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;