:: TSEP_2 semantic presentation

theorem Th1: :: TSEP_2:1
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds (b2 ` ) \ (b3 ` ) = b3 \ b2
proof end;

definition
let c1 be TopSpace;
let c2, c3 be Subset of c1;
pred c2,c3 constitute_a_decomposition means :Def1: :: TSEP_2:def 1
( a2 misses a3 & a2 \/ a3 = the carrier of a1 );
symmetry
for b1, b2 being Subset of c1 holds
( b1 misses b2 & b1 \/ b2 = the carrier of c1 implies ( b2 misses b1 & b2 \/ b1 = the carrier of c1 ) )
;
end;

:: deftheorem Def1 defines constitute_a_decomposition TSEP_2:def 1 :
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 constitute_a_decomposition iff ( b2 misses b3 & b2 \/ b3 = the carrier of b1 ) );

notation
let c1 be TopSpace;
let c2, c3 be Subset of c1;
antonym c2,c3 do_not_constitute_a_decomposition for c2,c3 constitute_a_decomposition ;
end;

theorem Th2: :: TSEP_2:2
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 constitute_a_decomposition iff ( b2 misses b3 & b2 \/ b3 = [#] b1 ) ) by Def1;

theorem Th3: :: TSEP_2:3
canceled;

theorem Th4: :: TSEP_2:4
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 constitute_a_decomposition implies ( b2 = b3 ` & b3 = b2 ` ) )
proof end;

theorem Th5: :: TSEP_2:5
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( ( b2 = b3 ` or b3 = b2 ` ) implies b2,b3 constitute_a_decomposition )
proof end;

theorem Th6: :: TSEP_2:6
for b1 being non empty TopSpace
for b2 being Subset of b1 holds b2,b2 ` constitute_a_decomposition
proof end;

theorem Th7: :: TSEP_2:7
for b1 being non empty TopSpace holds {} b1, [#] b1 constitute_a_decomposition
proof end;

theorem Th8: :: TSEP_2:8
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
b2,b2 do_not_constitute_a_decomposition
proof end;

definition
let c1 be non empty TopSpace;
let c2, c3 be Subset of c1;
redefine pred constitute_a_decomposition as c2,c3 constitute_a_decomposition ;
irreflexivity
for b1 being Subset of c1 holds
not b1,b1 constitute_a_decomposition
by Th8;
end;

theorem Th9: :: TSEP_2:9
for b1 being non empty TopSpace
for b2, b3, b4 being Subset of b1 holds
( b2,b3 constitute_a_decomposition & b3,b4 constitute_a_decomposition implies b2 = b4 )
proof end;

theorem Th10: :: TSEP_2:10
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 constitute_a_decomposition implies ( Cl b2, Int b3 constitute_a_decomposition & Int b2, Cl b3 constitute_a_decomposition ) )
proof end;

theorem Th11: :: TSEP_2:11
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( Cl b2, Int (b2 ` ) constitute_a_decomposition & Cl (b2 ` ), Int b2 constitute_a_decomposition & Int b2, Cl (b2 ` ) constitute_a_decomposition & Int (b2 ` ), Cl b2 constitute_a_decomposition )
proof end;

theorem Th12: :: TSEP_2:12
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 constitute_a_decomposition implies ( b2 is open iff b3 is closed ) )
proof end;

theorem Th13: :: TSEP_2:13
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 constitute_a_decomposition implies ( b2 is closed iff b3 is open ) ) by Th12;

theorem Th14: :: TSEP_2:14
for b1 being non empty TopSpace
for b2, b3, b4, b5 being Subset of b1 holds
( b2,b3 constitute_a_decomposition & b4,b5 constitute_a_decomposition implies b2 /\ b4,b3 \/ b5 constitute_a_decomposition )
proof end;

theorem Th15: :: TSEP_2:15
for b1 being non empty TopSpace
for b2, b3, b4, b5 being Subset of b1 holds
( b2,b3 constitute_a_decomposition & b4,b5 constitute_a_decomposition implies b2 \/ b4,b3 /\ b5 constitute_a_decomposition ) by Th14;

theorem Th16: :: TSEP_2:16
for b1 being non empty TopSpace
for b2, b3, b4, b5 being Subset of b1 holds
( b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition implies ( b2,b3 are_weakly_separated iff b4,b5 are_weakly_separated ) )
proof end;

theorem Th17: :: TSEP_2:17
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 are_weakly_separated iff b2 ` ,b3 ` are_weakly_separated )
proof end;

theorem Th18: :: TSEP_2:18
for b1 being non empty TopSpace
for b2, b3, b4, b5 being Subset of b1 holds
( b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b2,b3 are_separated implies b4,b5 are_weakly_separated )
proof end;

theorem Th19: :: TSEP_2:19
for b1 being non empty TopSpace
for b2, b3, b4, b5 being Subset of b1 holds
( b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b2 misses b3 & b4,b5 are_weakly_separated implies b2,b3 are_separated )
proof end;

theorem Th20: :: TSEP_2:20
for b1 being non empty TopSpace
for b2, b3, b4, b5 being Subset of b1 holds
( b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b4 \/ b5 = the carrier of b1 & b4,b5 are_weakly_separated implies b2,b3 are_separated )
proof end;

theorem Th21: :: TSEP_2:21
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 constitute_a_decomposition implies ( b2,b3 are_weakly_separated iff b2,b3 are_separated ) )
proof end;

theorem Th22: :: TSEP_2:22
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 are_weakly_separated iff (b2 \/ b3) \ b2,(b2 \/ b3) \ b3 are_separated )
proof end;

theorem Th23: :: TSEP_2:23
for b1 being non empty TopSpace
for b2, b3, b4, b5 being Subset of b1 holds
( b4 c= b2 & b5 c= b3 & b4 \/ b5 = b2 \/ b3 & b4,b5 are_weakly_separated implies b2,b3 are_weakly_separated )
proof end;

theorem Th24: :: TSEP_2:24
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 are_weakly_separated iff b2 \ (b2 /\ b3),b3 \ (b2 /\ b3) are_separated )
proof end;

theorem Th25: :: TSEP_2:25
for b1 being non empty TopSpace
for b2, b3, b4, b5 being Subset of b1 holds
( b4 c= b2 & b5 c= b3 & b4 /\ b5 = b2 /\ b3 & b2,b3 are_weakly_separated implies b4,b5 are_weakly_separated )
proof end;

theorem Th26: :: TSEP_2:26
for b1 being non empty TopSpace
for b2, b3 being Subset of b1
for b4 being non empty SubSpace of b1
for b5, b6 being Subset of b4 holds
( b5 = b2 & b6 = b3 implies ( b2,b3 are_separated iff b5,b6 are_separated ) )
proof end;

theorem Th27: :: TSEP_2:27
for b1 being non empty TopSpace
for b2, b3 being Subset of b1
for b4 being non empty SubSpace of b1
for b5, b6 being Subset of b4 holds
( b5 = the carrier of b4 /\ b2 & b6 = the carrier of b4 /\ b3 & b2,b3 are_separated implies b5,b6 are_separated )
proof end;

theorem Th28: :: TSEP_2:28
for b1 being non empty TopSpace
for b2, b3 being Subset of b1
for b4 being non empty SubSpace of b1
for b5, b6 being Subset of b4 holds
( b5 = b2 & b6 = b3 implies ( b2,b3 are_weakly_separated iff b5,b6 are_weakly_separated ) )
proof end;

theorem Th29: :: TSEP_2:29
for b1 being non empty TopSpace
for b2, b3 being Subset of b1
for b4 being non empty SubSpace of b1
for b5, b6 being Subset of b4 holds
( b5 = the carrier of b4 /\ b2 & b6 = the carrier of b4 /\ b3 & b2,b3 are_weakly_separated implies b5,b6 are_weakly_separated )
proof end;

definition
let c1 be non empty TopSpace;
let c2, c3 be SubSpace of c1;
pred c2,c3 constitute_a_decomposition means :Def2: :: TSEP_2:def 2
for b1, b2 being Subset of a1 holds
( b1 = the carrier of a2 & b2 = the carrier of a3 implies b1,b2 constitute_a_decomposition );
symmetry
for b1, b2 being SubSpace of c1 holds
( ( for b3, b4 being Subset of c1 holds
( b3 = the carrier of b1 & b4 = the carrier of b2 implies b3,b4 constitute_a_decomposition ) ) implies for b3, b4 being Subset of c1 holds
( b3 = the carrier of b2 & b4 = the carrier of b1 implies b3,b4 constitute_a_decomposition ) )
;
end;

:: deftheorem Def2 defines constitute_a_decomposition TSEP_2:def 2 :
for b1 being non empty TopSpace
for b2, b3 being SubSpace of b1 holds
( b2,b3 constitute_a_decomposition iff for b4, b5 being Subset of b1 holds
( b4 = the carrier of b2 & b5 = the carrier of b3 implies b4,b5 constitute_a_decomposition ) );

notation
let c1 be non empty TopSpace;
let c2, c3 be SubSpace of c1;
antonym c2,c3 do_not_constitute_a_decomposition for c2,c3 constitute_a_decomposition ;
end;

theorem Th30: :: TSEP_2:30
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2,b3 constitute_a_decomposition iff ( b2 misses b3 & TopStruct(# the carrier of b1,the topology of b1 #) = b2 union b3 ) )
proof end;

theorem Th31: :: TSEP_2:31
canceled;

theorem Th32: :: TSEP_2:32
for b1 being non empty TopSpace
for b2 being non empty SubSpace of b1 holds
b2,b2 do_not_constitute_a_decomposition
proof end;

definition
let c1 be non empty TopSpace;
let c2, c3 be non empty SubSpace of c1;
redefine pred constitute_a_decomposition as c2,c3 constitute_a_decomposition ;
irreflexivity
for b1 being non empty SubSpace of c1 holds
not b1,b1 constitute_a_decomposition
by Th32;
end;

theorem Th33: :: TSEP_2:33
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds
( b2,b3 constitute_a_decomposition & b3,b4 constitute_a_decomposition implies TopStruct(# the carrier of b2,the topology of b2 #) = TopStruct(# the carrier of b4,the topology of b4 #) )
proof end;

theorem Th34: :: TSEP_2:34
for b1 being non empty TopSpace
for b2, b3, b4, b5 being non empty SubSpace of b1 holds
( b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition implies ( b4 union b5 = TopStruct(# the carrier of b1,the topology of b1 #) iff b2 misses b3 ) )
proof end;

theorem Th35: :: TSEP_2:35
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2,b3 constitute_a_decomposition implies ( b2 is open iff b3 is closed ) )
proof end;

theorem Th36: :: TSEP_2:36
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2,b3 constitute_a_decomposition implies ( b2 is closed iff b3 is open ) ) by Th35;

theorem Th37: :: TSEP_2:37
for b1 being non empty TopSpace
for b2, b3, b4, b5 being non empty SubSpace of b1 holds
( b2 meets b3 & b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition implies b2 meet b3,b4 union b5 constitute_a_decomposition )
proof end;

theorem Th38: :: TSEP_2:38
for b1 being non empty TopSpace
for b2, b3, b4, b5 being non empty SubSpace of b1 holds
( b2 meets b3 & b4,b2 constitute_a_decomposition & b5,b3 constitute_a_decomposition implies b4 union b5,b2 meet b3 constitute_a_decomposition ) by Th37;

theorem Th39: :: TSEP_2:39
for b1 being non empty TopSpace
for b2, b3, b4, b5 being SubSpace of b1 holds
( b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b2,b3 are_weakly_separated implies b4,b5 are_weakly_separated )
proof end;

theorem Th40: :: TSEP_2:40
for b1 being non empty TopSpace
for b2, b3, b4, b5 being non empty SubSpace of b1 holds
( b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b2,b3 are_separated implies b4,b5 are_weakly_separated )
proof end;

theorem Th41: :: TSEP_2:41
for b1 being non empty TopSpace
for b2, b3, b4, b5 being non empty SubSpace of b1 holds
( b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b2 misses b3 & b4,b5 are_weakly_separated implies b2,b3 are_separated )
proof end;

theorem Th42: :: TSEP_2:42
for b1 being non empty TopSpace
for b2, b3, b4, b5 being non empty SubSpace of b1 holds
( b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b4 union b5 = TopStruct(# the carrier of b1,the topology of b1 #) & b4,b5 are_weakly_separated implies b2,b3 are_separated )
proof end;

theorem Th43: :: TSEP_2:43
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2,b3 constitute_a_decomposition implies ( b2,b3 are_weakly_separated iff b2,b3 are_separated ) )
proof end;

theorem Th44: :: TSEP_2:44
for b1 being non empty TopSpace
for b2, b3, b4, b5 being non empty SubSpace of b1 holds
( b4 is SubSpace of b2 & b5 is SubSpace of b3 & b4 union b5 = b2 union b3 & b4,b5 are_weakly_separated implies b2,b3 are_weakly_separated )
proof end;

theorem Th45: :: TSEP_2:45
for b1 being non empty TopSpace
for b2, b3, b4, b5 being non empty SubSpace of b1 holds
( b4 is SubSpace of b2 & b5 is SubSpace of b3 & b4 meets b5 & b4 meet b5 = b2 meet b3 & b2,b3 are_weakly_separated implies b4,b5 are_weakly_separated )
proof end;

theorem Th46: :: TSEP_2:46
for b1 being non empty TopSpace
for b2 being non empty SubSpace of b1
for b3, b4 being SubSpace of b1
for b5, b6 being SubSpace of b2 holds
( the carrier of b3 = the carrier of b5 & the carrier of b4 = the carrier of b6 implies ( b3,b4 are_separated iff b5,b6 are_separated ) )
proof end;

theorem Th47: :: TSEP_2:47
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds
( b3 meets b2 & b4 meets b2 implies for b5, b6 being SubSpace of b2 holds
( b5 = b3 meet b2 & b6 = b4 meet b2 & b3,b4 are_separated implies b5,b6 are_separated ) )
proof end;

theorem Th48: :: TSEP_2:48
for b1 being non empty TopSpace
for b2 being non empty SubSpace of b1
for b3, b4 being SubSpace of b1
for b5, b6 being SubSpace of b2 holds
( the carrier of b3 = the carrier of b5 & the carrier of b4 = the carrier of b6 implies ( b3,b4 are_weakly_separated iff b5,b6 are_weakly_separated ) )
proof end;

theorem Th49: :: TSEP_2:49
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds
( b3 meets b2 & b4 meets b2 implies for b5, b6 being SubSpace of b2 holds
( b5 = b3 meet b2 & b6 = b4 meet b2 & b3,b4 are_weakly_separated implies b5,b6 are_weakly_separated ) )
proof end;