SET007 Axioms: SET007+730.ax
%------------------------------------------------------------------------------
% File : SET007+730 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Minimal Distance Between Sets in Euclidean Space
% 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 : jordan1k [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 82 ( 0 unt; 0 def)
% Number of atoms : 530 ( 50 equ)
% Maximal formula atoms : 17 ( 6 avg)
% Number of connectives : 534 ( 86 ~; 1 |; 214 &)
% ( 3 <=>; 230 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 9 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 31 ( 30 usr; 0 prp; 1-3 aty)
% Number of functors : 64 ( 64 usr; 6 con; 0-5 aty)
% Number of variables : 235 ( 227 !; 8 ?)
% 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_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( ~ v1_xboole_0(k1_jordan2c(np__2,A))
& v3_pre_topc(k1_jordan2c(np__2,A),k15_euclid(np__2)) ) ) ).
fof(t1_jordan1k,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( v2_funct_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,B)
=> ? [E] :
( r2_hidden(E,A)
& D = k1_funct_1(C,E) ) ) ) ) ) ).
fof(t2_jordan1k,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( v2_funct_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,B)
=> ? [E] :
( m1_subset_1(E,A)
& D = k1_funct_1(C,E) ) ) ) ) ) ).
fof(t3_jordan1k,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( v2_funct_2(C,A,B)
=> r1_tarski(k3_subset_1(B,k2_funct_2(A,B,C,D)),k2_funct_2(A,B,C,k3_subset_1(A,D))) ) ) ) ) ).
fof(t4_jordan1k,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( v2_funct_1(C)
=> r1_tarski(k2_funct_2(A,B,C,k3_subset_1(A,D)),k3_subset_1(B,k2_funct_2(A,B,C,D))) ) ) ) ) ).
fof(t5_jordan1k,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( v3_funct_2(C,A,B)
=> k3_subset_1(B,k2_funct_2(A,B,C,D)) = k2_funct_2(A,B,C,k3_subset_1(A,D)) ) ) ) ) ).
fof(t6_jordan1k,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( r4_connsp_1(A,k1_pre_topc(A),B)
<=> v1_xboole_0(B) ) ) ) ).
fof(t7_jordan1k,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)))
=> ! [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(B,C)
& r4_connsp_1(A,D,B)
& r4_connsp_1(A,D,C) )
=> B = C ) ) ) ) ) ).
fof(t8_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k14_euclid(A))))
=> ~ ( v6_tbsp_1(B,k14_euclid(A))
& v6_tbsp_1(k3_subset_1(u1_struct_0(k14_euclid(A)),B),k14_euclid(A)) ) ) ) ) ).
fof(t9_jordan1k,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(k5_pcomps_1(A)))
=> r1_xreal_0(np__0,k8_funct_2(u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k3_topmetr),k8_weierstr(A,B),C)) ) ) ) ).
fof(t10_jordan1k,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [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) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B)))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ( r2_hidden(E,C)
=> r1_xreal_0(A,k4_metric_1(B,D,E)) ) )
=> r1_xreal_0(A,k1_funct_1(k8_weierstr(B,C),D)) ) ) ) ) ) ).
fof(t11_jordan1k,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)))) )
=> r1_xreal_0(np__0,k9_weierstr(A,B,C)) ) ) ) ).
fof(t12_jordan1k,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] :
( ( v6_compts_1(B,k5_pcomps_1(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ! [C] :
( ( v6_compts_1(C,k5_pcomps_1(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> ( ~ r1_xboole_0(B,C)
=> k9_weierstr(A,B,C) = np__0 ) ) ) ) ).
fof(t13_jordan1k,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [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) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B)))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B)))) )
=> ( ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( r2_hidden(E,C)
& r2_hidden(F,D) )
=> r1_xreal_0(A,k4_metric_1(B,E,F)) ) ) )
=> r1_xreal_0(A,k9_weierstr(B,C,D)) ) ) ) ) ) ).
fof(t14_jordan1k,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))))
=> ~ ( r4_connsp_1(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),C),B)
& ~ r1_jordan2c(A,C,B)
& ~ r2_jordan2c(A,C,B) ) ) ) ) ).
fof(t15_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> k1_jordan2c(A,k1_pre_topc(k15_euclid(A))) = k1_pre_topc(k15_euclid(A)) ) ) ).
fof(t16_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_jordan2c(A,k2_pre_topc(k15_euclid(A))) = k1_pre_topc(k15_euclid(A)) ) ).
fof(t17_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> k2_jordan2c(A,k1_pre_topc(k15_euclid(A))) = k2_pre_topc(k15_euclid(A)) ) ) ).
fof(t18_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_jordan2c(A,k2_pre_topc(k15_euclid(A))) = k1_pre_topc(k15_euclid(A)) ) ).
fof(t19_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_connsp_1(B,k15_euclid(A))
& 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))))
=> ~ ( r1_xboole_0(B,C)
& ~ r1_tarski(B,k2_jordan2c(A,C))
& ~ r1_tarski(B,k1_jordan2c(A,C)) ) ) ) ) ).
fof(t20_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k1_gobrd14(np__2,k23_euclid(np__0,np__0),k18_euclid(B,np__2,A)) = k4_real_1(k18_complex1(B),k1_gobrd14(np__2,k23_euclid(np__0,np__0),A)) ) ) ).
fof(t21_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> k1_gobrd14(np__2,k17_euclid(np__2,A,B),k17_euclid(np__2,C,B)) = k1_gobrd14(np__2,A,C) ) ) ) ).
fof(t22_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( A != B
& r1_xreal_0(k1_gobrd14(np__2,A,B),np__0) ) ) ) ).
fof(t23_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> k1_gobrd14(np__2,k20_euclid(np__2,A,B),k20_euclid(np__2,C,B)) = k1_gobrd14(np__2,A,C) ) ) ) ).
fof(t24_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> k1_gobrd14(np__2,A,B) = k1_gobrd14(np__2,k19_euclid(np__2,A),k19_euclid(np__2,B)) ) ) ).
fof(t25_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> k1_gobrd14(np__2,k20_euclid(np__2,A,B),k20_euclid(np__2,A,C)) = k1_gobrd14(np__2,B,C) ) ) ) ).
fof(t26_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k1_gobrd14(np__2,k18_euclid(C,np__2,A),k18_euclid(C,np__2,B)) = k4_real_1(k18_complex1(C),k1_gobrd14(np__2,A,B)) ) ) ) ).
fof(t27_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r1_xreal_0(C,np__1)
=> k1_gobrd14(np__2,A,k17_euclid(np__2,k18_euclid(C,np__2,A),k18_euclid(k5_real_1(np__1,C),np__2,B))) = k4_real_1(k5_real_1(np__1,C),k1_gobrd14(np__2,A,B)) ) ) ) ) ).
fof(t28_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r1_xreal_0(np__0,C)
=> k1_gobrd14(np__2,A,k17_euclid(np__2,k18_euclid(C,np__2,B),k18_euclid(k5_real_1(np__1,C),np__2,A))) = k4_real_1(C,k1_gobrd14(np__2,B,A)) ) ) ) ) ).
fof(t29_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(A,k3_topreal1(np__2,B,C))
=> k3_real_1(k1_gobrd14(np__2,B,A),k1_gobrd14(np__2,A,C)) = k1_gobrd14(np__2,B,C) ) ) ) ) ).
fof(t30_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r2_hidden(A,k3_topreal1(np__2,B,C))
& A != B
& r1_xreal_0(k1_gobrd14(np__2,B,C),k1_gobrd14(np__2,A,C)) ) ) ) ) ).
fof(t32_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k2_jgraph_2(B,C,B,D),A) = k17_euclid(np__2,k18_euclid(B,np__2,A),k23_euclid(C,D)) ) ) ) ) ).
fof(t33_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k2_jgraph_2(C,k21_euclid(A),C,k22_euclid(A)),B) = k17_euclid(np__2,k18_euclid(C,np__2,B),A) ) ) ) ).
fof(t34_jordan1k,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,np__0)
& k7_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k2_jgraph_2(k6_real_1(np__1,A),k1_real_1(k6_real_1(C,A)),k6_real_1(np__1,B),k1_real_1(k6_real_1(D,B))),k2_jgraph_2(A,C,B,D)) != k6_partfun1(k1_euclid(np__2)) ) ) ) ) ) ).
fof(t35_jordan1k,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k14_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k14_euclid(np__2)))
=> ( ( C = k23_euclid(np__0,np__0)
& D = A )
=> ( r1_xreal_0(B,np__0)
| k4_pre_topc(k15_euclid(np__2),k15_euclid(np__2),k2_jgraph_2(B,k21_euclid(A),B,k22_euclid(A)),k9_metric_1(k14_euclid(np__2),C,np__1)) = k9_metric_1(k14_euclid(np__2),D,B) ) ) ) ) ) ) ).
fof(t36_jordan1k,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(C,np__0)
& ~ v2_funct_2(k2_jgraph_2(A,B,C,D),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2))) ) ) ) ) ) ).
fof(t37_jordan1k,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
=> ( v2_connsp_1(k3_subset_1(u1_struct_0(k14_euclid(np__2)),k9_metric_1(k14_euclid(np__2),B,A)),k15_euclid(np__2))
& m1_subset_1(k3_subset_1(u1_struct_0(k14_euclid(np__2)),k9_metric_1(k14_euclid(np__2),B,A)),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ) ) ).
fof(d1_jordan1k,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 = k1_jordan1k(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 = k9_weierstr(k14_euclid(A),E,F) ) ) ) ) ) ) ) ).
fof(t38_jordan1k,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)))) )
=> r1_xreal_0(np__0,k1_jordan1k(A,B,C)) ) ) ) ).
fof(t39_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v6_compts_1(B,k15_euclid(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ! [C] :
( ( v6_compts_1(C,k15_euclid(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ( ~ r1_xboole_0(B,C)
=> k1_jordan1k(A,B,C) = np__0 ) ) ) ) ).
fof(t40_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [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)))) )
=> ( ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
=> ( ( r2_hidden(E,C)
& r2_hidden(F,D) )
=> r1_xreal_0(B,k1_gobrd14(A,E,F)) ) ) )
=> r1_xreal_0(B,k1_jordan1k(A,C,D)) ) ) ) ) ) ).
fof(t41_jordan1k,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] :
( ( ~ 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)))) )
=> ( r1_tarski(D,B)
=> r1_xreal_0(k1_jordan1k(A,C,B),k1_jordan1k(A,C,D)) ) ) ) ) ) ).
fof(t42_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v6_compts_1(C,k15_euclid(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ? [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
& ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
& r2_hidden(D,B)
& r2_hidden(E,C)
& k4_jordan1k(A,B,C) = k1_gobrd14(A,D,E) ) ) ) ) ) ).
fof(t43_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> k4_jordan1k(A,k1_struct_0(k15_euclid(A),B),k1_struct_0(k15_euclid(A),C)) = k1_gobrd14(A,B,C) ) ) ) ).
fof(d2_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k5_jordan1k(A,B,C) = k1_jordan1k(A,k1_struct_0(k15_euclid(A),B),C) ) ) ) ).
fof(t44_jordan1k,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] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> r1_xreal_0(np__0,k5_jordan1k(A,C,B)) ) ) ) ).
fof(t45_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v6_compts_1(B,k15_euclid(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ( r2_hidden(C,B)
=> k5_jordan1k(A,C,B) = np__0 ) ) ) ) ).
fof(t46_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
& r2_hidden(D,B)
& k5_jordan1k(A,C,B) = k1_gobrd14(A,C,D) ) ) ) ) ).
fof(t47_jordan1k,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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r1_tarski(B,C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> r1_xreal_0(k5_jordan1k(A,D,C),k5_jordan1k(A,D,B)) ) ) ) ) ) ).
fof(t48_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ( ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ( r2_hidden(E,C)
=> r1_xreal_0(B,k1_gobrd14(A,D,E)) ) )
=> r1_xreal_0(B,k5_jordan1k(A,D,C)) ) ) ) ) ) ).
fof(t49_jordan1k,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> k5_jordan1k(A,B,k1_struct_0(k15_euclid(A),C)) = k1_gobrd14(A,B,C) ) ) ) ).
fof(t50_jordan1k,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] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ( r2_hidden(D,B)
=> r1_xreal_0(k5_jordan1k(A,C,B),k1_gobrd14(A,C,D)) ) ) ) ) ) ).
fof(t51_jordan1k,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v6_compts_1(A,k15_euclid(np__2))
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( ( v3_pre_topc(B,k15_euclid(np__2))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_tarski(A,B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( ~ r2_hidden(C,B)
& r1_xreal_0(k5_jordan1k(np__2,C,A),k5_jordan1k(np__2,C,B)) ) ) ) ) ) ).
fof(t52_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ r2_subset_1(k2_jordan2c(np__2,A),k2_jordan2c(np__2,B)) ) ) ).
fof(t53_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r2_hidden(B,k2_jordan2c(np__2,A))
& r2_hidden(C,k1_jordan2c(np__2,A))
& r1_xreal_0(k1_gobrd14(np__2,B,C),k5_jordan1k(np__2,B,A)) ) ) ) ) ).
fof(t54_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ~ r2_hidden(B,k1_jordan2c(np__2,A))
=> r1_xreal_0(k5_jordan1k(np__2,B,A),k5_jordan1k(np__2,B,k1_jordan2c(np__2,A))) ) ) ) ).
fof(t55_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_tarski(A,k1_jordan2c(np__2,B))
& r1_tarski(B,k1_jordan2c(np__2,A)) ) ) ) ).
fof(t56_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_tarski(A,k1_jordan2c(np__2,B))
=> r1_tarski(B,k2_jordan2c(np__2,A)) ) ) ) ).
fof(t57_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_tarski(k5_topreal1(np__2,k1_jordan9(A,B)),k2_jordan2c(np__2,A)) ) ) ).
fof(d3_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k6_jordan1k(A) = k1_jordan5c(k9_jordan6(A),k6_jordan6(k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),np__2)),k30_pscomp_1(A),k34_pscomp_1(A)) ) ).
fof(d4_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k7_jordan1k(A) = k1_jordan5c(k8_jordan6(A),k6_jordan6(k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),np__2)),k30_pscomp_1(A),k34_pscomp_1(A)) ) ).
fof(t58_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ r1_xboole_0(k9_jordan6(A),k6_jordan6(k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),np__2))) ) ).
fof(t59_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ r1_xboole_0(k8_jordan6(A),k6_jordan6(k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),np__2))) ) ).
fof(t60_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k21_euclid(k6_jordan1k(A)) = k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),np__2) ) ).
fof(t61_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k21_euclid(k7_jordan1k(A)) = k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),np__2) ) ).
fof(t62_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r2_hidden(k6_jordan1k(A),k9_jordan6(A)) ) ).
fof(t63_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r2_hidden(k7_jordan1k(A),k8_jordan6(A)) ) ).
fof(dt_k1_jordan1k,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(k1_jordan1k(A,B,C),k1_numbers) ) ).
fof(dt_k2_jordan1k,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_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_xboole_0(C)
& v6_compts_1(C,k5_pcomps_1(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> m1_subset_1(k2_jordan1k(A,B,C),k1_numbers) ) ).
fof(commutativity_k2_jordan1k,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_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_xboole_0(C)
& v6_compts_1(C,k5_pcomps_1(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> k2_jordan1k(A,B,C) = k2_jordan1k(A,C,B) ) ).
fof(redefinition_k2_jordan1k,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_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_xboole_0(C)
& v6_compts_1(C,k5_pcomps_1(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> k2_jordan1k(A,B,C) = k9_weierstr(A,B,C) ) ).
fof(dt_k3_jordan1k,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_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_xboole_0(C)
& v6_compts_1(C,k5_pcomps_1(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> m1_subset_1(k3_jordan1k(A,B,C),k1_numbers) ) ).
fof(commutativity_k3_jordan1k,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_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_xboole_0(C)
& v6_compts_1(C,k5_pcomps_1(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> k3_jordan1k(A,B,C) = k3_jordan1k(A,C,B) ) ).
fof(redefinition_k3_jordan1k,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_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_xboole_0(C)
& v6_compts_1(C,k5_pcomps_1(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
=> k3_jordan1k(A,B,C) = k12_weierstr(A,B,C) ) ).
fof(dt_k4_jordan1k,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& ~ v1_xboole_0(C)
& v6_compts_1(C,k15_euclid(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k4_jordan1k(A,B,C),k1_numbers) ) ).
fof(commutativity_k4_jordan1k,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& ~ v1_xboole_0(C)
& v6_compts_1(C,k15_euclid(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> k4_jordan1k(A,B,C) = k4_jordan1k(A,C,B) ) ).
fof(redefinition_k4_jordan1k,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& ~ v1_xboole_0(C)
& v6_compts_1(C,k15_euclid(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> k4_jordan1k(A,B,C) = k1_jordan1k(A,B,C) ) ).
fof(dt_k5_jordan1k,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k5_jordan1k(A,B,C),k1_numbers) ) ).
fof(dt_k6_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> m1_subset_1(k6_jordan1k(A),u1_struct_0(k15_euclid(np__2))) ) ).
fof(dt_k7_jordan1k,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> m1_subset_1(k7_jordan1k(A),u1_struct_0(k15_euclid(np__2))) ) ).
fof(t31_jordan1k,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
=> ( B = k23_euclid(np__0,np__0)
=> k9_metric_1(k14_euclid(np__2),B,A) = a_1_0_jordan1k(A) ) ) ) ).
fof(fraenkel_a_1_0_jordan1k,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_0_jordan1k(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& A = C
& ~ r1_xreal_0(B,k5_toprns_1(np__2,C)) ) ) ) ).
%------------------------------------------------------------------------------