SET007 Axioms: SET007+327.ax
%------------------------------------------------------------------------------
% File : SET007+327 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Separated and 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 117 ( 9 unt; 0 def)
% Number of atoms : 1210 ( 61 equ)
% Maximal formula atoms : 26 ( 10 avg)
% Number of connectives : 1387 ( 294 ~; 10 |; 614 &)
% ( 43 <=>; 426 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 20 ( 18 usr; 1 prp; 0-3 aty)
% Number of functors : 14 ( 14 usr; 0 con; 1-3 aty)
% Number of variables : 407 ( 372 !; 35 ?)
% 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(rc1_tsep_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_pre_topc(B,A)
& v1_pre_topc(B)
& v2_pre_topc(B)
& v1_tsep_1(B,A) ) ) ).
fof(rc2_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_pre_topc(B,A)
& ~ v3_struct_0(B)
& v1_pre_topc(B)
& v2_pre_topc(B)
& v1_tsep_1(B,A) ) ) ).
fof(t1_tsep_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> m1_subset_1(u1_struct_0(B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(t2_tsep_1,axiom,
! [A] :
( l1_pre_topc(A)
=> m1_pre_topc(A,A) ) ).
fof(t3_tsep_1,axiom,
! [A] :
( ( v1_pre_topc(A)
& l1_pre_topc(A) )
=> k3_pre_topc(A,k2_pre_topc(A)) = A ) ).
fof(t4_tsep_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_pre_topc(C,A)
=> ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
<=> m1_pre_topc(B,C) ) ) ) ) ).
fof(t5_tsep_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_pre_topc(C,A)
=> ( u1_struct_0(B) = u1_struct_0(C)
=> g1_pre_topc(u1_struct_0(B),u1_pre_topc(B)) = g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) ) ) ) ) ).
fof(t6_tsep_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_pre_topc(C,A)
=> ( ( m1_pre_topc(B,C)
& m1_pre_topc(C,B) )
=> g1_pre_topc(u1_struct_0(B),u1_pre_topc(B)) = g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) ) ) ) ) ).
fof(t7_tsep_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_pre_topc(C,B)
=> m1_pre_topc(C,A) ) ) ) ).
fof(t8_tsep_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,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(B)))
=> ( ( v4_pre_topc(C,A)
& r1_tarski(C,u1_struct_0(B))
& r1_tarski(D,C)
& D = E )
=> ( v4_pre_topc(E,B)
<=> v4_pre_topc(D,A) ) ) ) ) ) ) ) ).
fof(t9_tsep_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,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(B)))
=> ( ( v3_pre_topc(C,A)
& r1_tarski(C,u1_struct_0(B))
& r1_tarski(D,C)
& D = E )
=> ( v3_pre_topc(E,B)
<=> v3_pre_topc(D,A) ) ) ) ) ) ) ) ).
fof(t10_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] :
( ~ v3_struct_0(C)
& v1_pre_topc(C)
& m1_pre_topc(C,A)
& B = u1_struct_0(C) ) ) ) ).
fof(t11_tsep_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = u1_struct_0(B)
=> ( ( v1_borsuk_1(B,A)
& m1_pre_topc(B,A) )
<=> v4_pre_topc(C,A) ) ) ) ) ) ).
fof(t12_tsep_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_borsuk_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( C = D
=> ( v4_pre_topc(D,B)
<=> v4_pre_topc(C,A) ) ) ) ) ) ) ).
fof(t13_tsep_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_borsuk_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( v1_borsuk_1(C,B)
& m1_pre_topc(C,B) )
=> ( v1_borsuk_1(C,A)
& m1_pre_topc(C,A) ) ) ) ) ).
fof(t14_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
=> ( ~ v3_struct_0(B)
& v1_borsuk_1(B,C)
& m1_pre_topc(B,C) ) ) ) ) ) ).
fof(t15_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v4_pre_topc(B,A)
& ! [C] :
( ( ~ v3_struct_0(C)
& v1_pre_topc(C)
& v1_borsuk_1(C,A)
& m1_pre_topc(C,A) )
=> B != u1_struct_0(C) ) ) ) ) ).
fof(d1_tsep_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> ( v1_tsep_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = u1_struct_0(B)
=> v3_pre_topc(C,A) ) ) ) ) ) ).
fof(t16_tsep_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = u1_struct_0(B)
=> ( ( v1_tsep_1(B,A)
& m1_pre_topc(B,A) )
<=> v3_pre_topc(C,A) ) ) ) ) ) ).
fof(t17_tsep_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_tsep_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( C = D
=> ( v3_pre_topc(D,B)
<=> v3_pre_topc(C,A) ) ) ) ) ) ) ).
fof(t18_tsep_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_tsep_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( v1_tsep_1(C,B)
& m1_pre_topc(C,B) )
=> ( v1_tsep_1(C,A)
& m1_pre_topc(C,A) ) ) ) ) ).
fof(t19_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_tsep_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
=> ( v1_tsep_1(B,C)
& m1_pre_topc(B,C) ) ) ) ) ) ).
fof(t20_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v3_pre_topc(B,A)
& ! [C] :
( ( ~ v3_struct_0(C)
& v1_pre_topc(C)
& v1_tsep_1(C,A)
& m1_pre_topc(C,A) )
=> B != u1_struct_0(C) ) ) ) ) ).
fof(d2_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(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)
& v1_pre_topc(D)
& m1_pre_topc(D,A) )
=> ( D = k1_tsep_1(A,B,C)
<=> u1_struct_0(D) = k2_xboole_0(u1_struct_0(B),u1_struct_0(C)) ) ) ) ) ) ).
fof(t21_tsep_1,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) )
=> k1_tsep_1(A,k1_tsep_1(A,B,C),D) = k1_tsep_1(A,B,k1_tsep_1(A,C,D)) ) ) ) ) ).
fof(t22_tsep_1,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) )
=> m1_pre_topc(B,k1_tsep_1(A,B,C)) ) ) ) ).
fof(t23_tsep_1,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) )
=> ( m1_pre_topc(B,C)
<=> k1_tsep_1(A,B,C) = g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) ) ) ) ) ).
fof(t24_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_borsuk_1(C,A)
& m1_pre_topc(C,A) )
=> ( v1_borsuk_1(k1_tsep_1(A,B,C),A)
& m1_pre_topc(k1_tsep_1(A,B,C),A) ) ) ) ) ).
fof(t25_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_tsep_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_tsep_1(C,A)
& m1_pre_topc(C,A) )
=> ( v1_tsep_1(k1_tsep_1(A,B,C),A)
& m1_pre_topc(k1_tsep_1(A,B,C),A) ) ) ) ) ).
fof(d3_tsep_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_pre_topc(C,A)
=> ( r1_tsep_1(A,B,C)
<=> r1_xboole_0(u1_struct_0(B),u1_struct_0(C)) ) ) ) ) ).
fof(d4_tsep_1,axiom,
$true ).
fof(d5_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(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) )
=> ( ~ r1_tsep_1(A,B,C)
=> ! [D] :
( ( ~ v3_struct_0(D)
& v1_pre_topc(D)
& m1_pre_topc(D,A) )
=> ( D = k2_tsep_1(A,B,C)
<=> u1_struct_0(D) = k3_xboole_0(u1_struct_0(B),u1_struct_0(C)) ) ) ) ) ) ) ).
fof(t26_tsep_1,axiom,
$true ).
fof(t27_tsep_1,axiom,
$true ).
fof(t28_tsep_1,axiom,
$true ).
fof(t29_tsep_1,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,B,C)
=> k2_tsep_1(A,B,C) = k2_tsep_1(A,C,B) )
& ( ( ( ~ r1_tsep_1(A,B,C)
& ~ r1_tsep_1(A,k2_tsep_1(A,B,C),D) )
| ( ~ r1_tsep_1(A,C,D)
& ~ r1_tsep_1(A,B,k2_tsep_1(A,C,D)) ) )
=> k2_tsep_1(A,k2_tsep_1(A,B,C),D) = k2_tsep_1(A,B,k2_tsep_1(A,C,D)) ) ) ) ) ) ) ).
fof(t30_tsep_1,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) )
=> ( ~ r1_tsep_1(A,B,C)
=> ( m1_pre_topc(k2_tsep_1(A,B,C),B)
& m1_pre_topc(k2_tsep_1(A,B,C),C) ) ) ) ) ) ).
fof(t31_tsep_1,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) )
=> ( ~ r1_tsep_1(A,B,C)
=> ( ( m1_pre_topc(B,C)
=> k2_tsep_1(A,B,C) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B)) )
& ( k2_tsep_1(A,B,C) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B))
=> m1_pre_topc(B,C) )
& ( m1_pre_topc(C,B)
=> k2_tsep_1(A,B,C) = g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) )
& ( k2_tsep_1(A,B,C) = g1_pre_topc(u1_struct_0(C),u1_pre_topc(C))
=> m1_pre_topc(C,B) ) ) ) ) ) ) ).
fof(t32_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_borsuk_1(C,A)
& m1_pre_topc(C,A) )
=> ( ~ r1_tsep_1(A,B,C)
=> ( v1_borsuk_1(k2_tsep_1(A,B,C),A)
& m1_pre_topc(k2_tsep_1(A,B,C),A) ) ) ) ) ) ).
fof(t33_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_tsep_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_tsep_1(C,A)
& m1_pre_topc(C,A) )
=> ( ~ r1_tsep_1(A,B,C)
=> ( v1_tsep_1(k2_tsep_1(A,B,C),A)
& m1_pre_topc(k2_tsep_1(A,B,C),A) ) ) ) ) ) ).
fof(t34_tsep_1,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) )
=> ( ~ r1_tsep_1(A,B,C)
=> m1_pre_topc(k2_tsep_1(A,B,C),k1_tsep_1(A,B,C)) ) ) ) ) ).
fof(t35_tsep_1,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,B,D)
& ~ r1_tsep_1(A,D,C)
& ~ ( k2_tsep_1(A,k1_tsep_1(A,B,C),D) = k1_tsep_1(A,k2_tsep_1(A,B,D),k2_tsep_1(A,C,D))
& k2_tsep_1(A,D,k1_tsep_1(A,B,C)) = k1_tsep_1(A,k2_tsep_1(A,D,B),k2_tsep_1(A,D,C)) ) ) ) ) ) ) ).
fof(t36_tsep_1,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,B,C)
=> ( k1_tsep_1(A,k2_tsep_1(A,B,C),D) = k2_tsep_1(A,k1_tsep_1(A,B,D),k1_tsep_1(A,C,D))
& k1_tsep_1(A,D,k2_tsep_1(A,B,C)) = k2_tsep_1(A,k1_tsep_1(A,D,B),k1_tsep_1(A,D,C)) ) ) ) ) ) ) ).
fof(t37_tsep_1,axiom,
$true ).
fof(t38_tsep_1,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)))
=> ( ( v4_pre_topc(B,A)
& v4_pre_topc(C,A) )
=> ( r1_xboole_0(B,C)
<=> r1_connsp_1(A,B,C) ) ) ) ) ) ).
fof(t39_tsep_1,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)))
=> ( ( v4_pre_topc(k4_subset_1(u1_struct_0(A),B,C),A)
& r1_connsp_1(A,B,C) )
=> ( v4_pre_topc(B,A)
& v4_pre_topc(C,A) ) ) ) ) ) ).
fof(t40_tsep_1,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_xboole_0(B,C)
& v3_pre_topc(B,A) )
=> r1_xboole_0(B,k6_pre_topc(A,C)) ) ) ) ) ).
fof(t41_tsep_1,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)))
=> ( ( v3_pre_topc(B,A)
& v3_pre_topc(C,A) )
=> ( r1_xboole_0(B,C)
<=> r1_connsp_1(A,B,C) ) ) ) ) ) ).
fof(t42_tsep_1,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)))
=> ( ( v3_pre_topc(k4_subset_1(u1_struct_0(A),B,C),A)
& r1_connsp_1(A,B,C) )
=> ( v3_pre_topc(B,A)
& v3_pre_topc(C,A) ) ) ) ) ) ).
fof(t43_tsep_1,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)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_connsp_1(A,B,C)
=> r1_connsp_1(A,k5_subset_1(u1_struct_0(A),B,D),k5_subset_1(u1_struct_0(A),C,D)) ) ) ) ) ) ).
fof(t44_tsep_1,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)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r1_connsp_1(A,B,D)
| r1_connsp_1(A,C,D) )
=> r1_connsp_1(A,k5_subset_1(u1_struct_0(A),B,C),D) ) ) ) ) ) ).
fof(t45_tsep_1,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)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r1_connsp_1(A,B,D)
& r1_connsp_1(A,C,D) )
<=> r1_connsp_1(A,k4_subset_1(u1_struct_0(A),B,C),D) ) ) ) ) ) ).
fof(t46_tsep_1,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_connsp_1(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)))
& r1_tarski(B,D)
& r1_tarski(C,E)
& r1_xboole_0(D,C)
& r1_xboole_0(E,B)
& v4_pre_topc(D,A)
& v4_pre_topc(E,A) ) ) ) ) ) ) ).
fof(t47_tsep_1,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_connsp_1(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)))
& r1_tarski(B,D)
& r1_tarski(C,E)
& r1_xboole_0(k5_subset_1(u1_struct_0(A),D,E),k4_subset_1(u1_struct_0(A),B,C))
& v4_pre_topc(D,A)
& v4_pre_topc(E,A) ) ) ) ) ) ) ).
fof(t48_tsep_1,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_connsp_1(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)))
& r1_tarski(B,D)
& r1_tarski(C,E)
& r1_xboole_0(D,C)
& r1_xboole_0(E,B)
& v3_pre_topc(D,A)
& v3_pre_topc(E,A) ) ) ) ) ) ) ).
fof(t49_tsep_1,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_connsp_1(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)))
& r1_tarski(B,D)
& r1_tarski(C,E)
& r1_xboole_0(k5_subset_1(u1_struct_0(A),D,E),k4_subset_1(u1_struct_0(A),B,C))
& v3_pre_topc(D,A)
& v3_pre_topc(E,A) ) ) ) ) ) ) ).
fof(d6_tsep_1,axiom,
$true ).
fof(d7_tsep_1,axiom,
! [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,C),k6_subset_1(u1_struct_0(A),C,B)) ) ) ) ) ).
fof(t50_tsep_1,axiom,
$true ).
fof(t51_tsep_1,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_xboole_0(B,C)
& r2_tsep_1(A,B,C) )
<=> r1_connsp_1(A,B,C) ) ) ) ) ).
fof(t52_tsep_1,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_tarski(B,C)
=> r2_tsep_1(A,B,C) ) ) ) ) ).
fof(t53_tsep_1,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)))
=> ( ( v4_pre_topc(B,A)
& v4_pre_topc(C,A) )
=> r2_tsep_1(A,B,C) ) ) ) ) ).
fof(t54_tsep_1,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)))
=> ( ( v3_pre_topc(B,A)
& v3_pre_topc(C,A) )
=> r2_tsep_1(A,B,C) ) ) ) ) ).
fof(t55_tsep_1,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)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_tsep_1(A,B,C)
=> r2_tsep_1(A,k4_subset_1(u1_struct_0(A),B,D),k4_subset_1(u1_struct_0(A),C,D)) ) ) ) ) ) ).
fof(t56_tsep_1,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)))
=> ! [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)
& r2_tsep_1(A,C,B) )
=> r2_tsep_1(A,k4_subset_1(u1_struct_0(A),C,D),k4_subset_1(u1_struct_0(A),B,E)) ) ) ) ) ) ) ).
fof(t57_tsep_1,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)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_tsep_1(A,B,D)
& r2_tsep_1(A,C,D) )
=> r2_tsep_1(A,k5_subset_1(u1_struct_0(A),B,C),D) ) ) ) ) ) ).
fof(t58_tsep_1,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)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_tsep_1(A,B,D)
& r2_tsep_1(A,C,D) )
=> r2_tsep_1(A,k4_subset_1(u1_struct_0(A),B,C),D) ) ) ) ) ) ).
fof(t59_tsep_1,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)))
=> ( r2_tsep_1(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)))
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
& r1_tarski(k5_subset_1(u1_struct_0(A),D,k4_subset_1(u1_struct_0(A),B,C)),B)
& r1_tarski(k5_subset_1(u1_struct_0(A),E,k4_subset_1(u1_struct_0(A),B,C)),C)
& r1_tarski(k5_subset_1(u1_struct_0(A),F,k4_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),B,C))
& u1_struct_0(A) = k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),D,E),F)
& v4_pre_topc(D,A)
& v4_pre_topc(E,A)
& v3_pre_topc(F,A) ) ) ) ) ) ) ) ).
fof(t60_tsep_1,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_tarski(B,C)
& ~ r1_tarski(C,B)
& ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v4_pre_topc(D,A)
& v4_pre_topc(E,A)
& r1_tarski(k5_subset_1(u1_struct_0(A),D,k4_subset_1(u1_struct_0(A),B,C)),B)
& r1_tarski(k5_subset_1(u1_struct_0(A),E,k4_subset_1(u1_struct_0(A),B,C)),C)
& ~ ( ~ r1_tarski(k4_subset_1(u1_struct_0(A),B,C),k4_subset_1(u1_struct_0(A),D,E))
& ! [F] :
( ( ~ v1_xboole_0(F)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v3_pre_topc(F,A)
& r1_tarski(k5_subset_1(u1_struct_0(A),F,k4_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),B,C))
& u1_struct_0(A) = k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),D,E),F) ) ) ) ) ) ) ) ) ) ) ).
fof(t61_tsep_1,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)))
=> ( k4_subset_1(u1_struct_0(A),B,C) = u1_struct_0(A)
=> ( r2_tsep_1(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)))
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
& k4_subset_1(u1_struct_0(A),B,C) = k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),D,E),F)
& r1_tarski(D,B)
& r1_tarski(E,C)
& r1_tarski(F,k5_subset_1(u1_struct_0(A),B,C))
& v4_pre_topc(D,A)
& v4_pre_topc(E,A)
& v3_pre_topc(F,A) ) ) ) ) ) ) ) ) ).
fof(t62_tsep_1,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)))
=> ~ ( k4_subset_1(u1_struct_0(A),B,C) = u1_struct_0(A)
& r2_tsep_1(A,B,C)
& ~ r1_tarski(B,C)
& ~ r1_tarski(C,B)
& ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v4_pre_topc(D,A)
& v4_pre_topc(E,A)
& r1_tarski(D,B)
& r1_tarski(E,C)
& ~ ( k4_subset_1(u1_struct_0(A),B,C) != k4_subset_1(u1_struct_0(A),D,E)
& ! [F] :
( ( ~ v1_xboole_0(F)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( k4_subset_1(u1_struct_0(A),B,C) = k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),D,E),F)
& r1_tarski(F,k5_subset_1(u1_struct_0(A),B,C))
& v3_pre_topc(F,A) ) ) ) ) ) ) ) ) ) ) ).
fof(t63_tsep_1,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)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
& r1_tarski(k5_subset_1(u1_struct_0(A),D,k4_subset_1(u1_struct_0(A),B,C)),B)
& r1_tarski(k5_subset_1(u1_struct_0(A),E,k4_subset_1(u1_struct_0(A),B,C)),C)
& r1_tarski(k5_subset_1(u1_struct_0(A),F,k4_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),B,C))
& u1_struct_0(A) = k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),D,E),F)
& v3_pre_topc(D,A)
& v3_pre_topc(E,A)
& v4_pre_topc(F,A) ) ) ) ) ) ) ) ).
fof(t64_tsep_1,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_tarski(B,C)
& ~ r1_tarski(C,B)
& ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v3_pre_topc(D,A)
& v3_pre_topc(E,A)
& r1_tarski(k5_subset_1(u1_struct_0(A),D,k4_subset_1(u1_struct_0(A),B,C)),B)
& r1_tarski(k5_subset_1(u1_struct_0(A),E,k4_subset_1(u1_struct_0(A),B,C)),C)
& ~ ( ~ r1_tarski(k4_subset_1(u1_struct_0(A),B,C),k4_subset_1(u1_struct_0(A),D,E))
& ! [F] :
( ( ~ v1_xboole_0(F)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v4_pre_topc(F,A)
& r1_tarski(k5_subset_1(u1_struct_0(A),F,k4_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),B,C))
& u1_struct_0(A) = k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),D,E),F) ) ) ) ) ) ) ) ) ) ) ).
fof(t65_tsep_1,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)))
=> ( k4_subset_1(u1_struct_0(A),B,C) = u1_struct_0(A)
=> ( r2_tsep_1(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)))
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
& k4_subset_1(u1_struct_0(A),B,C) = k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),D,E),F)
& r1_tarski(D,B)
& r1_tarski(E,C)
& r1_tarski(F,k5_subset_1(u1_struct_0(A),B,C))
& v3_pre_topc(D,A)
& v3_pre_topc(E,A)
& v4_pre_topc(F,A) ) ) ) ) ) ) ) ) ).
fof(t66_tsep_1,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)))
=> ~ ( k4_subset_1(u1_struct_0(A),B,C) = u1_struct_0(A)
& r2_tsep_1(A,B,C)
& ~ r1_tarski(B,C)
& ~ r1_tarski(C,B)
& ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v3_pre_topc(D,A)
& v3_pre_topc(E,A)
& r1_tarski(D,B)
& r1_tarski(E,C)
& ~ ( k4_subset_1(u1_struct_0(A),B,C) != k4_subset_1(u1_struct_0(A),D,E)
& ! [F] :
( ( ~ v1_xboole_0(F)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( k4_subset_1(u1_struct_0(A),B,C) = k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),D,E),F)
& r1_tarski(F,k5_subset_1(u1_struct_0(A),B,C))
& v4_pre_topc(F,A) ) ) ) ) ) ) ) ) ) ) ).
fof(t67_tsep_1,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_connsp_1(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)))
& r2_tsep_1(A,D,E)
& r1_tarski(B,D)
& r1_tarski(C,E)
& r1_xboole_0(k5_subset_1(u1_struct_0(A),D,E),k4_subset_1(u1_struct_0(A),B,C)) ) ) ) ) ) ) ).
fof(d8_tsep_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_pre_topc(C,A)
=> ( r3_tsep_1(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) )
=> r1_connsp_1(A,D,E) ) ) ) ) ) ) ) ).
fof(t68_tsep_1,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_1(A,B,C)
=> r1_tsep_1(A,B,C) ) ) ) ) ).
fof(t69_tsep_1,axiom,
$true ).
fof(t70_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_borsuk_1(C,A)
& m1_pre_topc(C,A) )
=> ( r1_tsep_1(A,B,C)
<=> r3_tsep_1(A,B,C) ) ) ) ) ).
fof(t71_tsep_1,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) )
=> ( ( A = k1_tsep_1(A,B,C)
& r3_tsep_1(A,B,C) )
=> ( v1_borsuk_1(B,A)
& m1_pre_topc(B,A) ) ) ) ) ) ).
fof(t72_tsep_1,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) )
=> ( ( v1_borsuk_1(k1_tsep_1(A,B,C),A)
& m1_pre_topc(k1_tsep_1(A,B,C),A)
& r3_tsep_1(A,B,C) )
=> ( v1_borsuk_1(B,A)
& m1_pre_topc(B,A) ) ) ) ) ) ).
fof(t73_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_tsep_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_tsep_1(C,A)
& m1_pre_topc(C,A) )
=> ( r1_tsep_1(A,B,C)
<=> r3_tsep_1(A,B,C) ) ) ) ) ).
fof(t74_tsep_1,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) )
=> ( ( A = k1_tsep_1(A,B,C)
& r3_tsep_1(A,B,C) )
=> ( v1_tsep_1(B,A)
& m1_pre_topc(B,A) ) ) ) ) ) ).
fof(t75_tsep_1,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) )
=> ( ( v1_tsep_1(k1_tsep_1(A,B,C),A)
& m1_pre_topc(k1_tsep_1(A,B,C),A)
& r3_tsep_1(A,B,C) )
=> ( v1_tsep_1(B,A)
& m1_pre_topc(B,A) ) ) ) ) ) ).
fof(t76_tsep_1,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) )
=> ( r3_tsep_1(A,C,D)
=> ( r1_tsep_1(A,C,B)
| r1_tsep_1(A,D,B)
| ( r3_tsep_1(A,k2_tsep_1(A,C,B),k2_tsep_1(A,D,B))
& r3_tsep_1(A,k2_tsep_1(A,B,C),k2_tsep_1(A,B,D)) ) ) ) ) ) ) ) ).
fof(t77_tsep_1,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] :
( m1_pre_topc(D,A)
=> ! [E] :
( m1_pre_topc(E,A)
=> ( ( m1_pre_topc(D,B)
& m1_pre_topc(E,C)
& r3_tsep_1(A,B,C) )
=> r3_tsep_1(A,D,E) ) ) ) ) ) ) ).
fof(t78_tsep_1,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) )
=> ( r3_tsep_1(A,B,D)
=> ( r1_tsep_1(A,B,C)
| r3_tsep_1(A,k2_tsep_1(A,B,C),D) ) ) ) ) ) ) ).
fof(t79_tsep_1,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) )
=> ( ( r3_tsep_1(A,B,D)
& r3_tsep_1(A,C,D) )
<=> r3_tsep_1(A,k1_tsep_1(A,B,C),D) ) ) ) ) ) ).
fof(t80_tsep_1,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_1(A,B,C)
<=> ? [D] :
( ~ v3_struct_0(D)
& v1_borsuk_1(D,A)
& m1_pre_topc(D,A)
& ? [E] :
( ~ v3_struct_0(E)
& v1_borsuk_1(E,A)
& m1_pre_topc(E,A)
& m1_pre_topc(B,D)
& m1_pre_topc(C,E)
& r1_tsep_1(A,D,C)
& r1_tsep_1(A,E,B) ) ) ) ) ) ) ).
fof(t81_tsep_1,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_1(A,B,C)
<=> ? [D] :
( ~ v3_struct_0(D)
& v1_borsuk_1(D,A)
& m1_pre_topc(D,A)
& ? [E] :
( ~ v3_struct_0(E)
& v1_borsuk_1(E,A)
& m1_pre_topc(E,A)
& m1_pre_topc(B,D)
& m1_pre_topc(C,E)
& ( r1_tsep_1(A,D,E)
| r1_tsep_1(A,k2_tsep_1(A,D,E),k1_tsep_1(A,B,C)) ) ) ) ) ) ) ) ).
fof(t82_tsep_1,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_1(A,B,C)
<=> ? [D] :
( ~ v3_struct_0(D)
& v1_tsep_1(D,A)
& m1_pre_topc(D,A)
& ? [E] :
( ~ v3_struct_0(E)
& v1_tsep_1(E,A)
& m1_pre_topc(E,A)
& m1_pre_topc(B,D)
& m1_pre_topc(C,E)
& r1_tsep_1(A,D,C)
& r1_tsep_1(A,E,B) ) ) ) ) ) ) ).
fof(t83_tsep_1,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_1(A,B,C)
<=> ? [D] :
( ~ v3_struct_0(D)
& v1_tsep_1(D,A)
& m1_pre_topc(D,A)
& ? [E] :
( ~ v3_struct_0(E)
& v1_tsep_1(E,A)
& m1_pre_topc(E,A)
& m1_pre_topc(B,D)
& m1_pre_topc(C,E)
& ( r1_tsep_1(A,D,E)
| r1_tsep_1(A,k2_tsep_1(A,D,E),k1_tsep_1(A,B,C)) ) ) ) ) ) ) ) ).
fof(d9_tsep_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_pre_topc(C,A)
=> ( r4_tsep_1(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_1(A,D,E) ) ) ) ) ) ) ) ).
fof(t84_tsep_1,axiom,
$true ).
fof(t85_tsep_1,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) )
=> ( ( r1_tsep_1(A,B,C)
& r4_tsep_1(A,B,C) )
<=> r3_tsep_1(A,B,C) ) ) ) ) ).
fof(t86_tsep_1,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) )
=> ( m1_pre_topc(B,C)
=> r4_tsep_1(A,B,C) ) ) ) ) ).
fof(t87_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_borsuk_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( v1_borsuk_1(C,A)
& m1_pre_topc(C,A) )
=> r4_tsep_1(A,B,C) ) ) ) ).
fof(t88_tsep_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_tsep_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( v1_tsep_1(C,A)
& m1_pre_topc(C,A) )
=> r4_tsep_1(A,B,C) ) ) ) ).
fof(t89_tsep_1,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_1(A,B,C)
=> r4_tsep_1(A,k1_tsep_1(A,B,D),k1_tsep_1(A,C,D)) ) ) ) ) ) ).
fof(t90_tsep_1,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)
& r4_tsep_1(A,C,B) )
=> ( r4_tsep_1(A,k1_tsep_1(A,C,D),k1_tsep_1(A,B,E))
& r4_tsep_1(A,k1_tsep_1(A,D,C),k1_tsep_1(A,E,B)) ) ) ) ) ) ) ) ).
fof(t91_tsep_1,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,D)
=> ( ( ( r4_tsep_1(A,C,B)
& r4_tsep_1(A,D,B) )
=> r4_tsep_1(A,k2_tsep_1(A,C,D),B) )
& ( ( r4_tsep_1(A,B,C)
& r4_tsep_1(A,B,D) )
=> r4_tsep_1(A,B,k2_tsep_1(A,C,D)) ) ) ) ) ) ) ) ).
fof(t92_tsep_1,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_1(A,B,D)
& r4_tsep_1(A,C,D) )
=> r4_tsep_1(A,k1_tsep_1(A,B,C),D) )
& ( ( r4_tsep_1(A,D,B)
& r4_tsep_1(A,D,C) )
=> r4_tsep_1(A,D,k1_tsep_1(A,B,C)) ) ) ) ) ) ) ).
fof(t93_tsep_1,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) )
=> ( ~ r1_tsep_1(A,B,C)
=> ( r4_tsep_1(A,B,C)
<=> ~ ( ~ m1_pre_topc(B,C)
& ~ m1_pre_topc(C,B)
& ! [D] :
( ( ~ v3_struct_0(D)
& v1_borsuk_1(D,A)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v1_borsuk_1(E,A)
& m1_pre_topc(E,A) )
=> ~ ( m1_pre_topc(k2_tsep_1(A,D,k1_tsep_1(A,B,C)),B)
& m1_pre_topc(k2_tsep_1(A,E,k1_tsep_1(A,B,C)),C)
& ~ ( ~ m1_pre_topc(k1_tsep_1(A,B,C),k1_tsep_1(A,D,E))
& ! [F] :
( ( ~ v3_struct_0(F)
& v1_tsep_1(F,A)
& m1_pre_topc(F,A) )
=> ~ ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = k1_tsep_1(A,k1_tsep_1(A,D,E),F)
& m1_pre_topc(k2_tsep_1(A,F,k1_tsep_1(A,B,C)),k2_tsep_1(A,B,C)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t94_tsep_1,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) )
=> ( A = k1_tsep_1(A,B,C)
=> ( r1_tsep_1(A,B,C)
| ( r4_tsep_1(A,B,C)
<=> ~ ( ~ m1_pre_topc(B,C)
& ~ m1_pre_topc(C,B)
& ! [D] :
( ( ~ v3_struct_0(D)
& v1_borsuk_1(D,A)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v1_borsuk_1(E,A)
& m1_pre_topc(E,A) )
=> ~ ( m1_pre_topc(D,B)
& m1_pre_topc(E,C)
& ~ ( A != k1_tsep_1(A,D,E)
& ! [F] :
( ( ~ v3_struct_0(F)
& v1_tsep_1(F,A)
& m1_pre_topc(F,A) )
=> ~ ( A = k1_tsep_1(A,k1_tsep_1(A,D,E),F)
& m1_pre_topc(F,k2_tsep_1(A,B,C)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t95_tsep_1,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) )
=> ( ( A = k1_tsep_1(A,B,C)
& r1_tsep_1(A,B,C) )
=> ( r4_tsep_1(A,B,C)
<=> ( v1_borsuk_1(B,A)
& m1_pre_topc(B,A)
& v1_borsuk_1(C,A)
& m1_pre_topc(C,A) ) ) ) ) ) ) ).
fof(t96_tsep_1,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) )
=> ( ~ r1_tsep_1(A,B,C)
=> ( r4_tsep_1(A,B,C)
<=> ~ ( ~ m1_pre_topc(B,C)
& ~ m1_pre_topc(C,B)
& ! [D] :
( ( ~ v3_struct_0(D)
& v1_tsep_1(D,A)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v1_tsep_1(E,A)
& m1_pre_topc(E,A) )
=> ~ ( m1_pre_topc(k2_tsep_1(A,D,k1_tsep_1(A,B,C)),B)
& m1_pre_topc(k2_tsep_1(A,E,k1_tsep_1(A,B,C)),C)
& ~ ( ~ m1_pre_topc(k1_tsep_1(A,B,C),k1_tsep_1(A,D,E))
& ! [F] :
( ( ~ v3_struct_0(F)
& v1_borsuk_1(F,A)
& m1_pre_topc(F,A) )
=> ~ ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = k1_tsep_1(A,k1_tsep_1(A,D,E),F)
& m1_pre_topc(k2_tsep_1(A,F,k1_tsep_1(A,B,C)),k2_tsep_1(A,B,C)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t97_tsep_1,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) )
=> ( A = k1_tsep_1(A,B,C)
=> ( r1_tsep_1(A,B,C)
| ( r4_tsep_1(A,B,C)
<=> ~ ( ~ m1_pre_topc(B,C)
& ~ m1_pre_topc(C,B)
& ! [D] :
( ( ~ v3_struct_0(D)
& v1_tsep_1(D,A)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v1_tsep_1(E,A)
& m1_pre_topc(E,A) )
=> ~ ( m1_pre_topc(D,B)
& m1_pre_topc(E,C)
& ~ ( A != k1_tsep_1(A,D,E)
& ! [F] :
( ( ~ v3_struct_0(F)
& v1_borsuk_1(F,A)
& m1_pre_topc(F,A) )
=> ~ ( A = k1_tsep_1(A,k1_tsep_1(A,D,E),F)
& m1_pre_topc(F,k2_tsep_1(A,B,C)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t98_tsep_1,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) )
=> ( ( A = k1_tsep_1(A,B,C)
& r1_tsep_1(A,B,C) )
=> ( r4_tsep_1(A,B,C)
<=> ( v1_tsep_1(B,A)
& m1_pre_topc(B,A)
& v1_tsep_1(C,A)
& m1_pre_topc(C,A) ) ) ) ) ) ) ).
fof(t99_tsep_1,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_1(A,B,C)
<=> ? [D] :
( ~ v3_struct_0(D)
& m1_pre_topc(D,A)
& ? [E] :
( ~ v3_struct_0(E)
& m1_pre_topc(E,A)
& r4_tsep_1(A,D,E)
& m1_pre_topc(B,D)
& m1_pre_topc(C,E)
& ( r1_tsep_1(A,D,E)
| r1_tsep_1(A,k2_tsep_1(A,D,E),k1_tsep_1(A,B,C)) ) ) ) ) ) ) ) ).
fof(symmetry_r1_tsep_1,axiom,
! [A,B,C] :
( ( l1_pre_topc(A)
& m1_pre_topc(B,A)
& m1_pre_topc(C,A) )
=> ( r1_tsep_1(A,B,C)
=> r1_tsep_1(A,C,B) ) ) ).
fof(symmetry_r2_tsep_1,axiom,
! [A,B,C] :
( ( 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_1(A,B,C)
=> r2_tsep_1(A,C,B) ) ) ).
fof(symmetry_r3_tsep_1,axiom,
! [A,B,C] :
( ( l1_pre_topc(A)
& m1_pre_topc(B,A)
& m1_pre_topc(C,A) )
=> ( r3_tsep_1(A,B,C)
=> r3_tsep_1(A,C,B) ) ) ).
fof(symmetry_r4_tsep_1,axiom,
! [A,B,C] :
( ( l1_pre_topc(A)
& m1_pre_topc(B,A)
& m1_pre_topc(C,A) )
=> ( r4_tsep_1(A,B,C)
=> r4_tsep_1(A,C,B) ) ) ).
fof(dt_k1_tsep_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& m1_pre_topc(B,A)
& ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ( ~ v3_struct_0(k1_tsep_1(A,B,C))
& v1_pre_topc(k1_tsep_1(A,B,C))
& m1_pre_topc(k1_tsep_1(A,B,C),A) ) ) ).
fof(commutativity_k1_tsep_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& m1_pre_topc(B,A)
& ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> k1_tsep_1(A,B,C) = k1_tsep_1(A,C,B) ) ).
fof(dt_k2_tsep_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& m1_pre_topc(B,A)
& ~ v3_struct_0(C)
& m1_pre_topc(C,A) )
=> ( ~ v3_struct_0(k2_tsep_1(A,B,C))
& v1_pre_topc(k2_tsep_1(A,B,C))
& m1_pre_topc(k2_tsep_1(A,B,C),A) ) ) ).
%------------------------------------------------------------------------------