:: 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
x,
y being
R_eal st
x is
Real holds
(
(y - x) + x = y &
(y + x) - x = y )
theorem Th9: :: MEASURE6:9
canceled;
theorem Th10: :: MEASURE6:10
theorem Th11: :: MEASURE6:11
theorem Th12: :: MEASURE6:12
:: deftheorem Def1 defines R_EAL MEASURE6:def 1 :
theorem Th13: :: MEASURE6:13
theorem Th14: :: MEASURE6:14
theorem Th15: :: MEASURE6:15
for
x,
y,
z being
R_eal st
x < y &
y < z holds
y is
Real
theorem Th16: :: MEASURE6:16
theorem Th17: :: MEASURE6:17
theorem Th18: :: MEASURE6:18
theorem Th19: :: MEASURE6:19
theorem Th20: :: MEASURE6:20
theorem Th21: :: MEASURE6:21
theorem Th22: :: MEASURE6:22
theorem Th23: :: MEASURE6:23
theorem Th24: :: MEASURE6:24
theorem Th25: :: MEASURE6:25
theorem Th26: :: MEASURE6:26
:: 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
theorem Th47: :: MEASURE6:47
definition
let A be
Interval;
assume E18:
A <> {}
;
func ^^ c1 -> R_eal means :
Def4:
:: MEASURE6:def 4
ex
b being
R_eal st
(
it <= b & (
A = ].it,b.[ or
A = ].it,b.] or
A = [.it,b.] or
A = [.it,b.[ ) );
existence
ex b1, b being R_eal st
( b1 <= b & ( A = ].b1,b.[ or A = ].b1,b.] or A = [.b1,b.] or A = [.b1,b.[ ) )
uniqueness
for b1, b2 being R_eal st ex b being R_eal st
( b1 <= b & ( A = ].b1,b.[ or A = ].b1,b.] or A = [.b1,b.] or A = [.b1,b.[ ) ) & ex b being R_eal st
( b2 <= b & ( A = ].b2,b.[ or A = ].b2,b.] or A = [.b2,b.] or A = [.b2,b.[ ) ) holds
b1 = b2
end;
:: deftheorem Def4 defines ^^ MEASURE6:def 4 :
definition
let A be
Interval;
assume E18:
A <> {}
;
func c1 ^^ -> R_eal means :
Def5:
:: MEASURE6:def 5
ex
a being
R_eal st
(
a <= it & (
A = ].a,it.[ or
A = ].a,it.] or
A = [.a,it.] or
A = [.a,it.[ ) );
existence
ex b1, a being R_eal st
( a <= b1 & ( A = ].a,b1.[ or A = ].a,b1.] or A = [.a,b1.] or A = [.a,b1.[ ) )
uniqueness
for b1, b2 being R_eal st ex a being R_eal st
( a <= b1 & ( A = ].a,b1.[ or A = ].a,b1.] or A = [.a,b1.] or A = [.a,b1.[ ) ) & ex a being R_eal st
( a <= b2 & ( A = ].a,b2.[ or A = ].a,b2.] or A = [.a,b2.] or A = [.a,b2.[ ) ) holds
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