:: BORSUK_4 semantic presentation

registration
cluster -> non trivial Element of K18(the carrier of (TOP-REAL 2));
coherence
for b1 being Simple_closed_curve holds
not b1 is trivial
proof end;
end;

theorem Th1: :: BORSUK_4:1
for b1 being non empty set
for b2, b3 being non empty Subset of b1 holds
not ( b2 c< b3 & ( for b4 being Element of b1 holds
not ( b4 in b3 & b2 c= b3 \ {b4} ) ) )
proof end;

theorem Th2: :: BORSUK_4:2
for b1 being non empty set
for b2 being non empty Subset of b1 holds
( b2 is trivial iff ex b3 being Element of b1 st b2 = {b3} )
proof end;

registration
let c1 be non trivial 1-sorted ;
cluster non trivial Element of K18(the carrier of a1);
existence
not for b1 being Subset of c1 holds b1 is trivial
proof end;
end;

theorem Th3: :: BORSUK_4:3
for b1 being non trivial set
for b2 being set holds
not for b3 being Element of b1 holds not b3 <> b2
proof end;

registration
let c1 be non trivial set ;
cluster non trivial Element of K18(a1);
existence
not for b1 being Subset of c1 holds b1 is trivial
proof end;
end;

theorem Th4: :: BORSUK_4:4
for b1 being non trivial set
for b2 being non trivial Subset of b1
for b3 being set holds
ex b4 being Element of b1 st
( b4 in b2 & b4 <> b3 )
proof end;

theorem Th5: :: BORSUK_4:5
for b1, b2 being Function
for b3 being set holds
( b1 is one-to-one & b2 is one-to-one & (dom b1) /\ (dom b2) = {b3} & (rng b1) /\ (rng b2) = {(b1 . b3)} implies b1 +* b2 is one-to-one )
proof end;

theorem Th6: :: BORSUK_4:6
for b1, b2 being Function
for b3 being set holds
( b1 is one-to-one & b2 is one-to-one & (dom b1) /\ (dom b2) = {b3} & (rng b1) /\ (rng b2) = {(b1 . b3)} & b1 . b3 = b2 . b3 implies (b1 +* b2) " = (b1 " ) +* (b2 " ) )
proof end;

theorem Th7: :: BORSUK_4:7
for b1 being Nat
for b2 being Subset of (TOP-REAL b1)
for b3, b4 being Point of (TOP-REAL b1) holds
not ( b2 is_an_arc_of b3,b4 & b2 \ {b3} is empty )
proof end;

theorem Th8: :: BORSUK_4:8
for b1 being Nat
for b2, b3 being Point of (TOP-REAL b1) holds LSeg b2,b3 is convex
proof end;

theorem Th9: :: BORSUK_4:9
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b1 < b3 & 0 <= b4 & b4 <= 1 implies b1 <= ((1 - b4) * b2) + (b4 * b3) )
proof end;

theorem Th10: :: BORSUK_4:10
for b1 being set
for b2, b3 being real number holds
not ( b1 in [.b2,b3.] & not b1 in ].b2,b3.[ & not b1 = b2 & not b1 = b3 )
proof end;

theorem Th11: :: BORSUK_4:11
for b1, b2, b3, b4 being real number holds
not ( ].b1,b2.[ meets [.b3,b4.] & not b2 > b3 )
proof end;

theorem Th12: :: BORSUK_4:12
for b1, b2, b3, b4 being real number holds
( b2 <= b3 implies [.b1,b2.] misses ].b3,b4.[ )
proof end;

theorem Th13: :: BORSUK_4:13
for b1, b2, b3, b4 being real number holds
( b2 <= b3 implies ].b1,b2.[ misses [.b3,b4.] )
proof end;

theorem Th14: :: BORSUK_4:14
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & [.b1,b2.] c= [.b3,b4.] implies ( b3 <= b1 & b2 <= b4 ) )
proof end;

theorem Th15: :: BORSUK_4:15
for b1, b2, b3, b4 being real number holds
( b1 < b2 & ].b1,b2.[ c= [.b3,b4.] implies ( b3 <= b1 & b2 <= b4 ) )
proof end;

theorem Th16: :: BORSUK_4:16
for b1, b2, b3, b4 being real number holds
( b1 < b2 & ].b1,b2.[ c= [.b3,b4.] implies [.b1,b2.] c= [.b3,b4.] )
proof end;

theorem Th17: :: BORSUK_4:17
for b1 being Subset of I[01]
for b2, b3 being real number holds
( b2 < b3 & b1 = ].b2,b3.[ implies [.b2,b3.] c= the carrier of I[01] )
proof end;

theorem Th18: :: BORSUK_4:18
for b1 being Subset of I[01]
for b2, b3 being real number holds
( b2 < b3 & b1 = ].b2,b3.] implies [.b2,b3.] c= the carrier of I[01] )
proof end;

theorem Th19: :: BORSUK_4:19
for b1 being Subset of I[01]
for b2, b3 being real number holds
( b2 < b3 & b1 = [.b2,b3.[ implies [.b2,b3.] c= the carrier of I[01] )
proof end;

theorem Th20: :: BORSUK_4:20
for b1, b2 being real number holds
( b1 <> b2 implies Cl ].b1,b2.] = [.b1,b2.] )
proof end;

theorem Th21: :: BORSUK_4:21
for b1, b2 being real number holds
( b1 <> b2 implies Cl [.b1,b2.[ = [.b1,b2.] )
proof end;

theorem Th22: :: BORSUK_4:22
for b1 being Subset of I[01]
for b2, b3 being real number holds
( b2 < b3 & b1 = ].b2,b3.[ implies Cl b1 = [.b2,b3.] )
proof end;

theorem Th23: :: BORSUK_4:23
for b1 being Subset of I[01]
for b2, b3 being real number holds
( b2 < b3 & b1 = ].b2,b3.] implies Cl b1 = [.b2,b3.] )
proof end;

theorem Th24: :: BORSUK_4:24
for b1 being Subset of I[01]
for b2, b3 being real number holds
( b2 < b3 & b1 = [.b2,b3.[ implies Cl b1 = [.b2,b3.] )
proof end;

theorem Th25: :: BORSUK_4:25
for b1, b2 being real number holds
not ( b1 < b2 & not [.b1,b2.] <> ].b1,b2.] )
proof end;

theorem Th26: :: BORSUK_4:26
for b1, b2 being real number holds
( [.b1,b2.[ misses {b2} & ].b1,b2.] misses {b1} )
proof end;

Lemma21: for b1, b2 being real number holds
( ].b1,b2.[ misses {b2} & ].b1,b2.[ misses {b1} )
proof end;

theorem Th27: :: BORSUK_4:27
for b1, b2 being real number holds
( b1 <= b2 implies [.b1,b2.] \ {b1} = ].b1,b2.] )
proof end;

theorem Th28: :: BORSUK_4:28
for b1, b2 being real number holds
( b1 <= b2 implies [.b1,b2.] \ {b2} = [.b1,b2.[ )
proof end;

theorem Th29: :: BORSUK_4:29
for b1, b2, b3 being real number holds
( b1 < b2 & b2 < b3 implies ].b1,b2.] /\ [.b2,b3.[ = {b2} )
proof end;

theorem Th30: :: BORSUK_4:30
for b1, b2, b3 being real number holds
( [.b1,b2.[ misses [.b2,b3.] & [.b1,b2.] misses ].b2,b3.] )
proof end;

theorem Th31: :: BORSUK_4:31
for b1, b2, b3 being real number holds
( b1 <= b2 & b2 <= b3 implies [.b1,b3.] \ {b2} = [.b1,b2.[ \/ ].b2,b3.] )
proof end;

theorem Th32: :: BORSUK_4:32
for b1 being Subset of I[01]
for b2, b3 being real number holds
( b2 <= b3 & b1 = [.b2,b3.] implies ( 0 <= b2 & b3 <= 1 ) )
proof end;

theorem Th33: :: BORSUK_4:33
for b1, b2 being Subset of I[01]
for b3, b4, b5 being real number holds
( b3 < b4 & b4 < b5 & b1 = [.b3,b4.[ & b2 = ].b4,b5.] implies b1,b2 are_separated )
proof end;

theorem Th34: :: BORSUK_4:34
for b1, b2 being real number holds
( b1 <= b2 implies [.b1,b2.] = [.b1,b2.[ \/ {b2} )
proof end;

theorem Th35: :: BORSUK_4:35
for b1, b2 being real number holds
( b1 <= b2 implies [.b1,b2.] = {b1} \/ ].b1,b2.] )
proof end;

theorem Th36: :: BORSUK_4:36
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b2 < b3 & b3 <= b4 implies [.b1,b4.] = ([.b1,b2.] \/ ].b2,b3.[) \/ [.b3,b4.] )
proof end;

theorem Th37: :: BORSUK_4:37
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b2 < b3 & b3 <= b4 implies [.b1,b4.] \ ([.b1,b2.] \/ [.b3,b4.]) = ].b2,b3.[ )
proof end;

theorem Th38: :: BORSUK_4:38
for b1, b2, b3 being real number holds
( b1 < b2 & b2 < b3 implies ].b1,b2.] \/ ].b2,b3.[ = ].b1,b3.[ )
proof end;

theorem Th39: :: BORSUK_4:39
for b1, b2, b3 being real number holds
( b1 < b2 & b2 < b3 implies [.b2,b3.[ c= ].b1,b3.[ )
proof end;

theorem Th40: :: BORSUK_4:40
for b1, b2, b3 being real number holds
( b1 < b2 & b2 < b3 implies ].b1,b2.] \/ [.b2,b3.[ = ].b1,b3.[ )
proof end;

theorem Th41: :: BORSUK_4:41
for b1, b2, b3 being real number holds
( b1 < b2 & b2 < b3 implies ].b1,b3.[ \ ].b1,b2.] = ].b2,b3.[ )
proof end;

theorem Th42: :: BORSUK_4:42
for b1, b2, b3 being real number holds
( b1 < b2 & b2 < b3 implies ].b1,b3.[ \ [.b2,b3.[ = ].b1,b2.[ )
proof end;

theorem Th43: :: BORSUK_4:43
for b1, b2 being Point of I[01] holds
[.b1,b2.] is Subset of I[01] by BORSUK_1:83, RCOMP_1:16;

theorem Th44: :: BORSUK_4:44
for b1, b2 being Point of I[01] holds
].b1,b2.[ is Subset of I[01]
proof end;

theorem Th45: :: BORSUK_4:45
for b1 being real number holds
{b1} is closed-interval Subset of REAL
proof end;

theorem Th46: :: BORSUK_4:46
for b1 being non empty connected Subset of I[01]
for b2, b3, b4 being Point of I[01] holds
( b2 <= b3 & b3 <= b4 & b2 in b1 & b4 in b1 implies b3 in b1 )
proof end;

theorem Th47: :: BORSUK_4:47
for b1 being non empty connected Subset of I[01]
for b2, b3 being real number holds
( b2 in b1 & b3 in b1 implies [.b2,b3.] c= b1 )
proof end;

theorem Th48: :: BORSUK_4:48
for b1, b2 being real number
for b3 being Subset of I[01] holds
( b3 = [.b1,b2.] implies b3 is closed )
proof end;

theorem Th49: :: BORSUK_4:49
for b1, b2 being Point of I[01] holds
( b1 <= b2 implies [.b1,b2.] is non empty connected compact Subset of I[01] )
proof end;

theorem Th50: :: BORSUK_4:50
for b1 being Subset of I[01]
for b2 being Subset of REAL holds
( b2 = b1 implies ( b2 is bounded_above & b2 is bounded_below ) )
proof end;

theorem Th51: :: BORSUK_4:51
for b1 being Subset of I[01]
for b2 being Subset of REAL
for b3 being real number holds
( b3 in b2 & b2 = b1 implies ( inf b2 <= b3 & b3 <= sup b2 ) )
proof end;

Lemma45: I[01] is closed SubSpace of R^1
by TOPMETR:27, TREAL_1:5;

theorem Th52: :: BORSUK_4:52
for b1 being Subset of REAL
for b2 being Subset of I[01] holds
( b1 = b2 implies ( b1 is closed iff b2 is closed ) )
proof end;

theorem Th53: :: BORSUK_4:53
for b1 being closed-interval Subset of REAL holds inf b1 <= sup b1
proof end;

theorem Th54: :: BORSUK_4:54
for b1 being non empty connected compact Subset of I[01]
for b2 being Subset of REAL holds
( b1 = b2 & [.(inf b2),(sup b2).] c= b2 implies [.(inf b2),(sup b2).] = b2 )
proof end;

theorem Th55: :: BORSUK_4:55
for b1 being non empty connected compact Subset of I[01] holds
b1 is closed-interval Subset of REAL
proof end;

theorem Th56: :: BORSUK_4:56
for b1 being non empty connected compact Subset of I[01] holds
ex b2, b3 being Point of I[01] st
( b2 <= b3 & b1 = [.b2,b3.] )
proof end;

definition
func I(01) -> non empty strict SubSpace of I[01] means :Def1: :: BORSUK_4:def 1
the carrier of a1 = ].0,1.[;
existence
ex b1 being non empty strict SubSpace of I[01] st the carrier of b1 = ].0,1.[
proof end;
uniqueness
for b1, b2 being non empty strict SubSpace of I[01] holds
( the carrier of b1 = ].0,1.[ & the carrier of b2 = ].0,1.[ implies b1 = b2 )
by TSEP_1:5;
end;

:: deftheorem Def1 defines I(01) BORSUK_4:def 1 :
for b1 being non empty strict SubSpace of I[01] holds
( b1 = I(01) iff the carrier of b1 = ].0,1.[ );

theorem Th57: :: BORSUK_4:57
for b1 being Subset of I[01] holds
( b1 = the carrier of I(01) implies I(01) = I[01] | b1 )
proof end;

theorem Th58: :: BORSUK_4:58
the carrier of I(01) = the carrier of I[01] \ {0,1}
proof end;

theorem Th59: :: BORSUK_4:59
I(01) is open SubSpace of I[01] by Th58, JORDAN6:41, TSEP_1:def 1;

theorem Th60: :: BORSUK_4:60
for b1 being real number holds
( b1 in the carrier of I(01) iff ( 0 < b1 & b1 < 1 ) )
proof end;

theorem Th61: :: BORSUK_4:61
for b1, b2 being Point of I[01] holds
( b1 < b2 & b2 <> 1 implies ].b1,b2.] is non empty Subset of I(01) )
proof end;

theorem Th62: :: BORSUK_4:62
for b1, b2 being Point of I[01] holds
( b1 < b2 & b1 <> 0 implies [.b1,b2.[ is non empty Subset of I(01) )
proof end;

theorem Th63: :: BORSUK_4:63
for b1 being Simple_closed_curve holds (TOP-REAL 2) | R^2-unit_square ,(TOP-REAL 2) | b1 are_homeomorphic
proof end;

theorem Th64: :: BORSUK_4:64
for b1 being Nat
for b2 being non empty Subset of (TOP-REAL b1)
for b3, b4 being Point of (TOP-REAL b1) holds
( b2 is_an_arc_of b3,b4 implies I(01) ,(TOP-REAL b1) | (b2 \ {b3,b4}) are_homeomorphic )
proof end;

theorem Th65: :: BORSUK_4:65
for b1 being Nat
for b2 being Subset of (TOP-REAL b1)
for b3, b4 being Point of (TOP-REAL b1) holds
( b2 is_an_arc_of b3,b4 implies I[01] ,(TOP-REAL b1) | b2 are_homeomorphic )
proof end;

theorem Th66: :: BORSUK_4:66
for b1 being Nat
for b2, b3 being Point of (TOP-REAL b1) holds
( b2 <> b3 implies I[01] ,(TOP-REAL b1) | (LSeg b2,b3) are_homeomorphic )
proof end;

theorem Th67: :: BORSUK_4:67
for b1 being Subset of I(01) holds
( ex b2, b3 being Point of I[01] st
( b2 < b3 & b1 = [.b2,b3.] ) implies I[01] ,I(01) | b1 are_homeomorphic )
proof end;

theorem Th68: :: BORSUK_4:68
for b1 being Nat
for b2 being Subset of (TOP-REAL b1)
for b3, b4 being Point of (TOP-REAL b1)
for b5, b6 being Point of I[01] holds
not ( b2 is_an_arc_of b3,b4 & b5 < b6 & ( for b7 being non empty Subset of I[01]
for b8 being Function of (I[01] | b7),((TOP-REAL b1) | b2) holds
not ( b7 = [.b5,b6.] & b8 is_homeomorphism & b8 . b5 = b3 & b8 . b6 = b4 ) ) )
proof end;

theorem Th69: :: BORSUK_4:69
for b1 being TopSpace
for b2 being non empty TopSpace
for b3 being Function of b1,b2
for b4 being TopSpace
for b5 being Subset of b1 holds
( b3 is continuous & b4 is SubSpace of b2 implies for b6 being Function of (b1 | b5),b4 holds
( b6 = b3 | b5 implies b6 is continuous ) )
proof end;

theorem Th70: :: BORSUK_4:70
for b1 being Subset of I[01]
for b2, b3 being Point of I[01] holds
( b1 = ].b2,b3.[ implies b1 is open )
proof end;

theorem Th71: :: BORSUK_4:71
for b1 being Subset of I(01)
for b2, b3 being Point of I[01] holds
( b1 = ].b2,b3.[ implies b1 is open )
proof end;

theorem Th72: :: BORSUK_4:72
for b1 being Subset of I(01)
for b2 being Point of I[01] holds
( b1 = ].0,b2.] implies b1 is closed )
proof end;

theorem Th73: :: BORSUK_4:73
for b1 being Subset of I(01)
for b2 being Point of I[01] holds
( b1 = [.b2,1.[ implies b1 is closed )
proof end;

theorem Th74: :: BORSUK_4:74
for b1 being Nat
for b2 being Subset of (TOP-REAL b1)
for b3, b4 being Point of (TOP-REAL b1)
for b5, b6 being Point of I[01] holds
not ( b2 is_an_arc_of b3,b4 & b5 < b6 & b6 <> 1 & ( for b7 being non empty Subset of I(01)
for b8 being Function of (I(01) | b7),((TOP-REAL b1) | (b2 \ {b3})) holds
not ( b7 = ].b5,b6.] & b8 is_homeomorphism & b8 . b6 = b4 ) ) )
proof end;

theorem Th75: :: BORSUK_4:75
for b1 being Nat
for b2 being Subset of (TOP-REAL b1)
for b3, b4 being Point of (TOP-REAL b1)
for b5, b6 being Point of I[01] holds
not ( b2 is_an_arc_of b3,b4 & b5 < b6 & b5 <> 0 & ( for b7 being non empty Subset of I(01)
for b8 being Function of (I(01) | b7),((TOP-REAL b1) | (b2 \ {b4})) holds
not ( b7 = [.b5,b6.[ & b8 is_homeomorphism & b8 . b5 = b3 ) ) )
proof end;

theorem Th76: :: BORSUK_4:76
for b1 being Nat
for b2, b3 being Subset of (TOP-REAL b1)
for b4, b5 being Point of (TOP-REAL b1) holds
( b2 is_an_arc_of b4,b5 & b3 is_an_arc_of b5,b4 & b2 /\ b3 = {b4,b5} & b4 <> b5 implies I(01) ,(TOP-REAL b1) | ((b2 \ {b4}) \/ (b3 \ {b4})) are_homeomorphic )
proof end;

theorem Th77: :: BORSUK_4:77
for b1 being Simple_closed_curve
for b2 being Point of (TOP-REAL 2) holds
( b2 in b1 implies (TOP-REAL 2) | (b1 \ {b2}), I(01) are_homeomorphic )
proof end;

theorem Th78: :: BORSUK_4:78
for b1 being Simple_closed_curve
for b2, b3 being Point of (TOP-REAL 2) holds
( b2 in b1 & b3 in b1 implies (TOP-REAL 2) | (b1 \ {b2}),(TOP-REAL 2) | (b1 \ {b3}) are_homeomorphic )
proof end;

theorem Th79: :: BORSUK_4:79
for b1 being Nat
for b2 being non empty Subset of (TOP-REAL b1)
for b3 being Subset of I(01) holds
not ( ex b4, b5 being Point of I[01] st
( b4 < b5 & b3 = [.b4,b5.] ) & I(01) | b3,(TOP-REAL b1) | b2 are_homeomorphic & ( for b4, b5 being Point of (TOP-REAL b1) holds
not b2 is_an_arc_of b4,b5 ) )
proof end;

theorem Th80: :: BORSUK_4:80
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Function of ((TOP-REAL 2) | b1),I(01)
for b3 being non empty Subset of (TOP-REAL 2) holds
not ( b2 is_homeomorphism & b3 c= b1 & ex b4, b5 being Point of I[01] st
( b4 < b5 & b2 .: b3 = [.b4,b5.] ) & ( for b4, b5 being Point of (TOP-REAL 2) holds
not b3 is_an_arc_of b4,b5 ) )
proof end;

theorem Th81: :: BORSUK_4:81
for b1 being Simple_closed_curve
for b2 being non empty connected compact Subset of (TOP-REAL 2) holds
not ( b2 c= b1 & not b2 = b1 & ( for b3, b4 being Point of (TOP-REAL 2) holds
not b2 is_an_arc_of b3,b4 ) & ( for b3 being Point of (TOP-REAL 2) holds
not b2 = {b3} ) )
proof end;

registration
cluster the carrier of I[01] -> real-membered ;
coherence
the carrier of I[01] is real-membered
by BORSUK_1:83;
end;

Lemma72: for b1, b2, b3, b4 being real number holds
( ( b1 <= b2 or b3 <= b4 ) & [.b1,b2.] = [.b3,b4.] implies ( b1 = b3 & b2 = b4 ) )
proof end;

theorem Th82: :: BORSUK_4:82
for b1 being non empty compact Subset of I[01] holds
not ( b1 c= ].0,1.[ & ( for b2 being closed-interval Subset of REAL holds
not ( b1 c= b2 & b2 c= ].0,1.[ & inf b1 = inf b2 & sup b1 = sup b2 ) ) )
proof end;

theorem Th83: :: BORSUK_4:83
for b1 being non empty compact Subset of I[01] holds
not ( b1 c= ].0,1.[ & ( for b2, b3 being Point of I[01] holds
not ( b2 <= b3 & b1 c= [.b2,b3.] & [.b2,b3.] c= ].0,1.[ ) ) )
proof end;

theorem Th84: :: BORSUK_4:84
for b1 being Simple_closed_curve
for b2 being closed Subset of (TOP-REAL 2) holds
not ( b2 c= b1 & b2 <> b1 & ( for b3, b4 being Point of (TOP-REAL 2)
for b5 being Subset of (TOP-REAL 2) holds
not ( b5 is_an_arc_of b3,b4 & b2 c= b5 & b5 c= b1 ) ) )
proof end;