Article BORSUK_5, MML version 4.99.1005
:: BORSUK_5:th 2
theorem
for b1, b2, b3 being set
st b1 c= b2 & b2 c= b1 \/ {b3} & b1 \/ {b3} <> b2
holds b1 = b2;
:: BORSUK_5:th 3
theorem
for b1, b2, b3, b4, b5, b6 being set holds
{b1,b2,b3,b4,b5,b6} = {b1,b3,b6} \/ {b2,b4,b5};
:: BORSUK_5:th 4
theorem
for b1, b2, b3, b4, b5, b6 being set
st b1,b2,b3,b4,b5,b6 are_mutually_different
holds card {b1,b2,b3,b4,b5,b6} = 6;
:: BORSUK_5:th 5
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
st b1,b2,b3,b4,b5,b6,b7 are_mutually_different
holds card {b1,b2,b3,b4,b5,b6,b7} = 7;
:: BORSUK_5:th 6
theorem
for b1, b2, b3, b4, b5, b6 being set
st {b1,b2,b3} misses {b4,b5,b6}
holds b1 <> b4 & b1 <> b5 & b1 <> b6 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b3 <> b4 & b3 <> b5 & b3 <> b6;
:: BORSUK_5:th 7
theorem
for b1, b2, b3, b4, b5, b6 being set
st b1,b2,b3 are_mutually_different &
b4,b5,b6 are_mutually_different &
{b1,b2,b3} misses {b4,b5,b6}
holds b1,b2,b3,b4,b5,b6 are_mutually_different;
:: BORSUK_5:th 8
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
st b1,b2,b3,b4,b5,b6 are_mutually_different &
{b1,b2,b3,b4,b5,b6} misses {b7}
holds b1,b2,b3,b4,b5,b6,b7 are_mutually_different;
:: BORSUK_5:th 9
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
st b1,b2,b3,b4,b5,b6,b7 are_mutually_different
holds b7,b1,b2,b3,b4,b5,b6 are_mutually_different;
:: BORSUK_5:th 10
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
st b1,b2,b3,b4,b5,b6,b7 are_mutually_different
holds b1,b2,b5,b3,b6,b7,b4 are_mutually_different;
:: BORSUK_5:funcreg 1
registration
cluster R^1 -> strict TopSpace-like arcwise_connected;
end;
:: BORSUK_5:exreg 1
registration
cluster non empty TopSpace-like connected TopStruct;
end;
:: BORSUK_5:th 14
theorem
for b1, b2 being real set holds
b1 in ].b2,+infty.[
iff
b2 < b1;
:: BORSUK_5:th 15
theorem
for b1, b2 being real set holds
b1 in [.b2,+infty.[
iff
b2 <= b1;
:: BORSUK_5:th 16
theorem
for b1, b2 being real set holds
b1 in ].-infty,b2.]
iff
b1 <= b2;
:: BORSUK_5:th 17
theorem
for b1, b2 being real set holds
b1 in ].-infty,b2.[
iff
b1 < b2;
:: BORSUK_5:th 18
theorem
for b1 being real set holds
REAL \ {b1} = ].-infty,b1.[ \/ ].b1,+infty.[;
:: BORSUK_5:th 19
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & b2 <= b3
holds [.b1,b2.] misses ].b3,b4.];
:: BORSUK_5:th 20
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & b2 <= b3
holds [.b1,b2.[ misses [.b3,b4.];
:: BORSUK_5:th 21
theorem
for b1, b2 being Element of bool the carrier of R^1
for b3, b4, b5, b6 being real set
st b3 < b4 & b4 <= b5 & b5 < b6 & b1 = [.b3,b4.[ & b2 = ].b5,b6.]
holds b1,b2 are_separated;
:: BORSUK_5:th 26
theorem
for b1 being real set holds
].-infty,b1.] misses ].b1,+infty.[;
:: BORSUK_5:th 27
theorem
for b1 being real set holds
].-infty,b1.[ misses [.b1,+infty.[;
:: BORSUK_5:th 28
theorem
for b1, b2, b3 being real set
st b1 <= b3 & b3 <= b2
holds [.b1,b2.] \/ [.b3,+infty.[ = [.b1,+infty.[;
:: BORSUK_5:th 29
theorem
for b1, b2, b3 being real set
st b1 <= b3 & b3 <= b2
holds ].-infty,b3.] \/ [.b1,b2.] = ].-infty,b2.];
:: BORSUK_5:condreg 1
registration
cluster -> real (Element of RAT);
end;
:: BORSUK_5:condreg 2
registration
cluster -> real (Element of the carrier of RealSpace);
end;
:: BORSUK_5:th 33
theorem
for b1 being Element of bool the carrier of R^1
for b2 being Element of the carrier of RealSpace holds
b2 in Cl b1
iff
for b3 being real set
st 0 < b3
holds Ball(b2,b3) meets b1;
:: BORSUK_5:th 34
theorem
for b1, b2 being Element of the carrier of RealSpace
st b1 <= b2
holds dist(b1,b2) = b2 - b1;
:: BORSUK_5:th 35
theorem
for b1 being Element of bool the carrier of R^1
st b1 = RAT
holds Cl b1 = the carrier of R^1;
:: BORSUK_5:th 36
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
st b1 = ].b2,b3.[ & b2 <> b3
holds Cl b1 = [.b2,b3.];
:: BORSUK_5:funcreg 2
registration
cluster number_e -> real non rational;
end;
:: BORSUK_5:funcnot 1 => BORSUK_5:func 1
definition
func IRRAT -> Element of bool REAL equals
REAL \ RAT;
end;
:: BORSUK_5:def 3
theorem
IRRAT = REAL \ RAT;
:: BORSUK_5:funcnot 2 => BORSUK_5:func 2
definition
let a1, a2 be real set;
func RAT(A1,A2) -> Element of bool REAL equals
RAT /\ ].a1,a2.[;
end;
:: BORSUK_5:def 4
theorem
for b1, b2 being real set holds
RAT(b1,b2) = RAT /\ ].b1,b2.[;
:: BORSUK_5:funcnot 3 => BORSUK_5:func 3
definition
let a1, a2 be real set;
func IRRAT(A1,A2) -> Element of bool REAL equals
IRRAT /\ ].a1,a2.[;
end;
:: BORSUK_5:def 5
theorem
for b1, b2 being real set holds
IRRAT(b1,b2) = IRRAT /\ ].b1,b2.[;
:: BORSUK_5:th 37
theorem
for b1 being real set holds
b1 is not rational
iff
b1 in IRRAT;
:: BORSUK_5:exreg 2
registration
cluster complex real ext-real non rational set;
end;
:: BORSUK_5:funcreg 3
registration
cluster IRRAT -> non empty;
end;
:: BORSUK_5:th 38
theorem
for b1 being rational set
for b2 being real non rational set holds
b1 + b2 is not rational;
:: BORSUK_5:th 39
theorem
for b1 being real non rational set holds
- b1 is not rational;
:: BORSUK_5:th 40
theorem
for b1 being rational set
for b2 being real non rational set holds
b1 - b2 is not rational;
:: BORSUK_5:th 41
theorem
for b1 being rational set
for b2 being real non rational set holds
b2 - b1 is not rational;
:: BORSUK_5:th 42
theorem
for b1 being rational set
for b2 being real non rational set
st b1 <> 0
holds b1 * b2 is not rational;
:: BORSUK_5:th 43
theorem
for b1 being rational set
for b2 being real non rational set
st b1 <> 0
holds b2 / b1 is not rational;
:: BORSUK_5:condreg 3
registration
cluster real non rational -> non empty (set);
end;
:: BORSUK_5:th 44
theorem
for b1 being rational set
for b2 being real non rational set
st b1 <> 0
holds b1 / b2 is not rational;
:: BORSUK_5:th 45
theorem
for b1 being real non rational set holds
frac b1 is not rational;
:: BORSUK_5:funcreg 4
registration
let a1 be real non rational set;
cluster frac a1 -> non rational;
end;
:: BORSUK_5:funcreg 5
registration
let a1 be real non rational set;
cluster - a1 -> complex non rational;
end;
:: BORSUK_5:funcreg 6
registration
let a1 be rational set;
let a2 be real non rational set;
cluster a1 + a2 -> non rational;
end;
:: BORSUK_5:funcreg 7
registration
let a1 be rational set;
let a2 be real non rational set;
cluster a2 + a1 -> non rational;
end;
:: BORSUK_5:funcreg 8
registration
let a1 be rational set;
let a2 be real non rational set;
cluster a1 - a2 -> non rational;
end;
:: BORSUK_5:funcreg 9
registration
let a1 be rational set;
let a2 be real non rational set;
cluster a2 - a1 -> non rational;
end;
:: BORSUK_5:exreg 3
registration
cluster non empty complex real ext-real rational set;
end;
:: BORSUK_5:funcreg 10
registration
let a1 be non empty rational set;
let a2 be real non rational set;
cluster a1 * a2 -> non rational;
end;
:: BORSUK_5:funcreg 11
registration
let a1 be non empty rational set;
let a2 be real non rational set;
cluster a2 * a1 -> non rational;
end;
:: BORSUK_5:funcreg 12
registration
let a1 be non empty rational set;
let a2 be real non rational set;
cluster a1 / a2 -> non rational;
end;
:: BORSUK_5:funcreg 13
registration
let a1 be non empty rational set;
let a2 be real non rational set;
cluster a2 / a1 -> non rational;
end;
:: BORSUK_5:th 47
theorem
for b1, b2 being real set
st b1 < b2
holds ex b3, b4 being rational set st
b1 < b3 & b3 < b4 & b4 < b2;
:: BORSUK_5:th 50
theorem
for b1, b2 being real set
st b1 < b2
holds ex b3 being real non rational set st
b1 < b3 & b3 < b2;
:: BORSUK_5:th 51
theorem
for b1 being Element of bool the carrier of R^1
st b1 = IRRAT
holds Cl b1 = the carrier of R^1;
:: BORSUK_5:th 52
theorem
for b1, b2, b3 being real set
st b1 < b2
holds b3 in RAT(b1,b2)
iff
b3 is rational & b1 < b3 & b3 < b2;
:: BORSUK_5:th 53
theorem
for b1, b2, b3 being real set
st b1 < b2
holds b3 in IRRAT(b1,b2)
iff
b3 is not rational & b1 < b3 & b3 < b2;
:: BORSUK_5:th 54
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
st b2 < b3 & b1 = RAT(b2,b3)
holds Cl b1 = [.b2,b3.];
:: BORSUK_5:th 55
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
st b2 < b3 & b1 = IRRAT(b2,b3)
holds Cl b1 = [.b2,b3.];
:: BORSUK_5:th 56
theorem
for b1 being TopSpace-like connected TopStruct
for b2 being open closed Element of bool the carrier of b1
st b2 <> {}
holds b2 = [#] b1;
:: BORSUK_5:th 57
theorem
for b1 being Element of bool the carrier of R^1
st b1 is closed(R^1) & b1 is open(R^1) & b1 <> {}
holds b1 = REAL;
:: BORSUK_5:th 58
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
st b1 = [.b2,b3.[ & b2 <> b3
holds Cl b1 = [.b2,b3.];
:: BORSUK_5:th 59
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
st b1 = ].b2,b3.] & b2 <> b3
holds Cl b1 = [.b2,b3.];
:: BORSUK_5:th 60
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3, b4 being real set
st b1 = [.b2,b3.[ \/ ].b3,b4.] &
b2 < b3 &
b3 < b4
holds Cl b1 = [.b2,b4.];
:: BORSUK_5:th 61
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
st b1 = {b2}
holds Cl b1 = {b2};
:: BORSUK_5:th 62
theorem
for b1 being Element of bool REAL
for b2 being Element of bool the carrier of R^1
st b1 = b2
holds b1 is open
iff
b2 is open(R^1);
:: BORSUK_5:th 63
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
st b1 = ].b2,+infty.[
holds b1 is open(R^1);
:: BORSUK_5:th 64
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
st b1 = ].-infty,b2.[
holds b1 is open(R^1);
:: BORSUK_5:th 65
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
st b1 = ].-infty,b2.]
holds b1 is closed(R^1);
:: BORSUK_5:th 66
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
st b1 = [.b2,+infty.[
holds b1 is closed(R^1);
:: BORSUK_5:th 67
theorem
for b1 being real set holds
[.b1,+infty.[ = {b1} \/ ].b1,+infty.[;
:: BORSUK_5:th 68
theorem
for b1 being real set holds
].-infty,b1.] = {b1} \/ ].-infty,b1.[;
:: BORSUK_5:funcreg 14
registration
let a1 be real set;
cluster ].a1,+infty.[ -> non empty;
end;
:: BORSUK_5:funcreg 15
registration
let a1 be real set;
cluster ].-infty,a1.] -> non empty;
end;
:: BORSUK_5:funcreg 16
registration
let a1 be real set;
cluster ].-infty,a1.[ -> non empty;
end;
:: BORSUK_5:funcreg 17
registration
let a1 be real set;
cluster [.a1,+infty.[ -> non empty;
end;
:: BORSUK_5:th 71
theorem
for b1 being real set holds
].b1,+infty.[ <> REAL;
:: BORSUK_5:th 72
theorem
for b1 being real set holds
[.b1,+infty.[ <> REAL;
:: BORSUK_5:th 73
theorem
for b1 being real set holds
].-infty,b1.] <> REAL;
:: BORSUK_5:th 74
theorem
for b1 being real set holds
].-infty,b1.[ <> REAL;
:: BORSUK_5:th 75
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
st b1 = ].b2,+infty.[
holds Cl b1 = [.b2,+infty.[;
:: BORSUK_5:th 76
theorem
for b1 being real set holds
Cl ].b1,+infty.[ = [.b1,+infty.[;
:: BORSUK_5:th 77
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
st b1 = ].-infty,b2.[
holds Cl b1 = ].-infty,b2.];
:: BORSUK_5:th 78
theorem
for b1 being real set holds
Cl ].-infty,b1.[ = ].-infty,b1.];
:: BORSUK_5:th 79
theorem
for b1, b2 being Element of bool the carrier of R^1
for b3 being real set
st b1 = ].-infty,b3.[ & b2 = ].b3,+infty.[
holds b1,b2 are_separated;
:: BORSUK_5:th 80
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
st b2 < b3 &
b1 = [.b2,b3.[ \/ ].b3,+infty.[
holds Cl b1 = [.b2,+infty.[;
:: BORSUK_5:th 81
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
st b2 < b3 &
b1 = ].b2,b3.[ \/ ].b3,+infty.[
holds Cl b1 = [.b2,+infty.[;
:: BORSUK_5:th 82
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3, b4 being real set
st b2 < b3 &
b3 < b4 &
b1 = ((RAT(b2,b3)) \/ ].b3,b4.[) \/ ].b4,+infty.[
holds Cl b1 = [.b2,+infty.[;
:: BORSUK_5:th 83
theorem
for b1 being Element of bool the carrier of R^1 holds
b1 ` = REAL \ b1;
:: BORSUK_5:th 84
theorem
for b1, b2 being real set
st b1 < b2
holds IRRAT(b1,b2) misses RAT(b1,b2);
:: BORSUK_5:th 85
theorem
for b1, b2 being real set holds
REAL \ RAT(b1,b2) = (].-infty,b1.] \/ IRRAT(b1,b2)) \/ [.b2,+infty.[;
:: BORSUK_5:th 86
theorem
for b1, b2, b3 being real set
st b1 <= b2 & b2 < b3
holds not b1 in ].b2,b3.[ \/ ].b3,+infty.[;
:: BORSUK_5:th 87
theorem
for b1, b2 being real set
st b1 < b2
holds not b2 in ].b1,b2.[ \/ ].b2,+infty.[;
:: BORSUK_5:th 88
theorem
for b1, b2 being real set
st b1 < b2
holds [.b1,+infty.[ \ (].b1,b2.[ \/ ].b2,+infty.[) = {b1} \/ {b2};
:: BORSUK_5:th 89
theorem
for b1 being Element of bool the carrier of R^1
st b1 = ((RAT(2,4)) \/ ].4,5.[) \/ ].5,+infty.[
holds b1 ` = ((].-infty,2.] \/ IRRAT(2,4)) \/ {4}) \/ {5};
:: BORSUK_5:th 90
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
st b1 = {b2}
holds b1 ` = ].-infty,b2.[ \/ ].b2,+infty.[;
:: BORSUK_5:th 91
theorem
for b1, b2 being real set
st b1 < b2
holds ].b1,+infty.[ /\ ].-infty,b2.] = ].b1,b2.];
:: BORSUK_5:th 92
theorem
(].-infty,1.[ \/ ].1,+infty.[) /\ (((].-infty,2.] \/ IRRAT(2,4)) \/ {4}) \/ {5}) = (((].-infty,1.[ \/ ].1,2.]) \/ IRRAT(2,4)) \/ {4}) \/ {5};
:: BORSUK_5:th 93
theorem
for b1, b2 being real set
st b1 <= b2
holds ].-infty,b2.[ \ {b1} = ].-infty,b1.[ \/ ].b1,b2.[;
:: BORSUK_5:th 94
theorem
for b1, b2 being real set
st b1 <= b2
holds ].b1,+infty.[ \ {b2} = ].b1,b2.[ \/ ].b2,+infty.[;
:: BORSUK_5:th 95
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
st b2 <= b3 &
b1 = {b2} \/ [.b3,+infty.[
holds b1 ` = ].-infty,b2.[ \/ ].b2,b3.[;
:: BORSUK_5:th 96
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
st b2 < b3 &
b1 = ].-infty,b2.[ \/ ].b2,b3.[
holds Cl b1 = ].-infty,b3.];
:: BORSUK_5:th 97
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
st b2 < b3 &
b1 = ].-infty,b2.[ \/ ].b2,b3.]
holds Cl b1 = ].-infty,b3.];
:: BORSUK_5:th 98
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
st b1 = ].-infty,b2.]
holds b1 ` = ].b2,+infty.[;
:: BORSUK_5:th 99
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
st b1 = [.b2,+infty.[
holds b1 ` = ].-infty,b2.[;
:: BORSUK_5:th 100
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3, b4 being real set
st b2 < b3 &
b3 < b4 &
b1 = ((].-infty,b2.[ \/ ].b2,b3.]) \/ IRRAT(b3,b4)) \/ {b4}
holds Cl b1 = ].-infty,b4.];
:: BORSUK_5:th 101
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3, b4, b5 being real set
st b2 < b3 &
b3 < b4 &
b1 = (((].-infty,b2.[ \/ ].b2,b3.]) \/ IRRAT(b3,b4)) \/ {b4}) \/ {b5}
holds Cl b1 = ].-infty,b4.] \/ {b5};
:: BORSUK_5:th 102
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
st b2 <= b3 &
b1 = ].-infty,b2.] \/ {b3}
holds b1 ` = ].b2,b3.[ \/ ].b3,+infty.[;
:: BORSUK_5:th 103
theorem
for b1, b2 being real set holds
[.b1,+infty.[ \/ {b2} <> REAL;
:: BORSUK_5:th 104
theorem
for b1, b2 being real set holds
].-infty,b1.] \/ {b2} <> REAL;
:: BORSUK_5:th 105
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 <> b3
holds b2 ` <> b3 `;
:: BORSUK_5:th 106
theorem
for b1 being Element of bool the carrier of R^1
st REAL = b1 `
holds b1 = {};
:: BORSUK_5:th 107
theorem
for b1 being compact Element of bool the carrier of R^1
for b2 being Element of bool REAL
st b2 = b1
holds b2 is bounded_above & b2 is bounded_below;
:: BORSUK_5:th 108
theorem
for b1 being compact Element of bool the carrier of R^1
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_5:th 109
theorem
for b1 being non empty connected compact Element of bool the carrier of R^1
for b2 being Element of bool REAL
st b1 = b2 & [.inf b2,sup b2.] c= b2
holds [.inf b2,sup b2.] = b2;
:: BORSUK_5:th 110
theorem
for b1 being connected Element of bool the carrier of R^1
for b2, b3, b4 being real set
st b2 <= b3 & b3 <= b4 & b2 in b1 & b4 in b1
holds b3 in b1;
:: BORSUK_5:th 111
theorem
for b1 being connected Element of bool the carrier of R^1
for b2, b3 being real set
st b2 in b1 & b3 in b1
holds [.b2,b3.] c= b1;
:: BORSUK_5:th 112
theorem
for b1 being non empty connected compact Element of bool the carrier of R^1 holds
b1 is non empty closed-interval Element of bool REAL;
:: BORSUK_5:th 113
theorem
for b1 being non empty connected compact Element of bool the carrier of R^1 holds
ex b2, b3 being real set st
b2 <= b3 & b1 = [.b2,b3.];
:: BORSUK_5:attrnot 1 => BORSUK_5:attr 1
definition
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
attr a2 is with_proper_subsets means
not the carrier of a1 in a2;
end;
:: BORSUK_5:dfs 4
definiens
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is with_proper_subsets
it is sufficient to prove
thus not the carrier of a1 in a2;
:: BORSUK_5:def 6
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is with_proper_subsets(b1)
iff
not the carrier of b1 in b2;
:: BORSUK_5:th 114
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is with_proper_subsets(b1) & b3 c= b2
holds b3 is with_proper_subsets(b1);
:: BORSUK_5:exreg 4
registration
let a1 be non empty TopStruct;
cluster with_proper_subsets Element of bool bool the carrier of a1;
end;
:: BORSUK_5:th 115
theorem
for b1 being non empty TopStruct
for b2, b3 being with_proper_subsets Element of bool bool the carrier of b1 holds
b2 \/ b3 is with_proper_subsets(b1);
:: BORSUK_5:exreg 5
registration
let a1 be TopSpace-like TopStruct;
cluster non empty open closed Element of bool bool the carrier of a1;
end;