Article XXREAL_1, MML version 4.99.1005

:: XXREAL_1:sch 1
scheme XXREAL_1:sch 1
ex b1 being ext-real set st
   (for b2 being ext-real set
          st P1[b2]
       holds b2 <= b1) &
    (for b2 being ext-real set
          st P2[b2]
       holds b1 <= b2)
provided
   for b1, b2 being ext-real set
         st P1[b1] & P2[b2]
      holds b1 <= b2;


:: XXREAL_1:funcnot 1 => XXREAL_1:func 1
definition
  let a1, a2 be ext-real set;
  func [.A1,A2.] -> set equals
    {b1 where b1 is Element of ExtREAL: a1 <= b1 & b1 <= a2};
end;

:: XXREAL_1:def 1
theorem
for b1, b2 being ext-real set holds
[.b1,b2.] = {b3 where b3 is Element of ExtREAL: b1 <= b3 & b3 <= b2};

:: XXREAL_1:funcnot 2 => XXREAL_1:func 2
definition
  let a1, a2 be ext-real set;
  func [.A1,A2.[ -> set equals
    {b1 where b1 is Element of ExtREAL: a1 <= b1 & b1 < a2};
end;

:: XXREAL_1:def 2
theorem
for b1, b2 being ext-real set holds
[.b1,b2.[ = {b3 where b3 is Element of ExtREAL: b1 <= b3 & b3 < b2};

:: XXREAL_1:funcnot 3 => XXREAL_1:func 3
definition
  let a1, a2 be ext-real set;
  func ].A1,A2.] -> set equals
    {b1 where b1 is Element of ExtREAL: a1 < b1 & b1 <= a2};
end;

:: XXREAL_1:def 3
theorem
for b1, b2 being ext-real set holds
].b1,b2.] = {b3 where b3 is Element of ExtREAL: b1 < b3 & b3 <= b2};

:: XXREAL_1:funcnot 4 => XXREAL_1:func 4
definition
  let a1, a2 be ext-real set;
  func ].A1,A2.[ -> set equals
    {b1 where b1 is Element of ExtREAL: a1 < b1 & b1 < a2};
end;

:: XXREAL_1:def 4
theorem
for b1, b2 being ext-real set holds
].b1,b2.[ = {b3 where b3 is Element of ExtREAL: b1 < b3 & b3 < b2};

:: XXREAL_1:th 1
theorem
for b1, b2, b3 being ext-real set holds
   b1 in [.b2,b3.]
iff
   b2 <= b1 & b1 <= b3;

:: XXREAL_1:th 2
theorem
for b1, b2, b3 being ext-real set holds
   b1 in ].b2,b3.]
iff
   b2 < b1 & b1 <= b3;

:: XXREAL_1:th 3
theorem
for b1, b2, b3 being ext-real set holds
   b1 in [.b2,b3.[
iff
   b2 <= b1 & b1 < b3;

:: XXREAL_1:th 4
theorem
for b1, b2, b3 being ext-real set holds
   b1 in ].b2,b3.[
iff
   b2 < b1 & b1 < b3;

:: XXREAL_1:funcreg 1
registration
  let a1, a2 be ext-real set;
  cluster [.a1,a2.] -> ext-real-membered;
end;

:: XXREAL_1:funcreg 2
registration
  let a1, a2 be ext-real set;
  cluster [.a1,a2.[ -> ext-real-membered;
end;

:: XXREAL_1:funcreg 3
registration
  let a1, a2 be ext-real set;
  cluster ].a1,a2.] -> ext-real-membered;
end;

:: XXREAL_1:funcreg 4
registration
  let a1, a2 be ext-real set;
  cluster ].a1,a2.[ -> ext-real-membered;
end;

:: XXREAL_1:th 5
theorem
for b1 being set
for b2, b3 being ext-real set
      st b1 in [.b2,b3.] & not b1 in ].b2,b3.[ & b1 <> b2
   holds b1 = b3;

:: XXREAL_1:th 6
theorem
for b1 being set
for b2, b3 being ext-real set
      st b1 in [.b2,b3.] & not b1 in ].b2,b3.]
   holds b1 = b2;

:: XXREAL_1:th 7
theorem
for b1 being set
for b2, b3 being ext-real set
      st b1 in [.b2,b3.] & not b1 in [.b2,b3.[
   holds b1 = b3;

:: XXREAL_1:th 8
theorem
for b1 being set
for b2, b3 being ext-real set
      st b1 in [.b2,b3.[ & not b1 in ].b2,b3.[
   holds b1 = b2;

:: XXREAL_1:th 9
theorem
for b1 being set
for b2, b3 being ext-real set
      st b1 in ].b2,b3.] & not b1 in ].b2,b3.[
   holds b1 = b3;

:: XXREAL_1:th 10
theorem
for b1 being set
for b2, b3 being ext-real set
      st b1 in [.b2,b3.[ & (b1 in ].b2,b3.] implies b1 = b3)
   holds b1 = b2;

:: XXREAL_1:th 11
theorem
for b1 being set
for b2, b3 being ext-real set
      st b1 in ].b2,b3.] & (b1 in [.b2,b3.[ implies b1 = b2)
   holds b1 = b3;

:: XXREAL_1:th 12
theorem
for b1 being set
for b2, b3 being ext-real set
      st b1 in ].b2,b3.]
   holds b1 in [.b2,b3.] & b1 <> b2;

:: XXREAL_1:th 13
theorem
for b1 being set
for b2, b3 being ext-real set
      st b1 in [.b2,b3.[
   holds b1 in [.b2,b3.] & b1 <> b3;

:: XXREAL_1:th 14
theorem
for b1 being set
for b2, b3 being ext-real set
      st b1 in ].b2,b3.[
   holds b1 in [.b2,b3.[ & b1 <> b2;

:: XXREAL_1:th 15
theorem
for b1 being set
for b2, b3 being ext-real set
      st b1 in ].b2,b3.[
   holds b1 in ].b2,b3.] & b1 <> b3;

:: XXREAL_1:th 16
theorem
for b1 being set
for b2, b3 being ext-real set
      st b1 in ].b2,b3.[
   holds b1 in [.b2,b3.] & b1 <> b2 & b1 <> b3;

:: XXREAL_1:th 17
theorem
for b1 being ext-real set holds
   [.b1,b1.] = {b1};

:: XXREAL_1:th 18
theorem
for b1 being ext-real set holds
   [.b1,b1.[ = {};

:: XXREAL_1:th 19
theorem
for b1 being ext-real set holds
   ].b1,b1.] = {};

:: XXREAL_1:th 20
theorem
for b1 being ext-real set holds
   ].b1,b1.[ = {};

:: XXREAL_1:funcreg 5
registration
  let a1 be ext-real set;
  cluster [.a1,a1.] -> non empty;
end;

:: XXREAL_1:funcreg 6
registration
  let a1 be ext-real set;
  cluster [.a1,a1.[ -> empty;
end;

:: XXREAL_1:funcreg 7
registration
  let a1 be ext-real set;
  cluster ].a1,a1.] -> empty;
end;

:: XXREAL_1:funcreg 8
registration
  let a1 be ext-real set;
  cluster ].a1,a1.[ -> empty;
end;

:: XXREAL_1:th 21
theorem
for b1, b2 being ext-real set holds
].b1,b2.[ c= ].b1,b2.];

:: XXREAL_1:th 22
theorem
for b1, b2 being ext-real set holds
].b1,b2.[ c= [.b1,b2.[;

:: XXREAL_1:th 23
theorem
for b1, b2 being ext-real set holds
].b1,b2.] c= [.b1,b2.];

:: XXREAL_1:th 24
theorem
for b1, b2 being ext-real set holds
[.b1,b2.[ c= [.b1,b2.];

:: XXREAL_1:th 25
theorem
for b1, b2 being ext-real set holds
].b1,b2.[ c= [.b1,b2.];

:: XXREAL_1:th 26
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds ].b2,b1.] = {};

:: XXREAL_1:th 27
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds [.b2,b1.[ = {};

:: XXREAL_1:th 28
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds ].b2,b1.[ = {};

:: XXREAL_1:th 29
theorem
for b1, b2 being ext-real set
      st b1 < b2
   holds [.b2,b1.] = {};

:: XXREAL_1:th 30
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds [.b1,b2.] is not empty;

:: XXREAL_1:th 31
theorem
for b1, b2 being ext-real set
      st b1 < b2
   holds [.b1,b2.[ is not empty;

:: XXREAL_1:th 32
theorem
for b1, b2 being ext-real set
      st b1 < b2
   holds ].b1,b2.] is not empty;

:: XXREAL_1:th 33
theorem
for b1, b2 being ext-real set
      st b1 < b2
   holds ].b1,b2.[ is not empty;

:: XXREAL_1:th 34
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds [.b2,b3.] c= [.b1,b4.];

:: XXREAL_1:th 35
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds [.b2,b3.[ c= [.b1,b4.];

:: XXREAL_1:th 36
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.] c= [.b1,b4.];

:: XXREAL_1:th 37
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.[ c= [.b1,b4.];

:: XXREAL_1:th 38
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds [.b2,b3.[ c= [.b1,b4.[;

:: XXREAL_1:th 39
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 <= b4
   holds [.b2,b3.] c= ].b1,b4.];

:: XXREAL_1:th 40
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 <= b4
   holds [.b2,b3.[ c= ].b1,b4.];

:: XXREAL_1:th 41
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.[ c= ].b1,b4.];

:: XXREAL_1:th 42
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.] c= ].b1,b4.];

:: XXREAL_1:th 43
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 < b4
   holds [.b2,b3.] c= [.b1,b4.[;

:: XXREAL_1:th 44
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 < b4
   holds ].b2,b3.] c= [.b1,b4.[;

:: XXREAL_1:th 45
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.[ c= [.b1,b4.[;

:: XXREAL_1:th 46
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.[ c= ].b1,b4.[;

:: XXREAL_1:th 47
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b4
   holds [.b2,b3.] c= ].b1,b4.[;

:: XXREAL_1:th 48
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 <= b4
   holds [.b2,b3.[ c= ].b1,b4.[;

:: XXREAL_1:th 49
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 < b4
   holds ].b2,b3.] c= ].b1,b4.[;

:: XXREAL_1:th 50
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & [.b1,b2.] c= [.b3,b4.]
   holds b3 <= b1 & b2 <= b4;

:: XXREAL_1:th 51
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & ].b1,b2.[ c= [.b3,b4.]
   holds b3 <= b1 & b2 <= b4;

:: XXREAL_1:th 52
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & [.b1,b2.[ c= [.b3,b4.]
   holds b3 <= b1 & b2 <= b4;

:: XXREAL_1:th 53
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & ].b1,b2.] c= [.b3,b4.]
   holds b3 <= b1 & b2 <= b4;

:: XXREAL_1:th 54
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & [.b1,b2.] c= [.b3,b4.[
   holds b3 <= b1 & b2 < b4;

:: XXREAL_1:th 55
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & [.b1,b2.[ c= [.b3,b4.[
   holds b3 <= b1 & b2 <= b4;

:: XXREAL_1:th 56
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & ].b1,b2.[ c= [.b3,b4.[
   holds b3 <= b1 & b2 <= b4;

:: XXREAL_1:th 57
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & ].b1,b2.] c= [.b3,b4.[
   holds b3 <= b1 & b2 < b4;

:: XXREAL_1:th 58
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & [.b1,b2.] c= ].b3,b4.]
   holds b3 < b1 & b2 <= b4;

:: XXREAL_1:th 59
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & ].b1,b2.[ c= ].b3,b4.]
   holds b3 <= b1 & b2 <= b4;

:: XXREAL_1:th 60
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & [.b1,b2.[ c= ].b3,b4.]
   holds b3 < b1 & b2 <= b4;

:: XXREAL_1:th 61
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & ].b1,b2.] c= ].b3,b4.]
   holds b3 <= b1 & b2 <= b4;

:: XXREAL_1:th 62
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & [.b1,b2.] c= ].b3,b4.[
   holds b3 < b1 & b2 < b4;

:: XXREAL_1:th 63
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & ].b1,b2.[ c= ].b3,b4.[
   holds b3 <= b1 & b2 <= b4;

:: XXREAL_1:th 64
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & [.b1,b2.[ c= ].b3,b4.[
   holds b3 < b1 & b2 <= b4;

:: XXREAL_1:th 65
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & ].b1,b2.] c= ].b3,b4.[
   holds b3 <= b1 & b2 < b4;

:: XXREAL_1:th 66
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & [.b1,b2.] = [.b3,b4.]
   holds b1 = b3 & b2 = b4;

:: XXREAL_1:th 67
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & ].b1,b2.[ = ].b3,b4.[
   holds b1 = b3 & b2 = b4;

:: XXREAL_1:th 68
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & ].b1,b2.] = ].b3,b4.]
   holds b1 = b3 & b2 = b4;

:: XXREAL_1:th 69
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & [.b1,b2.[ = [.b3,b4.[
   holds b1 = b3 & b2 = b4;

:: XXREAL_1:th 70
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2
   holds [.b1,b2.] <> ].b3,b4.];

:: XXREAL_1:th 71
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2
   holds [.b1,b2.] <> [.b3,b4.[;

:: XXREAL_1:th 72
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2
   holds [.b1,b2.] <> ].b3,b4.[;

:: XXREAL_1:th 73
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2
   holds [.b1,b2.[ <> [.b3,b4.];

:: XXREAL_1:th 74
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2
   holds [.b1,b2.[ <> ].b3,b4.];

:: XXREAL_1:th 75
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2
   holds [.b1,b2.[ <> ].b3,b4.[;

:: XXREAL_1:th 76
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2
   holds ].b1,b2.] <> [.b3,b4.];

:: XXREAL_1:th 77
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2
   holds ].b1,b2.] <> [.b3,b4.[;

:: XXREAL_1:th 78
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2
   holds ].b1,b2.] <> ].b3,b4.[;

:: XXREAL_1:th 79
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2
   holds ].b1,b2.[ <> [.b3,b4.];

:: XXREAL_1:th 80
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2
   holds ].b1,b2.[ <> ].b3,b4.];

:: XXREAL_1:th 81
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2
   holds ].b1,b2.[ <> [.b3,b4.[;

:: XXREAL_1:th 82
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & [.b1,b2.] c< [.b3,b4.] & b1 <= b3
   holds b2 < b4;

:: XXREAL_1:th 83
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & ].b1,b2.[ c= [.b3,b4.]
   holds [.b1,b2.] c= [.b3,b4.];

:: XXREAL_1:th 84
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2
   holds [.b2,b3.[ c= ].b1,b3.[;

:: XXREAL_1:th 85
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds [.b2,b1.] c= {b2} & [.b2,b1.] c= {b1};

:: XXREAL_1:th 86
theorem
for b1, b2 being ext-real set holds
].b1,b2.[ misses {b1,b2};

:: XXREAL_1:th 87
theorem
for b1, b2 being ext-real set holds
[.b1,b2.[ misses {b2};

:: XXREAL_1:th 88
theorem
for b1, b2 being ext-real set holds
].b1,b2.] misses {b1};

:: XXREAL_1:th 89
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2
   holds [.b3,b1.] misses ].b2,b4.[;

:: XXREAL_1:th 90
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2
   holds [.b3,b1.] misses ].b2,b4.];

:: XXREAL_1:th 91
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2
   holds ].b3,b1.] misses ].b2,b4.[;

:: XXREAL_1:th 92
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2
   holds ].b3,b1.] misses ].b2,b4.];

:: XXREAL_1:th 93
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2
   holds ].b3,b1.[ misses [.b2,b4.];

:: XXREAL_1:th 94
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2
   holds ].b3,b1.[ misses [.b2,b4.[;

:: XXREAL_1:th 95
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2
   holds [.b3,b1.[ misses [.b2,b4.];

:: XXREAL_1:th 96
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2
   holds [.b3,b1.[ misses [.b2,b4.[;

:: XXREAL_1:th 97
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 < b3
   holds not ].b1,b3.[ c= [.b2,b4.];

:: XXREAL_1:th 98
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 < b3
   holds not [.b1,b3.[ c= [.b2,b4.];

:: XXREAL_1:th 99
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 < b3
   holds not ].b1,b3.] c= [.b2,b4.];

:: XXREAL_1:th 100
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 <= b3
   holds not [.b1,b3.] c= [.b2,b4.];

:: XXREAL_1:th 101
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 < b3
   holds not ].b1,b3.[ c= [.b2,b4.[;

:: XXREAL_1:th 102
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 < b3
   holds not ].b1,b3.] c= [.b2,b4.[;

:: XXREAL_1:th 103
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 < b3
   holds not [.b1,b3.[ c= [.b2,b4.[;

:: XXREAL_1:th 104
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 <= b3
   holds not [.b1,b3.] c= [.b2,b4.[;

:: XXREAL_1:th 105
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 < b3
   holds not ].b1,b3.[ c= ].b2,b4.];

:: XXREAL_1:th 106
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b1 < b3
   holds not [.b1,b3.[ c= ].b2,b4.];

:: XXREAL_1:th 107
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 < b3
   holds not ].b1,b3.] c= ].b2,b4.];

:: XXREAL_1:th 108
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b1 <= b3
   holds not [.b1,b3.] c= ].b2,b4.];

:: XXREAL_1:th 109
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b1 <= b3
   holds not [.b1,b3.] c= ].b2,b4.[;

:: XXREAL_1:th 110
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b1 < b3
   holds not [.b1,b3.[ c= ].b2,b4.[;

:: XXREAL_1:th 111
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 < b3
   holds not ].b1,b3.] c= ].b2,b4.[;

:: XXREAL_1:th 112
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 < b3
   holds not ].b1,b3.[ c= ].b2,b4.[;

:: XXREAL_1:th 113
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b2
   holds not ].b3,b2.[ c= [.b4,b1.];

:: XXREAL_1:th 114
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b2
   holds not [.b3,b2.[ c= [.b4,b1.];

:: XXREAL_1:th 115
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b2
   holds not ].b3,b2.] c= [.b4,b1.];

:: XXREAL_1:th 116
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 <= b2
   holds not [.b3,b2.] c= [.b4,b1.];

:: XXREAL_1:th 117
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b2
   holds not ].b3,b2.[ c= [.b4,b1.[;

:: XXREAL_1:th 118
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 < b2
   holds not ].b3,b2.] c= [.b4,b1.[;

:: XXREAL_1:th 119
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b2
   holds not [.b3,b2.[ c= [.b4,b1.[;

:: XXREAL_1:th 120
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b2
   holds not ].b3,b2.[ c= ].b4,b1.];

:: XXREAL_1:th 121
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 <= b2
   holds not [.b3,b2.] c= ].b4,b1.];

:: XXREAL_1:th 122
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b2
   holds not [.b3,b2.[ c= ].b4,b1.];

:: XXREAL_1:th 123
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b2
   holds not ].b3,b2.] c= ].b4,b1.];

:: XXREAL_1:th 124
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b2
   holds not [.b3,b2.] c= ].b4,b1.[;

:: XXREAL_1:th 125
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b2
   holds not [.b3,b2.[ c= ].b4,b1.[;

:: XXREAL_1:th 126
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 < b2
   holds not ].b3,b2.] c= ].b4,b1.[;

:: XXREAL_1:th 127
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b2
   holds not ].b3,b2.[ c= ].b4,b1.[;

:: XXREAL_1:th 128
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds [.b1,b2.] = ].b1,b2.[ \/ {b1,b2};

:: XXREAL_1:th 129
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds [.b1,b2.] = [.b1,b2.[ \/ {b2};

:: XXREAL_1:th 130
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds [.b1,b2.] = {b1} \/ ].b1,b2.];

:: XXREAL_1:th 131
theorem
for b1, b2 being ext-real set
      st b1 < b2
   holds [.b1,b2.[ = {b1} \/ ].b1,b2.[;

:: XXREAL_1:th 132
theorem
for b1, b2 being ext-real set
      st b1 < b2
   holds ].b1,b2.] = ].b1,b2.[ \/ {b2};

:: XXREAL_1:th 133
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds [.b1,b2.] \ {b1,b2} = ].b1,b2.[;

:: XXREAL_1:th 134
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds [.b1,b2.] \ {b1} = ].b1,b2.];

:: XXREAL_1:th 135
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds [.b1,b2.] \ {b2} = [.b1,b2.[;

:: XXREAL_1:th 136
theorem
for b1, b2 being ext-real set
      st b1 < b2
   holds [.b1,b2.[ \ {b1} = ].b1,b2.[;

:: XXREAL_1:th 137
theorem
for b1, b2 being ext-real set
      st b1 < b2
   holds ].b1,b2.] \ {b2} = ].b1,b2.[;

:: XXREAL_1:th 138
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2 & b2 < b3
   holds ].b1,b2.] /\ [.b2,b3.[ = {b2};

:: XXREAL_1:th 139
theorem
for b1, b2, b3, b4 being ext-real set holds
[.b1,b2.[ /\ [.b3,b4.[ = [.max(b1,b3),min(b2,b4).[;

:: XXREAL_1:th 140
theorem
for b1, b2, b3, b4 being ext-real set holds
[.b1,b2.] /\ [.b3,b4.] = [.max(b1,b3),min(b2,b4).];

:: XXREAL_1:th 141
theorem
for b1, b2, b3, b4 being ext-real set holds
].b1,b2.] /\ ].b3,b4.] = ].max(b1,b3),min(b2,b4).];

:: XXREAL_1:th 142
theorem
for b1, b2, b3, b4 being ext-real set holds
].b1,b2.[ /\ ].b3,b4.[ = ].max(b1,b3),min(b2,b4).[;

:: XXREAL_1:th 143
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds [.b1,b3.] /\ [.b2,b4.] = [.b2,b3.];

:: XXREAL_1:th 144
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds [.b1,b3.[ /\ [.b2,b4.] = [.b2,b3.[;

:: XXREAL_1:th 145
theorem
for b1, b2, b3, b4 being ext-real set
      st b2 <= b1 & b4 < b3
   holds [.b1,b3.[ /\ [.b2,b4.] = [.b1,b4.];

:: XXREAL_1:th 146
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 <= b4
   holds ].b1,b3.] /\ [.b2,b4.] = [.b2,b3.];

:: XXREAL_1:th 147
theorem
for b1, b2, b3, b4 being ext-real set
      st b2 <= b1 & b4 <= b3
   holds ].b1,b3.] /\ [.b2,b4.] = ].b1,b4.];

:: XXREAL_1:th 148
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 <= b4
   holds ].b1,b3.[ /\ [.b2,b4.] = [.b2,b3.[;

:: XXREAL_1:th 149
theorem
for b1, b2, b3, b4 being ext-real set
      st b2 <= b1 & b4 < b3
   holds ].b1,b3.[ /\ [.b2,b4.] = ].b1,b4.];

:: XXREAL_1:th 150
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds [.b1,b3.[ /\ [.b2,b4.[ = [.b2,b3.[;

:: XXREAL_1:th 151
theorem
for b1, b2, b3, b4 being ext-real set
      st b2 <= b1 & b4 <= b3
   holds [.b1,b3.[ /\ [.b2,b4.[ = [.b1,b4.[;

:: XXREAL_1:th 152
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b4
   holds ].b1,b3.] /\ [.b2,b4.[ = [.b2,b3.];

:: XXREAL_1:th 153
theorem
for b1, b2, b3, b4 being ext-real set
      st b2 <= b1 & b4 <= b3
   holds ].b1,b3.] /\ [.b2,b4.[ = ].b1,b4.[;

:: XXREAL_1:th 154
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 <= b4
   holds ].b1,b3.[ /\ [.b2,b4.[ = [.b2,b3.[;

:: XXREAL_1:th 155
theorem
for b1, b2, b3, b4 being ext-real set
      st b2 <= b1 & b4 <= b3
   holds ].b1,b3.[ /\ [.b2,b4.[ = ].b1,b4.[;

:: XXREAL_1:th 156
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds ].b1,b3.] /\ ].b2,b4.] = ].b2,b3.];

:: XXREAL_1:th 157
theorem
for b1, b2, b3, b4 being ext-real set
      st b2 <= b1 & b4 <= b3
   holds ].b1,b3.] /\ ].b2,b4.] = ].b1,b4.];

:: XXREAL_1:th 158
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds ].b1,b3.[ /\ ].b2,b4.] = ].b2,b3.[;

:: XXREAL_1:th 159
theorem
for b1, b2, b3, b4 being ext-real set
      st b2 <= b1 & b4 < b3
   holds ].b1,b3.[ /\ ].b2,b4.] = ].b1,b4.];

:: XXREAL_1:th 160
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds ].b1,b3.[ /\ ].b2,b4.[ = ].b2,b3.[;

:: XXREAL_1:th 161
theorem
for b1, b2, b3, b4 being ext-real set holds
[.b1,b2.[ \/ [.b3,b4.[ c= [.min(b1,b3),max(b2,b4).[;

:: XXREAL_1:th 162
theorem
for b1, b2, b3, b4 being ext-real set
      st [.b1,b2.[ meets [.b3,b4.[
   holds [.b1,b2.[ \/ [.b3,b4.[ = [.min(b1,b3),max(b2,b4).[;

:: XXREAL_1:th 163
theorem
for b1, b2, b3, b4 being ext-real set holds
].b1,b2.] \/ ].b3,b4.] c= ].min(b1,b3),max(b2,b4).];

:: XXREAL_1:th 164
theorem
for b1, b2, b3, b4 being ext-real set
      st ].b1,b2.] meets ].b3,b4.]
   holds ].b1,b2.] \/ ].b3,b4.] = ].min(b1,b3),max(b2,b4).];

:: XXREAL_1:th 165
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2 & b2 <= b3
   holds [.b1,b2.] \/ [.b2,b3.] = [.b1,b3.];

:: XXREAL_1:th 166
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2 & b2 <= b3
   holds [.b1,b2.[ \/ [.b2,b3.] = [.b1,b3.];

:: XXREAL_1:th 167
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2 & b2 <= b3
   holds [.b1,b2.] \/ ].b2,b3.] = [.b1,b3.];

:: XXREAL_1:th 168
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2 & b2 <= b3
   holds [.b1,b2.[ \/ [.b2,b3.[ = [.b1,b3.[;

:: XXREAL_1:th 169
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2 & b2 < b3
   holds [.b1,b2.] \/ ].b2,b3.[ = [.b1,b3.[;

:: XXREAL_1:th 170
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2 & b2 <= b3
   holds ].b1,b2.] \/ ].b2,b3.] = ].b1,b3.];

:: XXREAL_1:th 171
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2 & b2 < b3
   holds ].b1,b2.] \/ ].b2,b3.[ = ].b1,b3.[;

:: XXREAL_1:th 172
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2 & b2 < b3
   holds ].b1,b2.] \/ [.b2,b3.[ = ].b1,b3.[;

:: XXREAL_1:th 173
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2 & b2 < b3
   holds ].b1,b2.[ \/ [.b2,b3.[ = ].b1,b3.[;

:: XXREAL_1:th 174
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4 & b2 <= b3
   holds [.b1,b3.] \/ [.b2,b4.] = [.b1,b4.];

:: XXREAL_1:th 175
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4 & b2 < b3
   holds [.b1,b3.[ \/ ].b2,b4.] = [.b1,b4.];

:: XXREAL_1:th 176
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b2 <= b3 & b3 < b4
   holds [.b1,b3.] \/ [.b2,b4.[ = [.b1,b4.[;

:: XXREAL_1:th 177
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 <= b4 & b2 <= b3
   holds ].b1,b3.] \/ [.b2,b4.] = ].b1,b4.];

:: XXREAL_1:th 178
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b3 < b4 & b2 <= b3
   holds ].b1,b3.] \/ [.b2,b4.[ = ].b1,b4.[;

:: XXREAL_1:th 179
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b1 <= b3 & b2 <= b4 & b3 <= b4
   holds ([.b1,b2.[ \/ [.b2,b3.]) \/ ].b3,b4.] = [.b1,b4.];

:: XXREAL_1:th 180
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 < b2 & b1 < b3 & b2 < b4 & b3 < b4
   holds (].b1,b2.] \/ ].b2,b3.[) \/ [.b3,b4.[ = ].b1,b4.[;

:: XXREAL_1:th 181
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b2 <= b3 & b3 <= b4
   holds ([.b1,b2.] \/ ].b2,b3.[) \/ [.b3,b4.] = [.b1,b4.];

:: XXREAL_1:th 182
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2
   holds [.b1,b3.] \ [.b1,b2.] = ].b2,b3.];

:: XXREAL_1:th 183
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2
   holds [.b1,b3.[ \ [.b1,b2.] = ].b2,b3.[;

:: XXREAL_1:th 184
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2
   holds [.b1,b3.] \ [.b1,b2.[ = [.b2,b3.];

:: XXREAL_1:th 185
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2
   holds [.b1,b3.[ \ [.b1,b2.[ = [.b2,b3.[;

:: XXREAL_1:th 186
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2
   holds [.b1,b3.] \ [.b1,b2.] = ].b2,b3.];

:: XXREAL_1:th 187
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2
   holds ].b1,b3.[ \ ].b1,b2.] = ].b2,b3.[;

:: XXREAL_1:th 188
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2
   holds ].b1,b3.] \ ].b1,b2.[ = [.b2,b3.];

:: XXREAL_1:th 189
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2
   holds ].b1,b3.[ \ ].b1,b2.[ = [.b2,b3.[;

:: XXREAL_1:th 190
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2
   holds [.b3,b2.] \ [.b1,b2.] = [.b3,b1.[;

:: XXREAL_1:th 191
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2
   holds ].b3,b2.] \ [.b1,b2.] = ].b3,b1.[;

:: XXREAL_1:th 192
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2
   holds [.b3,b2.] \ ].b1,b2.] = [.b3,b1.];

:: XXREAL_1:th 193
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2
   holds ].b3,b2.] \ ].b1,b2.] = ].b3,b1.];

:: XXREAL_1:th 194
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2
   holds [.b3,b2.[ \ [.b1,b2.[ = [.b3,b1.[;

:: XXREAL_1:th 195
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2
   holds ].b3,b2.[ \ [.b1,b2.[ = ].b3,b1.[;

:: XXREAL_1:th 196
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2
   holds [.b3,b2.[ \ ].b1,b2.[ = [.b3,b1.];

:: XXREAL_1:th 197
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2
   holds ].b3,b2.[ \ ].b1,b2.[ = ].b3,b1.];

:: XXREAL_1:th 198
theorem
for b1, b2, b3, b4 being ext-real set
      st [.b1,b2.[ meets [.b3,b4.[
   holds [.b1,b2.[ \ [.b3,b4.[ = [.b1,b3.[ \/ [.b4,b2.[;

:: XXREAL_1:th 199
theorem
for b1, b2, b3, b4 being ext-real set
      st ].b1,b2.] meets ].b3,b4.]
   holds ].b1,b2.] \ ].b3,b4.] = ].b1,b3.] \/ ].b4,b2.];

:: XXREAL_1:th 200
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b3 <= b4
   holds [.b1,b4.] \ ([.b1,b2.] \/ [.b3,b4.]) = ].b2,b3.[;

:: XXREAL_1:th 201
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2 & b2 <= b3
   holds [.b1,b3.] \ {b2} = [.b1,b2.[ \/ ].b2,b3.];

:: XXREAL_1:th 202
theorem
for b1, b2, b3 being ext-real set
      st b1 <= b2 & b2 < b3
   holds [.b1,b3.[ \ {b2} = [.b1,b2.[ \/ ].b2,b3.[;

:: XXREAL_1:th 203
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2 & b2 <= b3
   holds ].b1,b3.] \ {b2} = ].b1,b2.[ \/ ].b2,b3.];

:: XXREAL_1:th 204
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2 & b2 < b3
   holds ].b1,b3.[ \ {b2} = ].b1,b2.[ \/ ].b2,b3.[;

:: XXREAL_1:th 205
theorem
for b1, b2, b3 being ext-real set holds
not b1 in ].b2,b1.[ \/ ].b1,b3.[;

:: XXREAL_1:th 206
theorem
for b1, b2, b3 being ext-real set holds
not b1 in [.b2,b1.[ \/ ].b1,b3.[;

:: XXREAL_1:th 207
theorem
for b1, b2, b3 being ext-real set holds
not b1 in ].b2,b1.[ \/ ].b1,b3.];

:: XXREAL_1:th 208
theorem
for b1, b2, b3 being ext-real set holds
not b1 in [.b2,b1.[ \/ ].b1,b3.];

:: XXREAL_1:th 209
theorem
[.-infty,+infty.] = ExtREAL;

:: XXREAL_1:th 210
theorem
for b1 being ext-real set holds
   ].b1,-infty.[ = {};

:: XXREAL_1:th 211
theorem
for b1 being ext-real set holds
   [.b1,-infty.[ = {};

:: XXREAL_1:th 212
theorem
for b1 being ext-real set holds
   ].b1,-infty.] = {};

:: XXREAL_1:th 213
theorem
for b1 being ext-real set
      st b1 <> -infty
   holds [.b1,-infty.] = {};

:: XXREAL_1:th 214
theorem
for b1 being ext-real set holds
   ].+infty,b1.[ = {};

:: XXREAL_1:th 215
theorem
for b1 being ext-real set holds
   [.+infty,b1.[ = {};

:: XXREAL_1:th 216
theorem
for b1 being ext-real set holds
   ].+infty,b1.] = {};

:: XXREAL_1:th 217
theorem
for b1 being ext-real set
      st b1 <> +infty
   holds [.+infty,b1.] = {};

:: XXREAL_1:th 218
theorem
for b1, b2 being ext-real set
      st b2 < b1
   holds b1 in ].b2,+infty.];

:: XXREAL_1:th 219
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds b2 in [.b1,+infty.];

:: XXREAL_1:th 220
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds b1 in [.-infty,b2.];

:: XXREAL_1:th 221
theorem
for b1, b2 being ext-real set
      st b1 < b2
   holds b1 in [.-infty,b2.[;

:: XXREAL_1:th 222
theorem
for b1, b2 being ext-real set
      st b1 <= b2
   holds [.b1,b2.] = [.b1,b2.] \/ [.b2,b1.];

:: XXREAL_1:th 223
theorem
for b1, b2, b3, b4 being ext-real set
      st b1 <= b2 & b2 <= b3
   holds not b1 in ].b2,b3.[ \/ ].b3,b4.[;