Article BORSUK_4, MML version 4.99.1005
:: BORSUK_4:condreg 1
registration
cluster being_simple_closed_curve -> non trivial (Element of bool the carrier of TOP-REAL 2);
end;
:: BORSUK_4:th 1
theorem
for b1 being non empty set
for b2, b3 being non empty Element of bool b1
st b2 c< b3
holds ex b4 being Element of b1 st
b4 in b3 & b2 c= b3 \ {b4};
:: BORSUK_4:th 2
theorem
for b1 being non empty set
for b2 being non empty Element of bool b1 holds
b2 is trivial
iff
ex b3 being Element of b1 st
b2 = {b3};
:: BORSUK_4:exreg 1
registration
let a1 be non trivial 1-sorted;
cluster non trivial Element of bool the carrier of a1;
end;
:: BORSUK_4:th 3
theorem
for b1 being non trivial set
for b2 being set holds
ex b3 being Element of b1 st
b3 <> b2;
:: BORSUK_4:exreg 2
registration
let a1 be non trivial set;
cluster non trivial Element of bool a1;
end;
:: BORSUK_4:th 4
theorem
for b1 being non trivial set
for b2 being non trivial Element of bool b1
for b3 being set holds
ex b4 being Element of b1 st
b4 in b2 & b4 <> b3;
:: BORSUK_4:th 5
theorem
for b1, b2 being Relation-like Function-like set
for b3 being set
st b1 is one-to-one &
b2 is one-to-one &
(proj1 b1) /\ proj1 b2 = {b3} &
(proj2 b1) /\ proj2 b2 = {b1 . b3}
holds b1 +* b2 is one-to-one;
:: BORSUK_4:th 6
theorem
for b1, b2 being Relation-like Function-like set
for b3 being set
st b1 is one-to-one &
b2 is one-to-one &
(proj1 b1) /\ proj1 b2 = {b3} &
(proj2 b1) /\ proj2 b2 = {b1 . b3} &
b1 . b3 = b2 . b3
holds (b1 +* b2) " = b1 " +* (b2 ");
:: BORSUK_4:th 7
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1
st b2 is_an_arc_of b3,b4
holds b2 \ {b3} is not empty;
:: BORSUK_4:th 8
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
LSeg(b2,b3) is convex(b1);
:: BORSUK_4:th 9
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b1 < b3 & 0 <= b4 & b4 <= 1
holds b1 <= ((1 - b4) * b2) + (b4 * b3);
:: BORSUK_4:th 10
theorem
for b1 being set
for b2, b3 being real set
st b1 in [.b2,b3.] & not b1 in ].b2,b3.[ & b1 <> b2
holds b1 = b3;
:: BORSUK_4:th 11
theorem
for b1, b2, b3, b4 being real set
st ].b1,b2.[ meets [.b3,b4.]
holds b3 < b2;
:: BORSUK_4:th 12
theorem
for b1, b2, b3, b4 being real set
st b2 <= b3
holds [.b1,b2.] misses ].b3,b4.[;
:: BORSUK_4:th 13
theorem
for b1, b2, b3, b4 being real set
st b2 <= b3
holds ].b1,b2.[ misses [.b3,b4.];
:: BORSUK_4:th 14
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & [.b1,b2.] c= [.b3,b4.]
holds b3 <= b1 & b2 <= b4;
:: BORSUK_4:th 15
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & ].b1,b2.[ c= [.b3,b4.]
holds b3 <= b1 & b2 <= b4;
:: BORSUK_4:th 16
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & ].b1,b2.[ c= [.b3,b4.]
holds [.b1,b2.] c= [.b3,b4.];
:: BORSUK_4:th 17
theorem
for b1 being Element of bool the carrier of I[01]
for b2, b3 being real set
st b2 < b3 & b1 = ].b2,b3.[
holds [.b2,b3.] c= the carrier of I[01];
:: BORSUK_4:th 18
theorem
for b1 being Element of bool the carrier of I[01]
for b2, b3 being real set
st b2 < b3 & b1 = ].b2,b3.]
holds [.b2,b3.] c= the carrier of I[01];
:: BORSUK_4:th 19
theorem
for b1 being Element of bool the carrier of I[01]
for b2, b3 being real set
st b2 < b3 & b1 = [.b2,b3.[
holds [.b2,b3.] c= the carrier of I[01];
:: BORSUK_4:th 20
theorem
for b1, b2 being real set
st b1 <> b2
holds Cl ].b1,b2.] = [.b1,b2.];
:: BORSUK_4:th 21
theorem
for b1, b2 being real set
st b1 <> b2
holds Cl [.b1,b2.[ = [.b1,b2.];
:: BORSUK_4:th 22
theorem
for b1 being Element of bool the carrier of I[01]
for b2, b3 being real set
st b2 < b3 & b1 = ].b2,b3.[
holds Cl b1 = [.b2,b3.];
:: BORSUK_4:th 23
theorem
for b1 being Element of bool the carrier of I[01]
for b2, b3 being real set
st b2 < b3 & b1 = ].b2,b3.]
holds Cl b1 = [.b2,b3.];
:: BORSUK_4:th 24
theorem
for b1 being Element of bool the carrier of I[01]
for b2, b3 being real set
st b2 < b3 & b1 = [.b2,b3.[
holds Cl b1 = [.b2,b3.];
:: BORSUK_4:th 25
theorem
for b1, b2 being real set
st b1 < b2
holds [.b1,b2.] <> ].b1,b2.];
:: BORSUK_4:th 26
theorem
for b1, b2 being real set holds
[.b1,b2.[ misses {b2} & ].b1,b2.] misses {b1};
:: BORSUK_4:th 27
theorem
for b1, b2 being real set
st b1 <= b2
holds [.b1,b2.] \ {b1} = ].b1,b2.];
:: BORSUK_4:th 28
theorem
for b1, b2 being real set
st b1 <= b2
holds [.b1,b2.] \ {b2} = [.b1,b2.[;
:: BORSUK_4:th 29
theorem
for b1, b2, b3 being real set
st b1 < b2 & b2 < b3
holds ].b1,b2.] /\ [.b2,b3.[ = {b2};
:: BORSUK_4:th 30
theorem
for b1, b2, b3 being real set holds
[.b1,b2.[ misses [.b2,b3.] & [.b1,b2.] misses ].b2,b3.];
:: BORSUK_4:th 31
theorem
for b1, b2, b3 being real set
st b1 <= b2 & b2 <= b3
holds [.b1,b3.] \ {b2} = [.b1,b2.[ \/ ].b2,b3.];
:: BORSUK_4:th 32
theorem
for b1 being Element of bool the carrier of I[01]
for b2, b3 being real set
st b2 <= b3 & b1 = [.b2,b3.]
holds 0 <= b2 & b3 <= 1;
:: BORSUK_4:th 33
theorem
for b1, b2 being Element of bool the carrier of I[01]
for b3, b4, b5 being real set
st b3 < b4 & b4 < b5 & b1 = [.b3,b4.[ & b2 = ].b4,b5.]
holds b1,b2 are_separated;
:: BORSUK_4:th 34
theorem
for b1, b2 being real set
st b1 <= b2
holds [.b1,b2.] = [.b1,b2.[ \/ {b2};
:: BORSUK_4:th 35
theorem
for b1, b2 being real set
st b1 <= b2
holds [.b1,b2.] = {b1} \/ ].b1,b2.];
:: BORSUK_4:th 36
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b2 < b3 & b3 <= b4
holds [.b1,b4.] = ([.b1,b2.] \/ ].b2,b3.[) \/ [.b3,b4.];
:: BORSUK_4:th 37
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b2 < b3 & b3 <= b4
holds [.b1,b4.] \ ([.b1,b2.] \/ [.b3,b4.]) = ].b2,b3.[;
:: BORSUK_4:th 38
theorem
for b1, b2, b3 being real set
st b1 < b2 & b2 < b3
holds ].b1,b2.] \/ ].b2,b3.[ = ].b1,b3.[;
:: BORSUK_4:th 39
theorem
for b1, b2, b3 being real set
st b1 < b2 & b2 < b3
holds [.b2,b3.[ c= ].b1,b3.[;
:: BORSUK_4:th 40
theorem
for b1, b2, b3 being real set
st b1 < b2 & b2 < b3
holds ].b1,b2.] \/ [.b2,b3.[ = ].b1,b3.[;
:: BORSUK_4:th 41
theorem
for b1, b2, b3 being real set
st b1 < b2 & b2 < b3
holds ].b1,b3.[ \ ].b1,b2.] = ].b2,b3.[;
:: BORSUK_4:th 42
theorem
for b1, b2, b3 being real set
st b1 < b2 & b2 < b3
holds ].b1,b3.[ \ [.b2,b3.[ = ].b1,b2.[;
:: BORSUK_4:th 43
theorem
for b1, b2 being Element of the carrier of I[01] holds
[.b1,b2.] is Element of bool the carrier of I[01];
:: BORSUK_4:th 44
theorem
for b1, b2 being Element of the carrier of I[01] holds
].b1,b2.[ is Element of bool the carrier of I[01];
:: BORSUK_4:th 45
theorem
for b1 being real set holds
{b1} is closed-interval Element of bool REAL;
:: BORSUK_4:th 46
theorem
for b1 being non empty connected Element of bool the carrier of I[01]
for b2, b3, b4 being Element of the carrier of I[01]
st b2 <= b3 & b3 <= b4 & b2 in b1 & b4 in b1
holds b3 in b1;
:: BORSUK_4:th 47
theorem
for b1 being non empty connected Element of bool the carrier of I[01]
for b2, b3 being real set
st b2 in b1 & b3 in b1
holds [.b2,b3.] c= b1;
:: BORSUK_4:th 48
theorem
for b1, b2 being real set
for b3 being Element of bool the carrier of I[01]
st b3 = [.b1,b2.]
holds b3 is closed(I[01]);
:: BORSUK_4:th 49
theorem
for b1, b2 being Element of the carrier of I[01]
st b1 <= b2
holds [.b1,b2.] is non empty connected compact Element of bool the carrier of I[01];
:: BORSUK_4:th 50
theorem
for b1 being Element of bool the carrier of I[01]
for b2 being Element of bool REAL
st b2 = b1
holds b2 is bounded_above & b2 is bounded_below;
:: BORSUK_4:th 51
theorem
for b1 being Element of bool the carrier of I[01]
for b2 being Element of bool REAL
for b3 being real set
st b3 in b2 & b2 = b1
holds inf b2 <= b3 & b3 <= sup b2;
:: BORSUK_4:th 52
theorem
for b1 being Element of bool REAL
for b2 being Element of bool the carrier of I[01]
st b1 = b2
holds b1 is closed
iff
b2 is closed(I[01]);
:: BORSUK_4:th 53
theorem
for b1 being closed-interval Element of bool REAL holds
inf b1 <= sup b1;
:: BORSUK_4:th 54
theorem
for b1 being non empty connected compact Element of bool the carrier of I[01]
for b2 being Element of bool REAL
st b1 = b2 & [.inf b2,sup b2.] c= b2
holds [.inf b2,sup b2.] = b2;
:: BORSUK_4:th 55
theorem
for b1 being non empty connected compact Element of bool the carrier of I[01] holds
b1 is closed-interval Element of bool REAL;
:: BORSUK_4:th 56
theorem
for b1 being non empty connected compact Element of bool the carrier of I[01] holds
ex b2, b3 being Element of the carrier of I[01] st
b2 <= b3 & b1 = [.b2,b3.];
:: BORSUK_4:funcnot 1 => BORSUK_4:func 1
definition
func I(01) -> non empty strict SubSpace of I[01] means
the carrier of it = ].0,1.[;
end;
:: BORSUK_4:def 1
theorem
for b1 being non empty strict SubSpace of I[01] holds
b1 = I(01)
iff
the carrier of b1 = ].0,1.[;
:: BORSUK_4:th 57
theorem
for b1 being Element of bool the carrier of I[01]
st b1 = the carrier of I(01)
holds I(01) = I[01] | b1;
:: BORSUK_4:th 58
theorem
the carrier of I(01) = (the carrier of I[01]) \ {0,1};
:: BORSUK_4:th 59
theorem
I(01) is open SubSpace of I[01];
:: BORSUK_4:th 60
theorem
for b1 being real set holds
b1 in the carrier of I(01)
iff
0 < b1 & b1 < 1;
:: BORSUK_4:th 61
theorem
for b1, b2 being Element of the carrier of I[01]
st b1 < b2 & b2 <> 1
holds ].b1,b2.] is non empty Element of bool the carrier of I(01);
:: BORSUK_4:th 62
theorem
for b1, b2 being Element of the carrier of I[01]
st b1 < b2 & b1 <> 0
holds [.b1,b2.[ is non empty Element of bool the carrier of I(01);
:: BORSUK_4:th 63
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
(TOP-REAL 2) | R^2-unit_square,(TOP-REAL 2) | b1 are_homeomorphic;
:: BORSUK_4:th 64
theorem
for b1 being Element of NAT
for b2 being non empty Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1
st b2 is_an_arc_of b3,b4
holds I(01),(TOP-REAL b1) | (b2 \ {b3,b4}) are_homeomorphic;
:: BORSUK_4:th 65
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1
st b2 is_an_arc_of b3,b4
holds I[01],(TOP-REAL b1) | b2 are_homeomorphic;
:: BORSUK_4:th 66
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
st b2 <> b3
holds I[01],(TOP-REAL b1) | LSeg(b2,b3) are_homeomorphic;
:: BORSUK_4:th 67
theorem
for b1 being Element of bool the carrier of I(01)
st ex b2, b3 being Element of the carrier of I[01] st
b2 < b3 & b1 = [.b2,b3.]
holds I[01],I(01) | b1 are_homeomorphic;
:: BORSUK_4:th 68
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1
for b5, b6 being Element of the carrier of I[01]
st b2 is_an_arc_of b3,b4 & b5 < b6
holds ex b7 being non empty Element of bool the carrier of I[01] st
ex b8 being Function-like quasi_total Relation of the carrier of I[01] | b7,the carrier of (TOP-REAL b1) | b2 st
b7 = [.b5,b6.] &
b8 is being_homeomorphism(I[01] | b7, (TOP-REAL b1) | b2) &
b8 . b5 = b3 &
b8 . b6 = b4;
:: BORSUK_4:th 69
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being TopSpace-like TopStruct
for b5 being Element of bool the carrier of b1
st b3 is continuous(b1, b2) & b4 is SubSpace of b2
for b6 being Function-like quasi_total Relation of the carrier of b1 | b5,the carrier of b4
st b6 = b3 | b5
holds b6 is continuous(b1 | b5, b4);
:: BORSUK_4:th 70
theorem
for b1 being Element of bool the carrier of I[01]
for b2, b3 being Element of the carrier of I[01]
st b1 = ].b2,b3.[
holds b1 is open(I[01]);
:: BORSUK_4:th 71
theorem
for b1 being Element of bool the carrier of I(01)
for b2, b3 being Element of the carrier of I[01]
st b1 = ].b2,b3.[
holds b1 is open(I(01));
:: BORSUK_4:th 72
theorem
for b1 being Element of bool the carrier of I(01)
for b2 being Element of the carrier of I[01]
st b1 = ].0,b2.]
holds b1 is closed(I(01));
:: BORSUK_4:th 73
theorem
for b1 being Element of bool the carrier of I(01)
for b2 being Element of the carrier of I[01]
st b1 = [.b2,1.[
holds b1 is closed(I(01));
:: BORSUK_4:th 74
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1
for b5, b6 being Element of the carrier of I[01]
st b2 is_an_arc_of b3,b4 & b5 < b6 & b6 <> 1
holds ex b7 being non empty Element of bool the carrier of I(01) st
ex b8 being Function-like quasi_total Relation of the carrier of I(01) | b7,the carrier of (TOP-REAL b1) | (b2 \ {b3}) st
b7 = ].b5,b6.] &
b8 is being_homeomorphism(I(01) | b7, (TOP-REAL b1) | (b2 \ {b3})) &
b8 . b6 = b4;
:: BORSUK_4:th 75
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1
for b5, b6 being Element of the carrier of I[01]
st b2 is_an_arc_of b3,b4 & b5 < b6 & b5 <> 0
holds ex b7 being non empty Element of bool the carrier of I(01) st
ex b8 being Function-like quasi_total Relation of the carrier of I(01) | b7,the carrier of (TOP-REAL b1) | (b2 \ {b4}) st
b7 = [.b5,b6.[ &
b8 is being_homeomorphism(I(01) | b7, (TOP-REAL b1) | (b2 \ {b4})) &
b8 . b5 = b3;
:: BORSUK_4:th 76
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4, b5 being Element of the carrier of TOP-REAL b1
st b2 is_an_arc_of b4,b5 & b3 is_an_arc_of b5,b4 & b2 /\ b3 = {b4,b5} & b4 <> b5
holds I(01),(TOP-REAL b1) | ((b2 \ {b4}) \/ (b3 \ {b4})) are_homeomorphic;
:: BORSUK_4:th 77
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in b1
holds (TOP-REAL 2) | (b1 \ {b2}),I(01) are_homeomorphic;
:: BORSUK_4:th 78
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
st b2 in b1 & b3 in b1
holds (TOP-REAL 2) | (b1 \ {b2}),(TOP-REAL 2) | (b1 \ {b3}) are_homeomorphic;
:: BORSUK_4:th 79
theorem
for b1 being Element of NAT
for b2 being non empty Element of bool the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of I(01)
st (ex b4, b5 being Element of the carrier of I[01] st
b4 < b5 & b3 = [.b4,b5.]) &
I(01) | b3,(TOP-REAL b1) | b2 are_homeomorphic
holds ex b4, b5 being Element of the carrier of TOP-REAL b1 st
b2 is_an_arc_of b4,b5;
:: BORSUK_4:th 80
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of I(01)
for b3 being non empty Element of bool the carrier of TOP-REAL 2
st b2 is being_homeomorphism((TOP-REAL 2) | b1, I(01)) &
b3 c= b1 &
(ex b4, b5 being Element of the carrier of I[01] st
b4 < b5 & b2 .: b3 = [.b4,b5.])
holds ex b4, b5 being Element of the carrier of TOP-REAL 2 st
b3 is_an_arc_of b4,b5;
:: BORSUK_4:th 81
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being non empty connected compact Element of bool the carrier of TOP-REAL 2
st b2 c= b1 &
b2 <> b1 &
(for b3, b4 being Element of the carrier of TOP-REAL 2 holds
not b2 is_an_arc_of b3,b4)
holds ex b3 being Element of the carrier of TOP-REAL 2 st
b2 = {b3};
:: BORSUK_4:attrnot 1 => BORSUK_4:attr 1
definition
let a1 be 1-sorted;
attr a1 is real-membered means
the carrier of a1 is real-membered;
end;
:: BORSUK_4:dfs 2
definiens
let a1 be 1-sorted;
To prove
a1 is real-membered
it is sufficient to prove
thus the carrier of a1 is real-membered;
:: BORSUK_4:def 2
theorem
for b1 being 1-sorted holds
b1 is real-membered
iff
the carrier of b1 is real-membered;
:: BORSUK_4:funcreg 1
registration
cluster I[01] -> real-membered;
end;
:: BORSUK_4:exreg 3
registration
cluster real-membered 1-sorted;
end;
:: BORSUK_4:funcreg 2
registration
let a1 be real-membered 1-sorted;
cluster the carrier of a1 -> real-membered;
end;
:: BORSUK_4:th 82
theorem
for b1 being non empty compact Element of bool the carrier of I[01]
st b1 c= ].0,1.[
holds ex b2 being closed-interval Element of bool REAL st
b1 c= b2 & b2 c= ].0,1.[ & lower_bound b1 = inf b2 & upper_bound b1 = sup b2;
:: BORSUK_4:th 83
theorem
for b1 being non empty compact Element of bool the carrier of I[01]
st b1 c= ].0,1.[
holds ex b2, b3 being Element of the carrier of I[01] st
b2 <= b3 &
b1 c= [.b2,b3.] &
[.b2,b3.] c= ].0,1.[;
:: BORSUK_4:th 84
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being closed Element of bool the carrier of TOP-REAL 2
st b2 c< b1
holds ex b3, b4 being Element of the carrier of TOP-REAL 2 st
ex b5 being Element of bool the carrier of TOP-REAL 2 st
b5 is_an_arc_of b3,b4 & b2 c= b5 & b5 c= b1;