SET007 Axioms: SET007+586.ax
%------------------------------------------------------------------------------
% File : SET007+586 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Sequential Closure Operator in 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 : frechet2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 65 ( 1 unt; 0 def)
% Number of atoms : 627 ( 45 equ)
% Maximal formula atoms : 19 ( 9 avg)
% Number of connectives : 644 ( 82 ~; 1 |; 311 &)
% ( 13 <=>; 237 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 46 ( 44 usr; 1 prp; 0-3 aty)
% Number of functors : 40 ( 40 usr; 13 con; 0-5 aty)
% Number of variables : 225 ( 205 !; 20 ?)
% 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(t1_frechet2,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)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_seqm_3(C)
& v7_seqm_3(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k5_relat_1(C,B))
& v1_funct_2(k5_relat_1(C,B),k5_numbers,u1_struct_0(A))
& m2_relset_1(k5_relat_1(C,B),k5_numbers,u1_struct_0(A)) ) ) ) ) ).
fof(t2_frechet2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( A = k6_relat_1(k5_numbers)
=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_seqm_3(A)
& v7_seqm_3(A)
& m2_relset_1(A,k5_numbers,k1_numbers) ) ) ) ).
fof(t3_frechet2,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_bhsp_3(B,A,B) ) ) ).
fof(t4_frechet2,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)) )
=> ! [C] :
( m1_bhsp_3(C,A,B)
=> r1_tarski(k1_frechet(A,C),k1_frechet(A,B)) ) ) ) ).
fof(t5_frechet2,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)) )
=> ! [C] :
( m1_bhsp_3(C,A,B)
=> ! [D] :
( m1_bhsp_3(D,A,C)
=> m1_bhsp_3(D,A,B) ) ) ) ) ).
fof(t6_frechet2,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)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( ! [D] :
( m1_bhsp_3(D,A,B)
=> ~ r1_tarski(k1_frechet(A,D),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,B,E),C) ) ) ) ) ) ) ).
fof(t7_frechet2,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)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r1_tarski(k1_frechet(A,B),k4_subset_1(u1_struct_0(A),C,D))
& ! [E] :
( m1_bhsp_3(E,A,B)
=> ( ~ r1_tarski(k1_frechet(A,E),C)
& ~ r1_tarski(k1_frechet(A,E),D) ) ) ) ) ) ) ) ).
fof(t8_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(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))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,k2_frechet(A,B))
& r2_hidden(D,k2_frechet(A,B)) )
=> C = D ) ) ) )
=> v1_urysohn1(A) ) ) ).
fof(t9_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_compts_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] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,k2_frechet(A,B))
& r2_hidden(D,k2_frechet(A,B)) )
=> C = D ) ) ) ) ) ) ).
fof(t10_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_frechet(A)
=> ( v3_compts_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] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,k2_frechet(A,B))
& r2_hidden(D,k2_frechet(A,B)) )
=> C = D ) ) ) ) ) ) ) ).
fof(t11_frechet2,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)) )
=> ( ~ v2_frechet(B,A)
=> k2_frechet(A,B) = k1_xboole_0 ) ) ) ).
fof(t12_frechet2,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)) )
=> ( r1_tarski(k1_frechet(A,C),B)
=> r1_tarski(k2_frechet(A,C),B) ) ) ) ) ) ).
fof(t13_frechet2,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,u1_struct_0(A))
=> ~ ( ~ r1_frechet(A,B,C)
& ! [D] :
( m1_bhsp_3(D,A,B)
=> ? [E] :
( m1_bhsp_3(E,A,D)
& r1_frechet(A,E,C) ) ) ) ) ) ) ).
fof(t14_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(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)) )
=> ( v5_pre_topc(C,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)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(B))
& m2_relset_1(E,k5_numbers,u1_struct_0(B)) )
=> ( E = k7_funct_2(k5_numbers,u1_struct_0(A),u1_struct_0(B),D,C)
=> r1_tarski(k4_pre_topc(A,B,C,k2_frechet(A,D)),k2_frechet(B,E)) ) ) ) ) ) ) ) ).
fof(t15_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(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)) )
=> ( v4_frechet(A)
=> ( v5_pre_topc(C,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)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(B))
& m2_relset_1(E,k5_numbers,u1_struct_0(B)) )
=> ( E = k7_funct_2(k5_numbers,u1_struct_0(A),u1_struct_0(B),D,C)
=> r1_tarski(k4_pre_topc(A,B,C,k2_frechet(A,D)),k2_frechet(B,E)) ) ) ) ) ) ) ) ) ).
fof(d1_frechet2,axiom,
$true ).
fof(d2_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(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)))
=> ( C = k3_frechet2(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,C)
<=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(A))
& m2_relset_1(E,k5_numbers,u1_struct_0(A))
& r1_tarski(k1_frechet(A,E),B)
& r2_hidden(D,k2_frechet(A,E)) ) ) ) ) ) ) ) ).
fof(t16_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(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)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r1_tarski(k1_frechet(A,C),B)
& r2_hidden(D,k2_frechet(A,C)) )
=> r2_hidden(D,k6_pre_topc(A,B)) ) ) ) ) ) ).
fof(t17_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k3_frechet2(A,B),k6_pre_topc(A,B)) ) ) ).
fof(t18_frechet2,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_bhsp_3(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_frechet(A,B,D)
=> r1_frechet(A,C,D) ) ) ) ) ) ).
fof(t19_frechet2,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_bhsp_3(C,A,B)
=> r1_tarski(k2_frechet(A,B),k2_frechet(A,C)) ) ) ) ).
fof(t20_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> k3_frechet2(A,k1_pre_topc(A)) = k1_xboole_0 ) ).
fof(t21_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(B,k3_frechet2(A,B)) ) ) ).
fof(t22_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(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)))
=> k4_subset_1(u1_struct_0(A),k3_frechet2(A,B),k3_frechet2(A,C)) = k3_frechet2(A,k4_subset_1(u1_struct_0(A),B,C)) ) ) ) ).
fof(t23_frechet2,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)))
=> k6_pre_topc(A,B) = k3_frechet2(A,B) ) ) ) ).
fof(t24_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(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,k1_zfmisc_1(u1_struct_0(A)))
=> ( k3_frechet2(A,k1_pre_topc(A)) = k1_xboole_0
& r1_tarski(B,k3_frechet2(A,B))
& k3_frechet2(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k3_frechet2(A,B),k3_frechet2(A,C))
& k3_frechet2(A,k3_frechet2(A,B)) = k3_frechet2(A,B) ) ) ) ) ) ).
fof(t25_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ( v4_frechet(A)
& ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_frechet2(A,k3_frechet2(A,B)) = k3_frechet2(A,B) ) )
=> v3_frechet(A) ) ) ).
fof(t26_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v4_frechet(A)
=> ( v3_frechet(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)))
=> ( k3_frechet2(A,k1_pre_topc(A)) = k1_xboole_0
& r1_tarski(B,k3_frechet2(A,B))
& k3_frechet2(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k3_frechet2(A,B),k3_frechet2(A,C))
& k3_frechet2(A,k3_frechet2(A,B)) = k3_frechet2(A,B) ) ) ) ) ) ) ).
fof(d3_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(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))
& k2_frechet(A,B) = k1_struct_0(A,C) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C = k4_frechet2(A,B)
<=> r1_frechet(A,B,C) ) ) ) ) ) ).
fof(t27_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_compts_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_frechet(B,A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_frechet(A,B) != k1_struct_0(A,C) ) ) ) ) ) ).
fof(t28_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_compts_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_frechet(A,B,C)
<=> ( v2_frechet(B,A)
& C = k4_frechet2(A,B) ) ) ) ) ) ) ).
fof(t29_frechet2,axiom,
! [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_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k5_pcomps_1(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k5_pcomps_1(A))) ) ) ) ).
fof(t30_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k5_pcomps_1(A)))
& m2_relset_1(B,k5_numbers,u1_struct_0(k5_pcomps_1(A))) )
=> ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) ) ) ).
fof(t31_frechet2,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,u1_struct_0(k5_pcomps_1(A)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k5_pcomps_1(A))) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k5_pcomps_1(A)))
=> ( ( B = D
& C = E )
=> ( r1_metric_6(A,B,C)
<=> r1_frechet(k5_pcomps_1(A),D,E) ) ) ) ) ) ) ) ).
fof(t32_frechet2,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(k5_pcomps_1(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k5_pcomps_1(A))) )
=> ( B = C
=> ( v2_tbsp_1(B,A)
<=> v2_frechet(C,k5_pcomps_1(A)) ) ) ) ) ) ).
fof(t33_frechet2,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(k5_pcomps_1(A)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k5_pcomps_1(A))) )
=> ( ( B = C
& v2_tbsp_1(B,A) )
=> k1_tbsp_1(A,B) = k4_frechet2(k5_pcomps_1(A),C) ) ) ) ) ).
fof(d4_frechet2,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_frechet2(A,B,C)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( v3_pre_topc(D,A)
& r2_hidden(C,D)
& ! [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(t34_frechet2,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,u1_struct_0(A))
=> ( ? [D] :
( m1_bhsp_3(D,A,B)
& r1_frechet(A,D,C) )
=> r1_frechet2(A,B,C) ) ) ) ) ).
fof(t35_frechet2,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,u1_struct_0(A))
=> ( r1_frechet(A,B,C)
=> r1_frechet2(A,B,C) ) ) ) ) ).
fof(t37_frechet2,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,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( k2_normsp_1(A,B,E) = D
& r1_frechet(A,B,C) ) )
=> r2_hidden(C,k6_pre_topc(A,k1_struct_0(A,D))) ) ) ) ) ) ).
fof(t39_frechet2,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] :
( ( 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),k1_frechet(A,B))
& v2_funct_1(C)
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k5_numbers)
& v3_funct_2(D,k5_numbers,k5_numbers)
& m2_relset_1(D,k5_numbers,k5_numbers) )
=> ~ m1_bhsp_3(k1_frechet2(A,D,C),A,B) ) ) ) ) ) ).
fof(t40_frechet2,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] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k5_numbers)
& v3_funct_2(C,k5_numbers,k5_numbers)
& m2_relset_1(C,k5_numbers,k5_numbers) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_frechet(A,B,D)
=> r1_frechet(A,k1_frechet2(A,C,B),D) ) ) ) ) ) ).
fof(t41_frechet2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v1_seqm_3(B)
& v7_seqm_3(B)
& m2_relset_1(B,k5_numbers,k1_numbers)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seqm_3(B,C) = k1_nat_1(C,A) ) ) ) ).
fof(t42_frechet2,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)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( m1_bhsp_3(D,A,B)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_normsp_1(A,D,E) = k2_normsp_1(A,B,k1_nat_1(E,C)) ) ) ) ) ) ).
fof(t43_frechet2,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,u1_struct_0(A))
=> ! [D] :
( m1_bhsp_3(D,A,B)
=> ( r1_frechet2(A,B,C)
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k2_normsp_1(A,D,F) = k2_normsp_1(A,B,k1_nat_1(F,E)) ) )
| r1_frechet2(A,D,C) ) ) ) ) ) ) ).
fof(t44_frechet2,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,u1_struct_0(A))
=> ( r1_frechet2(A,B,C)
=> r2_hidden(C,k6_pre_topc(A,k1_frechet(A,B))) ) ) ) ) ).
fof(t45_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( v3_frechet(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_frechet2(A,B,C)
& ! [D] :
( m1_bhsp_3(D,A,B)
=> ~ r1_frechet(A,D,C) ) ) ) ) ) ) ).
fof(t46_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_frechet(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_yellow_8(C,A,B)
& ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& k1_relat_1(D) = k5_numbers
& k2_relat_1(D) = C
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_xreal_0(E,F)
=> r1_tarski(k1_funct_1(D,F),k1_funct_1(D,E)) ) ) ) ) ) ) ) ) ).
fof(t47_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_urysohn1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_pre_topc(A,k1_struct_0(A,B)) = k1_struct_0(A,B) ) ) ) ).
fof(t48_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_compts_1(A)
=> v1_urysohn1(A) ) ) ).
fof(t49_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ~ ( ~ v1_urysohn1(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)) )
=> ~ ( D = k3_finsop_1(u1_struct_0(A),k5_numbers,B)
& B != C
& r1_frechet(A,D,C) ) ) ) ) ) ) ).
fof(t50_frechet2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ~ ( ~ v1_finset_1(k1_relat_1(A))
& v2_funct_1(A)
& v1_finset_1(k2_relat_1(A)) ) ) ).
fof(s1_frechet2,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ? [C] :
( m1_subset_1(C,u1_struct_0(f1_s1_frechet2))
& r1_xreal_0(A,B)
& C = k2_normsp_1(f1_s1_frechet2,f2_s1_frechet2,B)
& p1_s1_frechet2(C) ) ) )
=> ? [A] :
( m1_bhsp_3(A,f1_s1_frechet2,f2_s1_frechet2)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> p1_s1_frechet2(k2_normsp_1(f1_s1_frechet2,A,B)) ) ) ) ).
fof(s2_frechet2,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ? [C] :
( m1_subset_1(C,u1_struct_0(f1_s2_frechet2))
& r1_xreal_0(A,B)
& C = k2_normsp_1(f1_s2_frechet2,f2_s2_frechet2,B)
& p1_s2_frechet2(C) ) ) )
=> ? [A] :
( m1_bhsp_3(A,f1_s2_frechet2,f2_s2_frechet2)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> p1_s2_frechet2(k2_normsp_1(f1_s2_frechet2,A,B)) ) ) ) ).
fof(s3_frechet2,axiom,
( ? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(f1_s3_frechet2))
=> ( ( r1_xreal_0(A,B)
& C = k2_normsp_1(f1_s3_frechet2,f2_s3_frechet2,B) )
=> p1_s3_frechet2(C) ) ) ) )
=> ? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> p1_s3_frechet2(k2_normsp_1(f1_s3_frechet2,k1_frechet2(f1_s3_frechet2,f3_s3_frechet2,f2_s3_frechet2),B)) ) ) ) ) ).
fof(s4_frechet2,axiom,
( ? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(f1_s4_frechet2))
=> ( ( r1_xreal_0(A,B)
& C = k2_normsp_1(f1_s4_frechet2,f2_s4_frechet2,B) )
=> p1_s4_frechet2(C) ) ) ) )
=> ? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> p1_s4_frechet2(k2_normsp_1(f1_s4_frechet2,k1_frechet2(f1_s4_frechet2,f3_s4_frechet2,f2_s4_frechet2),B)) ) ) ) ) ).
fof(dt_k1_frechet2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k5_numbers)
& m1_relset_1(B,k5_numbers,k5_numbers)
& 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(k1_frechet2(A,B,C))
& v1_funct_2(k1_frechet2(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k1_frechet2(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(redefinition_k1_frechet2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k5_numbers)
& m1_relset_1(B,k5_numbers,k5_numbers)
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> k1_frechet2(A,B,C) = k5_relat_1(B,C) ) ).
fof(dt_k2_frechet2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v1_seqm_3(B)
& v7_seqm_3(B)
& m1_relset_1(B,k5_numbers,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> m1_bhsp_3(k2_frechet2(A,B,C),A,C) ) ).
fof(redefinition_k2_frechet2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v1_seqm_3(B)
& v7_seqm_3(B)
& m1_relset_1(B,k5_numbers,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> k2_frechet2(A,B,C) = k5_relat_1(B,C) ) ).
fof(dt_k3_frechet2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k3_frechet2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k4_frechet2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(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(k4_frechet2(A,B),u1_struct_0(A)) ) ).
fof(t36_frechet2,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,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( D = a_2_0_frechet2(A,C)
& r1_tarski(k1_frechet(A,B),D) )
=> r1_frechet(A,B,C) ) ) ) ) ) ).
fof(t38_frechet2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(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)) )
=> ~ ( C = a_2_0_frechet2(A,B)
& r1_xboole_0(k1_frechet(A,D),C)
& r1_frechet(A,D,B)
& ! [E] :
( m1_bhsp_3(E,A,D)
=> ~ v2_funct_1(E) ) ) ) ) ) ) ).
fof(fraenkel_a_2_0_frechet2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_0_frechet2(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r2_hidden(C,k6_pre_topc(B,k1_struct_0(B,D))) ) ) ) ).
%------------------------------------------------------------------------------