SET007 Axioms: SET007+811.ax
%------------------------------------------------------------------------------
% File : SET007+811 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Behaviour of an Arc Crossing a Line
% 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 : jordan20 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 42 ( 1 unt; 0 def)
% Number of atoms : 521 ( 58 equ)
% Maximal formula atoms : 26 ( 12 avg)
% Number of connectives : 552 ( 73 ~; 18 |; 219 &)
% ( 8 <=>; 234 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 16 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 30 ( 28 usr; 1 prp; 0-5 aty)
% Number of functors : 31 ( 31 usr; 7 con; 0-6 aty)
% Number of variables : 221 ( 207 !; 14 ?)
% 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(t1_jordan20,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) )
=> k5_jordan6(A,B,C,D,D) = k1_struct_0(k15_euclid(np__2),D) ) ) ) ) ) ).
fof(t2_jordan20,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)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r2_hidden(C,k3_topreal1(np__2,A,B))
& r1_xreal_0(k21_euclid(A),D)
& r1_xreal_0(k21_euclid(B),D) )
=> r1_xreal_0(k21_euclid(C),D) ) ) ) ) ) ).
fof(t3_jordan20,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)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r2_hidden(C,k3_topreal1(np__2,A,B))
& r1_xreal_0(D,k21_euclid(A))
& r1_xreal_0(D,k21_euclid(B)) )
=> r1_xreal_0(D,k21_euclid(C)) ) ) ) ) ) ).
fof(t4_jordan20,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)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r2_hidden(C,k3_topreal1(np__2,A,B))
& ~ r1_xreal_0(D,k21_euclid(A))
& ~ r1_xreal_0(D,k21_euclid(B))
& r1_xreal_0(D,k21_euclid(C)) ) ) ) ) ) ).
fof(t5_jordan20,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)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r2_hidden(C,k3_topreal1(np__2,A,B))
& ~ r1_xreal_0(k21_euclid(A),D)
& ~ r1_xreal_0(k21_euclid(B),D)
& r1_xreal_0(k21_euclid(C),D) ) ) ) ) ) ).
fof(t6_jordan20,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v4_topreal1(B)
& m2_finseq_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_xreal_0(np__1,A)
& r2_hidden(C,k4_topreal1(np__2,B,A))
& r2_hidden(D,k4_topreal1(np__2,B,A))
& k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)))
& r1_jordan5c(k5_topreal1(np__2,B),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,k3_finseq_1(B)),C,D) )
=> ( r1_xreal_0(k3_finseq_1(B),A)
| r1_xreal_0(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)),k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))))
| r1_xreal_0(k21_euclid(D),k21_euclid(C)) ) ) ) ) ) ) ).
fof(t7_jordan20,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v4_topreal1(B)
& m2_finseq_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_xreal_0(np__1,A)
& r2_hidden(C,k4_topreal1(np__2,B,A))
& r2_hidden(D,k4_topreal1(np__2,B,A))
& k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)))
& r1_jordan5c(k5_topreal1(np__2,B),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,k3_finseq_1(B)),C,D) )
=> ( r1_xreal_0(k3_finseq_1(B),A)
| r1_xreal_0(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))),k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)))
| r1_xreal_0(k21_euclid(C),k21_euclid(D)) ) ) ) ) ) ) ).
fof(d1_jordan20,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,k1_numbers)
=> ( r1_jordan20(A,B,C,D,E)
<=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r2_hidden(D,A)
& k21_euclid(D) = E
& ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
& ~ r1_xreal_0(E,k21_euclid(F))
& r1_jordan5c(A,B,C,F,D)
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_jordan5c(A,B,C,F,G)
& r1_jordan5c(A,B,C,G,D) )
=> r1_xreal_0(k21_euclid(G),E) ) ) ) ) ) ) ) ) ) ) ).
fof(d2_jordan20,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,k1_numbers)
=> ( r2_jordan20(A,B,C,D,E)
<=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r2_hidden(D,A)
& k21_euclid(D) = E
& ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
& ~ r1_xreal_0(k21_euclid(F),E)
& r1_jordan5c(A,B,C,F,D)
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_jordan5c(A,B,C,F,G)
& r1_jordan5c(A,B,C,G,D) )
=> r1_xreal_0(E,k21_euclid(G)) ) ) ) ) ) ) ) ) ) ) ).
fof(d3_jordan20,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,k1_numbers)
=> ( r3_jordan20(A,B,C,D,E)
<=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r2_hidden(D,A)
& k21_euclid(D) = E
& ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
& ~ r1_xreal_0(E,k21_euclid(F))
& r1_jordan5c(A,B,C,D,F)
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_jordan5c(A,B,C,G,F)
& r1_jordan5c(A,B,C,D,G) )
=> r1_xreal_0(k21_euclid(G),E) ) ) ) ) ) ) ) ) ) ) ).
fof(d4_jordan20,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,k1_numbers)
=> ( r4_jordan20(A,B,C,D,E)
<=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r2_hidden(D,A)
& k21_euclid(D) = E
& ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
& ~ r1_xreal_0(k21_euclid(F),E)
& r1_jordan5c(A,B,C,D,F)
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_jordan5c(A,B,C,G,F)
& r1_jordan5c(A,B,C,D,G) )
=> r1_xreal_0(E,k21_euclid(G)) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_jordan20,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,k1_numbers)
=> ( r5_jordan20(A,B,C,D,E)
<=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r2_hidden(D,A)
& k21_euclid(D) = E
& ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
& r1_jordan5c(A,B,C,F,D)
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_jordan5c(A,B,C,F,G)
& r1_jordan5c(A,B,C,G,D) )
=> k21_euclid(G) = E ) )
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( r1_jordan5c(A,B,C,G,F)
=> ( G = F
| ( ? [H] :
( m1_subset_1(H,u1_struct_0(k15_euclid(np__2)))
& r1_jordan5c(A,B,C,G,H)
& r1_jordan5c(A,B,C,H,F)
& ~ r1_xreal_0(k21_euclid(H),E) )
& ? [H] :
( m1_subset_1(H,u1_struct_0(k15_euclid(np__2)))
& r1_jordan5c(A,B,C,G,H)
& r1_jordan5c(A,B,C,H,F)
& ~ r1_xreal_0(E,k21_euclid(H)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d6_jordan20,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,k1_numbers)
=> ( r6_jordan20(A,B,C,D,E)
<=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r2_hidden(D,A)
& k21_euclid(D) = E
& ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
& r1_jordan5c(A,B,C,D,F)
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_jordan5c(A,B,C,G,F)
& r1_jordan5c(A,B,C,D,G) )
=> k21_euclid(G) = E ) )
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( r1_jordan5c(A,B,C,F,G)
=> ( G = F
| ( ? [H] :
( m1_subset_1(H,u1_struct_0(k15_euclid(np__2)))
& r1_jordan5c(A,B,C,H,G)
& r1_jordan5c(A,B,C,F,H)
& ~ r1_xreal_0(k21_euclid(H),E) )
& ? [H] :
( m1_subset_1(H,u1_struct_0(k15_euclid(np__2)))
& r1_jordan5c(A,B,C,H,G)
& r1_jordan5c(A,B,C,F,H)
& ~ r1_xreal_0(E,k21_euclid(H)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t8_jordan20,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,k1_numbers)
=> ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r1_xreal_0(k21_euclid(B),E)
& r1_xreal_0(E,k21_euclid(C))
& ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r2_hidden(F,A)
& k21_euclid(F) = E ) ) ) ) ) ) ) ) ).
fof(t9_jordan20,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,k1_numbers)
=> ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
& ~ r1_xreal_0(E,k21_euclid(B))
& ~ r1_xreal_0(k21_euclid(C),E)
& r2_hidden(D,A)
& k21_euclid(D) = E
& ~ r1_jordan20(A,B,C,D,E)
& ~ r2_jordan20(A,B,C,D,E)
& ~ r5_jordan20(A,B,C,D,E) ) ) ) ) ) ) ).
fof(t10_jordan20,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,k1_numbers)
=> ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
& ~ r1_xreal_0(E,k21_euclid(B))
& ~ r1_xreal_0(k21_euclid(C),E)
& r2_hidden(D,A)
& k21_euclid(D) = E
& ~ r3_jordan20(A,B,C,D,E)
& ~ r4_jordan20(A,B,C,D,E)
& ~ r6_jordan20(A,B,C,D,E) ) ) ) ) ) ) ).
fof(t11_jordan20,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( A = k1_rcomp_2(np__0,B)
=> v3_pre_topc(A,k5_topmetr) ) ) ) ).
fof(t12_jordan20,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( A = k2_rcomp_2(B,np__1)
=> v3_pre_topc(A,k5_topmetr) ) ) ) ).
fof(t17_jordan20,axiom,
! [A] :
( ( ~ v3_struct_0(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,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( k5_subset_1(u1_struct_0(A),B,C) = k1_xboole_0
& k4_subset_1(u1_struct_0(A),B,C) = u1_struct_0(A)
& r2_hidden(D,B)
& r2_hidden(E,C)
& v3_pre_topc(B,A)
& v3_pre_topc(C,A) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(A))
& m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(A)) )
=> ~ ( v5_pre_topc(F,k5_topmetr,A)
& k1_funct_1(F,np__0) = D
& k1_funct_1(F,np__1) = E ) ) ) ) ) ) ) ) ).
fof(t18_jordan20,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,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))))
=> ! [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),C,D,A)
& r2_hidden(E,A)
& B = k6_subset_1(u1_struct_0(k15_euclid(np__2)),A,k1_struct_0(k15_euclid(np__2),E)) )
=> ( E = C
| E = D
| ( ~ v2_connsp_1(B,k3_pre_topc(k15_euclid(np__2),A))
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(np__2),A),B)))
& m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(np__2),A),B))) )
=> ~ ( v5_pre_topc(F,k5_topmetr,k3_pre_topc(k3_pre_topc(k15_euclid(np__2),A),B))
& k1_funct_1(F,np__0) = C
& k1_funct_1(F,np__1) = D ) ) ) ) ) ) ) ) ) ) ).
fof(t19_jordan20,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)))
=> ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r2_hidden(D,A)
& r2_hidden(E,A)
& ~ r1_jordan5c(A,B,C,D,E)
& ~ r1_jordan5c(A,B,C,E,D) ) ) ) ) ) ) ).
fof(t20_jordan20,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)))
=> ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r2_hidden(D,A) )
=> ( B = D
| r1_topreal1(k15_euclid(np__2),B,D,k5_jordan6(A,B,C,B,D)) ) ) ) ) ) ) ).
fof(t21_jordan20,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)))
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ( ( r1_topreal1(k15_euclid(A),B,C,D)
& r1_topreal1(k15_euclid(A),B,C,E)
& r1_tarski(E,D) )
=> E = D ) ) ) ) ) ) ).
fof(t22_jordan20,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)))
=> ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r2_hidden(D,A) )
=> ( C = D
| r1_topreal1(k15_euclid(np__2),D,C,k5_jordan6(A,B,C,D,C)) ) ) ) ) ) ) ).
fof(t23_jordan20,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] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r1_jordan5c(A,B,C,D,E)
& r1_jordan5c(A,B,C,E,F) )
=> k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_jordan6(A,B,C,D,E),k5_jordan6(A,B,C,E,F)) = k5_jordan6(A,B,C,D,F) ) ) ) ) ) ) ) ).
fof(t24_jordan20,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] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
& r1_jordan5c(A,B,C,D,E)
& r1_jordan5c(A,B,C,E,F) )
=> k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_jordan6(A,B,C,D,E),k5_jordan6(A,B,C,E,F)) = k1_struct_0(k15_euclid(np__2),E) ) ) ) ) ) ) ) ).
fof(t25_jordan20,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)))
=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
=> k5_jordan6(A,B,C,B,C) = A ) ) ) ) ).
fof(t26_jordan20,axiom,
$true ).
fof(t27_jordan20,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_borsuk_2(E,A,B,C)
=> ! [F] :
( m1_borsuk_2(F,A,C,D)
=> ( ( v5_pre_topc(E,k5_topmetr,A)
& v5_pre_topc(F,k5_topmetr,A)
& k1_funct_1(E,np__0) = B
& k1_funct_1(E,np__1) = C
& k1_funct_1(F,np__0) = C
& k1_funct_1(F,np__1) = D )
=> ( v5_pre_topc(k1_borsuk_2(A,B,C,D,E,F),k5_topmetr,A)
& k1_funct_1(k1_borsuk_2(A,B,C,D,E,F),np__0) = B
& k1_funct_1(k1_borsuk_2(A,B,C,D,E,F),np__1) = D ) ) ) ) ) ) ) ) ).
fof(t28_jordan20,axiom,
! [A] :
( ( ~ v1_xboole_0(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)))) )
=> ! [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)
& r1_topreal1(k15_euclid(np__2),E,F,B)
& r1_jordan5c(A,C,D,E,F)
& r1_tarski(B,A) )
=> B = k5_jordan6(A,C,D,E,F) ) ) ) ) ) ) ) ).
fof(t29_jordan20,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] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ( ( r1_jordan20(A,B,C,D,G)
& k21_euclid(E) = G
& r1_tarski(k3_topreal1(np__2,D,E),A)
& r2_hidden(F,k3_topreal1(np__2,D,E)) )
=> ( r1_xreal_0(G,k21_euclid(B))
| r1_xreal_0(k21_euclid(C),G)
| r1_jordan20(A,B,C,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t30_jordan20,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] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ( ( r2_jordan20(A,B,C,D,G)
& k21_euclid(E) = G
& r1_tarski(k3_topreal1(np__2,D,E),A)
& r2_hidden(F,k3_topreal1(np__2,D,E)) )
=> ( r1_xreal_0(G,k21_euclid(B))
| r1_xreal_0(k21_euclid(C),G)
| r2_jordan20(A,B,C,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t31_jordan20,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] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ( ( r3_jordan20(A,B,C,D,G)
& k21_euclid(E) = G
& r1_tarski(k3_topreal1(np__2,D,E),A)
& r2_hidden(F,k3_topreal1(np__2,D,E)) )
=> ( r1_xreal_0(G,k21_euclid(B))
| r1_xreal_0(k21_euclid(C),G)
| r3_jordan20(A,B,C,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t32_jordan20,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] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ( ( r4_jordan20(A,B,C,D,G)
& k21_euclid(E) = G
& r1_tarski(k3_topreal1(np__2,D,E),A)
& r2_hidden(F,k3_topreal1(np__2,D,E)) )
=> ( r1_xreal_0(G,k21_euclid(B))
| r1_xreal_0(k21_euclid(C),G)
| r4_jordan20(A,B,C,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t33_jordan20,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,k1_numbers)
=> ~ ( r1_topreal4(A,B,C)
& ~ r1_xreal_0(E,k21_euclid(B))
& ~ r1_xreal_0(k21_euclid(C),E)
& r2_hidden(D,A)
& k21_euclid(D) = E
& ~ r1_jordan20(A,B,C,D,E)
& ~ r2_jordan20(A,B,C,D,E) ) ) ) ) ) ) ).
fof(t34_jordan20,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,k1_numbers)
=> ~ ( r1_topreal4(A,B,C)
& ~ r1_xreal_0(E,k21_euclid(B))
& ~ r1_xreal_0(k21_euclid(C),E)
& r2_hidden(D,A)
& k21_euclid(D) = E
& ~ r3_jordan20(A,B,C,D,E)
& ~ r4_jordan20(A,B,C,D,E) ) ) ) ) ) ) ).
fof(t13_jordan20,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,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topmetr)))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
& m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r1_xreal_0(E,np__1)
& B = a_3_0_jordan20(A,D,E)
& C = k1_rcomp_2(np__0,E) )
=> k4_pre_topc(k5_topmetr,k3_pre_topc(k15_euclid(np__2),A),D,C) = B ) ) ) ) ) ) ).
fof(t14_jordan20,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,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topmetr)))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
& m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r1_xreal_0(np__0,E)
& B = a_3_1_jordan20(A,D,E)
& C = k2_rcomp_2(E,np__1) )
=> k4_pre_topc(k5_topmetr,k3_pre_topc(k15_euclid(np__2),A),D,C) = B ) ) ) ) ) ) ).
fof(t15_jordan20,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,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
& m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r1_xreal_0(D,np__1)
& v3_tops_2(C,k5_topmetr,k3_pre_topc(k15_euclid(np__2),A))
& B = a_3_0_jordan20(A,C,D) )
=> v3_pre_topc(B,k3_pre_topc(k15_euclid(np__2),A)) ) ) ) ) ) ).
fof(t16_jordan20,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,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
& m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r1_xreal_0(np__0,D)
& v3_tops_2(C,k5_topmetr,k3_pre_topc(k15_euclid(np__2),A))
& B = a_3_1_jordan20(A,C,D) )
=> v3_pre_topc(B,k3_pre_topc(k15_euclid(np__2),A)) ) ) ) ) ) ).
fof(fraenkel_a_3_0_jordan20,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),B)))
& m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),B)))
& m1_subset_1(D,k1_numbers) )
=> ( r2_hidden(A,a_3_0_jordan20(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
& A = E
& ? [F] :
( m1_subset_1(F,k1_numbers)
& r1_xreal_0(np__0,F)
& ~ r1_xreal_0(D,F)
& E = k1_funct_1(C,F) ) ) ) ) ).
fof(fraenkel_a_3_1_jordan20,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),B)))
& m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),B)))
& m1_subset_1(D,k1_numbers) )
=> ( r2_hidden(A,a_3_1_jordan20(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
& A = E
& ? [F] :
( m1_subset_1(F,k1_numbers)
& ~ r1_xreal_0(F,D)
& r1_xreal_0(F,np__1)
& E = k1_funct_1(C,F) ) ) ) ) ).
%------------------------------------------------------------------------------