:: PROB_1 semantic presentation
theorem Th1: :: PROB_1:1
canceled;
theorem Th2: :: PROB_1:2
canceled;
theorem Th3: :: PROB_1:3
:: deftheorem Def1 defines compl-closed PROB_1:def 1 :
theorem Th4: :: PROB_1:4
theorem Th5: :: PROB_1:5
canceled;
theorem Th6: :: PROB_1:6
theorem Th7: :: PROB_1:7
canceled;
theorem Th8: :: PROB_1:8
canceled;
theorem Th9: :: PROB_1:9
theorem Th10: :: PROB_1:10
theorem Th11: :: PROB_1:11
theorem Th12: :: PROB_1:12
theorem Th13: :: PROB_1:13
theorem Th14: :: PROB_1:14
theorem Th15: :: PROB_1:15
theorem Th16: :: PROB_1:16
theorem Th17: :: PROB_1:17
canceled;
theorem Th18: :: PROB_1:18
theorem Th19: :: PROB_1:19
Lemma58:
for X being set
for A1 being SetSequence of X holds
( dom A1 = NAT & ( for n being Element of NAT holds A1 . n in bool X ) )
by FUNCT_2:def 1;
theorem Th20: :: PROB_1:20
canceled;
theorem Th21: :: PROB_1:21
theorem Th22: :: PROB_1:22
theorem Th23: :: PROB_1:23
theorem Th24: :: PROB_1:24
canceled;
theorem Th25: :: PROB_1:25
theorem Th26: :: PROB_1:26
:: deftheorem Def2 PROB_1:def 2 :
canceled;
:: deftheorem Def3 PROB_1:def 3 :
canceled;
:: deftheorem Def4 defines Complement PROB_1:def 4 :
:: deftheorem Def5 defines Intersection PROB_1:def 5 :
theorem Th27: :: PROB_1:27
canceled;
theorem Th28: :: PROB_1:28
canceled;
theorem Th29: :: PROB_1:29
theorem Th30: :: PROB_1:30
:: deftheorem Def6 defines non-increasing PROB_1:def 6 :
:: deftheorem Def7 defines non-decreasing PROB_1:def 7 :
:: deftheorem Def8 defines sigma-multiplicative PROB_1:def 8 :
theorem Th31: :: PROB_1:31
canceled;
theorem Th32: :: PROB_1:32
theorem Th33: :: PROB_1:33
canceled;
theorem Th34: :: PROB_1:34
canceled;
theorem Th35: :: PROB_1:35
theorem Th36: :: PROB_1:36
canceled;
theorem Th37: :: PROB_1:37
canceled;
theorem Th38: :: PROB_1:38
theorem Th39: :: PROB_1:39
canceled;
theorem Th40: :: PROB_1:40
canceled;
theorem Th41: :: PROB_1:41
theorem Th42: :: PROB_1:42
theorem Th43: :: PROB_1:43
theorem Th44: :: PROB_1:44
:: deftheorem Def9 defines SetSequence PROB_1:def 9 :
theorem Th45: :: PROB_1:45
canceled;
theorem Th46: :: PROB_1:46
:: deftheorem Def10 defines Event PROB_1:def 10 :
theorem Th47: :: PROB_1:47
canceled;
theorem Th48: :: PROB_1:48
theorem Th49: :: PROB_1:49
theorem Th50: :: PROB_1:50
theorem Th51: :: PROB_1:51
theorem Th52: :: PROB_1:52
theorem Th53: :: PROB_1:53
theorem Th54: :: PROB_1:54
:: deftheorem Def11 defines [#] PROB_1:def 11 :
theorem Th55: :: PROB_1:55
canceled;
theorem Th56: :: PROB_1:56
canceled;
theorem Th57: :: PROB_1:57
theorem Th58: :: PROB_1:58
theorem Th59: :: PROB_1:59
theorem Th60: :: PROB_1:60
theorem Th61: :: PROB_1:61
canceled;
theorem Th62: :: PROB_1:62
:: deftheorem Def12 PROB_1:def 12 :
canceled;
:: deftheorem Def13 defines Probability PROB_1:def 13 :
theorem Th63: :: PROB_1:63
canceled;
theorem Th64: :: PROB_1:64
theorem Th65: :: PROB_1:65
canceled;
theorem Th66: :: PROB_1:66
theorem Th67: :: PROB_1:67
theorem Th68: :: PROB_1:68
theorem Th69: :: PROB_1:69
theorem Th70: :: PROB_1:70
theorem Th71: :: PROB_1:71
theorem Th72: :: PROB_1:72
theorem Th73: :: PROB_1:73
theorem Th74: :: PROB_1:74
theorem Th75: :: PROB_1:75
theorem Th76: :: PROB_1:76
Lemma120:
for Omega being non empty set
for X being Subset-Family of Omega ex Y being SigmaField of Omega st
( X c= Y & ( for Z being set st X c= Z & Z is SigmaField of Omega holds
Y c= Z ) )
:: deftheorem Def14 defines sigma PROB_1:def 14 :
:: deftheorem Def15 defines halfline PROB_1:def 15 :
:: deftheorem Def16 defines Family_of_halflines PROB_1:def 16 :
:: deftheorem Def17 defines Borel_Sets PROB_1:def 17 :