Article MEASURE5, MML version 4.99.1005

:: MEASURE5:th 1
theorem
for b1, b2 being Element of ExtREAL
      st b1 <> -infty & b1 <> +infty & b1 <= b2
   holds 0. <= b2 - b1;

:: MEASURE5:th 2
theorem
for b1, b2 being Element of ExtREAL
      st (b1 = -infty implies b2 <> -infty) & (b1 = +infty implies b2 <> +infty) & b1 <= b2
   holds 0. <= b2 - b1;

:: MEASURE5:th 8
theorem
for b1, b2, b3 being Element of ExtREAL
      st b2 <> -infty & b2 <> +infty & (b1 = -infty implies b3 <> -infty) & (b1 = +infty implies b3 <> +infty)
   holds (b3 - b2) + (b2 - b1) = b3 - b1;

:: MEASURE5:th 9
theorem
for b1, b2 being Element of ExtREAL holds
inf {b1,b2} <= b1 & inf {b1,b2} <= b2 & b1 <= sup {b1,b2} & b2 <= sup {b1,b2};

:: MEASURE5:sch 1
scheme MEASURE5:sch 1
for b1, b2 being Element of bool REAL
      st (for b3 being Element of ExtREAL holds
               b3 in b1
            iff
               P1[b3]) &
         (for b3 being Element of ExtREAL holds
               b3 in b2
            iff
               P1[b3])
   holds b1 = b2


:: MEASURE5:funcnot 1 => MEASURE5:func 1
definition
  let a1, a2 be Element of ExtREAL;
  func [.A1,A2.] -> Element of bool REAL means
    for b1 being Element of ExtREAL holds
          b1 in it
       iff
          a1 <= b1 & b1 <= a2 & b1 in REAL;
end;

:: MEASURE5:def 1
theorem
for b1, b2 being Element of ExtREAL
for b3 being Element of bool REAL holds
      b3 = [.b1,b2.]
   iff
      for b4 being Element of ExtREAL holds
            b4 in b3
         iff
            b1 <= b4 & b4 <= b2 & b4 in REAL;

:: MEASURE5:funcnot 2 => MEASURE5:func 2
definition
  let a1, a2 be Element of ExtREAL;
  redefine func ].A1,A2.[ -> Element of bool REAL means
    for b1 being Element of ExtREAL holds
          b1 in it
       iff
          a1 < b1 & b1 < a2 & b1 in REAL;
end;

:: MEASURE5:def 2
theorem
for b1, b2 being Element of ExtREAL
for b3 being Element of bool REAL holds
      b3 = ].b1,b2.[
   iff
      for b4 being Element of ExtREAL holds
            b4 in b3
         iff
            b1 < b4 & b4 < b2 & b4 in REAL;

:: MEASURE5:funcnot 3 => MEASURE5:func 3
definition
  let a1, a2 be Element of ExtREAL;
  func ].A1,A2.] -> Element of bool REAL means
    for b1 being Element of ExtREAL holds
          b1 in it
       iff
          a1 < b1 & b1 <= a2 & b1 in REAL;
end;

:: MEASURE5:def 3
theorem
for b1, b2 being Element of ExtREAL
for b3 being Element of bool REAL holds
      b3 = ].b1,b2.]
   iff
      for b4 being Element of ExtREAL holds
            b4 in b3
         iff
            b1 < b4 & b4 <= b2 & b4 in REAL;

:: MEASURE5:funcnot 4 => MEASURE5:func 4
definition
  let a1, a2 be Element of ExtREAL;
  func [.A1,A2.[ -> Element of bool REAL means
    for b1 being Element of ExtREAL holds
          b1 in it
       iff
          a1 <= b1 & b1 < a2 & b1 in REAL;
end;

:: MEASURE5:def 4
theorem
for b1, b2 being Element of ExtREAL
for b3 being Element of bool REAL holds
      b3 = [.b1,b2.[
   iff
      for b4 being Element of ExtREAL holds
            b4 in b3
         iff
            b1 <= b4 & b4 < b2 & b4 in REAL;

:: MEASURE5:attrnot 1 => MEASURE5:attr 1
definition
  let a1 be Element of bool REAL;
  attr a1 is open_interval means
    ex b1, b2 being Element of ExtREAL st
       b1 <= b2 & a1 = ].b1,b2.[;
end;

:: MEASURE5:dfs 5
definiens
  let a1 be Element of bool REAL;
To prove
     a1 is open_interval
it is sufficient to prove
  thus ex b1, b2 being Element of ExtREAL st
       b1 <= b2 & a1 = ].b1,b2.[;

:: MEASURE5:def 5
theorem
for b1 being Element of bool REAL holds
      b1 is open_interval
   iff
      ex b2, b3 being Element of ExtREAL st
         b2 <= b3 & b1 = ].b2,b3.[;

:: MEASURE5:attrnot 2 => MEASURE5:attr 2
definition
  let a1 be Element of bool REAL;
  attr a1 is closed_interval means
    ex b1, b2 being Element of ExtREAL st
       b1 <= b2 & a1 = [.b1,b2.];
end;

:: MEASURE5:dfs 6
definiens
  let a1 be Element of bool REAL;
To prove
     a1 is closed_interval
it is sufficient to prove
  thus ex b1, b2 being Element of ExtREAL st
       b1 <= b2 & a1 = [.b1,b2.];

:: MEASURE5:def 6
theorem
for b1 being Element of bool REAL holds
      b1 is closed_interval
   iff
      ex b2, b3 being Element of ExtREAL st
         b2 <= b3 & b1 = [.b2,b3.];

:: MEASURE5:exreg 1
registration
  cluster complex-membered ext-real-membered real-membered open_interval Element of bool REAL;
end;

:: MEASURE5:exreg 2
registration
  cluster complex-membered ext-real-membered real-membered closed_interval Element of bool REAL;
end;

:: MEASURE5:attrnot 3 => MEASURE5:attr 3
definition
  let a1 be Element of bool REAL;
  attr a1 is right_open_interval means
    ex b1, b2 being Element of ExtREAL st
       b1 <= b2 & a1 = [.b1,b2.[;
end;

:: MEASURE5:dfs 7
definiens
  let a1 be Element of bool REAL;
To prove
     a1 is right_open_interval
it is sufficient to prove
  thus ex b1, b2 being Element of ExtREAL st
       b1 <= b2 & a1 = [.b1,b2.[;

:: MEASURE5:def 7
theorem
for b1 being Element of bool REAL holds
      b1 is right_open_interval
   iff
      ex b2, b3 being Element of ExtREAL st
         b2 <= b3 & b1 = [.b2,b3.[;

:: MEASURE5:attrnot 4 => MEASURE5:attr 3
notation
  let a1 be Element of bool REAL;
  synonym left_closed_interval for right_open_interval;
end;

:: MEASURE5:attrnot 5 => MEASURE5:attr 4
definition
  let a1 be Element of bool REAL;
  attr a1 is left_open_interval means
    ex b1, b2 being Element of ExtREAL st
       b1 <= b2 & a1 = ].b1,b2.];
end;

:: MEASURE5:dfs 8
definiens
  let a1 be Element of bool REAL;
To prove
     a1 is left_open_interval
it is sufficient to prove
  thus ex b1, b2 being Element of ExtREAL st
       b1 <= b2 & a1 = ].b1,b2.];

:: MEASURE5:def 8
theorem
for b1 being Element of bool REAL holds
      b1 is left_open_interval
   iff
      ex b2, b3 being Element of ExtREAL st
         b2 <= b3 & b1 = ].b2,b3.];

:: MEASURE5:attrnot 6 => MEASURE5:attr 4
notation
  let a1 be Element of bool REAL;
  synonym right_closed_interval for left_open_interval;
end;

:: MEASURE5:exreg 3
registration
  cluster complex-membered ext-real-membered real-membered right_open_interval Element of bool REAL;
end;

:: MEASURE5:exreg 4
registration
  cluster complex-membered ext-real-membered real-membered left_open_interval Element of bool REAL;
end;

:: MEASURE5:attrnot 7 => MEASURE5:attr 5
definition
  let a1 be Element of bool REAL;
  attr a1 is interval means
    (a1 is not open_interval & a1 is not closed_interval & a1 is not right_open_interval) implies a1 is left_open_interval;
end;

:: MEASURE5:dfs 9
definiens
  let a1 be Element of bool REAL;
To prove
     a1 is interval
it is sufficient to prove
  thus (a1 is not open_interval & a1 is not closed_interval & a1 is not right_open_interval) implies a1 is left_open_interval;

:: MEASURE5:def 9
theorem
for b1 being Element of bool REAL holds
      b1 is interval
   iff
      (b1 is not open_interval & b1 is not closed_interval & b1 is not right_open_interval implies b1 is left_open_interval);

:: MEASURE5:exreg 5
registration
  cluster complex-membered ext-real-membered real-membered interval Element of bool REAL;
end;

:: MEASURE5:modenot 1
definition
  mode Interval is interval Element of bool REAL;
end;

:: MEASURE5:condreg 1
registration
  cluster open_interval -> interval (Element of bool REAL);
end;

:: MEASURE5:condreg 2
registration
  cluster closed_interval -> interval (Element of bool REAL);
end;

:: MEASURE5:condreg 3
registration
  cluster right_open_interval -> interval (Element of bool REAL);
end;

:: MEASURE5:condreg 4
registration
  cluster left_open_interval -> interval (Element of bool REAL);
end;

:: MEASURE5:th 11
theorem
for b1 being set
for b2, b3 being Element of ExtREAL
      st (not b1 in ].b2,b3.[ & not b1 in [.b2,b3.] & not b1 in [.b2,b3.[ implies b1 in ].b2,b3.])
   holds b1 is Element of ExtREAL;

:: MEASURE5:th 12
theorem
for b1, b2 being Element of ExtREAL
      st b2 < b1
   holds [.b1,b2.] = {} & [.b1,b2.[ = {} & ].b1,b2.] = {};

:: MEASURE5:th 13
theorem
for b1 being Element of ExtREAL holds
   [.b1,b1.[ = {} & ].b1,b1.] = {};

:: MEASURE5:th 14
theorem
for b1 being Element of ExtREAL holds
   (b1 <> -infty & b1 <> +infty or [.b1,b1.] = {}) &
    (b1 <> -infty & b1 <> +infty implies [.b1,b1.] = {b1});

:: MEASURE5:th 15
theorem
for b1, b2 being Element of ExtREAL
      st b2 <= b1
   holds ].b1,b2.[ = {} & [.b1,b2.[ = {} & ].b1,b2.] = {} & [.b1,b2.] c= {b1} & [.b1,b2.] c= {b2};

:: MEASURE5:th 16
theorem
for b1, b2, b3 being Element of ExtREAL
      st b1 < b2 & b2 < b3
   holds b2 in REAL;

:: MEASURE5:th 17
theorem
for b1, b2 being Element of ExtREAL
      st b1 < b2
   holds ex b3 being Element of ExtREAL st
      b1 < b3 & b3 < b2 & b3 in REAL;

:: MEASURE5:th 18
theorem
for b1, b2, b3 being Element of ExtREAL
      st b1 < b2 & b1 < b3
   holds ex b4 being Element of ExtREAL st
      b1 < b4 & b4 < b2 & b4 < b3 & b4 in REAL;

:: MEASURE5:th 19
theorem
for b1, b2, b3 being Element of ExtREAL
      st b1 < b3 & b2 < b3
   holds ex b4 being Element of ExtREAL st
      b1 < b4 & b2 < b4 & b4 < b3 & b4 in REAL;

:: MEASURE5:th 20
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b1,b3.[ & not b5 in ].b2,b4.[ or not b5 in ].b1,b3.[ & b5 in ].b2,b4.[);

:: MEASURE5:th 21
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b3,b1.[ & not b5 in ].b4,b2.[ or not b5 in ].b3,b1.[ & b5 in ].b4,b2.[);

:: MEASURE5:th 22
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b1,b3.] & not b5 in ].b2,b4.[ or not b5 in [.b1,b3.] & b5 in ].b2,b4.[);

:: MEASURE5:th 23
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b3,b1.] & not b5 in ].b4,b2.[ or not b5 in [.b3,b1.] & b5 in ].b4,b2.[);

:: MEASURE5:th 24
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b1,b3.[ & not b5 in [.b2,b4.] or not b5 in ].b1,b3.[ & b5 in [.b2,b4.]);

:: MEASURE5:th 25
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b3,b1.[ & not b5 in [.b4,b2.] or not b5 in ].b3,b1.[ & b5 in [.b4,b2.]);

:: MEASURE5:th 26
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b1,b3.[ & not b5 in [.b2,b4.[ or not b5 in ].b1,b3.[ & b5 in [.b2,b4.[);

:: MEASURE5:th 27
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b3,b1.[ & not b5 in [.b4,b2.[ or not b5 in ].b3,b1.[ & b5 in [.b4,b2.[);

:: MEASURE5:th 28
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b1,b3.[ & not b5 in ].b2,b4.[ or not b5 in [.b1,b3.[ & b5 in ].b2,b4.[);

:: MEASURE5:th 29
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b3,b1.[ & not b5 in ].b4,b2.[ or not b5 in [.b3,b1.[ & b5 in ].b4,b2.[);

:: MEASURE5:th 30
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b1,b3.[ & not b5 in ].b2,b4.] or not b5 in ].b1,b3.[ & b5 in ].b2,b4.]);

:: MEASURE5:th 31
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b3,b1.[ & not b5 in ].b4,b2.] or not b5 in ].b3,b1.[ & b5 in ].b4,b2.]);

:: MEASURE5:th 32
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b1,b3.] & not b5 in ].b2,b4.[ or not b5 in ].b1,b3.] & b5 in ].b2,b4.[);

:: MEASURE5:th 33
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b3,b1.] & not b5 in ].b4,b2.[ or not b5 in ].b3,b1.] & b5 in ].b4,b2.[);

:: MEASURE5:th 34
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b1,b3.] & not b5 in [.b2,b4.] or not b5 in [.b1,b3.] & b5 in [.b2,b4.]);

:: MEASURE5:th 35
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b3,b1.] & not b5 in [.b4,b2.] or not b5 in [.b3,b1.] & b5 in [.b4,b2.]);

:: MEASURE5:th 36
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b1,b3.] & not b5 in [.b2,b4.[ or not b5 in [.b1,b3.] & b5 in [.b2,b4.[);

:: MEASURE5:th 37
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b3,b1.] & not b5 in [.b4,b2.[ or not b5 in [.b3,b1.] & b5 in [.b4,b2.[);

:: MEASURE5:th 38
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b1,b3.[ & not b5 in [.b2,b4.] or not b5 in [.b1,b3.[ & b5 in [.b2,b4.]);

:: MEASURE5:th 39
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b3,b1.[ & not b5 in [.b4,b2.] or not b5 in [.b3,b1.[ & b5 in [.b4,b2.]);

:: MEASURE5:th 40
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b1,b3.] & not b5 in ].b2,b4.] or not b5 in [.b1,b3.] & b5 in ].b2,b4.]);

:: MEASURE5:th 41
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b3,b1.] & not b5 in ].b4,b2.] or not b5 in [.b3,b1.] & b5 in ].b4,b2.]);

:: MEASURE5:th 42
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b1,b3.] & not b5 in [.b2,b4.] or not b5 in ].b1,b3.] & b5 in [.b2,b4.]);

:: MEASURE5:th 43
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b3,b1.] & not b5 in [.b4,b2.] or not b5 in ].b3,b1.] & b5 in [.b4,b2.]);

:: MEASURE5:th 44
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b1,b3.[ & not b5 in [.b2,b4.[ or not b5 in [.b1,b3.[ & b5 in [.b2,b4.[);

:: MEASURE5:th 45
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b3,b1.[ & not b5 in [.b4,b2.[ or not b5 in [.b3,b1.[ & b5 in [.b4,b2.[);

:: MEASURE5:th 46
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b1,b3.[ & not b5 in ].b2,b4.] or not b5 in [.b1,b3.[ & b5 in ].b2,b4.]);

:: MEASURE5:th 47
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in [.b3,b1.[ & not b5 in ].b4,b2.] or not b5 in [.b3,b1.[ & b5 in ].b4,b2.]);

:: MEASURE5:th 48
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b1,b3.] & not b5 in [.b2,b4.[ or not b5 in ].b1,b3.] & b5 in [.b2,b4.[);

:: MEASURE5:th 49
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b3,b1.] & not b5 in [.b4,b2.[ or not b5 in ].b3,b1.] & b5 in [.b4,b2.[);

:: MEASURE5:th 50
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b3 <= b1 implies b2 < b4)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b1,b3.] & not b5 in ].b2,b4.] or not b5 in ].b1,b3.] & b5 in ].b2,b4.]);

:: MEASURE5:th 51
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st b1 < b2 & (b1 <= b3 implies b4 < b2)
   holds ex b5 being Element of ExtREAL st
      (b5 in ].b3,b1.] & not b5 in ].b4,b2.] or not b5 in ].b3,b1.] & b5 in ].b4,b2.]);

:: MEASURE5:th 52
theorem
for b1, b2, b3, b4 being Element of ExtREAL
for b5 being interval Element of bool REAL
      st b1 < b2 &
         (b5 <> ].b1,b2.[ & b5 <> [.b1,b2.] & b5 <> [.b1,b2.[ implies b5 = ].b1,b2.]) &
         (b5 <> ].b3,b4.[ & b5 <> [.b3,b4.] & b5 <> [.b3,b4.[ implies b5 = ].b3,b4.])
   holds b1 = b3 & b2 = b4;

:: MEASURE5:funcnot 5 => MEASURE5:func 5
definition
  let a1 be interval Element of bool REAL;
  func vol A1 -> Element of ExtREAL means
    ex b1, b2 being Element of ExtREAL st
       (a1 <> ].b1,b2.[ & a1 <> [.b1,b2.] & a1 <> [.b1,b2.[ implies a1 = ].b1,b2.]) &
        (b2 <= b1 or it = b2 - b1) &
        (b2 <= b1 implies it = 0.);
end;

:: MEASURE5:def 10
theorem
for b1 being interval Element of bool REAL
for b2 being Element of ExtREAL holds
      b2 = vol b1
   iff
      ex b3, b4 being Element of ExtREAL st
         (b1 <> ].b3,b4.[ & b1 <> [.b3,b4.] & b1 <> [.b3,b4.[ implies b1 = ].b3,b4.]) &
          (b4 <= b3 or b2 = b4 - b3) &
          (b4 <= b3 implies b2 = 0.);

:: MEASURE5:th 53
theorem
for b1 being open_interval Element of bool REAL
for b2, b3 being Element of ExtREAL
      st b1 = ].b2,b3.[
   holds (b3 <= b2 or vol b1 = b3 - b2) & (b3 <= b2 implies vol b1 = 0.);

:: MEASURE5:th 54
theorem
for b1 being closed_interval Element of bool REAL
for b2, b3 being Element of ExtREAL
      st b1 = [.b2,b3.]
   holds (b3 <= b2 or vol b1 = b3 - b2) & (b3 <= b2 implies vol b1 = 0.);

:: MEASURE5:th 55
theorem
for b1 being right_open_interval Element of bool REAL
for b2, b3 being Element of ExtREAL
      st b1 = [.b2,b3.[
   holds (b3 <= b2 or vol b1 = b3 - b2) & (b3 <= b2 implies vol b1 = 0.);

:: MEASURE5:th 56
theorem
for b1 being left_open_interval Element of bool REAL
for b2, b3 being Element of ExtREAL
      st b1 = ].b2,b3.]
   holds (b3 <= b2 or vol b1 = b3 - b2) & (b3 <= b2 implies vol b1 = 0.);

:: MEASURE5:th 57
theorem
for b1 being interval Element of bool REAL
for b2, b3, b4 being Element of ExtREAL
      st b2 = -infty &
         b3 in REAL &
         b4 = +infty &
         (b1 <> ].b2,b3.[ & b1 <> ].b3,b4.[ & b1 <> [.b2,b3.] & b1 <> [.b3,b4.] & b1 <> [.b2,b3.[ & b1 <> [.b3,b4.[ & b1 <> ].b2,b3.] implies b1 = ].b3,b4.])
   holds vol b1 = +infty;

:: MEASURE5:th 58
theorem
for b1 being interval Element of bool REAL
for b2, b3 being Element of ExtREAL
      st b2 = -infty &
         b3 = +infty &
         (b1 <> ].b2,b3.[ & b1 <> [.b2,b3.] & b1 <> [.b2,b3.[ implies b1 = ].b2,b3.])
   holds vol b1 = +infty;

:: MEASURE5:exreg 6
registration
  cluster empty complex-membered ext-real-membered real-membered interval Element of bool REAL;
end;

:: MEASURE5:funcnot 6 => MEASURE5:func 6
definition
  redefine func {} -> interval Element of bool REAL;
end;

:: MEASURE5:th 60
theorem
vol {} = 0.;

:: MEASURE5:th 61
theorem
for b1, b2 being Element of ExtREAL
for b3, b4 being interval Element of bool REAL
      st b3 c= b4 & b4 = [.b1,b2.] & b2 <= b1
   holds vol b3 = 0. & vol b4 = 0.;

:: MEASURE5:th 62
theorem
for b1, b2 being interval Element of bool REAL
      st b1 c= b2
   holds vol b1 <= vol b2;

:: MEASURE5:th 63
theorem
for b1 being interval Element of bool REAL holds
   0. <= vol b1;