SET007 Axioms: SET007+757.ax
%------------------------------------------------------------------------------
% File : SET007+757 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Hausdorff Distance Between Compact Subsets
% 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 : hausdorf [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 58 ( 0 unt; 0 def)
% Number of atoms : 626 ( 39 equ)
% Maximal formula atoms : 16 ( 10 avg)
% Number of connectives : 684 ( 116 ~; 1 |; 362 &)
% ( 7 <=>; 198 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 10 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 28 ( 27 usr; 0 prp; 1-3 aty)
% Number of functors : 35 ( 35 usr; 4 con; 0-4 aty)
% Number of variables : 189 ( 182 !; 7 ?)
% 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(fc1_hausdorf,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) )
=> ( ~ v3_struct_0(k5_pcomps_1(A))
& v1_pre_topc(k5_pcomps_1(A))
& v2_pre_topc(k5_pcomps_1(A))
& v3_compts_1(k5_pcomps_1(A)) ) ) ).
fof(fc2_hausdorf,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)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_relat_1(k6_weierstr(A,B))
& v1_funct_1(k6_weierstr(A,B))
& v1_funct_2(k6_weierstr(A,B),u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k3_topmetr))
& v5_pre_topc(k6_weierstr(A,B),k5_pcomps_1(A),k3_topmetr) ) ) ).
fof(fc3_hausdorf,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_xboole_0(B)
& v6_compts_1(B,k5_pcomps_1(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ( v1_relat_1(k7_weierstr(A,B))
& v1_funct_1(k7_weierstr(A,B))
& v1_funct_2(k7_weierstr(A,B),u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k3_topmetr))
& v5_pre_topc(k7_weierstr(A,B),k5_pcomps_1(A),k3_topmetr) ) ) ).
fof(fc4_hausdorf,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_xboole_0(B)
& v6_compts_1(B,k5_pcomps_1(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ( v1_relat_1(k8_weierstr(A,B))
& v1_funct_1(k8_weierstr(A,B))
& v1_funct_2(k8_weierstr(A,B),u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k3_topmetr))
& v5_pre_topc(k8_weierstr(A,B),k5_pcomps_1(A),k3_topmetr) ) ) ).
fof(t1_hausdorf,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B)
& k2_square_1(A,B) = np__0 )
=> A = np__0 ) ) ) ).
fof(t2_hausdorf,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_funct_1(k6_weierstr(A,B),B) = np__0 ) ) ).
fof(t3_hausdorf,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(k5_pcomps_1(A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> r2_hidden(np__0,k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k6_weierstr(A,C),B)) ) ) ) ) ).
fof(t4_hausdorf,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(k5_pcomps_1(A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( v1_xreal_0(D)
=> ( r2_hidden(D,k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k6_weierstr(A,C),B))
=> r1_xreal_0(np__0,D) ) ) ) ) ) ).
fof(t5_hausdorf,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(k5_pcomps_1(A))))
=> ! [C] :
( r2_hidden(C,B)
=> k1_funct_1(k8_weierstr(A,B),C) = np__0 ) ) ) ).
fof(t6_hausdorf,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(k5_pcomps_1(A)))
=> ! [D] :
( v1_xreal_0(D)
=> ( B = C
=> ( r1_xreal_0(D,np__0)
| m1_connsp_2(k9_metric_1(A,B,D),k5_pcomps_1(A),C) ) ) ) ) ) ) ).
fof(t7_hausdorf,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(k5_pcomps_1(A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,k6_pre_topc(k5_pcomps_1(A),B))
<=> ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_xboole_0(k9_metric_1(A,C,D),B) ) ) ) ) ) ) ).
fof(t8_hausdorf,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,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ( r2_hidden(B,k6_pre_topc(k5_pcomps_1(A),C))
<=> ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(E,C)
& ~ r1_xreal_0(D,k4_metric_1(A,B,E)) ) ) ) ) ) ) ) ) ).
fof(t9_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_funct_1(k8_weierstr(A,B),C) = np__0
<=> ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(E,B)
& ~ r1_xreal_0(D,k4_metric_1(A,C,E)) ) ) ) ) ) ) ) ) ).
fof(t10_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,k6_pre_topc(k5_pcomps_1(A),B))
<=> k1_funct_1(k8_weierstr(A,B),C) = np__0 ) ) ) ) ).
fof(t11_hausdorf,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_xboole_0(B)
& v4_pre_topc(B,k5_pcomps_1(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
<=> k1_funct_1(k8_weierstr(A,B),C) = np__0 ) ) ) ) ).
fof(t12_hausdorf,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) )
=> ? [B] :
( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers))
& A = B
& k5_weierstr(A) = k4_pscomp_1(B) ) ) ).
fof(t13_hausdorf,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) )
=> ? [B] :
( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers))
& A = B
& k4_weierstr(A) = k3_pscomp_1(B) ) ) ).
fof(t14_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_numbers))
=> ( D = k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k6_weierstr(A,C),B)
=> v2_seq_4(D) ) ) ) ) ) ).
fof(t15_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,B)
=> r1_xreal_0(k1_funct_1(k8_weierstr(A,B),C),k4_metric_1(A,C,D)) ) ) ) ) ) ).
fof(t16_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,B)
=> r1_xreal_0(C,k4_metric_1(A,D,E)) ) )
=> r1_xreal_0(C,k1_funct_1(k8_weierstr(A,B),D)) ) ) ) ) ) ).
fof(t17_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r1_xreal_0(k1_funct_1(k8_weierstr(A,B),C),k2_xcmplx_0(k4_metric_1(A,C,D),k1_funct_1(k8_weierstr(A,B),D))) ) ) ) ) ).
fof(t18_hausdorf,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(k5_pcomps_1(A))))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( B = C
=> k3_pre_topc(k5_pcomps_1(A),B) = k5_pcomps_1(k1_topmetr(A,C)) ) ) ) ) ).
fof(t19_hausdorf,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_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k1_topmetr(A,C))))
=> ( ( r1_tarski(B,C)
& B = D
& v6_tbsp_1(D,k1_topmetr(A,C)) )
=> v6_tbsp_1(B,A) ) ) ) ) ) ).
fof(t20_hausdorf,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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ( ( C = B
& v6_compts_1(C,k5_pcomps_1(A)) )
=> v6_tbsp_1(B,A) ) ) ) ) ).
fof(t21_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& r2_hidden(D,B)
& r1_xreal_0(k1_funct_1(k8_weierstr(A,B),C),k4_metric_1(A,D,C)) ) ) ) ) ).
fof(t22_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(D,B)
& v6_compts_1(B,k5_pcomps_1(A)) )
=> r1_xreal_0(k4_metric_1(A,C,D),k1_funct_1(k7_weierstr(A,B),C)) ) ) ) ) ) ).
fof(t23_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( v6_compts_1(B,k5_pcomps_1(A))
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(D,B)
& r1_xreal_0(k4_metric_1(A,D,C),k1_funct_1(k7_weierstr(A,B),C)) ) ) ) ) ) ) ).
fof(t24_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A))
& r2_hidden(D,C) )
=> r1_xreal_0(k1_funct_1(k8_weierstr(A,B),D),k12_weierstr(A,B,C)) ) ) ) ) ) ).
fof(t25_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A))
& r2_hidden(D,C) )
=> r1_xreal_0(k1_funct_1(k7_weierstr(A,B),D),k12_weierstr(A,B,C)) ) ) ) ) ) ).
fof(t26_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_numbers))
=> ( ( D = k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k7_weierstr(A,B),C)
& v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A)) )
=> v1_seq_4(D) ) ) ) ) ) ).
fof(t27_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_numbers))
=> ( ( D = k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k8_weierstr(A,B),C)
& v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A)) )
=> v1_seq_4(D) ) ) ) ) ) ).
fof(t28_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( v6_compts_1(B,k5_pcomps_1(A))
=> r1_xreal_0(k1_funct_1(k8_weierstr(A,B),C),k1_funct_1(k7_weierstr(A,B),C)) ) ) ) ) ).
fof(t29_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k8_weierstr(A,B),B) = k1_hausdorf(np__0) ) ) ).
fof(t30_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A)) )
=> r1_xreal_0(np__0,k10_weierstr(A,B,C)) ) ) ) ) ).
fof(t31_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> k10_weierstr(A,B,B) = np__0 ) ) ).
fof(t32_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A)) )
=> r1_xreal_0(np__0,k11_weierstr(A,B,C)) ) ) ) ) ).
fof(t33_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A))
& r2_hidden(D,B) )
=> r1_xreal_0(k1_funct_1(k8_weierstr(A,C),D),k10_weierstr(A,C,B)) ) ) ) ) ) ).
fof(d1_hausdorf,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(k5_pcomps_1(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> k2_hausdorf(A,B,C) = k4_square_1(k10_weierstr(A,B,C),k10_weierstr(A,C,B)) ) ) ) ).
fof(t34_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A))
& r2_hidden(D,B) )
=> r1_xreal_0(k1_funct_1(k8_weierstr(A,C),D),k2_hausdorf(A,B,C)) ) ) ) ) ) ).
fof(t35_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A))
& v6_compts_1(D,k5_pcomps_1(A)) )
=> r1_xreal_0(k10_weierstr(A,B,D),k3_real_1(k2_hausdorf(A,B,C),k2_hausdorf(A,C,D))) ) ) ) ) ) ).
fof(t36_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A))
& v6_compts_1(D,k5_pcomps_1(A)) )
=> r1_xreal_0(k10_weierstr(A,D,B),k3_real_1(k2_hausdorf(A,B,C),k2_hausdorf(A,C,D))) ) ) ) ) ) ).
fof(t37_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A)) )
=> r1_xreal_0(np__0,k2_hausdorf(A,B,C)) ) ) ) ) ).
fof(t38_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> k2_hausdorf(A,B,B) = np__0 ) ) ).
fof(t39_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A))
& k2_hausdorf(A,B,C) = np__0 )
=> B = C ) ) ) ) ).
fof(t40_hausdorf,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A))
& v6_compts_1(D,k5_pcomps_1(A)) )
=> r1_xreal_0(k2_hausdorf(A,B,D),k3_real_1(k2_hausdorf(A,B,C),k2_hausdorf(A,C,D))) ) ) ) ) ) ).
fof(d2_hausdorf,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( D = k3_hausdorf(A,B,C)
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(k14_euclid(A)))))
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(k14_euclid(A)))))
& B = E
& C = F
& D = k10_weierstr(k14_euclid(A),E,F) ) ) ) ) ) ) ) ).
fof(d3_hausdorf,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( D = k4_hausdorf(A,B,C)
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(k14_euclid(A)))))
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(k14_euclid(A)))))
& B = E
& C = F
& D = k2_hausdorf(k14_euclid(A),E,F) ) ) ) ) ) ) ) ).
fof(t41_hausdorf,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ( ( v6_compts_1(B,k15_euclid(A))
& v6_compts_1(C,k15_euclid(A)) )
=> r1_xreal_0(np__0,k4_hausdorf(A,B,C)) ) ) ) ) ).
fof(t42_hausdorf,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> k4_hausdorf(A,B,B) = np__0 ) ) ).
fof(t43_hausdorf,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ( ( v6_compts_1(B,k15_euclid(A))
& v6_compts_1(C,k15_euclid(A))
& k4_hausdorf(A,B,C) = np__0 )
=> B = C ) ) ) ) ).
fof(t44_hausdorf,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ( ( v6_compts_1(B,k15_euclid(A))
& v6_compts_1(C,k15_euclid(A))
& v6_compts_1(D,k15_euclid(A)) )
=> r1_xreal_0(k4_hausdorf(A,B,D),k3_real_1(k4_hausdorf(A,B,C),k4_hausdorf(A,C,D))) ) ) ) ) ) ).
fof(dt_k1_hausdorf,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k1_hausdorf(A),k1_zfmisc_1(k1_numbers)) ) ).
fof(redefinition_k1_hausdorf,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_hausdorf(A) = k1_tarski(A) ) ).
fof(dt_k2_hausdorf,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)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> m1_subset_1(k2_hausdorf(A,B,C),k1_numbers) ) ).
fof(commutativity_k2_hausdorf,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)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> k2_hausdorf(A,B,C) = k2_hausdorf(A,C,B) ) ).
fof(dt_k3_hausdorf,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k3_hausdorf(A,B,C),k1_numbers) ) ).
fof(dt_k4_hausdorf,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k4_hausdorf(A,B,C),k1_numbers) ) ).
fof(commutativity_k4_hausdorf,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> k4_hausdorf(A,B,C) = k4_hausdorf(A,C,B) ) ).
%------------------------------------------------------------------------------