SET007 Axioms: SET007+791.ax
%------------------------------------------------------------------------------
% File : SET007+791 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Segmentation of a Simple Closed Curve
% 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 : jordan_a [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 61 ( 0 unt; 0 def)
% Number of atoms : 460 ( 49 equ)
% Maximal formula atoms : 30 ( 7 avg)
% Number of connectives : 450 ( 51 ~; 15 |; 199 &)
% ( 12 <=>; 173 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 9 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 32 ( 31 usr; 0 prp; 1-4 aty)
% Number of functors : 52 ( 52 usr; 6 con; 0-5 aty)
% Number of variables : 190 ( 172 !; 18 ?)
% 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(rc1_jordan_a,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)))
& ~ v1_xboole_0(B)
& v6_compts_1(B,A) ) ) ).
fof(fc1_jordan_a,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_relat_1(k1_jordan_a(A))
& v1_funct_1(k1_jordan_a(A))
& v1_funct_2(k1_jordan_a(A),u1_struct_0(k6_borsuk_1(k15_euclid(A),k15_euclid(A))),k1_numbers)
& v9_pscomp_1(k1_jordan_a(A),k6_borsuk_1(k15_euclid(A),k15_euclid(A))) ) ) ).
fof(cc1_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ( ~ v1_xboole_0(B)
& v1_relat_1(B)
& ~ v1_realset1(B) ) ) ) ).
fof(fc2_jordan_a,axiom,
! [A,B,C] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_jordan_a(B,A)
& m1_subset_1(C,k5_numbers) )
=> ( ~ v1_xboole_0(k2_jordan_a(A,C,B))
& v6_compts_1(k2_jordan_a(A,C,B),k15_euclid(np__2)) ) ) ).
fof(t1_jordan_a,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> k3_seq_4(k4_subset_1(k1_numbers,A,B)) = k1_square_1(k3_seq_4(A),k3_seq_4(B)) ) ) ).
fof(t2_jordan_a,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m2_relset_1(B,u1_struct_0(A),k1_numbers) )
=> ! [C] :
( ( v6_compts_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v1_rcomp_1(k2_funct_2(u1_struct_0(A),k1_numbers,B,C)) ) ) ) ).
fof(t3_jordan_a,axiom,
! [A] :
( ( v1_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( r1_tarski(B,A)
=> r2_hidden(k4_pscomp_1(B),A) ) ) ) ).
fof(t6_jordan_a,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,k9_jordan6(A))
=> ( B = k30_pscomp_1(A)
| r1_jordan6(A,k34_pscomp_1(A),B) ) ) ) ) ).
fof(t7_jordan_a,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,k8_jordan6(A))
=> r1_jordan6(A,B,k34_pscomp_1(A)) ) ) ) ).
fof(d1_jordan_a,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k6_borsuk_1(k15_euclid(A),k15_euclid(A))),k1_numbers)
& m2_relset_1(B,u1_struct_0(k6_borsuk_1(k15_euclid(A),k15_euclid(A))),k1_numbers) )
=> ( B = k1_jordan_a(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k1_binop_1(B,C,D) = k5_toprns_1(A,k20_euclid(A,C,D)) ) ) ) ) ) ).
fof(d2_jordan_a,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& m2_relset_1(B,u1_struct_0(A),k1_numbers) )
=> ( v9_pscomp_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_rcomp_1(D,k8_funct_2(u1_struct_0(A),k1_numbers,B,C))
=> ? [E] :
( m1_connsp_2(E,A,C)
& r1_tarski(k2_funct_2(u1_struct_0(A),k1_numbers,B,E),D) ) ) ) ) ) ) ).
fof(t8_jordan_a,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)))) )
=> ~ ( r1_subset_1(B,C)
& r1_xreal_0(k4_jordan1k(A,B,C),np__0) ) ) ) ) ).
fof(t9_jordan_a,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)))
=> ( ( r1_jordan6(A,B,C)
& r1_jordan6(A,C,k34_pscomp_1(A)) )
=> ( B = C
| k1_jordan7(A,B,C) = k5_jordan6(k8_jordan6(A),k30_pscomp_1(A),k34_pscomp_1(A),B,C) ) ) ) ) ) ).
fof(t10_jordan_a,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)))
=> ( r1_jordan6(A,k34_pscomp_1(A),B)
=> k1_jordan7(A,k34_pscomp_1(A),B) = k5_jordan6(k9_jordan6(A),k34_pscomp_1(A),k30_pscomp_1(A),k34_pscomp_1(A),B) ) ) ) ).
fof(t11_jordan_a,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)))
=> ( r1_jordan6(A,k34_pscomp_1(A),B)
=> k1_jordan7(A,B,k30_pscomp_1(A)) = k5_jordan6(k9_jordan6(A),k34_pscomp_1(A),k30_pscomp_1(A),B,k30_pscomp_1(A)) ) ) ) ).
fof(t12_jordan_a,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)))
=> ( ( r1_jordan6(A,B,C)
& r1_jordan6(A,k34_pscomp_1(A),B) )
=> k1_jordan7(A,B,C) = k5_jordan6(k9_jordan6(A),k34_pscomp_1(A),k30_pscomp_1(A),B,C) ) ) ) ) ).
fof(t13_jordan_a,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)))
=> ( ( r1_jordan6(A,B,k34_pscomp_1(A))
& r1_jordan6(A,k34_pscomp_1(A),C) )
=> k1_jordan7(A,B,C) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_jordan6(k8_jordan6(A),k30_pscomp_1(A),k34_pscomp_1(A),B),k3_jordan6(k9_jordan6(A),k34_pscomp_1(A),k30_pscomp_1(A),C)) ) ) ) ) ).
fof(t14_jordan_a,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)))
=> ( r1_jordan6(A,B,k34_pscomp_1(A))
=> k1_jordan7(A,B,k30_pscomp_1(A)) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_jordan6(k8_jordan6(A),k30_pscomp_1(A),k34_pscomp_1(A),B),k3_jordan6(k9_jordan6(A),k34_pscomp_1(A),k30_pscomp_1(A),k30_pscomp_1(A))) ) ) ) ).
fof(t15_jordan_a,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)))
=> k4_jordan6(k8_jordan6(A),k30_pscomp_1(A),k34_pscomp_1(A),B) = k5_jordan6(k8_jordan6(A),k30_pscomp_1(A),k34_pscomp_1(A),B,k34_pscomp_1(A)) ) ) ).
fof(t16_jordan_a,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)))
=> k3_jordan6(k9_jordan6(A),k34_pscomp_1(A),k30_pscomp_1(A),B) = k5_jordan6(k9_jordan6(A),k34_pscomp_1(A),k30_pscomp_1(A),k34_pscomp_1(A),B) ) ) ).
fof(t17_jordan_a,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,A)
=> ( B = k30_pscomp_1(A)
| r1_topreal1(k15_euclid(np__2),B,k30_pscomp_1(A),k1_jordan7(A,B,k30_pscomp_1(A))) ) ) ) ) ).
fof(t18_jordan_a,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)))
=> ( r1_jordan6(A,B,C)
=> ( B = C
| r1_topreal1(k15_euclid(np__2),B,C,k1_jordan7(A,B,C)) ) ) ) ) ) ).
fof(t19_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> A = k1_jordan7(A,k30_pscomp_1(A),k30_pscomp_1(A)) ) ).
fof(t20_jordan_a,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,A)
=> v6_compts_1(k1_jordan7(A,B,k30_pscomp_1(A)),k15_euclid(np__2)) ) ) ) ).
fof(t21_jordan_a,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)))
=> ( r1_jordan6(A,B,C)
=> v6_compts_1(k1_jordan7(A,B,C),k15_euclid(np__2)) ) ) ) ) ).
fof(d3_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( m1_jordan_a(B,A)
<=> ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1) = k30_pscomp_1(A)
& v2_funct_1(B)
& r1_xreal_0(np__8,k3_finseq_1(B))
& r1_tarski(k2_relat_1(B),A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k3_finseq_1(B),C)
| r1_jordan6(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(C,np__1))) ) ) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k3_finseq_1(B),k1_nat_1(C,np__1))
| k5_subset_1(u1_struct_0(k15_euclid(np__2)),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(C,np__1))),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(C,np__1)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(C,np__2)))) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(C,np__1))) ) ) )
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__2))) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1))
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k5_binarith(k3_finseq_1(B),np__1)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1))) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
& r1_xboole_0(k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k5_binarith(k3_finseq_1(B),np__1)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__2)))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(D,C)
| r1_xreal_0(k3_finseq_1(B),D)
| r1_gobrd10(C,D)
| r1_xboole_0(k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(C,np__1))),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,D),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(D,np__1)))) ) ) ) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(C,np__1)
& ~ r1_xreal_0(k3_finseq_1(B),k1_nat_1(C,np__1))
& ~ r1_xboole_0(k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(C,np__1)))) ) ) ) ) ) ) ).
fof(t22_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(B)) )
=> r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,C),A) ) ) ) ) ).
fof(d4_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( m1_jordan_a(C,A)
=> ( ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(k3_finseq_1(C),B)
| k2_jordan_a(A,B,C) = k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,B),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k2_xcmplx_0(B,np__1))) ) )
& ( ~ ( r1_xreal_0(np__1,B)
& ~ r1_xreal_0(k3_finseq_1(C),B) )
=> k2_jordan_a(A,B,C) = k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1)) ) ) ) ) ) ).
fof(t23_jordan_a,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)
=> ! [C] :
( m1_jordan_a(C,A)
=> ( r2_hidden(B,k4_finseq_1(C))
=> r1_tarski(k2_jordan_a(A,B,C),A) ) ) ) ) ).
fof(t24_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r2_hidden(C,A)
& ! [D] :
( v4_ordinal2(D)
=> ~ ( r2_hidden(D,k4_finseq_1(B))
& r2_hidden(C,k2_jordan_a(A,D,B)) ) ) ) ) ) ) ).
fof(t25_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(D,C)
| r1_xreal_0(k3_finseq_1(B),D)
| r1_gobrd10(C,D)
| r1_subset_1(k2_jordan_a(A,C,B),k2_jordan_a(A,D,B)) ) ) ) ) ) ) ).
fof(t26_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(C,np__1)
& ~ r1_xreal_0(k5_binarith(k3_finseq_1(B),np__1),C)
& ~ r1_subset_1(k2_jordan_a(A,k3_finseq_1(B),B),k2_jordan_a(A,C,B)) ) ) ) ) ).
fof(t27_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_gobrd10(C,D) )
=> ( r1_xreal_0(D,C)
| r1_xreal_0(k3_finseq_1(B),D)
| k5_subset_1(u1_struct_0(k15_euclid(np__2)),k2_jordan_a(A,C,B),k2_jordan_a(A,D,B)) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(C,np__1))) ) ) ) ) ) ) ).
fof(t28_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,C)
& ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(k3_finseq_1(B),D)
& r1_gobrd10(C,D)
& r2_subset_1(k2_jordan_a(A,C,B),k2_jordan_a(A,D,B)) ) ) ) ) ) ).
fof(t29_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> k5_subset_1(u1_struct_0(k15_euclid(np__2)),k2_jordan_a(A,k3_finseq_1(B),B),k2_jordan_a(A,np__1,B)) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)) ) ) ).
fof(t30_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ~ r2_subset_1(k2_jordan_a(A,k3_finseq_1(B),B),k2_jordan_a(A,np__1,B)) ) ) ).
fof(t31_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> k5_subset_1(u1_struct_0(k15_euclid(np__2)),k2_jordan_a(A,k3_finseq_1(B),B),k2_jordan_a(A,k5_binarith(k3_finseq_1(B),np__1),B)) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))) ) ) ).
fof(t32_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ~ r2_subset_1(k2_jordan_a(A,k3_finseq_1(B),B),k2_jordan_a(A,k5_binarith(k3_finseq_1(B),np__1),B)) ) ) ).
fof(d5_jordan_a,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_numbers)
=> ( C = k3_jordan_a(A,B)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k14_euclid(A))))
& D = B
& C = k2_tbsp_1(k14_euclid(A),D) ) ) ) ) ) ).
fof(t33_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(k3_jordan_a(np__2,k2_jordan_a(A,C,B)),k4_jordan_a(A,B)) ) ) ) ).
fof(t34_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(C,k3_jordan_a(np__2,k2_jordan_a(A,D,B))) )
& r1_xreal_0(C,k4_jordan_a(A,B)) ) ) ) ) ).
fof(t35_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ! [C] :
( m1_jordan_a(C,A)
=> r1_xreal_0(B,k4_jordan_a(A,C)) ) ) ) ) ).
fof(t37_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ~ r1_xreal_0(k5_jordan_a(A,B),np__0) ) ) ).
fof(dt_m1_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) ) ) ).
fof(existence_m1_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ? [B] : m1_jordan_a(B,A) ) ).
fof(dt_k1_jordan_a,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k1_jordan_a(A))
& v1_funct_2(k1_jordan_a(A),u1_struct_0(k6_borsuk_1(k15_euclid(A),k15_euclid(A))),k1_numbers)
& m2_relset_1(k1_jordan_a(A),u1_struct_0(k6_borsuk_1(k15_euclid(A),k15_euclid(A))),k1_numbers) ) ) ).
fof(dt_k2_jordan_a,axiom,
! [A,B,C] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& v4_ordinal2(B)
& m1_jordan_a(C,A) )
=> m1_subset_1(k2_jordan_a(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ).
fof(dt_k3_jordan_a,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k3_jordan_a(A,B),k1_numbers) ) ).
fof(dt_k4_jordan_a,axiom,
! [A,B] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_jordan_a(B,A) )
=> m1_subset_1(k4_jordan_a(A,B),k1_numbers) ) ).
fof(dt_k5_jordan_a,axiom,
! [A,B] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_jordan_a(B,A) )
=> m1_subset_1(k5_jordan_a(A,B),k1_numbers) ) ).
fof(t4_jordan_a,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] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(k15_euclid(A),k15_euclid(A))),k1_numbers)
& v9_pscomp_1(D,k6_borsuk_1(k15_euclid(A),k15_euclid(A)))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(k15_euclid(A),k15_euclid(A))),k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k15_euclid(A)),k1_numbers)
& m2_relset_1(E,u1_struct_0(k15_euclid(A)),k1_numbers) )
=> ( ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
=> ? [G] :
( m1_subset_1(G,k1_zfmisc_1(k1_numbers))
& G = a_4_0_jordan_a(A,C,D,F)
& k8_funct_2(u1_struct_0(k15_euclid(A)),k1_numbers,E,F) = k4_pscomp_1(G) ) )
=> k4_pscomp_1(k2_funct_2(u1_struct_0(k6_borsuk_1(k15_euclid(A),k15_euclid(A))),k1_numbers,D,k7_borsuk_1(k15_euclid(A),k15_euclid(A),B,C))) = k4_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(A)),k1_numbers,E,B)) ) ) ) ) ) ) ).
fof(t5_jordan_a,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] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(k15_euclid(A),k15_euclid(A))),k1_numbers)
& v9_pscomp_1(D,k6_borsuk_1(k15_euclid(A),k15_euclid(A)))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(k15_euclid(A),k15_euclid(A))),k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k15_euclid(A)),k1_numbers)
& m2_relset_1(E,u1_struct_0(k15_euclid(A)),k1_numbers) )
=> ( ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
=> ? [G] :
( m1_subset_1(G,k1_zfmisc_1(k1_numbers))
& G = a_4_1_jordan_a(A,B,D,F)
& k8_funct_2(u1_struct_0(k15_euclid(A)),k1_numbers,E,F) = k4_pscomp_1(G) ) )
=> k4_pscomp_1(k2_funct_2(u1_struct_0(k6_borsuk_1(k15_euclid(A),k15_euclid(A))),k1_numbers,D,k7_borsuk_1(k15_euclid(A),k15_euclid(A),B,C))) = k4_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(A)),k1_numbers,E,C)) ) ) ) ) ) ) ).
fof(d6_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k4_jordan_a(A,B)
<=> ? [D] :
( ~ v1_xboole_0(D)
& v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(k1_numbers))
& D = a_2_0_jordan_a(A,B)
& C = k1_pre_circ(D) ) ) ) ) ) ).
fof(d7_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k5_jordan_a(A,B)
<=> ? [D] :
( ~ v1_xboole_0(D)
& v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(k1_numbers))
& ? [E] :
( ~ v1_xboole_0(E)
& v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(k1_numbers))
& D = a_2_1_jordan_a(A,B)
& E = a_2_2_jordan_a(A,B)
& C = k1_square_1(k3_seq_4(D),k3_seq_4(E)) ) ) ) ) ) ) ).
fof(t36_jordan_a,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_jordan_a(B,A)
=> ? [C] :
( ~ v1_xboole_0(C)
& v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers))
& C = a_2_3_jordan_a(A,B)
& k5_jordan_a(A,B) = k3_seq_4(C) ) ) ) ).
fof(fraenkel_a_4_0_jordan_a,axiom,
! [A,B,C,D,E] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& ~ v1_xboole_0(C)
& v6_compts_1(C,k15_euclid(B))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(B))))
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(k15_euclid(B),k15_euclid(B))),k1_numbers)
& v9_pscomp_1(D,k6_borsuk_1(k15_euclid(B),k15_euclid(B)))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(k15_euclid(B),k15_euclid(B))),k1_numbers)
& m1_subset_1(E,u1_struct_0(k15_euclid(B))) )
=> ( r2_hidden(A,a_4_0_jordan_a(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(B)))
& A = k1_binop_1(D,E,F)
& r2_hidden(F,C) ) ) ) ).
fof(fraenkel_a_4_1_jordan_a,axiom,
! [A,B,C,D,E] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& ~ v1_xboole_0(C)
& v6_compts_1(C,k15_euclid(B))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(B))))
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(k15_euclid(B),k15_euclid(B))),k1_numbers)
& v9_pscomp_1(D,k6_borsuk_1(k15_euclid(B),k15_euclid(B)))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(k15_euclid(B),k15_euclid(B))),k1_numbers)
& m1_subset_1(E,u1_struct_0(k15_euclid(B))) )
=> ( r2_hidden(A,a_4_1_jordan_a(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(B)))
& A = k1_binop_1(D,F,E)
& r2_hidden(F,C) ) ) ) ).
fof(fraenkel_a_2_0_jordan_a,axiom,
! [A,B,C] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_jordan_a(C,B) )
=> ( r2_hidden(A,a_2_0_jordan_a(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k3_jordan_a(np__2,k2_jordan_a(B,D,C))
& r2_hidden(D,k4_finseq_1(C)) ) ) ) ).
fof(fraenkel_a_2_1_jordan_a,axiom,
! [A,B,C] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_jordan_a(C,B) )
=> ( r2_hidden(A,a_2_1_jordan_a(B,C))
<=> ? [D,E] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& m2_subset_1(E,k1_numbers,k5_numbers)
& A = k4_jordan1k(np__2,k2_jordan_a(B,D,C),k2_jordan_a(B,E,C))
& r1_xreal_0(np__1,D)
& ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(k3_finseq_1(C),E)
& ~ r1_gobrd10(D,E) ) ) ) ).
fof(fraenkel_a_2_2_jordan_a,axiom,
! [A,B,C] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_jordan_a(C,B) )
=> ( r2_hidden(A,a_2_2_jordan_a(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k4_jordan1k(np__2,k2_jordan_a(B,k3_finseq_1(C),C),k2_jordan_a(B,D,C))
& ~ r1_xreal_0(D,np__1)
& ~ r1_xreal_0(k5_binarith(k3_finseq_1(C),np__1),D) ) ) ) ).
fof(fraenkel_a_2_3_jordan_a,axiom,
! [A,B,C] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_jordan_a(C,B) )
=> ( r2_hidden(A,a_2_3_jordan_a(B,C))
<=> ? [D,E] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& m2_subset_1(E,k1_numbers,k5_numbers)
& A = k4_jordan1k(np__2,k2_jordan_a(B,D,C),k2_jordan_a(B,E,C))
& r1_xreal_0(np__1,D)
& ~ r1_xreal_0(E,D)
& r1_xreal_0(E,k3_finseq_1(C))
& r1_subset_1(k2_jordan_a(B,D,C),k2_jordan_a(B,E,C)) ) ) ) ).
%------------------------------------------------------------------------------