SET007 Axioms: SET007+555.ax
%------------------------------------------------------------------------------
% File : SET007+555 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Dividing Function of the Simple Closed Curve into Segments
% 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 : jordan7 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 26 ( 0 unt; 0 def)
% Number of atoms : 304 ( 54 equ)
% Maximal formula atoms : 42 ( 11 avg)
% Number of connectives : 322 ( 44 ~; 19 |; 150 &)
% ( 2 <=>; 107 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 12 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 23 ( 22 usr; 0 prp; 1-5 aty)
% Number of functors : 30 ( 30 usr; 8 con; 0-4 aty)
% Number of variables : 94 ( 92 !; 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(cc1_jordan7,axiom,
! [A] :
( m1_finseq_1(A,k1_numbers)
=> ( v1_goboard1(A)
=> v2_funct_1(A) ) ) ).
fof(t1_jordan7,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)))) )
=> ( v1_topreal2(A)
=> ( r2_hidden(k30_pscomp_1(A),k9_jordan6(A))
& r2_hidden(k34_pscomp_1(A),k9_jordan6(A))
& r2_hidden(k30_pscomp_1(A),k8_jordan6(A))
& r2_hidden(k34_pscomp_1(A),k8_jordan6(A)) ) ) ) ).
fof(t2_jordan7,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] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal2(A)
& r1_jordan6(A,B,k30_pscomp_1(A)) )
=> B = k30_pscomp_1(A) ) ) ) ).
fof(t3_jordan7,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] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal2(A)
& r2_hidden(B,A) )
=> r1_jordan6(A,k30_pscomp_1(A),B) ) ) ) ).
fof(t4_jordan7,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)))) )
=> ( v1_topreal2(A)
=> ( k1_jordan7(A,k30_pscomp_1(A),k34_pscomp_1(A)) = k8_jordan6(A)
& k1_jordan7(A,k34_pscomp_1(A),k30_pscomp_1(A)) = k9_jordan6(A) ) ) ) ).
fof(t5_jordan7,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] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal2(A)
& r1_jordan6(A,B,C) )
=> ( r2_hidden(B,A)
& r2_hidden(C,A) ) ) ) ) ) ).
fof(t6_jordan7,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] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal2(A)
& r1_jordan6(A,B,C) )
=> ( r2_hidden(B,k1_jordan7(A,B,C))
& r2_hidden(C,k1_jordan7(A,B,C)) ) ) ) ) ) ).
fof(t7_jordan7,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] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( r2_hidden(B,A)
& v1_topreal2(A) )
=> r2_hidden(B,k1_jordan7(A,B,k30_pscomp_1(A))) ) ) ) ).
fof(t8_jordan7,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] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal2(A)
& r2_hidden(B,A) )
=> ( B = k30_pscomp_1(A)
| k1_jordan7(A,B,B) = k1_struct_0(k15_euclid(np__2),B) ) ) ) ) ).
fof(t9_jordan7,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] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v1_topreal2(A)
& B != k30_pscomp_1(A)
& C != k30_pscomp_1(A)
& r2_hidden(k30_pscomp_1(A),k1_jordan7(A,B,C)) ) ) ) ) ).
fof(t10_jordan7,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] :
( 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)))
=> ( ( v1_topreal2(A)
& r1_jordan6(A,B,C)
& r1_jordan6(A,C,D) )
=> ( ( B = C
& B = k30_pscomp_1(A) )
| B = D
| ( C = D
& C = k30_pscomp_1(A) )
| k5_subset_1(u1_struct_0(k15_euclid(np__2)),k1_jordan7(A,B,C),k1_jordan7(A,C,D)) = k1_struct_0(k15_euclid(np__2),C) ) ) ) ) ) ) ).
fof(t11_jordan7,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] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal2(A)
& r1_jordan6(A,B,C) )
=> ( B = k30_pscomp_1(A)
| C = k30_pscomp_1(A)
| k5_subset_1(u1_struct_0(k15_euclid(np__2)),k1_jordan7(A,B,C),k1_jordan7(A,C,k30_pscomp_1(A))) = k1_struct_0(k15_euclid(np__2),C) ) ) ) ) ) ).
fof(t12_jordan7,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] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal2(A)
& r1_jordan6(A,B,C) )
=> ( B = C
| B = k30_pscomp_1(A)
| k5_subset_1(u1_struct_0(k15_euclid(np__2)),k1_jordan7(A,C,k30_pscomp_1(A)),k1_jordan7(A,k30_pscomp_1(A),B)) = k1_struct_0(k15_euclid(np__2),k30_pscomp_1(A)) ) ) ) ) ) ).
fof(t13_jordan7,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] :
( 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)))
=> ( ( v1_topreal2(A)
& r1_jordan6(A,B,C)
& r1_jordan6(A,C,D)
& r1_jordan6(A,D,E) )
=> ( B = C
| C = D
| r1_xboole_0(k1_jordan7(A,B,C),k1_jordan7(A,D,E)) ) ) ) ) ) ) ) ).
fof(t14_jordan7,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] :
( 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)))
=> ( ( v1_topreal2(A)
& r1_jordan6(A,B,C)
& r1_jordan6(A,C,D) )
=> ( B = k30_pscomp_1(A)
| B = C
| C = D
| r1_xboole_0(k1_jordan7(A,B,C),k1_jordan7(A,D,k30_pscomp_1(A))) ) ) ) ) ) ) ).
fof(t15_jordan7,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_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B)))
& m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) )
=> ~ ( B != k1_xboole_0
& v3_tops_2(C,k5_topmetr,k3_pre_topc(k15_euclid(A),B))
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)))
& m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) )
=> ~ ( C = D
& v5_pre_topc(D,k5_topmetr,k15_euclid(A))
& v2_funct_1(D) ) ) ) ) ) ) ).
fof(t16_jordan7,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_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) )
=> ~ ( v5_pre_topc(C,k5_topmetr,k15_euclid(A))
& v2_funct_1(C)
& k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)),C) = B
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B)))
& m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) )
=> ~ ( D = C
& v3_tops_2(D,k5_topmetr,k3_pre_topc(k15_euclid(A),B)) ) ) ) ) ) ) ).
fof(t17_jordan7,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( v1_goboard1(A)
=> v2_funct_1(A) ) ) ).
fof(t18_jordan7,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)))
=> ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)))
& m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2))) )
=> ~ ( v5_pre_topc(D,k5_topmetr,k15_euclid(np__2))
& v2_funct_1(D)
& k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),D) = A
& k1_funct_1(D,np__0) = B
& k1_funct_1(D,np__1) = C ) ) ) ) ) ) ).
fof(t19_jordan7,axiom,
! [A] :
( ( ~ v1_xboole_0(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)))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)))
& m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2))) )
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
& v5_pre_topc(F,k5_topmetr,k15_euclid(np__2))
& v2_funct_1(F)
& k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),F) = A
& k1_funct_1(F,np__0) = B
& k1_funct_1(F,np__1) = C
& k1_funct_1(F,G) = D
& r1_xreal_0(np__0,G)
& r1_xreal_0(G,np__1)
& k1_funct_1(F,H) = E
& r1_xreal_0(np__0,H)
& r1_xreal_0(H,np__1)
& r1_xreal_0(G,H) )
=> r1_jordan5c(A,B,C,D,E) ) ) ) ) ) ) ) ) ) ).
fof(t20_jordan7,axiom,
! [A] :
( ( ~ v1_xboole_0(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)))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)))
& m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2))) )
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ( ( v5_pre_topc(F,k5_topmetr,k15_euclid(np__2))
& v2_funct_1(F)
& k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),F) = A
& k1_funct_1(F,np__0) = B
& k1_funct_1(F,np__1) = C
& k1_funct_1(F,G) = D
& r1_xreal_0(np__0,G)
& r1_xreal_0(G,np__1)
& k1_funct_1(F,H) = E
& r1_xreal_0(np__0,H)
& r1_xreal_0(H,np__1)
& r1_jordan5c(A,B,C,D,E) )
=> r1_xreal_0(G,H) ) ) ) ) ) ) ) ) ) ).
fof(t21_jordan7,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] :
( m1_subset_1(B,k1_numbers)
=> ~ ( v1_topreal2(A)
& ~ r1_xreal_0(B,np__0)
& ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( k1_funct_1(C,np__1) = k30_pscomp_1(A)
& v2_funct_1(C)
& r1_xreal_0(np__8,k3_finseq_1(C))
& r1_tarski(k2_relat_1(C),A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,D)
=> ( r1_xreal_0(k3_finseq_1(C),D)
| r1_jordan6(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,D),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(D,np__1))) ) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k14_euclid(np__2))))
=> ~ ( r1_xreal_0(np__1,D)
& ~ r1_xreal_0(k3_finseq_1(C),D)
& E = k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,D),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(D,np__1)))
& r1_xreal_0(B,k2_tbsp_1(k14_euclid(np__2),E)) ) ) )
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k14_euclid(np__2))))
=> ~ ( D = 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))
& r1_xreal_0(B,k2_tbsp_1(k14_euclid(np__2),D)) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,D)
=> ( r1_xreal_0(k3_finseq_1(C),k1_nat_1(D,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)),C,D),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(D,np__1))),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(D,np__1)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(D,np__2)))) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(D,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)),C,k3_finseq_1(C)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1)),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__2))) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),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)),C,k5_binarith(k3_finseq_1(C),np__1)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(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))) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)))
& r1_xboole_0(k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k5_binarith(k3_finseq_1(C),np__1)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C))),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__2)))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,D)
=> ( r1_xreal_0(E,D)
| r1_xreal_0(k3_finseq_1(C),E)
| r1_gobrd10(D,E)
| r1_xboole_0(k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,D),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(D,np__1))),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,E),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(E,np__1)))) ) ) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(D,np__1)
& ~ r1_xreal_0(k3_finseq_1(C),k1_nat_1(D,np__1))
& ~ r1_xboole_0(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)),k1_jordan7(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,D),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(D,np__1)))) ) ) ) ) ) ) ) ).
fof(dt_k1_jordan7,axiom,
! [A,B,C] :
( ( ~ 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))))
& m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& m1_subset_1(C,u1_struct_0(k15_euclid(np__2))) )
=> m1_subset_1(k1_jordan7(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ).
fof(d1_jordan7,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] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( C != k30_pscomp_1(A)
=> k1_jordan7(A,B,C) = a_3_0_jordan7(A,B,C) )
& ( C = k30_pscomp_1(A)
=> k1_jordan7(A,B,C) = a_2_0_jordan7(A,B) ) ) ) ) ) ).
fof(fraenkel_a_3_0_jordan7,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(np__2))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& m1_subset_1(D,u1_struct_0(k15_euclid(np__2))) )
=> ( r2_hidden(A,a_3_0_jordan7(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
& A = E
& r1_jordan6(B,C,E)
& r1_jordan6(B,E,D) ) ) ) ).
fof(fraenkel_a_2_0_jordan7,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(np__2))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_subset_1(C,u1_struct_0(k15_euclid(np__2))) )
=> ( r2_hidden(A,a_2_0_jordan7(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
& A = D
& ( r1_jordan6(B,C,D)
| ( r2_hidden(C,B)
& D = k30_pscomp_1(B) ) ) ) ) ) ).
%------------------------------------------------------------------------------