:: Properties of the Intervals of Real Numbers
:: by J\'ozef Bia{\l}as
::
:: Received January 12, 1993
:: Copyright (c) 1993 Association of Mizar Users

:: MEASURE5 semantic presentation

theorem :: MEASURE5:1
canceled;

theorem :: MEASURE5:2
canceled;

theorem :: MEASURE5:3
canceled;

theorem :: MEASURE5:4
canceled;

theorem :: MEASURE5:5
canceled;

theorem :: MEASURE5:6
canceled;

theorem :: MEASURE5:7
canceled;

theorem :: MEASURE5:8
for a, b, c being R_eal st b <> -infty & b <> +infty & ( not a = -infty or not c = -infty ) & ( not a = +infty or not c = +infty ) holds
(c - b) + (b - a) = c - a
proof end;

scheme :: MEASURE5:sch 1
RSetEq{ P1[ set ] } :
for X1, X2 being Subset of REAL st ( for x being R_eal holds
( x in X1 iff P1[x] ) ) & ( for x being R_eal holds
( x in X2 iff P1[x] ) ) holds
X1 = X2
proof end;

definition
let a, b be ext-real number ;
defpred S1[ R_eal] means ( a <= $1 & $1 <= b & $1 in REAL );
func |[.a,b.]| -> Subset of REAL means :Def1: :: MEASURE5:def 1
for x being R_eal holds
( x in it iff ( a <= x & x <= b & x in REAL ) );
existence
ex b1 being Subset of REAL st
for x being R_eal holds
( x in b1 iff ( a <= x & x <= b & x in REAL ) )
proof end;
uniqueness
for b1, b2 being Subset of REAL st ( for x being R_eal holds
( x in b1 iff ( a <= x & x <= b & x in REAL ) ) ) & ( for x being R_eal holds
( x in b2 iff ( a <= x & x <= b & x in REAL ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines |[. MEASURE5:def 1 :
for a, b being ext-real number
for b3 being Subset of REAL holds
( b3 = |[.a,b.]| iff for x being R_eal holds
( x in b3 iff ( a <= x & x <= b & x in REAL ) ) );

Lx1: for a, b being ext-real number holds |[.a,b.]| = [.a,b.] /\ REAL
proof end;

registration
let a, b be real number ;
identify |[.a,b.]| with [.a,b.];
compatibility
|[.a,b.]| = [.a,b.]
proof end;
end;

definition
let a, b be R_eal;
:: original: ].
redefine func ].a,b.[ -> Subset of REAL means :Def2: :: MEASURE5:def 2
for x being R_eal holds
( x in it iff ( a < x & x < b ) );
coherence
].a,b.[ is Subset of REAL
proof end;
compatibility
for b1 being Subset of REAL holds
( b1 = ].a,b.[ iff for x being R_eal holds
( x in b1 iff ( a < x & x < b ) ) )
proof end;
end;

:: deftheorem Def2 defines ]. MEASURE5:def 2 :
for a, b being R_eal
for b3 being Subset of REAL holds
( b3 = ].a,b.[ iff for x being R_eal holds
( x in b3 iff ( a < x & x < b ) ) );

definition
let a, b be ext-real number ;
defpred S1[ R_eal] means ( a < $1 & $1 <= b & $1 in REAL );
func ].a,b.]| -> Subset of REAL means :Def3: :: MEASURE5:def 3
for x being R_eal holds
( x in it iff ( a < x & x <= b & x in REAL ) );
existence
ex b1 being Subset of REAL st
for x being R_eal holds
( x in b1 iff ( a < x & x <= b & x in REAL ) )
proof end;
uniqueness
for b1, b2 being Subset of REAL st ( for x being R_eal holds
( x in b1 iff ( a < x & x <= b & x in REAL ) ) ) & ( for x being R_eal holds
( x in b2 iff ( a < x & x <= b & x in REAL ) ) ) holds
b1 = b2
proof end;
defpred S2[ R_eal] means ( a <= $1 & $1 < b & $1 in REAL );
func |[.a,b.[ -> Subset of REAL means :Def4: :: MEASURE5:def 4
for x being R_eal holds
( x in it iff ( a <= x & x < b & x in REAL ) );
existence
ex b1 being Subset of REAL st
for x being R_eal holds
( x in b1 iff ( a <= x & x < b & x in REAL ) )
proof end;
uniqueness
for b1, b2 being Subset of REAL st ( for x being R_eal holds
( x in b1 iff ( a <= x & x < b & x in REAL ) ) ) & ( for x being R_eal holds
( x in b2 iff ( a <= x & x < b & x in REAL ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines ]. MEASURE5:def 3 :
for a, b being ext-real number
for b3 being Subset of REAL holds
( b3 = ].a,b.]| iff for x being R_eal holds
( x in b3 iff ( a < x & x <= b & x in REAL ) ) );

:: deftheorem Def4 defines |[. MEASURE5:def 4 :
for a, b being ext-real number
for b3 being Subset of REAL holds
( b3 = |[.a,b.[ iff for x being R_eal holds
( x in b3 iff ( a <= x & x < b & x in REAL ) ) );

Lx2: for a, b being ext-real number holds |[.a,b.[ = [.a,b.[ /\ REAL
proof end;

Lx3: for a, b being ext-real number holds ].a,b.]| = ].a,b.] /\ REAL
proof end;

registration
let a be real number ;
let b be ext-real number ;
identify |[.a,b.[ with [.a,b.[;
compatibility
|[.a,b.[ = [.a,b.[
proof end;
end;

registration
let a be ext-real number ;
let b be real number ;
identify ].a,b.]| with ].a,b.];
compatibility
].a,b.]| = ].a,b.]
proof end;
end;

definition
let IT be Subset of REAL ;
attr IT is open_interval means :Def5: :: MEASURE5:def 5
ex a, b being R_eal st
( a <= b & IT = ].a,b.[ );
attr IT is closed_interval means :: MEASURE5:def 6
ex a, b being R_eal st
( a <= b & IT = |[.a,b.]| );
end;

:: deftheorem Def5 defines open_interval MEASURE5:def 5 :
for IT being Subset of REAL holds
( IT is open_interval iff ex a, b being R_eal st
( a <= b & IT = ].a,b.[ ) );

:: deftheorem defines closed_interval MEASURE5:def 6 :
for IT being Subset of REAL holds
( IT is closed_interval iff ex a, b being R_eal st
( a <= b & IT = |[.a,b.]| ) );

registration
cluster open_interval Element of K18(REAL );
existence
ex b1 being Subset of REAL st b1 is open_interval
proof end;
cluster closed_interval Element of K18(REAL );
existence
ex b1 being Subset of REAL st b1 is closed_interval
proof end;
end;

definition
let IT be Subset of REAL ;
attr IT is right_open_interval means :: MEASURE5:def 7
ex a, b being R_eal st
( a <= b & IT = |[.a,b.[ );
end;

:: deftheorem defines right_open_interval MEASURE5:def 7 :
for IT being Subset of REAL holds
( IT is right_open_interval iff ex a, b being R_eal st
( a <= b & IT = |[.a,b.[ ) );

notation
let IT be Subset of REAL ;
synonym left_closed_interval IT for right_open_interval IT;
end;

definition
let IT be Subset of REAL ;
attr IT is left_open_interval means :: MEASURE5:def 8
ex a, b being R_eal st
( a <= b & IT = ].a,b.]| );
end;

:: deftheorem defines left_open_interval MEASURE5:def 8 :
for IT being Subset of REAL holds
( IT is left_open_interval iff ex a, b being R_eal st
( a <= b & IT = ].a,b.]| ) );

notation
let IT be Subset of REAL ;
synonym right_closed_interval IT for left_open_interval IT;
end;

registration
cluster right_open_interval Element of K18(REAL );
existence
ex b1 being Subset of REAL st b1 is right_open_interval
proof end;
cluster left_open_interval Element of K18(REAL );
existence
ex b1 being Subset of REAL st b1 is left_open_interval
proof end;
end;

definition
let IT be Subset of REAL ;
attr IT is interval means :Def9: :: MEASURE5:def 9
( IT is open_interval or IT is closed_interval or IT is right_open_interval or IT is left_open_interval );
end;

:: deftheorem Def9 defines interval MEASURE5:def 9 :
for IT being Subset of REAL holds
( IT is interval iff ( IT is open_interval or IT is closed_interval or IT is right_open_interval or IT is left_open_interval ) );

registration
cluster interval Element of K18(REAL );
existence
ex b1 being Subset of REAL st b1 is interval
proof end;
end;

definition
mode Interval is interval Subset of REAL ;
end;

registration
cluster open_interval -> interval Element of K18(REAL );
coherence
for b1 being Subset of REAL st b1 is open_interval holds
b1 is interval
by Def9;
cluster closed_interval -> interval Element of K18(REAL );
coherence
for b1 being Subset of REAL st b1 is closed_interval holds
b1 is interval
by Def9;
cluster right_open_interval -> interval Element of K18(REAL );
coherence
for b1 being Subset of REAL st b1 is right_open_interval holds
b1 is interval
by Def9;
cluster left_open_interval -> interval Element of K18(REAL );
coherence
for b1 being Subset of REAL st b1 is left_open_interval holds
b1 is interval
by Def9;
end;

theorem :: MEASURE5:9
canceled;

theorem :: MEASURE5:10
canceled;

theorem Th11: :: MEASURE5:11
for x being set
for a, b being R_eal st ( x in ].a,b.[ or x in |[.a,b.]| or x in |[.a,b.[ or x in ].a,b.]| ) holds
x is R_eal
proof end;

theorem Th12: :: MEASURE5:12
for a, b being R_eal st b < a holds
( |[.a,b.]| = {} & |[.a,b.[ = {} & ].a,b.]| = {} )
proof end;

theorem Th13: :: MEASURE5:13
for a being R_eal holds
( |[.a,a.[ = {} & ].a,a.]| = {} )
proof end;

theorem Th14: :: MEASURE5:14
for a being R_eal holds
( ( ( a = -infty or a = +infty ) implies |[.a,a.]| = {} ) & ( a <> -infty & a <> +infty implies |[.a,a.]| = {a} ) )
proof end;

theorem Th15: :: MEASURE5:15
for a, b being R_eal st b <= a holds
( |[.a,b.[ = {} & ].a,b.]| = {} & |[.a,b.]| c= {a} & |[.a,b.]| c= {b} )
proof end;

theorem :: MEASURE5:16
canceled;

theorem Th17: :: MEASURE5:17
for a, b being R_eal st a < b holds
ex x being R_eal st
( a < x & x < b & x in REAL )
proof end;

theorem Th18: :: MEASURE5:18
for a, b, c being R_eal st a < b & a < c holds
ex x being R_eal st
( a < x & x < b & x < c & x in REAL )
proof end;

theorem Th19: :: MEASURE5:19
for a, b, c being R_eal st a < c & b < c holds
ex x being R_eal st
( a < x & b < x & x < c & x in REAL )
proof end;

theorem Th20: :: MEASURE5:20
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.[ & not x in ].a2,b2.[ ) or ( not x in ].a1,b1.[ & x in ].a2,b2.[ ) )
proof end;

theorem Th21: :: MEASURE5:21
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.[ & not x in ].a2,b2.[ ) or ( not x in ].a1,b1.[ & x in ].a2,b2.[ ) )
proof end;

theorem Th22: :: MEASURE5:22
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.]| & not x in ].a2,b2.[ ) or ( not x in |[.a1,b1.]| & x in ].a2,b2.[ ) )
proof end;

theorem Th23: :: MEASURE5:23
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.]| & not x in ].a2,b2.[ ) or ( not x in |[.a1,b1.]| & x in ].a2,b2.[ ) )
proof end;

theorem Th24: :: MEASURE5:24
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.[ & not x in |[.a2,b2.]| ) or ( not x in ].a1,b1.[ & x in |[.a2,b2.]| ) )
proof end;

theorem Th25: :: MEASURE5:25
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.[ & not x in |[.a2,b2.]| ) or ( not x in ].a1,b1.[ & x in |[.a2,b2.]| ) )
proof end;

theorem Th26: :: MEASURE5:26
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.[ & not x in |[.a2,b2.[ ) or ( not x in ].a1,b1.[ & x in |[.a2,b2.[ ) )
proof end;

theorem Th27: :: MEASURE5:27
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.[ & not x in |[.a2,b2.[ ) or ( not x in ].a1,b1.[ & x in |[.a2,b2.[ ) )
proof end;

theorem Th28: :: MEASURE5:28
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.[ & not x in ].a2,b2.[ ) or ( not x in |[.a1,b1.[ & x in ].a2,b2.[ ) )
proof end;

theorem Th29: :: MEASURE5:29
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.[ & not x in ].a2,b2.[ ) or ( not x in |[.a1,b1.[ & x in ].a2,b2.[ ) )
proof end;

theorem Th30: :: MEASURE5:30
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.[ & not x in ].a2,b2.]| ) or ( not x in ].a1,b1.[ & x in ].a2,b2.]| ) )
proof end;

theorem Th31: :: MEASURE5:31
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.[ & not x in ].a2,b2.]| ) or ( not x in ].a1,b1.[ & x in ].a2,b2.]| ) )
proof end;

theorem Th32: :: MEASURE5:32
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.]| & not x in ].a2,b2.[ ) or ( not x in ].a1,b1.]| & x in ].a2,b2.[ ) )
proof end;

theorem Th33: :: MEASURE5:33
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.]| & not x in ].a2,b2.[ ) or ( not x in ].a1,b1.]| & x in ].a2,b2.[ ) )
proof end;

theorem Th34: :: MEASURE5:34
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.]| & not x in |[.a2,b2.]| ) or ( not x in |[.a1,b1.]| & x in |[.a2,b2.]| ) )
proof end;

theorem Th35: :: MEASURE5:35
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.]| & not x in |[.a2,b2.]| ) or ( not x in |[.a1,b1.]| & x in |[.a2,b2.]| ) )
proof end;

theorem Th36: :: MEASURE5:36
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.]| & not x in |[.a2,b2.[ ) or ( not x in |[.a1,b1.]| & x in |[.a2,b2.[ ) )
proof end;

theorem Th37: :: MEASURE5:37
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.]| & not x in |[.a2,b2.[ ) or ( not x in |[.a1,b1.]| & x in |[.a2,b2.[ ) )
proof end;

theorem Th38: :: MEASURE5:38
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.[ & not x in |[.a2,b2.]| ) or ( not x in |[.a1,b1.[ & x in |[.a2,b2.]| ) )
proof end;

theorem Th39: :: MEASURE5:39
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.[ & not x in |[.a2,b2.]| ) or ( not x in |[.a1,b1.[ & x in |[.a2,b2.]| ) )
proof end;

theorem Th40: :: MEASURE5:40
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.]| & not x in ].a2,b2.]| ) or ( not x in |[.a1,b1.]| & x in ].a2,b2.]| ) )
proof end;

theorem Th41: :: MEASURE5:41
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.]| & not x in ].a2,b2.]| ) or ( not x in |[.a1,b1.]| & x in ].a2,b2.]| ) )
proof end;

theorem Th42: :: MEASURE5:42
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.]| & not x in |[.a2,b2.]| ) or ( not x in ].a1,b1.]| & x in |[.a2,b2.]| ) )
proof end;

theorem Th43: :: MEASURE5:43
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.]| & not x in |[.a2,b2.]| ) or ( not x in ].a1,b1.]| & x in |[.a2,b2.]| ) )
proof end;

theorem Th44: :: MEASURE5:44
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.[ & not x in |[.a2,b2.[ ) or ( not x in |[.a1,b1.[ & x in |[.a2,b2.[ ) )
proof end;

theorem Th45: :: MEASURE5:45
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.[ & not x in |[.a2,b2.[ ) or ( not x in |[.a1,b1.[ & x in |[.a2,b2.[ ) )
proof end;

theorem Th46: :: MEASURE5:46
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.[ & not x in ].a2,b2.]| ) or ( not x in |[.a1,b1.[ & x in ].a2,b2.]| ) )
proof end;

theorem Th47: :: MEASURE5:47
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in |[.a1,b1.[ & not x in ].a2,b2.]| ) or ( not x in |[.a1,b1.[ & x in ].a2,b2.]| ) )
proof end;

theorem Th48: :: MEASURE5:48
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.]| & not x in |[.a2,b2.[ ) or ( not x in ].a1,b1.]| & x in |[.a2,b2.[ ) )
proof end;

theorem Th49: :: MEASURE5:49
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.]| & not x in |[.a2,b2.[ ) or ( not x in ].a1,b1.]| & x in |[.a2,b2.[ ) )
proof end;

theorem Th50: :: MEASURE5:50
for a1, a2, b1, b2 being R_eal st a1 < a2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.]| & not x in ].a2,b2.]| ) or ( not x in ].a1,b1.]| & x in ].a2,b2.]| ) )
proof end;

theorem Th51: :: MEASURE5:51
for b1, b2, a1, a2 being R_eal st b1 < b2 & ( a1 < b1 or a2 < b2 ) holds
ex x being R_eal st
( ( x in ].a1,b1.]| & not x in ].a2,b2.]| ) or ( not x in ].a1,b1.]| & x in ].a2,b2.]| ) )
proof end;

theorem :: MEASURE5:52
for a1, b1, a2, b2 being R_eal
for A being Interval st a1 < b1 & ( A = ].a1,b1.[ or A = |[.a1,b1.]| or A = |[.a1,b1.[ or A = ].a1,b1.]| ) & ( A = ].a2,b2.[ or A = |[.a2,b2.]| or A = |[.a2,b2.[ or A = ].a2,b2.]| ) holds
( a1 = a2 & b1 = b2 )
proof end;

definition
let A be ext-real-membered set ;
func vol A -> R_eal equals :Def10: :: MEASURE5:def 10
(sup A) - (inf A) if A <> {}
otherwise 0. ;
coherence
( ( A <> {} implies (sup A) - (inf A) is R_eal ) & ( not A <> {} implies 0. is R_eal ) )
;
consistency
for b1 being R_eal holds verum
;
end;

:: deftheorem Def10 defines vol MEASURE5:def 10 :
for A being ext-real-membered set holds
( ( A <> {} implies vol A = (sup A) - (inf A) ) & ( not A <> {} implies vol A = 0. ) );

theorem :: MEASURE5:53
canceled;

theorem :: MEASURE5:54
for a, b being R_eal holds
( ( a < b implies vol ].a,b.[ = b - a ) & ( b <= a implies vol ].a,b.[ = 0. ) )
proof end;

theorem :: MEASURE5:55
for a, b being R_eal holds
( ( a <= b implies vol [.a,b.] = b - a ) & ( b < a implies vol [.a,b.] = 0. ) )
proof end;

theorem :: MEASURE5:56
for a, b being R_eal holds
( ( a < b implies vol [.a,b.[ = b - a ) & ( b <= a implies vol [.a,b.[ = 0. ) )
proof end;

theorem :: MEASURE5:57
for a, b being R_eal holds
( ( a < b implies vol ].a,b.] = b - a ) & ( b <= a implies vol ].a,b.] = 0. ) )
proof end;

theorem :: MEASURE5:58
canceled;

theorem :: MEASURE5:59
for A being Interval
for a, b being R_eal st a = -infty & b = +infty & ( A = ].a,b.[ or A = [.a,b.] or A = [.a,b.[ or A = ].a,b.] ) holds
vol A = +infty
proof end;

registration
cluster empty Element of K18(REAL );
existence
ex b1 being Interval st b1 is empty
proof end;
end;

definition
:: original: {}
redefine func {} -> Interval;
coherence
{} is Interval
proof end;
end;

theorem Th60: :: MEASURE5:60
vol {} = 0. by Def10;

LX3: for A being Interval holds vol A >= 0
proof end;

LX4: for A, B being Interval st A c= B holds
vol A <= vol B
proof end;

theorem :: MEASURE5:61
for a, b being R_eal
for A, B being Interval st A c= B & B = [.a,b.] & b <= a holds
( vol A = 0. & vol B = 0. )
proof end;

theorem :: MEASURE5:62
for A, B being Interval st A c= B holds
vol A <= vol B by LX4;

theorem :: MEASURE5:63
for A being Interval holds 0. <= vol A by LX3;

theorem :: MEASURE5:64
for a, b being ext-real number holds |[.a,b.]| = [.a,b.] /\ REAL by Lx1;

theorem :: MEASURE5:65
for a, b being ext-real number holds |[.a,b.[ = [.a,b.[ /\ REAL by Lx2;

theorem :: MEASURE5:66
for a, b being ext-real number holds ].a,b.]| = ].a,b.] /\ REAL by Lx3;

theorem :: MEASURE5:67
for a, b being ext-real number st a in REAL & b in REAL holds
|[.a,b.]| = [.a,b.]
proof end;

theorem :: MEASURE5:68
for a, b being ext-real number st a in REAL holds
|[.a,b.[ = [.a,b.[
proof end;

theorem :: MEASURE5:69
for a, b being ext-real number st b in REAL holds
].a,b.]| = ].a,b.]
proof end;

theorem :: MEASURE5:70
for a being ext-real number holds ].a,+infty .]| = ].a,+infty .[
proof end;

theorem :: MEASURE5:71
for a being ext-real number st a in REAL holds
|[.a,+infty .]| = [.a,+infty .[
proof end;

theorem :: MEASURE5:72
for a being ext-real number st a in REAL holds
|[.-infty ,a.]| = ].-infty ,a.]
proof end;

theorem :: MEASURE5:73
for a being ext-real number holds |[.-infty ,a.[ = ].-infty ,a.[
proof end;