SET007 Axioms: SET007+732.ax
%------------------------------------------------------------------------------
% File : SET007+732 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Decomposition of a Simple Closed Curve into Two Arcs
% 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 : jordan16 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 44 ( 0 unt; 0 def)
% Number of atoms : 357 ( 40 equ)
% Maximal formula atoms : 21 ( 8 avg)
% Number of connectives : 361 ( 48 ~; 4 |; 140 &)
% ( 3 <=>; 166 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 32 ( 31 usr; 0 prp; 1-5 aty)
% Number of functors : 40 ( 40 usr; 7 con; 0-6 aty)
% Number of variables : 154 ( 152 !; 2 ?)
% 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_jordan16,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& v1_realset1(B) ) ) ).
fof(fc1_jordan16,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( ~ v1_xboole_0(k1_jordan16(A,B))
& v1_relat_1(k1_jordan16(A,B))
& v1_funct_1(k1_jordan16(A,B))
& v1_funct_2(k1_jordan16(A,B),k1_numbers,k1_numbers)
& v1_seq_1(k1_jordan16(A,B))
& v1_partfun1(k1_jordan16(A,B),k1_numbers,k1_numbers)
& v1_jordan16(k1_jordan16(A,B)) ) ) ).
fof(rc2_jordan16,axiom,
? [A] :
( m1_relset_1(A,k1_numbers,k1_numbers)
& ~ v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_funct_2(A,k1_numbers,k1_numbers)
& v1_seq_1(A)
& v1_partfun1(A,k1_numbers,k1_numbers)
& v1_jordan16(A) ) ).
fof(t1_jordan16,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r2_hidden(C,A)
& r1_xreal_0(B,C) ) )
& r1_xreal_0(B,k1_pre_circ(A)) ) ) ) ).
fof(t2_jordan16,axiom,
! [A,B,C,D] :
( ( r2_hidden(A,D)
& r2_hidden(B,D)
& r2_hidden(C,D) )
=> r1_tarski(k1_enumset1(A,B,C),D) ) ).
fof(t3_jordan16,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> v1_jordan2c(k1_pre_topc(k15_euclid(A)),A) ) ).
fof(t4_jordan16,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k9_jordan6(A) != k8_jordan6(A) ) ).
fof(t5_jordan16,axiom,
! [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)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> r1_tarski(k5_jordan6(A,B,C,D,E),A) ) ) ) ) ) ).
fof(t6_jordan16,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)))
=> ( r1_tarski(B,C)
=> m1_pre_topc(k3_pre_topc(A,B),k3_pre_topc(A,C)) ) ) ) ) ).
fof(t7_jordan16,axiom,
! [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)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r2_hidden(D,A) )
=> r2_hidden(D,k3_jordan6(A,B,C,D)) ) ) ) ) ) ).
fof(t8_jordan16,axiom,
! [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)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r2_hidden(D,A) )
=> r2_hidden(D,k4_jordan6(A,B,C,D)) ) ) ) ) ) ).
fof(t9_jordan16,axiom,
! [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)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r1_jordan5c(A,B,C,D,E) )
=> ( r2_hidden(D,k5_jordan6(A,B,C,D,E))
& r2_hidden(E,k5_jordan6(A,B,C,D,E)) ) ) ) ) ) ) ) ).
fof(t10_jordan16,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_tarski(k1_jordan7(A,B,C),A) ) ) ) ).
fof(t11_jordan16,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,A)
& r2_hidden(C,A)
& ~ r1_jordan6(A,B,C)
& ~ r1_jordan6(A,C,B) ) ) ) ) ).
fof(t12_jordan16,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] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(C))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(C)) )
=> ( ( D = E
& v5_pre_topc(D,A,B) )
=> v5_pre_topc(E,A,C) ) ) ) ) ) ) ).
fof(t13_jordan16,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] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(E,A,B)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(C),u1_struct_0(D))
& m2_relset_1(F,u1_struct_0(C),u1_struct_0(D)) )
=> ( ( F = k2_tmap_1(A,B,C,E)
& v2_funct_2(F,u1_struct_0(C),u1_struct_0(D)) )
=> v3_tops_2(F,C,D) ) ) ) ) ) ) ) ) ).
fof(t14_jordan16,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_topreal1(k15_euclid(np__2),D,E,A)
& r1_topreal1(k15_euclid(np__2),D,E,B)
& r1_topreal1(k15_euclid(np__2),D,E,C)
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),B,C) = k2_struct_0(k15_euclid(np__2),D,E)
& r1_tarski(A,k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C))
& A != B
& A != C ) ) ) ) ) ) ).
fof(t15_jordan16,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_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_topreal1(k15_euclid(np__2),D,E,B)
& r1_topreal1(k15_euclid(np__2),D,E,C)
& r1_tarski(B,A)
& r1_tarski(C,A) )
=> ( B = C
| ( k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C) = A
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),B,C) = k2_struct_0(k15_euclid(np__2),D,E) ) ) ) ) ) ) ) ) ).
fof(t16_jordan16,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_topreal1(k15_euclid(np__2),C,D,A)
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),A,B) = k2_struct_0(k15_euclid(np__2),E,F)
& A = B ) ) ) ) ) ) ) ).
fof(t17_jordan16,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_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_topreal1(k15_euclid(np__2),D,E,B)
& r1_topreal1(k15_euclid(np__2),D,E,C)
& r1_tarski(B,A)
& r1_tarski(C,A)
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),B,C) = k2_struct_0(k15_euclid(np__2),D,E) )
=> k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C) = A ) ) ) ) ) ) ).
fof(t18_jordan16,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_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_tarski(B,A)
& r1_tarski(C,A)
& r1_topreal1(k15_euclid(np__2),D,E,B)
& r1_topreal1(k15_euclid(np__2),D,E,C) )
=> ( B = C
| ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r1_topreal1(k15_euclid(np__2),D,E,F)
& r1_tarski(F,A)
& F != B
& F != C ) ) ) ) ) ) ) ) ) ).
fof(t19_jordan16,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_topreal1(k15_euclid(np__2),k30_pscomp_1(A),k34_pscomp_1(A),B)
& r1_tarski(B,A)
& B != k9_jordan6(A)
& B != k8_jordan6(A) ) ) ) ).
fof(t20_jordan16,axiom,
! [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)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r1_jordan5c(A,B,C,D,E)
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
& m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ~ ( v3_tops_2(F,k5_topmetr,k3_pre_topc(k15_euclid(np__2),A))
& k1_funct_1(F,np__0) = B
& k1_funct_1(F,np__1) = C
& k1_funct_1(F,G) = D
& k1_funct_1(F,H) = E
& r1_xreal_0(np__0,G)
& r1_xreal_0(G,H)
& r1_xreal_0(H,np__1) ) ) ) ) ) ) ) ) ) ) ).
fof(t21_jordan16,axiom,
! [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)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r1_jordan5c(A,B,C,D,E)
& D != E
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
& m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ~ ( v3_tops_2(F,k5_topmetr,k3_pre_topc(k15_euclid(np__2),A))
& k1_funct_1(F,np__0) = B
& k1_funct_1(F,np__1) = C
& k1_funct_1(F,G) = D
& k1_funct_1(F,H) = E
& r1_xreal_0(np__0,G)
& ~ r1_xreal_0(H,G)
& r1_xreal_0(H,np__1) ) ) ) ) ) ) ) ) ) ) ).
fof(t22_jordan16,axiom,
! [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)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r1_jordan5c(A,B,C,D,E)
& v1_xboole_0(k5_jordan6(A,B,C,D,E)) ) ) ) ) ) ) ).
fof(t23_jordan16,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)
=> ( r2_hidden(B,k1_jordan7(A,B,k30_pscomp_1(A)))
& r2_hidden(k30_pscomp_1(A),k1_jordan7(A,B,k30_pscomp_1(A))) ) ) ) ) ).
fof(d1_jordan16,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v1_jordan16(A)
<=> r2_fcont_1(A,k1_relat_1(A)) ) ) ).
fof(d2_jordan16,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k1_numbers,k1_numbers)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v1_jordan16(A)
<=> r2_fcont_1(A,k1_numbers) ) ) ).
fof(d3_jordan16,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_numbers,k1_numbers)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( C = k1_jordan16(A,B)
<=> ! [D] :
( v1_xreal_0(D)
=> k1_funct_1(C,D) = k2_xcmplx_0(k3_xcmplx_0(A,D),B) ) ) ) ) ) ).
fof(t24_jordan16,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_jordan16(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_jordan16(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( v1_funct_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,A,B))
& v1_jordan16(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,A,B))
& m2_relset_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,A,B),k1_numbers,k1_numbers) ) ) ) ).
fof(t25_jordan16,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k8_funct_2(k1_numbers,k1_numbers,k1_jordan16(A,B),np__0) = B ) ) ).
fof(t26_jordan16,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k8_funct_2(k1_numbers,k1_numbers,k1_jordan16(A,B),np__1) = k2_xcmplx_0(A,B) ) ) ).
fof(t27_jordan16,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( A != np__0
=> v2_funct_1(k1_jordan16(A,B)) ) ) ) ).
fof(t28_jordan16,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(D,C)
& r1_xreal_0(k1_funct_1(k1_jordan16(A,B),D),k1_funct_1(k1_jordan16(A,B),C)) ) ) ) ) ) ).
fof(t29_jordan16,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(np__0,A)
& ~ r1_xreal_0(D,C)
& r1_xreal_0(k1_funct_1(k1_jordan16(A,B),C),k1_funct_1(k1_jordan16(A,B),D)) ) ) ) ) ) ).
fof(t30_jordan16,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(C,D) )
=> r1_xreal_0(k1_funct_1(k1_jordan16(A,B),C),k1_funct_1(k1_jordan16(A,B),D)) ) ) ) ) ) ).
fof(t31_jordan16,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(A,np__0)
& r1_xreal_0(C,D) )
=> r1_xreal_0(k1_funct_1(k1_jordan16(A,B),D),k1_funct_1(k1_jordan16(A,B),C)) ) ) ) ) ) ).
fof(t32_jordan16,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( A != np__0
=> k1_pscomp_1(k1_numbers,k1_numbers,k1_jordan16(A,B)) = k1_numbers ) ) ) ).
fof(t33_jordan16,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( A != np__0
=> k2_funct_1(k1_jordan16(A,B)) = k1_jordan16(k5_xcmplx_0(A),k4_xcmplx_0(k7_xcmplx_0(B,A))) ) ) ) ).
fof(t34_jordan16,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(A,np__0)
=> k2_funct_2(k1_numbers,k1_numbers,k1_jordan16(A,B),k1_rcomp_1(np__0,np__1)) = k1_rcomp_1(B,k2_xcmplx_0(A,B)) ) ) ) ).
fof(t35_jordan16,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k3_topmetr),u1_struct_0(k3_topmetr))
& m2_relset_1(A,u1_struct_0(k3_topmetr),u1_struct_0(k3_topmetr)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( A = k1_jordan16(B,C)
=> ( B = np__0
| v3_tops_2(A,k3_topmetr,k3_topmetr) ) ) ) ) ) ).
fof(t36_jordan16,axiom,
! [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)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r1_jordan5c(A,B,C,D,E) )
=> ( D = E
| r1_topreal1(k15_euclid(np__2),D,E,k5_jordan6(A,B,C,D,E)) ) ) ) ) ) ) ) ).
fof(t37_jordan16,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)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r1_tarski(D,A)
& r1_topreal1(k15_euclid(np__2),B,C,D)
& r2_hidden(k30_pscomp_1(A),D)
& r2_hidden(k34_pscomp_1(A),D)
& ~ r1_tarski(k8_jordan6(A),D)
& ~ r1_tarski(k9_jordan6(A),D) ) ) ) ) ) ).
fof(dt_k1_jordan16,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_funct_1(k1_jordan16(A,B))
& v1_funct_2(k1_jordan16(A,B),k1_numbers,k1_numbers)
& m2_relset_1(k1_jordan16(A,B),k1_numbers,k1_numbers) ) ) ).
%------------------------------------------------------------------------------