:: 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
for b
1, b
2, b
3 being
R_eal holds
( b
1 < b
2 & b
2 < b
3 implies b
2 in REAL )
theorem Th17: :: MEASURE5:17
for b
1, b
2 being
R_eal holds
not ( b
1 < b
2 & ( for b
3 being
R_eal holds
not ( b
1 < b
3 & b
3 < b
2 & b
3 in REAL ) ) )
theorem Th18: :: MEASURE5:18
for b
1, b
2, b
3 being
R_eal holds
not ( b
1 < b
2 & b
1 < b
3 & ( for b
4 being
R_eal holds
not ( b
1 < b
4 & b
4 < b
2 & b
4 < b
3 & b
4 in REAL ) ) )
theorem Th19: :: MEASURE5:19
for b
1, b
2, b
3 being
R_eal holds
not ( b
1 < b
3 & b
2 < b
3 & ( for b
4 being
R_eal holds
not ( b
1 < b
4 & b
2 < b
4 & b
4 < b
3 & b
4 in REAL ) ) )
theorem Th20: :: MEASURE5:20
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b1,b3.[ & not b
5 in ].b2,b4.[ ) & not ( not b
5 in ].b1,b3.[ & b
5 in ].b2,b4.[ ) ) ) )
theorem Th21: :: MEASURE5:21
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b3,b1.[ & not b
5 in ].b4,b2.[ ) & not ( not b
5 in ].b3,b1.[ & b
5 in ].b4,b2.[ ) ) ) )
theorem Th22: :: MEASURE5:22
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b1,b3.] & not b
5 in ].b2,b4.[ ) & not ( not b
5 in [.b1,b3.] & b
5 in ].b2,b4.[ ) ) ) )
theorem Th23: :: MEASURE5:23
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b3,b1.] & not b
5 in ].b4,b2.[ ) & not ( not b
5 in [.b3,b1.] & b
5 in ].b4,b2.[ ) ) ) )
theorem Th24: :: MEASURE5:24
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b1,b3.[ & not b
5 in [.b2,b4.] ) & not ( not b
5 in ].b1,b3.[ & b
5 in [.b2,b4.] ) ) ) )
theorem Th25: :: MEASURE5:25
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b3,b1.[ & not b
5 in [.b4,b2.] ) & not ( not b
5 in ].b3,b1.[ & b
5 in [.b4,b2.] ) ) ) )
theorem Th26: :: MEASURE5:26
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b1,b3.[ & not b
5 in [.b2,b4.[ ) & not ( not b
5 in ].b1,b3.[ & b
5 in [.b2,b4.[ ) ) ) )
theorem Th27: :: MEASURE5:27
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b3,b1.[ & not b
5 in [.b4,b2.[ ) & not ( not b
5 in ].b3,b1.[ & b
5 in [.b4,b2.[ ) ) ) )
theorem Th28: :: MEASURE5:28
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b1,b3.[ & not b
5 in ].b2,b4.[ ) & not ( not b
5 in [.b1,b3.[ & b
5 in ].b2,b4.[ ) ) ) )
theorem Th29: :: MEASURE5:29
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b3,b1.[ & not b
5 in ].b4,b2.[ ) & not ( not b
5 in [.b3,b1.[ & b
5 in ].b4,b2.[ ) ) ) )
theorem Th30: :: MEASURE5:30
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b1,b3.[ & not b
5 in ].b2,b4.] ) & not ( not b
5 in ].b1,b3.[ & b
5 in ].b2,b4.] ) ) ) )
theorem Th31: :: MEASURE5:31
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b3,b1.[ & not b
5 in ].b4,b2.] ) & not ( not b
5 in ].b3,b1.[ & b
5 in ].b4,b2.] ) ) ) )
theorem Th32: :: MEASURE5:32
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b1,b3.] & not b
5 in ].b2,b4.[ ) & not ( not b
5 in ].b1,b3.] & b
5 in ].b2,b4.[ ) ) ) )
theorem Th33: :: MEASURE5:33
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b3,b1.] & not b
5 in ].b4,b2.[ ) & not ( not b
5 in ].b3,b1.] & b
5 in ].b4,b2.[ ) ) ) )
theorem Th34: :: MEASURE5:34
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b1,b3.] & not b
5 in [.b2,b4.] ) & not ( not b
5 in [.b1,b3.] & b
5 in [.b2,b4.] ) ) ) )
theorem Th35: :: MEASURE5:35
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b3,b1.] & not b
5 in [.b4,b2.] ) & not ( not b
5 in [.b3,b1.] & b
5 in [.b4,b2.] ) ) ) )
theorem Th36: :: MEASURE5:36
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b1,b3.] & not b
5 in [.b2,b4.[ ) & not ( not b
5 in [.b1,b3.] & b
5 in [.b2,b4.[ ) ) ) )
theorem Th37: :: MEASURE5:37
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b3,b1.] & not b
5 in [.b4,b2.[ ) & not ( not b
5 in [.b3,b1.] & b
5 in [.b4,b2.[ ) ) ) )
theorem Th38: :: MEASURE5:38
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b1,b3.[ & not b
5 in [.b2,b4.] ) & not ( not b
5 in [.b1,b3.[ & b
5 in [.b2,b4.] ) ) ) )
theorem Th39: :: MEASURE5:39
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b3,b1.[ & not b
5 in [.b4,b2.] ) & not ( not b
5 in [.b3,b1.[ & b
5 in [.b4,b2.] ) ) ) )
theorem Th40: :: MEASURE5:40
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b1,b3.] & not b
5 in ].b2,b4.] ) & not ( not b
5 in [.b1,b3.] & b
5 in ].b2,b4.] ) ) ) )
theorem Th41: :: MEASURE5:41
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b3,b1.] & not b
5 in ].b4,b2.] ) & not ( not b
5 in [.b3,b1.] & b
5 in ].b4,b2.] ) ) ) )
theorem Th42: :: MEASURE5:42
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b1,b3.] & not b
5 in [.b2,b4.] ) & not ( not b
5 in ].b1,b3.] & b
5 in [.b2,b4.] ) ) ) )
theorem Th43: :: MEASURE5:43
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b3,b1.] & not b
5 in [.b4,b2.] ) & not ( not b
5 in ].b3,b1.] & b
5 in [.b4,b2.] ) ) ) )
theorem Th44: :: MEASURE5:44
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b1,b3.[ & not b
5 in [.b2,b4.[ ) & not ( not b
5 in [.b1,b3.[ & b
5 in [.b2,b4.[ ) ) ) )
theorem Th45: :: MEASURE5:45
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b3,b1.[ & not b
5 in [.b4,b2.[ ) & not ( not b
5 in [.b3,b1.[ & b
5 in [.b4,b2.[ ) ) ) )
theorem Th46: :: MEASURE5:46
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b1,b3.[ & not b
5 in ].b2,b4.] ) & not ( not b
5 in [.b1,b3.[ & b
5 in ].b2,b4.] ) ) ) )
theorem Th47: :: MEASURE5:47
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in [.b3,b1.[ & not b
5 in ].b4,b2.] ) & not ( not b
5 in [.b3,b1.[ & b
5 in ].b4,b2.] ) ) ) )
theorem Th48: :: MEASURE5:48
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b1,b3.] & not b
5 in [.b2,b4.[ ) & not ( not b
5 in ].b1,b3.] & b
5 in [.b2,b4.[ ) ) ) )
theorem Th49: :: MEASURE5:49
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b3,b1.] & not b
5 in [.b4,b2.[ ) & not ( not b
5 in ].b3,b1.] & b
5 in [.b4,b2.[ ) ) ) )
theorem Th50: :: MEASURE5:50
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
1 < b
3 & not b
2 < b
4 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b1,b3.] & not b
5 in ].b2,b4.] ) & not ( not b
5 in ].b1,b3.] & b
5 in ].b2,b4.] ) ) ) )
theorem Th51: :: MEASURE5:51
for b
1, b
2, b
3, b
4 being
R_eal holds
not ( b
1 < b
2 & not ( not b
3 < b
1 & not b
4 < b
2 ) & ( for b
5 being
R_eal holds
( not ( b
5 in ].b3,b1.] & not b
5 in ].b4,b2.] ) & not ( not b
5 in ].b3,b1.] & b
5 in ].b4,b2.] ) ) ) )
theorem Th52: :: MEASURE5:52
for b
1, b
2, b
3, b
4 being
R_ealfor b
5 being
Interval holds
( b
1 < b
2 & not ( not b
5 = ].b1,b2.[ & not b
5 = [.b1,b2.] & not b
5 = [.b1,b2.[ & not b
5 = ].b1,b2.] ) & not ( not b
5 = ].b3,b4.[ & not b
5 = [.b3,b4.] & not b
5 = [.b3,b4.[ & not b
5 = ].b3,b4.] ) implies ( b
1 = b
3 & b
2 = b
4 ) )
definition
let c
1 be
Interval;
func vol c
1 -> R_eal means :
Def10:
:: MEASURE5:def 10
ex b
1, b
2 being
R_eal st
( not ( not a
1 = ].b1,b2.[ & not a
1 = [.b1,b2.] & not a
1 = [.b1,b2.[ & not a
1 = ].b1,b2.] ) & ( b
1 < b
2 implies a
2 = b
2 - b
1 ) & ( b
2 <= b
1 implies a
2 = 0. ) );
existence
ex b1, b2, b3 being R_eal st
( not ( not c1 = ].b2,b3.[ & not c1 = [.b2,b3.] & not c1 = [.b2,b3.[ & not c1 = ].b2,b3.] ) & ( b2 < b3 implies b1 = b3 - b2 ) & ( b3 <= b2 implies b1 = 0. ) )
uniqueness
for b1, b2 being R_eal holds
( ex b3, b4 being R_eal st
( not ( not c1 = ].b3,b4.[ & not c1 = [.b3,b4.] & not c1 = [.b3,b4.[ & not c1 = ].b3,b4.] ) & ( b3 < b4 implies b1 = b4 - b3 ) & ( b4 <= b3 implies b1 = 0. ) ) & ex b3, b4 being R_eal st
( not ( not c1 = ].b3,b4.[ & not c1 = [.b3,b4.] & not c1 = [.b3,b4.[ & not c1 = ].b3,b4.] ) & ( b3 < b4 implies b2 = b4 - b3 ) & ( b4 <= b3 implies b2 = 0. ) ) implies b1 = b2 )
end;
:: deftheorem Def10 defines vol MEASURE5:def 10 :
for b
1 being
Intervalfor b
2 being
R_eal holds
( b
2 = vol b
1 iff ex b
3, b
4 being
R_eal st
( not ( not b
1 = ].b3,b4.[ & not b
1 = [.b3,b4.] & not b
1 = [.b3,b4.[ & not b
1 = ].b3,b4.] ) & ( b
3 < b
4 implies b
2 = b
4 - b
3 ) & ( b
4 <= b
3 implies b
2 = 0. ) ) );
theorem Th53: :: MEASURE5:53
theorem Th54: :: MEASURE5:54
theorem Th55: :: MEASURE5:55
theorem Th56: :: MEASURE5:56
theorem Th57: :: MEASURE5:57
for b
1 being
Intervalfor b
2, b
3, b
4 being
R_eal holds
( b
2 = -infty & b
3 in REAL & b
4 = +infty & not ( not b
1 = ].b2,b3.[ & not b
1 = ].b3,b4.[ & not b
1 = [.b2,b3.] & not b
1 = [.b3,b4.] & not b
1 = [.b2,b3.[ & not b
1 = [.b3,b4.[ & not b
1 = ].b2,b3.] & not b
1 = ].b3,b4.] ) implies
vol b
1 = +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