),k1_metric_1(k2_metric_2(A),k2_metric_2(A),k10_metric_2(A),C,D))) ) ) ) ) ).
fof(d14_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> k11_metric_2(A) = g1_metric_1(k2_metric_2(A),k10_metric_2(A)) ) ).
fof(dt_m1_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_metric_2(B,A)
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(existence_m1_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ? [B] : m1_metric_2(B,A) ) ).
fof(reflexivity_r2_metric_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& l1_metric_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> r2_metric_2(A,B,B) ) ).
fof(redefinition_r2_metric_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& l1_metric_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r2_metric_2(A,B,C)
<=> r1_metric_2(A,B,C) ) ) ).
fof(symmetry_r3_metric_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v8_metric_1(A)
& l1_metric_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r3_metric_2(A,B,C)
=> r3_metric_2(A,C,B) ) ) ).
fof(redefinition_r3_metric_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v8_metric_1(A)
& l1_metric_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r3_metric_2(A,B,C)
<=> r1_metric_2(A,B,C) ) ) ).
fof(dt_k1_metric_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k1_metric_2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k2_metric_2,axiom,
$true ).
fof(dt_k3_metric_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& m1_subset_1(B,k2_metric_2(A))
& m1_subset_1(C,k2_metric_2(A)) )
=> m1_subset_1(k3_metric_2(A,B,C),k1_zfmisc_1(k1_numbers)) ) ).
fof(dt_k4_metric_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k4_metric_2(A,B),k1_zfmisc_1(k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)))) ) ).
fof(dt_k5_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> m1_subset_1(k5_metric_2(A),k1_zfmisc_1(k1_numbers)) ) ).
fof(dt_k6_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> m1_subset_1(k6_metric_2(A),k1_zfmisc_1(k2_metric_2(A))) ) ).
fof(dt_k7_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> m1_subset_1(k7_metric_2(A),k1_zfmisc_1(k2_metric_2(A))) ) ).
fof(dt_k8_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> m1_subset_1(k8_metric_2(A),k1_zfmisc_1(k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)))) ) ).
fof(dt_k9_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> m1_subset_1(k9_metric_2(A),k1_zfmisc_1(k3_zfmisc_1(k2_metric_2(A),k2_metric_2(A),k1_numbers))) ) ).
fof(dt_k10_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ( v1_funct_1(k10_metric_2(A))
& v1_funct_2(k10_metric_2(A),k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)),k1_numbers)
& m2_relset_1(k10_metric_2(A),k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)),k1_numbers) ) ) ).
fof(dt_k11_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ( v6_metric_1(k11_metric_2(A))
& v7_metric_1(k11_metric_2(A))
& v8_metric_1(k11_metric_2(A))
& v9_metric_1(k11_metric_2(A))
& l1_metric_1(k11_metric_2(A)) ) ) ).
fof(d2_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_metric_2(A,B) = a_2_0_metric_2(A,B) ) ) ).
fof(d4_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> k2_metric_2(A) = a_1_0_metric_2(A) ) ).
fof(d6_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,k2_metric_2(A))
=> ! [C] :
( m1_subset_1(C,k2_metric_2(A))
=> k3_metric_2(A,B,C) = a_3_0_metric_2(A,B,C) ) ) ) ).
fof(d7_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k4_metric_2(A,B) = a_2_1_metric_2(A,B) ) ) ).
fof(d8_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> k5_metric_2(A) = a_1_1_metric_2(A) ) ).
fof(d9_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> k6_metric_2(A) = a_1_2_metric_2(A) ) ).
fof(d10_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> k7_metric_2(A) = a_1_3_metric_2(A) ) ).
fof(d11_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> k8_metric_2(A) = a_1_4_metric_2(A) ) ).
fof(d12_metric_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> k9_metric_2(A) = a_1_5_metric_2(A) ) ).
fof(fraenkel_a_2_0_metric_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_0_metric_2(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r1_metric_2(B,C,D) ) ) ) ).
fof(fraenkel_a_1_0_metric_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ( r2_hidden(A,a_1_0_metric_2(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& A = C
& ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& k1_metric_2(B,D) = C ) ) ) ) ).
fof(fraenkel_a_3_0_metric_2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B)
& m1_subset_1(C,k2_metric_2(B))
& m1_subset_1(D,k2_metric_2(B)) )
=> ( r2_hidden(A,a_3_0_metric_2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_numbers)
& A = E
& r4_metric_2(B,C,D,E) ) ) ) ).
fof(fraenkel_a_2_1_metric_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B)
& m1_subset_1(C,k1_numbers) )
=> ( r2_hidden(A,a_2_1_metric_2(B,C))
<=> ? [D] :
( m1_subset_1(D,k2_zfmisc_1(k2_metric_2(B),k2_metric_2(B)))
& A = D
& ? [E] :
( m1_subset_1(E,k2_metric_2(B))
& ? [F] :
( m1_subset_1(F,k2_metric_2(B))
& D = k1_domain_1(k2_metric_2(B),k2_metric_2(B),E,F)
& r4_metric_2(B,E,F,C) ) ) ) ) ) ).
fof(fraenkel_a_1_1_metric_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ( r2_hidden(A,a_1_1_metric_2(B))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& A = C
& ? [D] :
( m1_subset_1(D,k2_metric_2(B))
& ? [E] :
( m1_subset_1(E,k2_metric_2(B))
& r4_metric_2(B,D,E,C) ) ) ) ) ) ).
fof(fraenkel_a_1_2_metric_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ( r2_hidden(A,a_1_2_metric_2(B))
<=> ? [C] :
( m1_subset_1(C,k2_metric_2(B))
& A = C
& ? [D] :
( m1_subset_1(D,k2_metric_2(B))
& ? [E] :
( m1_subset_1(E,k1_numbers)
& r4_metric_2(B,C,D,E) ) ) ) ) ) ).
fof(fraenkel_a_1_3_metric_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ( r2_hidden(A,a_1_3_metric_2(B))
<=> ? [C] :
( m1_subset_1(C,k2_metric_2(B))
& A = C
& ? [D] :
( m1_subset_1(D,k2_metric_2(B))
& ? [E] :
( m1_subset_1(E,k1_numbers)
& r4_metric_2(B,D,C,E) ) ) ) ) ) ).
fof(fraenkel_a_1_4_metric_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ( r2_hidden(A,a_1_4_metric_2(B))
<=> ? [C] :
( m1_subset_1(C,k2_zfmisc_1(k2_metric_2(B),k2_metric_2(B)))
& A = C
& ? [D] :
( m1_subset_1(D,k2_metric_2(B))
& ? [E] :
( m1_subset_1(E,k2_metric_2(B))
& ? [F] :
( m1_subset_1(F,k1_numbers)
& C = k1_domain_1(k2_metric_2(B),k2_metric_2(B),D,E)
& r4_metric_2(B,D,E,F) ) ) ) ) ) ) ).
fof(fraenkel_a_1_5_metric_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ( r2_hidden(A,a_1_5_metric_2(B))
<=> ? [C] :
( m1_subset_1(C,k3_zfmisc_1(k2_metric_2(B),k2_metric_2(B),k1_numbers))
& A = C
& ? [D] :
( m1_subset_1(D,k2_metric_2(B))
& ? [E] :
( m1_subset_1(E,k2_metric_2(B))
& ? [F] :
( m1_subset_1(F,k1_numbers)
& C = k3_mcart_1(D,E,F)
& r4_metric_2(B,D,E,F) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------