:: MEASURE6 semantic presentation
theorem Th1: :: MEASURE6:1
theorem Th2: :: MEASURE6:2
theorem Th3: :: MEASURE6:3
theorem Th4: :: MEASURE6:4
canceled;
theorem Th5: :: MEASURE6:5
canceled;
theorem Th6: :: MEASURE6:6
canceled;
theorem Th7: :: MEASURE6:7
canceled;
theorem Th8: :: MEASURE6:8
for b
1, b
2 being
R_eal holds
( b
1 is
Real implies (
(b2 - b1) + b
1 = b
2 &
(b2 + b1) - b
1 = b
2 ) )
theorem Th9: :: MEASURE6:9
canceled;
theorem Th10: :: MEASURE6:10
for b
1, b
2, b
3 being
R_eal holds
( b
3 in REAL & b
2 < b
1 implies
(b3 + b1) - (b3 + b2) = b
1 - b
2 )
theorem Th11: :: MEASURE6:11
for b
1, b
2, b
3 being
R_eal holds
( b
3 in REAL & b
1 <= b
2 implies ( b
3 + b
1 <= b
3 + b
2 & b
1 + b
3 <= b
2 + b
3 & b
1 - b
3 <= b
2 - b
3 ) )
theorem Th12: :: MEASURE6:12
for b
1, b
2, b
3 being
R_eal holds
( b
3 in REAL & b
1 < b
2 implies ( b
3 + b
1 < b
3 + b
2 & b
1 + b
3 < b
2 + b
3 & b
1 - b
3 < b
2 - b
3 ) )
:: deftheorem Def1 defines R_EAL MEASURE6:def 1 :
theorem Th13: :: MEASURE6:13
theorem Th14: :: MEASURE6:14
theorem Th15: :: MEASURE6:15
for b
1, b
2, b
3 being
R_eal holds
( b
1 < b
2 & b
2 < b
3 implies b
2 is
Real )
theorem Th16: :: MEASURE6:16
theorem Th17: :: MEASURE6:17
for b
1, b
2, b
3 being
R_eal holds
( b
1 is
Real & b
1 <= b
2 & b
2 < b
3 implies b
2 is
Real )
theorem Th18: :: MEASURE6:18
for b
1, b
2, b
3 being
R_eal holds
( b
1 < b
2 & b
2 <= b
3 & b
3 is
Real implies b
2 is
Real )
theorem Th19: :: MEASURE6:19
for b
1, b
2 being
R_eal holds
not (
0. < b
1 & b
1 < b
2 & not
0. < b
2 - b
1 )
theorem Th20: :: MEASURE6:20
for b
1, b
2, b
3 being
R_eal holds
not (
0. <= b
1 &
0. <= b
3 & b
3 + b
1 < b
2 & not b
3 < b
2 - b
1 )
theorem Th21: :: MEASURE6:21
theorem Th22: :: MEASURE6:22
for b
1, b
2, b
3 being
R_eal holds
(
0. <= b
1 &
0. <= b
3 & b
3 + b
1 < b
2 implies b
3 <= b
2 )
theorem Th23: :: MEASURE6:23
for b
1 being
R_eal holds
not (
0. < b
1 & ( for b
2 being
R_eal holds
not (
0. < b
2 & b
2 < b
1 ) ) )
theorem Th24: :: MEASURE6:24
for b
1, b
2 being
R_eal holds
not (
0. < b
1 & b
1 < b
2 & ( for b
3 being
R_eal holds
not (
0. < b
3 & b
1 + b
3 < b
2 & b
3 in REAL ) ) )
theorem Th25: :: MEASURE6:25
for b
1, b
2 being
R_eal holds
not (
0. <= b
1 & b
1 < b
2 & ( for b
3 being
R_eal holds
not (
0. < b
3 & b
1 + b
3 < b
2 & b
3 in REAL ) ) )
theorem Th26: :: MEASURE6:26
for b
1 being
R_eal holds
not (
0. < b
1 & ( for b
2 being
R_eal holds
not (
0. < b
2 & b
2 + b
2 < b
1 ) ) )
:: deftheorem Def2 defines Seg MEASURE6:def 2 :
:: deftheorem Def3 defines len MEASURE6:def 3 :
theorem Th27: :: MEASURE6:27
theorem Th28: :: MEASURE6:28
theorem Th29: :: MEASURE6:29
theorem Th30: :: MEASURE6:30
theorem Th31: :: MEASURE6:31
theorem Th32: :: MEASURE6:32
theorem Th33: :: MEASURE6:33
theorem Th34: :: MEASURE6:34
theorem Th35: :: MEASURE6:35
theorem Th36: :: MEASURE6:36
theorem Th37: :: MEASURE6:37
theorem Th38: :: MEASURE6:38
theorem Th39: :: MEASURE6:39
theorem Th40: :: MEASURE6:40
theorem Th41: :: MEASURE6:41
theorem Th42: :: MEASURE6:42
theorem Th43: :: MEASURE6:43
theorem Th44: :: MEASURE6:44
theorem Th45: :: MEASURE6:45
theorem Th46: :: MEASURE6:46
for b
1 being
Subset of
REAL holds
( b
1 is
Interval iff for b
2, b
3 being
Real holds
( b
2 in b
1 & b
3 in b
1 implies for b
4 being
Real holds
( b
2 <= b
4 & b
4 <= b
3 implies b
4 in b
1 ) ) )
theorem Th47: :: MEASURE6:47
definition
let c
1 be
Interval;
assume E30:
c
1 <> {}
;
func ^^ c
1 -> R_eal means :
Def4:
:: MEASURE6:def 4
ex b
1 being
R_eal st
( a
2 <= b
1 & not ( not a
1 = ].a2,b1.[ & not a
1 = ].a2,b1.] & not a
1 = [.a2,b1.] & not a
1 = [.a2,b1.[ ) );
existence
ex b1, b2 being R_eal st
( b1 <= b2 & not ( not c1 = ].b1,b2.[ & not c1 = ].b1,b2.] & not c1 = [.b1,b2.] & not c1 = [.b1,b2.[ ) )
uniqueness
for b1, b2 being R_eal holds
( ex b3 being R_eal st
( b1 <= b3 & not ( not c1 = ].b1,b3.[ & not c1 = ].b1,b3.] & not c1 = [.b1,b3.] & not c1 = [.b1,b3.[ ) ) & ex b3 being R_eal st
( b2 <= b3 & not ( not c1 = ].b2,b3.[ & not c1 = ].b2,b3.] & not c1 = [.b2,b3.] & not c1 = [.b2,b3.[ ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines ^^ MEASURE6:def 4 :
definition
let c
1 be
Interval;
assume E31:
c
1 <> {}
;
func c
1 ^^ -> R_eal means :
Def5:
:: MEASURE6:def 5
ex b
1 being
R_eal st
( b
1 <= a
2 & not ( not a
1 = ].b1,a2.[ & not a
1 = ].b1,a2.] & not a
1 = [.b1,a2.] & not a
1 = [.b1,a2.[ ) );
existence
ex b1, b2 being R_eal st
( b2 <= b1 & not ( not c1 = ].b2,b1.[ & not c1 = ].b2,b1.] & not c1 = [.b2,b1.] & not c1 = [.b2,b1.[ ) )
uniqueness
for b1, b2 being R_eal holds
( ex b3 being R_eal st
( b3 <= b1 & not ( not c1 = ].b3,b1.[ & not c1 = ].b3,b1.] & not c1 = [.b3,b1.] & not c1 = [.b3,b1.[ ) ) & ex b3 being R_eal st
( b3 <= b2 & not ( not c1 = ].b3,b2.[ & not c1 = ].b3,b2.] & not c1 = [.b3,b2.] & not c1 = [.b3,b2.[ ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines ^^ MEASURE6:def 5 :
theorem Th48: :: MEASURE6:48
theorem Th49: :: MEASURE6:49
theorem Th50: :: MEASURE6:50
theorem Th51: :: MEASURE6:51
theorem Th52: :: MEASURE6:52
theorem Th53: :: MEASURE6:53
canceled;
theorem Th54: :: MEASURE6:54
theorem Th55: :: MEASURE6:55
theorem Th56: :: MEASURE6:56
theorem Th57: :: MEASURE6:57
theorem Th58: :: MEASURE6:58
:: deftheorem Def6 defines + MEASURE6:def 6 :
theorem Th59: :: MEASURE6:59
theorem Th60: :: MEASURE6:60
theorem Th61: :: MEASURE6:61
theorem Th62: :: MEASURE6:62
theorem Th63: :: MEASURE6:63
theorem Th64: :: MEASURE6:64
theorem Th65: :: MEASURE6:65
theorem Th66: :: MEASURE6:66
theorem Th67: :: MEASURE6:67