:: PROB_4 semantic presentation
Lemma1:
for b1, b2, b3 being set holds
( b3 c= b2 implies b1 \ b3 = (b1 \ b2) \/ (b1 /\ (b2 \ b3)) )
theorem Th1: :: PROB_4:1
theorem Th2: :: PROB_4:2
theorem Th3: :: PROB_4:3
for b
1 being
set for b
2, b
3 being
Subset of b
1 holds
ex b
4 being
SetSequence of b
1 st
( b
4 . 0
= b
2 & b
4 . 1
= b
3 & ( for b
5 being
Nat holds
( b
5 > 1 implies b
4 . b
5 = {} ) ) )
theorem Th4: :: PROB_4:4
theorem Th5: :: PROB_4:5
theorem Th6: :: PROB_4:6
theorem Th7: :: PROB_4:7
theorem Th8: :: PROB_4:8
theorem Th9: :: PROB_4:9
theorem Th10: :: PROB_4:10
theorem Th11: :: PROB_4:11
theorem Th12: :: PROB_4:12
theorem Th13: :: PROB_4:13
theorem Th14: :: PROB_4:14
:: deftheorem Def1 defines P2M PROB_4:def 1 :
theorem Th15: :: PROB_4:15
:: deftheorem Def2 defines M2P PROB_4:def 2 :
Lemma17:
for b1 being set
for b2 being SetSequence of b1 holds
( b2 is non-decreasing implies for b3 being Nat holds (Partial_Union b2) . b3 = b2 . b3 )
theorem Th16: :: PROB_4:16
theorem Th17: :: PROB_4:17
theorem Th18: :: PROB_4:18
theorem Th19: :: PROB_4:19
theorem Th20: :: PROB_4:20
theorem Th21: :: PROB_4:21
theorem Th22: :: PROB_4:22
:: deftheorem Def3 defines is_complete PROB_4:def 3 :
theorem Th23: :: PROB_4:23
:: deftheorem Def4 defines thin PROB_4:def 4 :
theorem Th24: :: PROB_4:24
theorem Th25: :: PROB_4:25
theorem Th26: :: PROB_4:26
:: deftheorem Def5 defines COM PROB_4:def 5 :
theorem Th27: :: PROB_4:27
theorem Th28: :: PROB_4:28
:: deftheorem Def6 defines P_COM2M_COM PROB_4:def 6 :
theorem Th29: :: PROB_4:29
:: deftheorem Def7 defines ProbPart PROB_4:def 7 :
theorem Th30: :: PROB_4:30
theorem Th31: :: PROB_4:31
theorem Th32: :: PROB_4:32
theorem Th33: :: PROB_4:33
theorem Th34: :: PROB_4:34
theorem Th35: :: PROB_4:35
theorem Th36: :: PROB_4:36
theorem Th37: :: PROB_4:37
:: deftheorem Def8 defines COM PROB_4:def 8 :
theorem Th38: :: PROB_4:38
theorem Th39: :: PROB_4:39
theorem Th40: :: PROB_4:40
theorem Th41: :: PROB_4:41
theorem Th42: :: PROB_4:42