SET007 Axioms: SET007+320.ax
%------------------------------------------------------------------------------
% File : SET007+320 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Topological Space E^2_T
% 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 : topreal1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 69 ( 16 unt; 0 def)
% Number of atoms : 335 ( 66 equ)
% Maximal formula atoms : 15 ( 4 avg)
% Number of connectives : 289 ( 23 ~; 3 |; 122 &)
% ( 17 <=>; 124 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 30 ( 28 usr; 1 prp; 0-4 aty)
% Number of functors : 44 ( 44 usr; 17 con; 0-4 aty)
% Number of variables : 147 ( 131 !; 16 ?)
% 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(fc1_topreal1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> ~ v1_xboole_0(k1_topreal1(A,B,C)) ) ).
fof(fc2_topreal1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k15_euclid(A))
& v1_pre_topc(k15_euclid(A))
& v2_pre_topc(k15_euclid(A))
& v3_compts_1(k15_euclid(A)) ) ) ).
fof(fc3_topreal1,axiom,
~ v1_xboole_0(k2_topreal1) ).
fof(cc1_topreal1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v5_topreal1(A)
=> ~ v1_xboole_0(A) ) ) ).
fof(d1_topreal1,axiom,
$true ).
fof(d2_topreal1,axiom,
! [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,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_topreal1(A,B,C,D)
<=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(A,D)))
& m2_relset_1(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(A,D)))
& v3_tops_2(E,k5_topmetr,k3_pre_topc(A,D))
& k1_funct_1(E,np__0) = B
& k1_funct_1(E,np__1) = C ) ) ) ) ) ) ).
fof(t1_topreal1,axiom,
$true ).
fof(t2_topreal1,axiom,
$true ).
fof(t3_topreal1,axiom,
$true ).
fof(t4_topreal1,axiom,
! [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,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_topreal1(A,C,D,B)
=> ( r2_hidden(C,B)
& r2_hidden(D,B) ) ) ) ) ) ) ).
fof(t5_topreal1,axiom,
! [A] :
( ( v2_pre_topc(A)
& v3_compts_1(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))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( r1_topreal1(A,D,E,B)
& r1_topreal1(A,E,F,C)
& k5_subset_1(u1_struct_0(A),B,C) = k1_tarski(E) )
=> r1_topreal1(A,D,F,k4_subset_1(u1_struct_0(A),B,C)) ) ) ) ) ) ) ) ).
fof(d4_topreal1,axiom,
k2_topreal1 = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__0,np__1)),k1_topreal1(np__2,k23_euclid(np__0,np__1),k23_euclid(np__1,np__1))),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal1(np__2,k23_euclid(np__1,np__1),k23_euclid(np__1,np__0)),k1_topreal1(np__2,k23_euclid(np__1,np__0),k23_euclid(np__0,np__0)))) ).
fof(t6_topreal1,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)))
=> ( r2_hidden(B,k1_topreal1(A,B,C))
& r2_hidden(C,k1_topreal1(A,B,C)) ) ) ) ) ).
fof(t7_topreal1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> k1_topreal1(A,B,B) = k1_struct_0(k15_euclid(A),B) ) ) ).
fof(t8_topreal1,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)))
=> k1_topreal1(A,B,C) = k1_topreal1(A,C,B) ) ) ) ).
fof(t9_topreal1,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)))
=> ( ( r1_xreal_0(k21_euclid(A),k21_euclid(B))
& r2_hidden(C,k3_topreal1(np__2,A,B)) )
=> ( r1_xreal_0(k21_euclid(A),k21_euclid(C))
& r1_xreal_0(k21_euclid(C),k21_euclid(B)) ) ) ) ) ) ).
fof(t10_topreal1,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)))
=> ( ( r1_xreal_0(k22_euclid(A),k22_euclid(B))
& r2_hidden(C,k3_topreal1(np__2,A,B)) )
=> ( r1_xreal_0(k22_euclid(A),k22_euclid(C))
& r1_xreal_0(k22_euclid(C),k22_euclid(B)) ) ) ) ) ) ).
fof(t11_topreal1,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] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ( r2_hidden(B,k3_topreal1(A,C,D))
=> k3_topreal1(A,C,D) = k4_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,C,B),k3_topreal1(A,B,D)) ) ) ) ) ) ).
fof(t12_topreal1,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] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ( ( r2_hidden(D,k3_topreal1(A,B,C))
& r2_hidden(E,k3_topreal1(A,B,C)) )
=> r1_tarski(k3_topreal1(A,D,E),k3_topreal1(A,B,C)) ) ) ) ) ) ) ).
fof(t13_topreal1,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] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ( ( r2_hidden(B,k3_topreal1(A,D,E))
& r2_hidden(C,k3_topreal1(A,D,E)) )
=> k3_topreal1(A,D,E) = k4_subset_1(u1_struct_0(k15_euclid(A)),k4_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,D,B),k3_topreal1(A,B,C)),k3_topreal1(A,C,E)) ) ) ) ) ) ) ).
fof(t14_topreal1,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] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ( r2_hidden(B,k3_topreal1(A,C,D))
=> k5_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,C,B),k3_topreal1(A,B,D)) = k1_struct_0(k15_euclid(A),B) ) ) ) ) ) ).
fof(t15_topreal1,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)))
=> ( B != C
=> r1_topreal1(k15_euclid(A),B,C,k3_topreal1(A,B,C)) ) ) ) ) ).
fof(t16_topreal1,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,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ( ( r1_topreal1(k15_euclid(A),C,D,B)
& k5_subset_1(u1_struct_0(k15_euclid(A)),B,k3_topreal1(A,D,E)) = k1_struct_0(k15_euclid(A),D) )
=> r1_topreal1(k15_euclid(A),C,E,k4_subset_1(u1_struct_0(k15_euclid(A)),B,k3_topreal1(A,D,E))) ) ) ) ) ) ) ).
fof(t17_topreal1,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,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ( ( r1_topreal1(k15_euclid(A),D,C,B)
& k5_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,E,D),B) = k1_struct_0(k15_euclid(A),D) )
=> r1_topreal1(k15_euclid(A),E,C,k4_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,E,D),B)) ) ) ) ) ) ) ).
fof(t18_topreal1,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] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ( k5_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,B,C),k3_topreal1(A,C,D)) = k1_struct_0(k15_euclid(A),C)
=> ( ( B = C
& C = D )
| r1_topreal1(k15_euclid(A),B,D,k4_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,B,C),k3_topreal1(A,C,D))) ) ) ) ) ) ) ).
fof(t21_topreal1,axiom,
k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__0,np__1)),k3_topreal1(np__2,k23_euclid(np__0,np__1),k23_euclid(np__1,np__1))) = k1_struct_0(k15_euclid(np__2),k23_euclid(np__0,np__1)) ).
fof(t22_topreal1,axiom,
k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__1,np__0)),k3_topreal1(np__2,k23_euclid(np__1,np__0),k23_euclid(np__1,np__1))) = k1_struct_0(k15_euclid(np__2),k23_euclid(np__1,np__0)) ).
fof(t23_topreal1,axiom,
k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__0,np__1)),k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__1,np__0))) = k1_struct_0(k15_euclid(np__2),k23_euclid(np__0,np__0)) ).
fof(t24_topreal1,axiom,
k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k23_euclid(np__0,np__1),k23_euclid(np__1,np__1)),k3_topreal1(np__2,k23_euclid(np__1,np__0),k23_euclid(np__1,np__1))) = k1_struct_0(k15_euclid(np__2),k23_euclid(np__1,np__1)) ).
fof(t25_topreal1,axiom,
r1_subset_1(k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__1,np__0)),k3_topreal1(np__2,k23_euclid(np__0,np__1),k23_euclid(np__1,np__1))) ).
fof(t26_topreal1,axiom,
r1_subset_1(k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__0,np__1)),k3_topreal1(np__2,k23_euclid(np__1,np__0),k23_euclid(np__1,np__1))) ).
fof(d5_topreal1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(k1_nat_1(C,np__1),k3_finseq_1(B)) )
=> k4_topreal1(A,B,C) = k3_topreal1(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),B,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),B,k1_nat_1(C,np__1))) )
& ( ~ ( r1_xreal_0(np__1,C)
& r1_xreal_0(k1_nat_1(C,np__1),k3_finseq_1(B)) )
=> k4_topreal1(A,B,C) = k1_xboole_0 ) ) ) ) ) ).
fof(t27_topreal1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(A)))
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(C)) )
=> ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),C,B),k4_topreal1(A,C,B))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),C,k1_nat_1(B,np__1)),k4_topreal1(A,C,B)) ) ) ) ) ) ).
fof(t28_topreal1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
=> ( ( k3_finseq_1(B) = np__0
| k3_finseq_1(B) = np__1 )
<=> k5_topreal1(A,B) = k1_xboole_0 ) ) ) ).
fof(t29_topreal1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
=> ~ ( r1_xreal_0(np__2,k3_finseq_1(B))
& k5_topreal1(A,B) = k1_xboole_0 ) ) ) ).
fof(d7_topreal1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( v1_topreal1(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A))
& k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,B)) != k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(B,np__1)))
& k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,B)) != k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(B,np__1))) ) ) ) ) ).
fof(d8_topreal1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( v2_topreal1(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__2),k3_finseq_1(A)) )
=> k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,A,B),k4_topreal1(np__2,A,k1_nat_1(B,np__1))) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(B,np__1))) ) ) ) ) ).
fof(d9_topreal1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( v3_topreal1(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,k1_nat_1(B,np__1))
=> r1_xboole_0(k4_topreal1(np__2,A,B),k4_topreal1(np__2,A,C)) ) ) ) ) ) ).
fof(d10_topreal1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( v4_topreal1(A)
<=> ( v2_funct_1(A)
& r1_xreal_0(np__2,k3_finseq_1(A))
& v2_topreal1(A)
& v3_topreal1(A)
& v1_topreal1(A) ) ) ) ).
fof(t30_topreal1,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)
& v4_topreal1(B)
& k2_topreal1 = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k5_topreal1(np__2,B))
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k5_topreal1(np__2,B)) = k2_struct_0(k15_euclid(np__2),k23_euclid(np__0,np__0),k23_euclid(np__1,np__1))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1) = k23_euclid(np__0,np__0)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)) = k23_euclid(np__1,np__1)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1) = k23_euclid(np__0,np__0)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)) = k23_euclid(np__1,np__1) ) ) ).
fof(t31_topreal1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( v4_topreal1(A)
=> r1_topreal1(k15_euclid(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,k3_finseq_1(A)),k5_topreal1(np__2,A)) ) ) ).
fof(d11_topreal1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v5_topreal1(A)
<=> ? [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
& v4_topreal1(B)
& A = k5_topreal1(np__2,B) ) ) ) ).
fof(t32_topreal1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( v5_topreal1(A)
& A = k1_xboole_0 ) ) ).
fof(t33_topreal1,axiom,
$true ).
fof(t34_topreal1,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))))
& v5_topreal1(A)
& v5_topreal1(B)
& k2_topreal1 = k4_subset_1(u1_struct_0(k15_euclid(np__2)),A,B)
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),A,B) = k2_struct_0(k15_euclid(np__2),k23_euclid(np__0,np__0),k23_euclid(np__1,np__1)) ) ) ).
fof(t35_topreal1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( v5_topreal1(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)))
=> ~ r1_topreal1(k15_euclid(np__2),B,C,A) ) ) ) ) ).
fof(t36_topreal1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( v5_topreal1(A)
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
& m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
=> ~ v3_tops_2(B,k5_topmetr,k3_pre_topc(k15_euclid(np__2),A)) ) ) ) ).
fof(dt_k1_topreal1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k1_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k2_topreal1,axiom,
m1_subset_1(k2_topreal1,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ).
fof(dt_k3_topreal1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k3_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(commutativity_k3_topreal1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> k3_topreal1(A,B,C) = k3_topreal1(A,C,B) ) ).
fof(redefinition_k3_topreal1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> k3_topreal1(A,B,C) = k1_topreal1(A,B,C) ) ).
fof(dt_k4_topreal1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_finseq_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k4_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k5_topreal1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_finseq_1(B,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k5_topreal1(A,B),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(d3_topreal1,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)))
=> k1_topreal1(A,B,C) = a_3_0_topreal1(A,B,C) ) ) ) ).
fof(t19_topreal1,axiom,
( k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__0,np__1)) = a_0_0_topreal1
& k3_topreal1(np__2,k23_euclid(np__0,np__1),k23_euclid(np__1,np__1)) = a_0_1_topreal1
& k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__1,np__0)) = a_0_2_topreal1
& k3_topreal1(np__2,k23_euclid(np__1,np__0),k23_euclid(np__1,np__1)) = a_0_3_topreal1 ) ).
fof(t20_topreal1,axiom,
k2_topreal1 = a_0_4_topreal1 ).
fof(d6_topreal1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
=> k5_topreal1(A,B) = k3_tarski(a_2_0_topreal1(A,B)) ) ) ).
fof(s1_topreal1,axiom,
a_0_5_topreal1 = k2_xboole_0(a_0_6_topreal1,a_0_7_topreal1) ).
fof(fraenkel_a_3_0_topreal1,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_subset_1(C,u1_struct_0(k15_euclid(B)))
& m1_subset_1(D,u1_struct_0(k15_euclid(B))) )
=> ( r2_hidden(A,a_3_0_topreal1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_numbers)
& A = k17_euclid(B,k18_euclid(k5_real_1(np__1,E),B,C),k18_euclid(E,B,D))
& r1_xreal_0(np__0,E)
& r1_xreal_0(E,np__1) ) ) ) ).
fof(fraenkel_a_0_0_topreal1,axiom,
! [A] :
( r2_hidden(A,a_0_0_topreal1)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& A = B
& k21_euclid(B) = np__0
& r1_xreal_0(k22_euclid(B),np__1)
& r1_xreal_0(np__0,k22_euclid(B)) ) ) ).
fof(fraenkel_a_0_1_topreal1,axiom,
! [A] :
( r2_hidden(A,a_0_1_topreal1)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& A = B
& r1_xreal_0(k21_euclid(B),np__1)
& r1_xreal_0(np__0,k21_euclid(B))
& k22_euclid(B) = np__1 ) ) ).
fof(fraenkel_a_0_2_topreal1,axiom,
! [A] :
( r2_hidden(A,a_0_2_topreal1)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& A = B
& r1_xreal_0(k21_euclid(B),np__1)
& r1_xreal_0(np__0,k21_euclid(B))
& k22_euclid(B) = np__0 ) ) ).
fof(fraenkel_a_0_3_topreal1,axiom,
! [A] :
( r2_hidden(A,a_0_3_topreal1)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& A = B
& k21_euclid(B) = np__1
& r1_xreal_0(k22_euclid(B),np__1)
& r1_xreal_0(np__0,k22_euclid(B)) ) ) ).
fof(fraenkel_a_0_4_topreal1,axiom,
! [A] :
( r2_hidden(A,a_0_4_topreal1)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& A = B
& ~ ( ~ ( k21_euclid(B) = np__0
& r1_xreal_0(k22_euclid(B),np__1)
& r1_xreal_0(np__0,k22_euclid(B)) )
& ~ ( r1_xreal_0(k21_euclid(B),np__1)
& r1_xreal_0(np__0,k21_euclid(B))
& k22_euclid(B) = np__1 )
& ~ ( r1_xreal_0(k21_euclid(B),np__1)
& r1_xreal_0(np__0,k21_euclid(B))
& k22_euclid(B) = np__0 )
& ~ ( k21_euclid(B) = np__1
& r1_xreal_0(k22_euclid(B),np__1)
& r1_xreal_0(np__0,k22_euclid(B)) ) ) ) ) ).
fof(fraenkel_a_2_0_topreal1,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_finseq_1(C,u1_struct_0(k15_euclid(B))) )
=> ( r2_hidden(A,a_2_0_topreal1(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k4_topreal1(B,C,D)
& r1_xreal_0(np__1,D)
& r1_xreal_0(k1_nat_1(D,np__1),k3_finseq_1(C)) ) ) ) ).
fof(fraenkel_a_0_5_topreal1,axiom,
! [A] :
( r2_hidden(A,a_0_5_topreal1)
<=> ? [B] :
( m1_subset_1(B,f1_s1_topreal1)
& A = B
& ( p1_s1_topreal1(B)
| p2_s1_topreal1(B) ) ) ) ).
fof(fraenkel_a_0_6_topreal1,axiom,
! [A] :
( r2_hidden(A,a_0_6_topreal1)
<=> ? [B] :
( m1_subset_1(B,f1_s1_topreal1)
& A = B
& p1_s1_topreal1(B) ) ) ).
fof(fraenkel_a_0_7_topreal1,axiom,
! [A] :
( r2_hidden(A,a_0_7_topreal1)
<=> ? [B] :
( m1_subset_1(B,f1_s1_topreal1)
& A = B
& p2_s1_topreal1(B) ) ) ).
%------------------------------------------------------------------------------