:: 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
:: deftheorem Def1 defines |[. MEASURE5:def 1 :
Lx1:
for a, b being ext-real number holds |[.a,b.]| = [.a,b.] /\ REAL
:: deftheorem Def2 defines ]. MEASURE5:def 2 :
:: deftheorem Def3 defines ]. MEASURE5:def 3 :
:: deftheorem Def4 defines |[. MEASURE5:def 4 :
Lx2:
for a, b being ext-real number holds |[.a,b.[ = [.a,b.[ /\ REAL
Lx3:
for a, b being ext-real number holds ].a,b.]| = ].a,b.] /\ REAL
:: deftheorem Def5 defines open_interval MEASURE5:def 5 :
:: deftheorem defines closed_interval MEASURE5:def 6 :
:: deftheorem defines right_open_interval MEASURE5:def 7 :
:: deftheorem defines left_open_interval MEASURE5:def 8 :
:: deftheorem Def9 defines interval MEASURE5:def 9 :
theorem :: MEASURE5:9
canceled;
theorem :: MEASURE5:10
canceled;
theorem Th11: :: MEASURE5:11
theorem Th12: :: MEASURE5:12
theorem Th13: :: MEASURE5:13
theorem Th14: :: MEASURE5:14
theorem Th15: :: MEASURE5:15
theorem :: MEASURE5:16
canceled;
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 :: 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 )
:: deftheorem Def10 defines vol MEASURE5:def 10 :
theorem :: MEASURE5:53
canceled;
theorem :: MEASURE5:54
theorem :: MEASURE5:55
theorem :: MEASURE5:56
theorem :: MEASURE5:57
theorem :: MEASURE5:58
canceled;
theorem :: MEASURE5:59
theorem Th60: :: MEASURE5:60
LX3:
for A being Interval holds vol A >= 0
LX4:
for A, B being Interval st A c= B holds
vol A <= vol B
theorem :: MEASURE5:61
theorem :: MEASURE5:62
theorem :: MEASURE5:63
theorem :: MEASURE5:64
theorem :: MEASURE5:65
theorem :: MEASURE5:66
theorem :: MEASURE5:67
theorem :: MEASURE5:68
theorem :: MEASURE5:69
theorem :: MEASURE5:70
theorem :: MEASURE5:71
theorem :: MEASURE5:72
theorem :: MEASURE5:73