SET007 Axioms: SET007+358.ax
%------------------------------------------------------------------------------
% File : SET007+358 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Duality Between Weakly Separated Subspaces of Topological Spaces
% 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 : tsep_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 59 ( 2 unt; 0 def)
% Number of atoms : 589 ( 35 equ)
% Maximal formula atoms : 17 ( 9 avg)
% Number of connectives : 659 ( 129 ~; 4 |; 259 &)
% ( 21 <=>; 246 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-3 aty)
% Number of functors : 15 ( 15 usr; 0 con; 1-3 aty)
% Number of variables : 227 ( 223 !; 4 ?)
% 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_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k6_subset_1(u1_struct_0(A),k3_subset_1(u1_struct_0(A),B),k3_subset_1(u1_struct_0(A),C)) = k6_subset_1(u1_struct_0(A),C,B) ) ) ) ).
fof(d1_tsep_2,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tsep_2(A,B,C)
<=> ( r1_xboole_0(B,C)
& k4_subset_1(u1_struct_0(A),B,C) = u1_struct_0(A) ) ) ) ) ) ).
fof(t2_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tsep_2(A,B,C)
<=> ( r1_xboole_0(B,C)
& k4_subset_1(u1_struct_0(A),B,C) = k2_pre_topc(A) ) ) ) ) ) ).
fof(t3_tsep_2,axiom,
$true ).
fof(t4_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tsep_2(A,B,C)
=> ( B = k3_subset_1(u1_struct_0(A),C)
& C = k3_subset_1(u1_struct_0(A),B) ) ) ) ) ) ).
fof(t5_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( B = k3_subset_1(u1_struct_0(A),C)
| C = k3_subset_1(u1_struct_0(A),B) )
=> r1_tsep_2(A,B,C) ) ) ) ) ).
fof(t6_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tsep_2(A,B,k3_subset_1(u1_struct_0(A),B)) ) ) ).
fof(t7_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> r1_tsep_2(A,k1_pre_topc(A),k2_pre_topc(A)) ) ).
fof(t8_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ r1_tsep_2(A,B,B) ) ) ).
fof(t9_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_tsep_2(A,B,C)
& r2_tsep_2(A,C,D) )
=> B = D ) ) ) ) ) ).
fof(t10_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_tsep_2(A,B,C)
=> ( r2_tsep_2(A,k6_pre_topc(A,B),k1_tops_1(A,C))
& r2_tsep_2(A,k1_tops_1(A,B),k6_pre_topc(A,C)) ) ) ) ) ) ).
fof(t11_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_tsep_2(A,k6_pre_topc(A,B),k1_tops_1(A,k3_subset_1(u1_struct_0(A),B)))
& r2_tsep_2(A,k6_pre_topc(A,k3_subset_1(u1_struct_0(A),B)),k1_tops_1(A,B))
& r2_tsep_2(A,k1_tops_1(A,B),k6_pre_topc(A,k3_subset_1(u1_struct_0(A),B)))
& r2_tsep_2(A,k1_tops_1(A,k3_subset_1(u1_struct_0(A),B)),k6_pre_topc(A,B)) ) ) ) ).
fof(t12_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_tsep_2(A,B,C)
=> ( v3_pre_topc(B,A)
<=> v4_pre_topc(C,A) ) ) ) ) ) ).
fof(t13_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_tsep_2(A,B,C)
=> ( v4_pre_topc(B,A)
<=> v3_pre_topc(C,A) ) ) ) ) ) ).
fof(t14_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_tsep_2(A,B,C)
& r2_tsep_2(A,D,E) )
=> r2_tsep_2(A,k5_subset_1(u1_struct_0(A),B,D),k4_subset_1(u1_struct_0(A),C,E)) ) ) ) ) ) ) ).
fof(t15_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_tsep_2(A,B,C)
& r2_tsep_2(A,D,E) )
=> r2_tsep_2(A,k4_subset_1(u1_struct_0(A),B,D),k5_subset_1(u1_struct_0(A),C,E)) ) ) ) ) ) ) ).
fof(t16_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_tsep_2(A,B,D)
& r2_tsep_2(A,C,E) )
=> ( r2_tsep_1(A,B,C)
<=> r2_tsep_1(A,D,E) ) ) ) ) ) ) ) ).
fof(t17_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_tsep_1(A,B,C)
<=> r2_tsep_1(A,k3_subset_1(u1_struct_0(A),B),k3_subset_1(u1_struct_0(A),C)) ) ) ) ) ).
fof(t18_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_tsep_2(A,B,D)
& r2_tsep_2(A,C,E)
& r1_connsp_1(A,B,C) )
=> r2_tsep_1(A,D,E) ) ) ) ) ) ) ).
fof(t19_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_tsep_2(A,B,D)
& r2_tsep_2(A,C,E)
& r1_xboole_0(B,C)
& r2_tsep_1(A,D,E) )
=> r1_connsp_1(A,B,C) ) ) ) ) ) ) ).
fof(t20_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_tsep_2(A,B,D)
& r2_tsep_2(A,C,E)
& k4_subset_1(u1_struct_0(A),D,E) = u1_struct_0(A)
& r2_tsep_1(A,D,E) )
=> r1_connsp_1(A,B,C) ) ) ) ) ) ) ).
fof(t21_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_tsep_2(A,B,C)
=> ( r2_tsep_1(A,B,C)
<=> r1_connsp_1(A,B,C) ) ) ) ) ) ).
fof(t22_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_tsep_1(A,B,C)
<=> r1_connsp_1(A,k6_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),B,C),B),k6_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),B,C),C)) ) ) ) ) ).
fof(t23_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r1_tarski(D,B)
& r1_tarski(E,C)
& k4_subset_1(u1_struct_0(A),D,E) = k4_subset_1(u1_struct_0(A),B,C)
& r2_tsep_1(A,D,E) )
=> r2_tsep_1(A,B,C) ) ) ) ) ) ) ).
fof(t24_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_tsep_1(A,B,C)
<=> r1_connsp_1(A,k6_subset_1(u1_struct_0(A),B,k5_subset_1(u1_struct_0(A),B,C)),k6_subset_1(u1_struct_0(A),C,k5_subset_1(u1_struct_0(A),B,C))) ) ) ) ) ).
fof(t25_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r1_tarski(D,B)
& r1_tarski(E,C)
& k5_subset_1(u1_struct_0(A),D,E) = k5_subset_1(u1_struct_0(A),B,C)
& r2_tsep_1(A,B,C) )
=> r2_tsep_1(A,D,E) ) ) ) ) ) ) ).
fof(t26_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(D)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(D)))
=> ( ( E = B
& F = C )
=> ( r1_connsp_1(A,B,C)
<=> r1_connsp_1(D,E,F) ) ) ) ) ) ) ) ) ).
fof(t27_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(D)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(D)))
=> ( ( E = k3_xboole_0(u1_struct_0(D),B)
& F = k3_xboole_0(u1_struct_0(D),C)
& r1_connsp_1(A,B,C) )
=> r1_connsp_1(D,E,F) ) ) ) ) ) ) ) ).
fof(t28_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(D)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(D)))
=> ( ( E = B
& F = C )
=> ( r2_tsep_1(A,B,C)
<=> r2_tsep_1(D,E,F) ) ) ) ) ) ) ) ) ).
fof(t29_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(D)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(D)))
=> ( ( E = k3_xboole_0(u1_struct_0(D),B)
& F = k3_xboole_0(u1_struct_0(D),C)
& r2_tsep_1(A,B,C) )
=> r2_tsep_1(D,E,F) ) ) ) ) ) ) ) ).
fof(d2_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_pre_topc(C,A)
=> ( r3_tsep_2(A,B,C)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( D = u1_struct_0(B)
& E = u1_struct_0(C) )
=> r2_tsep_2(A,D,E) ) ) ) ) ) ) ) ).
fof(t30_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ( r3_tsep_2(A,B,C)
<=> ( r1_tsep_1(A,B,C)
& g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = k1_tsep_1(A,B,C) ) ) ) ) ) ).
fof(t31_tsep_2,axiom,
$true ).
fof(t32_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ~ r3_tsep_2(A,B,B) ) ) ).
fof(t33_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ( ( r4_tsep_2(A,B,C)
& r4_tsep_2(A,C,D) )
=> g1_pre_topc(u1_struct_0(B),u1_pre_topc(B)) = g1_pre_topc(u1_struct_0(D),u1_pre_topc(D)) ) ) ) ) ) ).
fof(t34_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& m1_pre_topc(E,A) )
=> ( ( r4_tsep_2(A,B,D)
& r4_tsep_2(A,C,E) )
=> ( k1_tsep_1(A,D,E) = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))
<=> r1_tsep_1(A,B,C) ) ) ) ) ) ) ) ).
fof(t35_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ( r4_tsep_2(A,B,C)
=> ( v1_tsep_1(B,A)
<=> v1_borsuk_1(C,A) ) ) ) ) ) ).
fof(t36_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ( r4_tsep_2(A,B,C)
=> ( v1_borsuk_1(B,A)
<=> v1_tsep_1(C,A) ) ) ) ) ) ).
fof(t37_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& m1_pre_topc(E,A) )
=> ( ( r4_tsep_2(A,B,D)
& r4_tsep_2(A,C,E) )
=> ( r1_tsep_1(A,B,C)
| r4_tsep_2(A,k2_tsep_1(A,B,C),k1_tsep_1(A,D,E)) ) ) ) ) ) ) ) ).
fof(t38_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& m1_pre_topc(E,A) )
=> ( ( r4_tsep_2(A,D,B)
& r4_tsep_2(A,E,C) )
=> ( r1_tsep_1(A,B,C)
| r4_tsep_2(A,k1_tsep_1(A,D,E),k2_tsep_1(A,B,C)) ) ) ) ) ) ) ) ).
fof(t39_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_pre_topc(C,A)
=> ! [D] :
( m1_pre_topc(D,A)
=> ! [E] :
( m1_pre_topc(E,A)
=> ( ( r3_tsep_2(A,B,D)
& r3_tsep_2(A,C,E)
& r4_tsep_1(A,B,C) )
=> r4_tsep_1(A,D,E) ) ) ) ) ) ) ).
fof(t40_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& m1_pre_topc(E,A) )
=> ( ( r4_tsep_2(A,B,D)
& r4_tsep_2(A,C,E)
& r3_tsep_1(A,B,C) )
=> r4_tsep_1(A,D,E) ) ) ) ) ) ) ).
fof(t41_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& m1_pre_topc(E,A) )
=> ( ( r4_tsep_2(A,B,D)
& r4_tsep_2(A,C,E)
& r1_tsep_1(A,B,C)
& r4_tsep_1(A,D,E) )
=> r3_tsep_1(A,B,C) ) ) ) ) ) ) ).
fof(t42_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& m1_pre_topc(E,A) )
=> ( ( r4_tsep_2(A,B,D)
& r4_tsep_2(A,C,E)
& k1_tsep_1(A,D,E) = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))
& r4_tsep_1(A,D,E) )
=> r3_tsep_1(A,B,C) ) ) ) ) ) ) ).
fof(t43_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ( r4_tsep_2(A,B,C)
=> ( r4_tsep_1(A,B,C)
<=> r3_tsep_1(A,B,C) ) ) ) ) ) ).
fof(t44_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& m1_pre_topc(E,A) )
=> ( ( m1_pre_topc(D,B)
& m1_pre_topc(E,C)
& k1_tsep_1(A,D,E) = k1_tsep_1(A,B,C)
& r4_tsep_1(A,D,E) )
=> r4_tsep_1(A,B,C) ) ) ) ) ) ) ).
fof(t45_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& m1_pre_topc(E,A) )
=> ( ( m1_pre_topc(D,B)
& m1_pre_topc(E,C)
& k2_tsep_1(A,D,E) = k2_tsep_1(A,B,C)
& r4_tsep_1(A,B,C) )
=> ( r1_tsep_1(A,D,E)
| r4_tsep_1(A,D,E) ) ) ) ) ) ) ) ).
fof(t46_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( m1_pre_topc(C,A)
=> ! [D] :
( m1_pre_topc(D,A)
=> ! [E] :
( m1_pre_topc(E,B)
=> ! [F] :
( m1_pre_topc(F,B)
=> ( ( u1_struct_0(C) = u1_struct_0(E)
& u1_struct_0(D) = u1_struct_0(F) )
=> ( r3_tsep_1(A,C,D)
<=> r3_tsep_1(B,E,F) ) ) ) ) ) ) ) ) ).
fof(t47_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ~ ( ~ r1_tsep_1(A,C,B)
& ~ r1_tsep_1(A,D,B)
& ? [E] :
( m1_pre_topc(E,B)
& ? [F] :
( m1_pre_topc(F,B)
& E = k2_tsep_1(A,C,B)
& F = k2_tsep_1(A,D,B)
& r3_tsep_1(A,C,D)
& ~ r3_tsep_1(B,E,F) ) ) ) ) ) ) ) ).
fof(t48_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( m1_pre_topc(C,A)
=> ! [D] :
( m1_pre_topc(D,A)
=> ! [E] :
( m1_pre_topc(E,B)
=> ! [F] :
( m1_pre_topc(F,B)
=> ( ( u1_struct_0(C) = u1_struct_0(E)
& u1_struct_0(D) = u1_struct_0(F) )
=> ( r4_tsep_1(A,C,D)
<=> r4_tsep_1(B,E,F) ) ) ) ) ) ) ) ) ).
fof(t49_tsep_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ~ ( ~ r1_tsep_1(A,C,B)
& ~ r1_tsep_1(A,D,B)
& ? [E] :
( m1_pre_topc(E,B)
& ? [F] :
( m1_pre_topc(F,B)
& E = k2_tsep_1(A,C,B)
& F = k2_tsep_1(A,D,B)
& r4_tsep_1(A,C,D)
& ~ r4_tsep_1(B,E,F) ) ) ) ) ) ) ) ).
fof(symmetry_r1_tsep_2,axiom,
! [A,B,C] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_tsep_2(A,B,C)
=> r1_tsep_2(A,C,B) ) ) ).
fof(symmetry_r2_tsep_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r2_tsep_2(A,B,C)
=> r2_tsep_2(A,C,B) ) ) ).
fof(irreflexivity_r2_tsep_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ r2_tsep_2(A,B,B) ) ).
fof(redefinition_r2_tsep_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r2_tsep_2(A,B,C)
<=> r1_tsep_2(A,B,C) ) ) ).
fof(symmetry_r3_tsep_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_pre_topc(B,A)
& m1_pre_topc(C,A) )
=> ( r3_tsep_2(A,B,C)
=> r3_tsep_2(A,C,B) ) ) ).
fof(symmetry_r4_tsep_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& m1_pre_topc(B,A)
& ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ( r4_tsep_2(A,B,C)
=> r4_tsep_2(A,C,B) ) ) ).
fof(irreflexivity_r4_tsep_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& m1_pre_topc(B,A)
& ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ~ r4_tsep_2(A,B,B) ) ).
fof(redefinition_r4_tsep_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& m1_pre_topc(B,A)
& ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ( r4_tsep_2(A,B,C)
<=> r3_tsep_2(A,B,C) ) ) ).
%------------------------------------------------------------------------------