:: TSEP_1 semantic presentation

Lemma1: for b1 being set
for b2, b3, b4 being Subset of b1 holds
( b2 \ b3 = {} implies b2 misses b4 \ b3 )
proof end;

Lemma2: for b1 being set
for b2, b3, b4 being Subset of b1 holds
( b2 misses b3 implies b2 misses b3 \ b4 )
proof end;

Lemma3: for b1, b2, b3 being set holds (b1 /\ b2) \ b3 = (b1 \ b3) /\ (b2 \ b3)
proof end;

theorem Th1: :: TSEP_1:1
for b1 being TopStruct
for b2 being SubSpace of b1 holds
the carrier of b2 is Subset of b1
proof end;

theorem Th2: :: TSEP_1:2
for b1 being TopStruct holds
b1 is SubSpace of b1
proof end;

theorem Th3: :: TSEP_1:3
for b1 being strict TopStruct holds b1 | ([#] b1) = b1
proof end;

theorem Th4: :: TSEP_1:4
for b1 being TopSpace
for b2, b3 being SubSpace of b1 holds
( the carrier of b2 c= the carrier of b3 iff b2 is SubSpace of b3 )
proof end;

Lemma7: for b1 being TopStruct
for b2 being SubSpace of b1 holds
TopStruct(# the carrier of b2,the topology of b2 #) is strict SubSpace of b1
proof end;

theorem Th5: :: TSEP_1:5
for b1 being TopStruct
for b2, b3 being SubSpace of b1 holds
( the carrier of b2 = the carrier of b3 implies TopStruct(# the carrier of b2,the topology of b2 #) = TopStruct(# the carrier of b3,the topology of b3 #) )
proof end;

theorem Th6: :: TSEP_1:6
for b1 being TopSpace
for b2, b3 being SubSpace of b1 holds
( b2 is SubSpace of b3 & b3 is SubSpace of b2 implies TopStruct(# the carrier of b2,the topology of b2 #) = TopStruct(# the carrier of b3,the topology of b3 #) )
proof end;

theorem Th7: :: TSEP_1:7
for b1 being TopSpace
for b2 being SubSpace of b1
for b3 being SubSpace of b2 holds
b3 is SubSpace of b1
proof end;

theorem Th8: :: TSEP_1:8
for b1 being TopSpace
for b2 being SubSpace of b1
for b3, b4 being Subset of b1
for b5 being Subset of b2 holds
( b3 is closed & b3 c= the carrier of b2 & b4 c= b3 & b4 = b5 implies ( b5 is closed iff b4 is closed ) )
proof end;

theorem Th9: :: TSEP_1:9
for b1 being TopSpace
for b2 being SubSpace of b1
for b3, b4 being Subset of b1
for b5 being Subset of b2 holds
( b3 is open & b3 c= the carrier of b2 & b4 c= b3 & b4 = b5 implies ( b5 is open iff b4 is open ) )
proof end;

theorem Th10: :: TSEP_1:10
for b1 being non empty TopStruct
for b2 being non empty Subset of b1 holds
ex b3 being non empty strict SubSpace of b1 st b2 = the carrier of b3
proof end;

theorem Th11: :: TSEP_1:11
for b1 being TopSpace
for b2 being SubSpace of b1
for b3 being Subset of b1 holds
( b3 = the carrier of b2 implies ( b2 is closed SubSpace of b1 iff b3 is closed ) )
proof end;

theorem Th12: :: TSEP_1:12
for b1 being TopSpace
for b2 being closed SubSpace of b1
for b3 being Subset of b1
for b4 being Subset of b2 holds
( b3 = b4 implies ( b4 is closed iff b3 is closed ) )
proof end;

theorem Th13: :: TSEP_1:13
for b1 being TopSpace
for b2 being closed SubSpace of b1
for b3 being closed SubSpace of b2 holds
b3 is closed SubSpace of b1
proof end;

theorem Th14: :: TSEP_1:14
for b1 being non empty TopSpace
for b2 being non empty closed SubSpace of b1
for b3 being non empty SubSpace of b1 holds
( the carrier of b2 c= the carrier of b3 implies b2 is non empty closed SubSpace of b3 )
proof end;

theorem Th15: :: TSEP_1:15
for b1 being non empty TopSpace
for b2 being non empty Subset of b1 holds
not ( b2 is closed & ( for b3 being non empty strict closed SubSpace of b1 holds
not b2 = the carrier of b3 ) )
proof end;

definition
let c1 be TopStruct ;
let c2 be SubSpace of c1;
attr a2 is open means :Def1: :: TSEP_1:def 1
for b1 being Subset of a1 holds
( b1 = the carrier of a2 implies b1 is open );
end;

:: deftheorem Def1 defines open TSEP_1:def 1 :
for b1 being TopStruct
for b2 being SubSpace of b1 holds
( b2 is open iff for b3 being Subset of b1 holds
( b3 = the carrier of b2 implies b3 is open ) );

Lemma16: for b1 being TopStruct holds
TopStruct(# the carrier of b1,the topology of b1 #) is SubSpace of b1
proof end;

registration
let c1 be TopSpace;
cluster strict open SubSpace of a1;
existence
ex b1 being SubSpace of c1 st
( b1 is strict & b1 is open )
proof end;
end;

registration
let c1 be non empty TopSpace;
cluster non empty strict open SubSpace of a1;
existence
ex b1 being SubSpace of c1 st
( b1 is strict & b1 is open & not b1 is empty )
proof end;
end;

theorem Th16: :: TSEP_1:16
for b1 being TopSpace
for b2 being SubSpace of b1
for b3 being Subset of b1 holds
( b3 = the carrier of b2 implies ( b2 is open SubSpace of b1 iff b3 is open ) )
proof end;

theorem Th17: :: TSEP_1:17
for b1 being TopSpace
for b2 being open SubSpace of b1
for b3 being Subset of b1
for b4 being Subset of b2 holds
( b3 = b4 implies ( b4 is open iff b3 is open ) )
proof end;

theorem Th18: :: TSEP_1:18
for b1 being TopSpace
for b2 being open SubSpace of b1
for b3 being open SubSpace of b2 holds
b3 is open SubSpace of b1
proof end;

theorem Th19: :: TSEP_1:19
for b1 being non empty TopSpace
for b2 being open SubSpace of b1
for b3 being non empty SubSpace of b1 holds
( the carrier of b2 c= the carrier of b3 implies b2 is open SubSpace of b3 )
proof end;

theorem Th20: :: TSEP_1:20
for b1 being non empty TopSpace
for b2 being non empty Subset of b1 holds
not ( b2 is open & ( for b3 being non empty strict open SubSpace of b1 holds
not b2 = the carrier of b3 ) )
proof end;

definition
let c1 be non empty TopStruct ;
let c2, c3 be non empty SubSpace of c1;
func c2 union c3 -> non empty strict SubSpace of a1 means :Def2: :: TSEP_1:def 2
the carrier of a4 = the carrier of a2 \/ the carrier of a3;
existence
ex b1 being non empty strict SubSpace of c1 st the carrier of b1 = the carrier of c2 \/ the carrier of c3
proof end;
uniqueness
for b1, b2 being non empty strict SubSpace of c1 holds
( the carrier of b1 = the carrier of c2 \/ the carrier of c3 & the carrier of b2 = the carrier of c2 \/ the carrier of c3 implies b1 = b2 )
by Th5;
commutativity
for b1 being non empty strict SubSpace of c1
for b2, b3 being non empty SubSpace of c1 holds
( the carrier of b1 = the carrier of b2 \/ the carrier of b3 implies the carrier of b1 = the carrier of b3 \/ the carrier of b2 )
;
end;

:: deftheorem Def2 defines union TSEP_1:def 2 :
for b1 being non empty TopStruct
for b2, b3 being non empty SubSpace of b1
for b4 being non empty strict SubSpace of b1 holds
( b4 = b2 union b3 iff the carrier of b4 = the carrier of b2 \/ the carrier of b3 );

theorem Th21: :: TSEP_1:21
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds (b2 union b3) union b4 = b2 union (b3 union b4)
proof end;

theorem Th22: :: TSEP_1:22
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
b2 is SubSpace of b2 union b3
proof end;

theorem Th23: :: TSEP_1:23
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2 is SubSpace of b3 iff b2 union b3 = TopStruct(# the carrier of b3,the topology of b3 #) )
proof end;

theorem Th24: :: TSEP_1:24
for b1 being non empty TopSpace
for b2, b3 being non empty closed SubSpace of b1 holds
b2 union b3 is closed SubSpace of b1
proof end;

theorem Th25: :: TSEP_1:25
for b1 being non empty TopSpace
for b2, b3 being non empty open SubSpace of b1 holds
b2 union b3 is open SubSpace of b1
proof end;

definition
let c1 be TopStruct ;
let c2, c3 be SubSpace of c1;
pred c2 misses c3 means :Def3: :: TSEP_1:def 3
the carrier of a2 misses the carrier of a3;
correctness
;
symmetry
for b1, b2 being SubSpace of c1 holds
( the carrier of b1 misses the carrier of b2 implies the carrier of b2 misses the carrier of b1 )
;
end;

:: deftheorem Def3 defines misses TSEP_1:def 3 :
for b1 being TopStruct
for b2, b3 being SubSpace of b1 holds
( b2 misses b3 iff the carrier of b2 misses the carrier of b3 );

notation
let c1 be TopStruct ;
let c2, c3 be SubSpace of c1;
antonym c2 meets c3 for c2 misses c3;
end;

definition
let c1 be non empty TopStruct ;
let c2, c3 be non empty SubSpace of c1;
assume E22: c2 meets c3 ;
canceled;
func c2 meet c3 -> non empty strict SubSpace of a1 means :Def5: :: TSEP_1:def 5
the carrier of a4 = the carrier of a2 /\ the carrier of a3;
existence
ex b1 being non empty strict SubSpace of c1 st the carrier of b1 = the carrier of c2 /\ the carrier of c3
proof end;
uniqueness
for b1, b2 being non empty strict SubSpace of c1 holds
( the carrier of b1 = the carrier of c2 /\ the carrier of c3 & the carrier of b2 = the carrier of c2 /\ the carrier of c3 implies b1 = b2 )
by Th5;
end;

:: deftheorem Def4 TSEP_1:def 4 :
canceled;

:: deftheorem Def5 defines meet TSEP_1:def 5 :
for b1 being non empty TopStruct
for b2, b3 being non empty SubSpace of b1 holds
( b2 meets b3 implies for b4 being non empty strict SubSpace of b1 holds
( b4 = b2 meet b3 iff the carrier of b4 = the carrier of b2 /\ the carrier of b3 ) );

theorem Th26: :: TSEP_1:26
canceled;

theorem Th27: :: TSEP_1:27
canceled;

theorem Th28: :: TSEP_1:28
canceled;

theorem Th29: :: TSEP_1:29
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds
( ( b2 meets b3 implies b2 meet b3 = b3 meet b2 ) & ( ( ( b2 meets b3 & b2 meet b3 meets b4 ) or ( b3 meets b4 & b2 meets b3 meet b4 ) ) implies (b2 meet b3) meet b4 = b2 meet (b3 meet b4) ) )
proof end;

theorem Th30: :: TSEP_1:30
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2 meets b3 implies ( b2 meet b3 is SubSpace of b2 & b2 meet b3 is SubSpace of b3 ) )
proof end;

theorem Th31: :: TSEP_1:31
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2 meets b3 implies ( ( b2 is SubSpace of b3 implies b2 meet b3 = TopStruct(# the carrier of b2,the topology of b2 #) ) & ( b2 meet b3 = TopStruct(# the carrier of b2,the topology of b2 #) implies b2 is SubSpace of b3 ) & ( b3 is SubSpace of b2 implies b2 meet b3 = TopStruct(# the carrier of b3,the topology of b3 #) ) & ( b2 meet b3 = TopStruct(# the carrier of b3,the topology of b3 #) implies b3 is SubSpace of b2 ) ) )
proof end;

theorem Th32: :: TSEP_1:32
for b1 being non empty TopSpace
for b2, b3 being non empty closed SubSpace of b1 holds
( b2 meets b3 implies b2 meet b3 is closed SubSpace of b1 )
proof end;

theorem Th33: :: TSEP_1:33
for b1 being non empty TopSpace
for b2, b3 being non empty open SubSpace of b1 holds
( b2 meets b3 implies b2 meet b3 is open SubSpace of b1 )
proof end;

theorem Th34: :: TSEP_1:34
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2 meets b3 implies b2 meet b3 is SubSpace of b2 union b3 )
proof end;

theorem Th35: :: TSEP_1:35
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds
( b2 meets b4 & b4 meets b3 implies ( (b2 union b3) meet b4 = (b2 meet b4) union (b3 meet b4) & b4 meet (b2 union b3) = (b4 meet b2) union (b4 meet b3) ) )
proof end;

theorem Th36: :: TSEP_1:36
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds
( b2 meets b3 implies ( (b2 meet b3) union b4 = (b2 union b4) meet (b3 union b4) & b4 union (b2 meet b3) = (b4 union b2) meet (b4 union b3) ) )
proof end;

notation
let c1 be TopStruct ;
let c2, c3 be Subset of c1;
antonym c2,c3 are_not_separated for c2,c3 are_separated ;
end;

theorem Th37: :: TSEP_1:37
canceled;

theorem Th38: :: TSEP_1:38
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2 is closed & b3 is closed implies ( b2 misses b3 iff b2,b3 are_separated ) )
proof end;

theorem Th39: :: TSEP_1:39
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2 \/ b3 is closed & b2,b3 are_separated implies ( b2 is closed & b3 is closed ) )
proof end;

theorem Th40: :: TSEP_1:40
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2 misses b3 & b2 is open implies b2 misses Cl b3 )
proof end;

theorem Th41: :: TSEP_1:41
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2 is open & b3 is open implies ( b2 misses b3 iff b2,b3 are_separated ) )
proof end;

theorem Th42: :: TSEP_1:42
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2 \/ b3 is open & b2,b3 are_separated implies ( b2 is open & b3 is open ) )
proof end;

theorem Th43: :: TSEP_1:43
for b1 being TopSpace
for b2, b3, b4 being Subset of b1 holds
( b2,b3 are_separated implies b2 /\ b4,b3 /\ b4 are_separated )
proof end;

theorem Th44: :: TSEP_1:44
for b1 being TopSpace
for b2, b3, b4 being Subset of b1 holds
( ( b2,b4 are_separated or b3,b4 are_separated ) implies b2 /\ b3,b4 are_separated )
proof end;

theorem Th45: :: TSEP_1:45
for b1 being TopSpace
for b2, b3, b4 being Subset of b1 holds
( ( b2,b4 are_separated & b3,b4 are_separated ) iff b2 \/ b3,b4 are_separated )
proof end;

theorem Th46: :: TSEP_1:46
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 are_separated iff ex b4, b5 being Subset of b1 st
( b2 c= b4 & b3 c= b5 & b4 misses b3 & b5 misses b2 & b4 is closed & b5 is closed ) )
proof end;

theorem Th47: :: TSEP_1:47
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 are_separated iff ex b4, b5 being Subset of b1 st
( b2 c= b4 & b3 c= b5 & b4 /\ b5 misses b2 \/ b3 & b4 is closed & b5 is closed ) )
proof end;

theorem Th48: :: TSEP_1:48
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 are_separated iff ex b4, b5 being Subset of b1 st
( b2 c= b4 & b3 c= b5 & b4 misses b3 & b5 misses b2 & b4 is open & b5 is open ) )
proof end;

theorem Th49: :: TSEP_1:49
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 are_separated iff ex b4, b5 being Subset of b1 st
( b2 c= b4 & b3 c= b5 & b4 /\ b5 misses b2 \/ b3 & b4 is open & b5 is open ) )
proof end;

definition
let c1 be TopStruct ;
let c2, c3 be Subset of c1;
canceled;
pred c2,c3 are_weakly_separated means :Def7: :: TSEP_1:def 7
a2 \ a3,a3 \ a2 are_separated ;
symmetry
for b1, b2 being Subset of c1 holds
( b1 \ b2,b2 \ b1 are_separated implies b2 \ b1,b1 \ b2 are_separated )
;
end;

:: deftheorem Def6 TSEP_1:def 6 :
canceled;

:: deftheorem Def7 defines are_weakly_separated TSEP_1:def 7 :
for b1 being TopStruct
for b2, b3 being Subset of b1 holds
( b2,b3 are_weakly_separated iff b2 \ b3,b3 \ b2 are_separated );

notation
let c1 be TopStruct ;
let c2, c3 be Subset of c1;
antonym c2,c3 are_not_weakly_separated for c2,c3 are_weakly_separated ;
end;

theorem Th50: :: TSEP_1:50
canceled;

theorem Th51: :: TSEP_1:51
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( ( b2 misses b3 & b2,b3 are_weakly_separated ) iff b2,b3 are_separated )
proof end;

theorem Th52: :: TSEP_1:52
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2 c= b3 implies b2,b3 are_weakly_separated )
proof end;

theorem Th53: :: TSEP_1:53
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2 is closed & b3 is closed implies b2,b3 are_weakly_separated )
proof end;

theorem Th54: :: TSEP_1:54
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2 is open & b3 is open implies b2,b3 are_weakly_separated )
proof end;

theorem Th55: :: TSEP_1:55
for b1 being TopSpace
for b2, b3, b4 being Subset of b1 holds
( b2,b3 are_weakly_separated implies b2 \/ b4,b3 \/ b4 are_weakly_separated )
proof end;

theorem Th56: :: TSEP_1:56
for b1 being TopSpace
for b2, b3, b4, b5 being Subset of b1 holds
( b4 c= b2 & b5 c= b3 & b3,b2 are_weakly_separated implies b3 \/ b4,b2 \/ b5 are_weakly_separated )
proof end;

theorem Th57: :: TSEP_1:57
for b1 being TopSpace
for b2, b3, b4 being Subset of b1 holds
( b2,b4 are_weakly_separated & b3,b4 are_weakly_separated implies b2 /\ b3,b4 are_weakly_separated )
proof end;

theorem Th58: :: TSEP_1:58
for b1 being TopSpace
for b2, b3, b4 being Subset of b1 holds
( b2,b4 are_weakly_separated & b3,b4 are_weakly_separated implies b2 \/ b3,b4 are_weakly_separated )
proof end;

theorem Th59: :: TSEP_1:59
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 are_weakly_separated iff ex b4, b5, b6 being Subset of b1 st
( b4 /\ (b2 \/ b3) c= b2 & b5 /\ (b2 \/ b3) c= b3 & b6 /\ (b2 \/ b3) c= b2 /\ b3 & the carrier of b1 = (b4 \/ b5) \/ b6 & b4 is closed & b5 is closed & b6 is open ) )
proof end;

theorem Th60: :: TSEP_1:60
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
not ( b2,b3 are_weakly_separated & not b2 c= b3 & not b3 c= b2 & ( for b4, b5 being non empty Subset of b1 holds
not ( b4 is closed & b5 is closed & b4 /\ (b2 \/ b3) c= b2 & b5 /\ (b2 \/ b3) c= b3 & not ( not b2 \/ b3 c= b4 \/ b5 & ( for b6 being non empty Subset of b1 holds
not ( b6 is open & b6 /\ (b2 \/ b3) c= b2 /\ b3 & the carrier of b1 = (b4 \/ b5) \/ b6 ) ) ) ) ) )
proof end;

theorem Th61: :: TSEP_1:61
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2 \/ b3 = the carrier of b1 implies ( b2,b3 are_weakly_separated iff ex b4, b5, b6 being Subset of b1 st
( b2 \/ b3 = (b4 \/ b5) \/ b6 & b4 c= b2 & b5 c= b3 & b6 c= b2 /\ b3 & b4 is closed & b5 is closed & b6 is open ) ) )
proof end;

theorem Th62: :: TSEP_1:62
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
not ( b2 \/ b3 = the carrier of b1 & b2,b3 are_weakly_separated & not b2 c= b3 & not b3 c= b2 & ( for b4, b5 being non empty Subset of b1 holds
not ( b4 is closed & b5 is closed & b4 c= b2 & b5 c= b3 & not ( not b2 \/ b3 = b4 \/ b5 & ( for b6 being non empty Subset of b1 holds
not ( b2 \/ b3 = (b4 \/ b5) \/ b6 & b6 c= b2 /\ b3 & b6 is open ) ) ) ) ) )
proof end;

theorem Th63: :: TSEP_1:63
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 are_weakly_separated iff ex b4, b5, b6 being Subset of b1 st
( b4 /\ (b2 \/ b3) c= b2 & b5 /\ (b2 \/ b3) c= b3 & b6 /\ (b2 \/ b3) c= b2 /\ b3 & the carrier of b1 = (b4 \/ b5) \/ b6 & b4 is open & b5 is open & b6 is closed ) )
proof end;

theorem Th64: :: TSEP_1:64
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
not ( b2,b3 are_weakly_separated & not b2 c= b3 & not b3 c= b2 & ( for b4, b5 being non empty Subset of b1 holds
not ( b4 is open & b5 is open & b4 /\ (b2 \/ b3) c= b2 & b5 /\ (b2 \/ b3) c= b3 & not ( not b2 \/ b3 c= b4 \/ b5 & ( for b6 being non empty Subset of b1 holds
not ( b6 is closed & b6 /\ (b2 \/ b3) c= b2 /\ b3 & the carrier of b1 = (b4 \/ b5) \/ b6 ) ) ) ) ) )
proof end;

theorem Th65: :: TSEP_1:65
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2 \/ b3 = the carrier of b1 implies ( b2,b3 are_weakly_separated iff ex b4, b5, b6 being Subset of b1 st
( b2 \/ b3 = (b4 \/ b5) \/ b6 & b4 c= b2 & b5 c= b3 & b6 c= b2 /\ b3 & b4 is open & b5 is open & b6 is closed ) ) )
proof end;

theorem Th66: :: TSEP_1:66
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
not ( b2 \/ b3 = the carrier of b1 & b2,b3 are_weakly_separated & not b2 c= b3 & not b3 c= b2 & ( for b4, b5 being non empty Subset of b1 holds
not ( b4 is open & b5 is open & b4 c= b2 & b5 c= b3 & not ( not b2 \/ b3 = b4 \/ b5 & ( for b6 being non empty Subset of b1 holds
not ( b2 \/ b3 = (b4 \/ b5) \/ b6 & b6 c= b2 /\ b3 & b6 is closed ) ) ) ) ) )
proof end;

theorem Th67: :: TSEP_1:67
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2,b3 are_separated iff ex b4, b5 being Subset of b1 st
( b4,b5 are_weakly_separated & b2 c= b4 & b3 c= b5 & b4 /\ b5 misses b2 \/ b3 ) )
proof end;

definition
let c1 be TopStruct ;
let c2, c3 be SubSpace of c1;
pred c2,c3 are_separated means :Def8: :: TSEP_1:def 8
for b1, b2 being Subset of a1 holds
( b1 = the carrier of a2 & b2 = the carrier of a3 implies b1,b2 are_separated );
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 are_separated ) ) implies for b3, b4 being Subset of c1 holds
( b3 = the carrier of b2 & b4 = the carrier of b1 implies b3,b4 are_separated ) )
;
end;

:: deftheorem Def8 defines are_separated TSEP_1:def 8 :
for b1 being TopStruct
for b2, b3 being SubSpace of b1 holds
( b2,b3 are_separated iff for b4, b5 being Subset of b1 holds
( b4 = the carrier of b2 & b5 = the carrier of b3 implies b4,b5 are_separated ) );

notation
let c1 be TopStruct ;
let c2, c3 be SubSpace of c1;
antonym c2,c3 are_not_separated for c2,c3 are_separated ;
end;

theorem Th68: :: TSEP_1:68
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2,b3 are_separated implies b2 misses b3 )
proof end;

theorem Th69: :: TSEP_1:69
canceled;

theorem Th70: :: TSEP_1:70
for b1 being non empty TopSpace
for b2, b3 being non empty closed SubSpace of b1 holds
( b2 misses b3 iff b2,b3 are_separated )
proof end;

theorem Th71: :: TSEP_1:71
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b1 = b2 union b3 & b2,b3 are_separated implies b2 is closed SubSpace of b1 )
proof end;

theorem Th72: :: TSEP_1:72
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2 union b3 is closed SubSpace of b1 & b2,b3 are_separated implies b2 is closed SubSpace of b1 )
proof end;

theorem Th73: :: TSEP_1:73
for b1 being non empty TopSpace
for b2, b3 being non empty open SubSpace of b1 holds
( b2 misses b3 iff b2,b3 are_separated )
proof end;

theorem Th74: :: TSEP_1:74
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b1 = b2 union b3 & b2,b3 are_separated implies b2 is open SubSpace of b1 )
proof end;

theorem Th75: :: TSEP_1:75
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2 union b3 is open SubSpace of b1 & b2,b3 are_separated implies b2 is open SubSpace of b1 )
proof end;

theorem Th76: :: TSEP_1:76
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds
( b3 meets b2 & b4 meets b2 & b3,b4 are_separated implies ( b3 meet b2,b4 meet b2 are_separated & b2 meet b3,b2 meet b4 are_separated ) )
proof end;

theorem Th77: :: TSEP_1:77
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1
for b4, b5 being SubSpace of b1 holds
( b4 is SubSpace of b2 & b5 is SubSpace of b3 & b2,b3 are_separated implies b4,b5 are_separated )
proof end;

theorem Th78: :: TSEP_1:78
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds
( b2 meets b3 & b2,b4 are_separated implies b2 meet b3,b4 are_separated )
proof end;

theorem Th79: :: TSEP_1:79
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds
( ( b2,b4 are_separated & b3,b4 are_separated ) iff b2 union b3,b4 are_separated )
proof end;

theorem Th80: :: TSEP_1:80
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2,b3 are_separated iff ex b4, b5 being non empty closed SubSpace of b1 st
( b2 is SubSpace of b4 & b3 is SubSpace of b5 & b4 misses b3 & b5 misses b2 ) )
proof end;

theorem Th81: :: TSEP_1:81
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2,b3 are_separated iff ex b4, b5 being non empty closed SubSpace of b1 st
( b2 is SubSpace of b4 & b3 is SubSpace of b5 & ( b4 misses b5 or b4 meet b5 misses b2 union b3 ) ) )
proof end;

theorem Th82: :: TSEP_1:82
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2,b3 are_separated iff ex b4, b5 being non empty open SubSpace of b1 st
( b2 is SubSpace of b4 & b3 is SubSpace of b5 & b4 misses b3 & b5 misses b2 ) )
proof end;

theorem Th83: :: TSEP_1:83
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2,b3 are_separated iff ex b4, b5 being non empty open SubSpace of b1 st
( b2 is SubSpace of b4 & b3 is SubSpace of b5 & ( b4 misses b5 or b4 meet b5 misses b2 union b3 ) ) )
proof end;

definition
let c1 be TopStruct ;
let c2, c3 be SubSpace of c1;
pred c2,c3 are_weakly_separated means :Def9: :: TSEP_1:def 9
for b1, b2 being Subset of a1 holds
( b1 = the carrier of a2 & b2 = the carrier of a3 implies b1,b2 are_weakly_separated );
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 are_weakly_separated ) ) implies for b3, b4 being Subset of c1 holds
( b3 = the carrier of b2 & b4 = the carrier of b1 implies b3,b4 are_weakly_separated ) )
;
end;

:: deftheorem Def9 defines are_weakly_separated TSEP_1:def 9 :
for b1 being TopStruct
for b2, b3 being SubSpace of b1 holds
( b2,b3 are_weakly_separated iff for b4, b5 being Subset of b1 holds
( b4 = the carrier of b2 & b5 = the carrier of b3 implies b4,b5 are_weakly_separated ) );

notation
let c1 be TopStruct ;
let c2, c3 be SubSpace of c1;
antonym c2,c3 are_not_weakly_separated for c2,c3 are_weakly_separated ;
end;

theorem Th84: :: TSEP_1:84
canceled;

theorem Th85: :: TSEP_1:85
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( ( b2 misses b3 & b2,b3 are_weakly_separated ) iff b2,b3 are_separated )
proof end;

theorem Th86: :: TSEP_1:86
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2 is SubSpace of b3 implies b2,b3 are_weakly_separated )
proof end;

theorem Th87: :: TSEP_1:87
for b1 being non empty TopSpace
for b2, b3 being closed SubSpace of b1 holds b2,b3 are_weakly_separated
proof end;

theorem Th88: :: TSEP_1:88
for b1 being non empty TopSpace
for b2, b3 being open SubSpace of b1 holds b2,b3 are_weakly_separated
proof end;

theorem Th89: :: TSEP_1:89
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds
( b2,b3 are_weakly_separated implies b2 union b4,b3 union b4 are_weakly_separated )
proof end;

theorem Th90: :: TSEP_1:90
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 & b3,b2 are_weakly_separated implies ( b3 union b4,b2 union b5 are_weakly_separated & b4 union b3,b5 union b2 are_weakly_separated ) )
proof end;

theorem Th91: :: TSEP_1:91
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds
( b3 meets b4 implies ( ( b3,b2 are_weakly_separated & b4,b2 are_weakly_separated implies b3 meet b4,b2 are_weakly_separated ) & ( b2,b3 are_weakly_separated & b2,b4 are_weakly_separated implies b2,b3 meet b4 are_weakly_separated ) ) )
proof end;

theorem Th92: :: TSEP_1:92
for b1 being non empty TopSpace
for b2, b3, b4 being non empty SubSpace of b1 holds
( ( b2,b4 are_weakly_separated & b3,b4 are_weakly_separated implies b2 union b3,b4 are_weakly_separated ) & ( b4,b2 are_weakly_separated & b4,b3 are_weakly_separated implies b4,b2 union b3 are_weakly_separated ) )
proof end;

theorem Th93: :: TSEP_1:93
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2 meets b3 implies ( b2,b3 are_weakly_separated iff not ( not b2 is SubSpace of b3 & not b3 is SubSpace of b2 & ( for b4, b5 being non empty closed SubSpace of b1 holds
not ( b4 meet (b2 union b3) is SubSpace of b2 & b5 meet (b2 union b3) is SubSpace of b3 & not ( not b2 union b3 is SubSpace of b4 union b5 & ( for b6 being non empty open SubSpace of b1 holds
not ( TopStruct(# the carrier of b1,the topology of b1 #) = (b4 union b5) union b6 & b6 meet (b2 union b3) is SubSpace of b2 meet b3 ) ) ) ) ) ) ) )
proof end;

theorem Th94: :: TSEP_1:94
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b1 = b2 union b3 & b2 meets b3 implies ( b2,b3 are_weakly_separated iff not ( not b2 is SubSpace of b3 & not b3 is SubSpace of b2 & ( for b4, b5 being non empty closed SubSpace of b1 holds
not ( b4 is SubSpace of b2 & b5 is SubSpace of b3 & not ( not b1 = b4 union b5 & ( for b6 being non empty open SubSpace of b1 holds
not ( b1 = (b4 union b5) union b6 & b6 is SubSpace of b2 meet b3 ) ) ) ) ) ) ) )
proof end;

theorem Th95: :: TSEP_1:95
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b1 = b2 union b3 & b2 misses b3 implies ( b2,b3 are_weakly_separated iff ( b2 is closed SubSpace of b1 & b3 is closed SubSpace of b1 ) ) )
proof end;

theorem Th96: :: TSEP_1:96
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2 meets b3 implies ( b2,b3 are_weakly_separated iff not ( not b2 is SubSpace of b3 & not b3 is SubSpace of b2 & ( for b4, b5 being non empty open SubSpace of b1 holds
not ( b4 meet (b2 union b3) is SubSpace of b2 & b5 meet (b2 union b3) is SubSpace of b3 & not ( not b2 union b3 is SubSpace of b4 union b5 & ( for b6 being non empty closed SubSpace of b1 holds
not ( TopStruct(# the carrier of b1,the topology of b1 #) = (b4 union b5) union b6 & b6 meet (b2 union b3) is SubSpace of b2 meet b3 ) ) ) ) ) ) ) )
proof end;

theorem Th97: :: TSEP_1:97
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b1 = b2 union b3 & b2 meets b3 implies ( b2,b3 are_weakly_separated iff not ( not b2 is SubSpace of b3 & not b3 is SubSpace of b2 & ( for b4, b5 being non empty open SubSpace of b1 holds
not ( b4 is SubSpace of b2 & b5 is SubSpace of b3 & not ( not b1 = b4 union b5 & ( for b6 being non empty closed SubSpace of b1 holds
not ( b1 = (b4 union b5) union b6 & b6 is SubSpace of b2 meet b3 ) ) ) ) ) ) ) )
proof end;

theorem Th98: :: TSEP_1:98
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b1 = b2 union b3 & b2 misses b3 implies ( b2,b3 are_weakly_separated iff ( b2 is open SubSpace of b1 & b3 is open SubSpace of b1 ) ) )
proof end;

theorem Th99: :: TSEP_1:99
for b1 being non empty TopSpace
for b2, b3 being non empty SubSpace of b1 holds
( b2,b3 are_separated iff ex b4, b5 being non empty SubSpace of b1 st
( b4,b5 are_weakly_separated & b2 is SubSpace of b4 & b3 is SubSpace of b5 & ( b4 misses b5 or b4 meet b5 misses b2 union b3 ) ) )
proof end;

theorem Th100: :: TSEP_1:100
for b1 being TopStruct holds b1 | ([#] b1) = TopStruct(# the carrier of b1,the topology of b1 #)
proof end;