SET007 Axioms: SET007+348.ax
%------------------------------------------------------------------------------
% File : SET007+348 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Connectedness Conditions Using Polygonal 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 : topreal4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 1 unt; 0 def)
% Number of atoms : 375 ( 99 equ)
% Maximal formula atoms : 19 ( 10 avg)
% Number of connectives : 370 ( 30 ~; 24 |; 170 &)
% ( 5 <=>; 141 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 14 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 18 ( 16 usr; 1 prp; 0-4 aty)
% Number of functors : 33 ( 33 usr; 4 con; 0-4 aty)
% Number of variables : 130 ( 123 !; 7 ?)
% 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(d1_topreal4,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_topreal4(A,B,C)
<=> ? [D] :
( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
& v4_topreal1(D)
& A = k5_topreal1(np__2,D)
& B = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),D,np__1)
& C = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),D,k3_finseq_1(D)) ) ) ) ) ) ).
fof(d2_topreal4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v1_topreal4(A)
<=> ? [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))))
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& B != C
& r2_hidden(B,A)
& r2_hidden(C,A)
& r1_topreal4(D,B,C)
& r1_topreal4(E,B,C)
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),D,E) = k2_struct_0(k15_euclid(np__2),B,C)
& A = k4_subset_1(u1_struct_0(k15_euclid(np__2)),D,E) ) ) ) ) ) ) ).
fof(d3_topreal4,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_topreal4(B,A)
<=> ( v3_pre_topc(B,A)
& v2_connsp_1(B,A) ) ) ) ) ).
fof(t1_topreal4,axiom,
$true ).
fof(t2_topreal4,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_topreal4(A,B,C)
=> v5_topreal1(A) ) ) ) ) ).
fof(t3_topreal4,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_topreal4(A,B,C)
=> r1_topreal1(k15_euclid(np__2),B,C,A) ) ) ) ) ).
fof(t4_topreal4,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_topreal4(A,B,C)
=> ( r2_hidden(B,A)
& r2_hidden(C,A) ) ) ) ) ) ).
fof(t5_topreal4,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_topreal4(A,B,C)
& B = C ) ) ) ) ).
fof(t6_topreal4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v1_topreal4(A)
=> v1_topreal2(A) ) ) ).
fof(t7_topreal4,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] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( k21_euclid(A) = k21_euclid(B)
& r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(B,k9_metric_1(k14_euclid(np__2),E,D))
& C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,k23_euclid(k21_euclid(A),k6_real_1(k3_real_1(k22_euclid(A),k22_euclid(B)),np__2)),B) )
=> ( k22_euclid(A) = k22_euclid(B)
| ( v4_topreal1(C)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = A
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = B
& r1_topreal4(k5_topreal1(np__2,C),A,B)
& r1_tarski(k5_topreal1(np__2,C),k9_metric_1(k14_euclid(np__2),E,D)) ) ) ) ) ) ) ) ) ).
fof(t8_topreal4,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] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( k22_euclid(A) = k22_euclid(B)
& r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(B,k9_metric_1(k14_euclid(np__2),E,D))
& C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,k23_euclid(k6_real_1(k3_real_1(k21_euclid(A),k21_euclid(B)),np__2),k22_euclid(A)),B) )
=> ( k21_euclid(A) = k21_euclid(B)
| ( v4_topreal1(C)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = A
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = B
& r1_topreal4(k5_topreal1(np__2,C),A,B)
& r1_tarski(k5_topreal1(np__2,C),k9_metric_1(k14_euclid(np__2),E,D)) ) ) ) ) ) ) ) ) ).
fof(t9_topreal4,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] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(B,k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(k23_euclid(k21_euclid(A),k22_euclid(B)),k9_metric_1(k14_euclid(np__2),E,D))
& C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,k23_euclid(k21_euclid(A),k22_euclid(B)),B) )
=> ( k21_euclid(A) = k21_euclid(B)
| k22_euclid(A) = k22_euclid(B)
| ( v4_topreal1(C)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = A
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = B
& r1_topreal4(k5_topreal1(np__2,C),A,B)
& r1_tarski(k5_topreal1(np__2,C),k9_metric_1(k14_euclid(np__2),E,D)) ) ) ) ) ) ) ) ) ).
fof(t10_topreal4,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] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(B,k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(k23_euclid(k21_euclid(B),k22_euclid(A)),k9_metric_1(k14_euclid(np__2),E,D))
& C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,k23_euclid(k21_euclid(B),k22_euclid(A)),B) )
=> ( k21_euclid(A) = k21_euclid(B)
| k22_euclid(A) = k22_euclid(B)
| ( v4_topreal1(C)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = A
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = B
& r1_topreal4(k5_topreal1(np__2,C),A,B)
& r1_tarski(k5_topreal1(np__2,C),k9_metric_1(k14_euclid(np__2),E,D)) ) ) ) ) ) ) ) ) ).
fof(t11_topreal4,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)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k14_euclid(np__2)))
=> ~ ( A != B
& r2_hidden(A,k9_metric_1(k14_euclid(np__2),D,C))
& r2_hidden(B,k9_metric_1(k14_euclid(np__2),D,C))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r1_topreal4(E,A,B)
& r1_tarski(E,k9_metric_1(k14_euclid(np__2),D,C)) ) ) ) ) ) ) ) ).
fof(t12_topreal4,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)) = k22_euclid(A)
& v4_topreal1(B)
& r2_hidden(A,k4_topreal1(np__2,B,np__1))
& C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k23_euclid(k6_real_1(k3_real_1(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)),k21_euclid(A)),np__2),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1))),A) )
=> ( A = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
| ( v4_topreal1(C)
& 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)),B,np__1)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = A
& r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
& r1_tarski(k5_topreal1(np__2,C),k5_topreal1(np__2,B))
& k5_topreal1(np__2,C) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),B,np__1)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)) ) ) ) ) ) ) ).
fof(t13_topreal4,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)) = k21_euclid(A)
& v4_topreal1(B)
& r2_hidden(A,k4_topreal1(np__2,B,np__1))
& C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k23_euclid(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)),k6_real_1(k3_real_1(k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)),k22_euclid(A)),np__2)),A) )
=> ( A = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
| ( v4_topreal1(C)
& 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)),B,np__1)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = A
& r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
& r1_tarski(k5_topreal1(np__2,C),k5_topreal1(np__2,B))
& k5_topreal1(np__2,C) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),B,np__1)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)) ) ) ) ) ) ) ).
fof(t14_topreal4,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( v4_topreal1(B)
& r2_hidden(D,k4_finseq_1(B))
& r2_hidden(k1_nat_1(D,np__1),k4_finseq_1(B))
& r2_hidden(A,k4_topreal1(np__2,B,D))
& C = k8_finseq_1(u1_struct_0(k15_euclid(np__2)),k16_finseq_1(u1_struct_0(k15_euclid(np__2)),B,D),k12_finseq_1(u1_struct_0(k15_euclid(np__2)),A)) )
=> ( A = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
| r1_xreal_0(D,np__1)
| A = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,D)
| A = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(D,np__1))
| ( v4_topreal1(C)
& 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)),B,np__1)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = A
& r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
& r1_tarski(k5_topreal1(np__2,C),k5_topreal1(np__2,B))
& k5_topreal1(np__2,C) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),B,D)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,D),A)) ) ) ) ) ) ) ) ).
fof(t15_topreal4,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v4_topreal1(A)
& k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2)) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1))
& B = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k23_euclid(k6_real_1(k3_real_1(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1)),k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2))),np__2),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1))),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2)) )
=> ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1)
| ( v4_topreal1(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)),A,np__1)
& 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)),A,np__2)
& r1_topreal4(k5_topreal1(np__2,B),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2))
& r1_tarski(k5_topreal1(np__2,B),k5_topreal1(np__2,A))
& k5_topreal1(np__2,B) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,np__1)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2)))
& k5_topreal1(np__2,B) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,np__2)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2))) ) ) ) ) ) ).
fof(t16_topreal4,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v4_topreal1(A)
& k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2)) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1))
& B = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k23_euclid(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1)),k6_real_1(k3_real_1(k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1)),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2))),np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2)) )
=> ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1)
| ( v4_topreal1(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)),A,np__1)
& 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)),A,np__2)
& r1_topreal4(k5_topreal1(np__2,B),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2))
& r1_tarski(k5_topreal1(np__2,B),k5_topreal1(np__2,A))
& k5_topreal1(np__2,B) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,np__1)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2)))
& k5_topreal1(np__2,B) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,np__2)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2))) ) ) ) ) ) ).
fof(t17_topreal4,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( v4_topreal1(A)
& r2_hidden(C,k4_finseq_1(A))
& B = k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,C) )
=> ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,C) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1)
| r1_xreal_0(C,np__2)
| ( v4_topreal1(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)),A,np__1)
& 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)),A,C)
& r1_topreal4(k5_topreal1(np__2,B),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,C))
& r1_tarski(k5_topreal1(np__2,B),k5_topreal1(np__2,A))
& k5_topreal1(np__2,B) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,C)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,C))) ) ) ) ) ) ) ).
fof(t18_topreal4,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( A != k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
& v4_topreal1(B)
& r2_hidden(A,k4_topreal1(np__2,B,C))
& ! [D] :
( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v4_topreal1(D)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),D,np__1) = 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)),D,k3_finseq_1(D)) = A
& r1_topreal4(k5_topreal1(np__2,D),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
& r1_tarski(k5_topreal1(np__2,D),k5_topreal1(np__2,B))
& k5_topreal1(np__2,D) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),B,C)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,C),A)) ) ) ) ) ) ) ).
fof(t19_topreal4,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( A != k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
& v4_topreal1(B)
& r2_hidden(A,k5_topreal1(np__2,B))
& ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v4_topreal1(C)
& 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)),B,np__1)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = A
& r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
& r1_tarski(k5_topreal1(np__2,C),k5_topreal1(np__2,B)) ) ) ) ) ) ).
fof(t20_topreal4,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
& v4_topreal1(B)
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),A),k5_topreal1(np__2,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)))
& C = k8_finseq_1(u1_struct_0(k15_euclid(np__2)),B,k12_finseq_1(u1_struct_0(k15_euclid(np__2)),A)) )
=> ( ( ~ ( k21_euclid(A) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
& k22_euclid(A) != k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))) )
& ~ ( k21_euclid(A) != k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
& k22_euclid(A) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))) ) )
| r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k9_metric_1(k14_euclid(np__2),E,D))
| ( v4_topreal1(C)
& r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
& r1_tarski(k5_topreal1(np__2,C),k2_xboole_0(k5_topreal1(np__2,B),k9_metric_1(k14_euclid(np__2),E,D))) ) ) ) ) ) ) ) ) ).
fof(t21_topreal4,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(k23_euclid(k21_euclid(A),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))),k9_metric_1(k14_euclid(np__2),E,D))
& v4_topreal1(B)
& C = k8_finseq_1(u1_struct_0(k15_euclid(np__2)),B,k2_finseq_4(u1_struct_0(k15_euclid(np__2)),k23_euclid(k21_euclid(A),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))),A))
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k23_euclid(k21_euclid(A),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))))),k3_topreal1(np__2,k23_euclid(k21_euclid(A),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))),A)),k5_topreal1(np__2,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))) )
=> ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k9_metric_1(k14_euclid(np__2),E,D))
| k21_euclid(A) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
| k22_euclid(A) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
| ( r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
& r1_tarski(k5_topreal1(np__2,C),k2_xboole_0(k5_topreal1(np__2,B),k9_metric_1(k14_euclid(np__2),E,D))) ) ) ) ) ) ) ) ) ).
fof(t22_topreal4,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(k23_euclid(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))),k22_euclid(A)),k9_metric_1(k14_euclid(np__2),E,D))
& v4_topreal1(B)
& C = k8_finseq_1(u1_struct_0(k15_euclid(np__2)),B,k2_finseq_4(u1_struct_0(k15_euclid(np__2)),k23_euclid(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))),k22_euclid(A)),A))
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k23_euclid(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))),k22_euclid(A))),k3_topreal1(np__2,k23_euclid(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))),k22_euclid(A)),A)),k5_topreal1(np__2,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))) )
=> ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k9_metric_1(k14_euclid(np__2),E,D))
| k21_euclid(A) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
| k22_euclid(A) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
| ( r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
& r1_tarski(k5_topreal1(np__2,C),k2_xboole_0(k5_topreal1(np__2,B),k9_metric_1(k14_euclid(np__2),E,D))) ) ) ) ) ) ) ) ) ).
fof(t23_topreal4,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k14_euclid(np__2)))
=> ~ ( ~ r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k9_metric_1(k14_euclid(np__2),D,C))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k9_metric_1(k14_euclid(np__2),D,C))
& r2_hidden(A,k9_metric_1(k14_euclid(np__2),D,C))
& v4_topreal1(B)
& ~ r2_hidden(A,k5_topreal1(np__2,B))
& ! [E] :
( m2_finseq_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_topreal4(k5_topreal1(np__2,E),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
& r1_tarski(k5_topreal1(np__2,E),k2_xboole_0(k5_topreal1(np__2,B),k9_metric_1(k14_euclid(np__2),D,C))) ) ) ) ) ) ) ) ).
fof(t24_topreal4,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_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k14_euclid(np__2)))
=> ~ ( B != C
& r1_topreal4(E,C,D)
& r1_tarski(E,A)
& r2_hidden(B,k9_metric_1(k14_euclid(np__2),G,F))
& r2_hidden(D,k9_metric_1(k14_euclid(np__2),G,F))
& r1_tarski(k9_metric_1(k14_euclid(np__2),G,F),A)
& ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r1_topreal4(H,C,B)
& r1_tarski(H,A) ) ) ) ) ) ) ) ) ) ) ).
fof(t30_topreal4,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_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( v2_topreal4(C,k15_euclid(np__2))
& r2_hidden(A,C)
& r2_hidden(B,C)
& A != B
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r1_topreal4(D,A,B)
& r1_tarski(D,C) ) ) ) ) ) ) ).
fof(t25_topreal4,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)))
=> ( ( v2_topreal4(A,k15_euclid(np__2))
& B = a_2_0_topreal4(A,C) )
=> v3_pre_topc(B,k15_euclid(np__2)) ) ) ) ) ).
fof(t26_topreal4,axiom,
! [A] :
( m1_subset_1(A,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))))
=> ( ( v2_topreal4(B,k15_euclid(np__2))
& r2_hidden(A,B)
& C = a_2_1_topreal4(A,B) )
=> v3_pre_topc(C,k15_euclid(np__2)) ) ) ) ) ).
fof(t27_topreal4,axiom,
! [A] :
( m1_subset_1(A,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))))
=> ( ( r2_hidden(A,B)
& C = a_2_1_topreal4(A,B) )
=> r1_tarski(C,B) ) ) ) ) ).
fof(t28_topreal4,axiom,
! [A] :
( m1_subset_1(A,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))))
=> ( ( v2_topreal4(B,k15_euclid(np__2))
& r2_hidden(A,B)
& C = a_2_1_topreal4(A,B) )
=> r1_tarski(B,C) ) ) ) ) ).
fof(t29_topreal4,axiom,
! [A] :
( m1_subset_1(A,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))))
=> ( ( v2_topreal4(B,k15_euclid(np__2))
& r2_hidden(A,B)
& C = a_2_1_topreal4(A,B) )
=> B = C ) ) ) ) ).
fof(fraenkel_a_2_0_topreal4,axiom,
! [A,B,C] :
( ( 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_topreal4(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
& A = D
& D != C
& r2_hidden(D,B)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r1_topreal4(E,C,D)
& r1_tarski(E,B) ) ) ) ) ) ).
fof(fraenkel_a_2_1_topreal4,axiom,
! [A,B,C] :
( ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_2_1_topreal4(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
& A = D
& ~ ( D != B
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r1_topreal4(E,B,D)
& r1_tarski(E,C) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------