:: MEASURE5 semantic presentation
theorem Th1: :: MEASURE5:1
theorem Th2: :: MEASURE5:2
theorem Th3: :: MEASURE5:3
canceled;
theorem Th4: :: MEASURE5:4
canceled;
theorem Th5: :: MEASURE5:5
canceled;
theorem Th6: :: MEASURE5:6
canceled;
theorem Th7: :: MEASURE5:7
canceled;
theorem Th8: :: MEASURE5:8
theorem Th9: :: MEASURE5:9
:: deftheorem Def1 defines [. MEASURE5:def 1 :
:: deftheorem Def2 defines ]. MEASURE5:def 2 :
:: deftheorem Def3 defines ]. MEASURE5:def 3 :
:: deftheorem Def4 defines [. MEASURE5:def 4 :
:: deftheorem Def5 defines open_interval MEASURE5:def 5 :
:: deftheorem Def6 defines closed_interval MEASURE5:def 6 :
:: deftheorem Def7 defines right_open_interval MEASURE5:def 7 :
:: deftheorem Def8 defines left_open_interval MEASURE5:def 8 :
:: deftheorem Def9 defines interval MEASURE5:def 9 :
theorem Th10: :: MEASURE5:10
canceled;
theorem Th11: :: MEASURE5:11
theorem Th12: :: MEASURE5:12
theorem Th13: :: MEASURE5:13
theorem Th14: :: MEASURE5:14
theorem Th15: :: MEASURE5:15
theorem Th16: :: MEASURE5:16
theorem Th17: :: MEASURE5:17
theorem Th18: :: MEASURE5:18
theorem Th19: :: MEASURE5:19
theorem Th20: :: MEASURE5:20
theorem Th21: :: MEASURE5:21
theorem Th22: :: MEASURE5:22
theorem Th23: :: MEASURE5:23
theorem Th24: :: MEASURE5:24
theorem Th25: :: MEASURE5:25
theorem Th26: :: MEASURE5:26
theorem Th27: :: MEASURE5:27
theorem Th28: :: MEASURE5:28
theorem Th29: :: MEASURE5:29
theorem Th30: :: MEASURE5:30
theorem Th31: :: MEASURE5:31
theorem Th32: :: MEASURE5:32
theorem Th33: :: MEASURE5:33
theorem Th34: :: MEASURE5:34
theorem Th35: :: MEASURE5:35
theorem Th36: :: MEASURE5:36
theorem Th37: :: MEASURE5:37
theorem Th38: :: MEASURE5:38
theorem Th39: :: MEASURE5:39
theorem Th40: :: MEASURE5:40
theorem Th41: :: MEASURE5:41
theorem Th42: :: MEASURE5:42
theorem Th43: :: MEASURE5:43
theorem Th44: :: MEASURE5:44
theorem Th45: :: MEASURE5:45
theorem Th46: :: MEASURE5:46
theorem Th47: :: MEASURE5:47
theorem Th48: :: MEASURE5:48
theorem Th49: :: MEASURE5:49
theorem Th50: :: MEASURE5:50
theorem Th51: :: MEASURE5:51
theorem Th52: :: 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 )
definition
let A be
Interval;
func vol c1 -> R_eal means :
Def10:
:: MEASURE5:def 10
ex
a,
b being
R_eal st
( (
A = ].a,b.[ or
A = [.a,b.] or
A = [.a,b.[ or
A = ].a,b.] ) & (
a < b implies
it = b - a ) & (
b <= a implies
it = 0. ) );
existence
ex b1, a, b being R_eal st
( ( A = ].a,b.[ or A = [.a,b.] or A = [.a,b.[ or A = ].a,b.] ) & ( a < b implies b1 = b - a ) & ( b <= a implies b1 = 0. ) )
uniqueness
for b1, b2 being R_eal st ex a, b being R_eal st
( ( A = ].a,b.[ or A = [.a,b.] or A = [.a,b.[ or A = ].a,b.] ) & ( a < b implies b1 = b - a ) & ( b <= a implies b1 = 0. ) ) & ex a, b being R_eal st
( ( A = ].a,b.[ or A = [.a,b.] or A = [.a,b.[ or A = ].a,b.] ) & ( a < b implies b2 = b - a ) & ( b <= a implies b2 = 0. ) ) holds
b1 = b2
end;
:: deftheorem Def10 defines vol MEASURE5:def 10 :
theorem Th53: :: MEASURE5:53
theorem Th54: :: MEASURE5:54
theorem Th55: :: MEASURE5:55
theorem Th56: :: MEASURE5:56
theorem Th57: :: MEASURE5:57
for
A being
Interval for
a,
b,
c being
R_eal st
a = -infty &
b in REAL &
c = +infty & (
A = ].a,b.[ or
A = ].b,c.[ or
A = [.a,b.] or
A = [.b,c.] or
A = [.a,b.[ or
A = [.b,c.[ or
A = ].a,b.] or
A = ].b,c.] ) holds
vol A = +infty
theorem Th58: :: MEASURE5:58
theorem Th59: :: MEASURE5:59
canceled;
theorem Th60: :: MEASURE5:60
theorem Th61: :: MEASURE5:61
theorem Th62: :: MEASURE5:62
theorem Th63: :: MEASURE5:63