:: CARD_4 semantic presentation
Lemma1:
( 0 = Card 0 & 1 = Card 1 & 2 = Card 2 )
by CARD_1:def 5;
theorem Th1: :: CARD_4:1
theorem Th2: :: CARD_4:2
theorem Th3: :: CARD_4:3
theorem Th4: :: CARD_4:4
theorem Th5: :: CARD_4:5
theorem Th6: :: CARD_4:6
theorem Th7: :: CARD_4:7
theorem Th8: :: CARD_4:8
theorem Th9: :: CARD_4:9
theorem Th10: :: CARD_4:10
canceled;
theorem Th11: :: CARD_4:11
theorem Th12: :: CARD_4:12
canceled;
theorem Th13: :: CARD_4:13
theorem Th14: :: CARD_4:14
theorem Th15: :: CARD_4:15
theorem Th16: :: CARD_4:16
canceled;
theorem Th17: :: CARD_4:17
theorem Th18: :: CARD_4:18
canceled;
theorem Th19: :: CARD_4:19
theorem Th20: :: CARD_4:20
theorem Th21: :: CARD_4:21
theorem Th22: :: CARD_4:22
theorem Th23: :: CARD_4:23
theorem Th24: :: CARD_4:24
theorem Th25: :: CARD_4:25
theorem Th26: :: CARD_4:26
set c1 = succ one ;
theorem Th27: :: CARD_4:27
theorem Th28: :: CARD_4:28
theorem Th29: :: CARD_4:29
theorem Th30: :: CARD_4:30
theorem Th31: :: CARD_4:31
theorem Th32: :: CARD_4:32
theorem Th33: :: CARD_4:33
theorem Th34: :: CARD_4:34
theorem Th35: :: CARD_4:35
theorem Th36: :: CARD_4:36
theorem Th37: :: CARD_4:37
theorem Th38: :: CARD_4:38
theorem Th39: :: CARD_4:39
theorem Th40: :: CARD_4:40
theorem Th41: :: CARD_4:41
theorem Th42: :: CARD_4:42
:: deftheorem Def1 defines countable CARD_4:def 1 :
theorem Th43: :: CARD_4:43
theorem Th44: :: CARD_4:44
theorem Th45: :: CARD_4:45
theorem Th46: :: CARD_4:46
theorem Th47: :: CARD_4:47
theorem Th48: :: CARD_4:48
theorem Th49: :: CARD_4:49
theorem Th50: :: CARD_4:50
theorem Th51: :: CARD_4:51
for
b1 being
Nat for
b2 being
Real holds
( (
b2 <> 0 or
b1 = 0 ) iff
b2 |^ b1 <> 0 )
Lemma33:
for b1, b2, b3, b4 being Nat st (2 |^ b1) * ((2 * b2) + 1) = (2 |^ b3) * ((2 * b4) + 1) holds
b1 <= b3
theorem Th52: :: CARD_4:52
for
b1,
b2,
b3,
b4 being
Nat st
(2 |^ b1) * ((2 * b2) + 1) = (2 |^ b3) * ((2 * b4) + 1) holds
(
b1 = b3 &
b2 = b4 )
Lemma35:
for b1 being set st b1 in [:NAT ,NAT :] holds
ex b2, b3 being Nat st b1 = [b2,b3]
theorem Th53: :: CARD_4:53
theorem Th54: :: CARD_4:54
theorem Th55: :: CARD_4:55
theorem Th56: :: CARD_4:56
theorem Th57: :: CARD_4:57
theorem Th58: :: CARD_4:58
theorem Th59: :: CARD_4:59
theorem Th60: :: CARD_4:60
theorem Th61: :: CARD_4:61
theorem Th62: :: CARD_4:62
theorem Th63: :: CARD_4:63
theorem Th64: :: CARD_4:64
canceled;
theorem Th65: :: CARD_4:65
theorem Th66: :: CARD_4:66
theorem Th67: :: CARD_4:67
theorem Th68: :: CARD_4:68
theorem Th69: :: CARD_4:69
theorem Th70: :: CARD_4:70
for
b1,
b2,
b3,
b4 being
Cardinal holds
( ( not (
b1 <` b2 &
b3 <` b4 ) & not (
b1 <=` b2 &
b3 <` b4 ) & not (
b1 <` b2 &
b3 <=` b4 ) & not (
b1 <=` b2 &
b3 <=` b4 ) ) or
b1 = 0 or
exp b1,
b3 <=` exp b2,
b4 )
theorem Th71: :: CARD_4:71
theorem Th72: :: CARD_4:72
theorem Th73: :: CARD_4:73
theorem Th74: :: CARD_4:74
theorem Th75: :: CARD_4:75
theorem Th76: :: CARD_4:76
theorem Th77: :: CARD_4:77
theorem Th78: :: CARD_4:78
theorem Th79: :: CARD_4:79
theorem Th80: :: CARD_4:80
theorem Th81: :: CARD_4:81
theorem Th82: :: CARD_4:82
theorem Th83: :: CARD_4:83
theorem Th84: :: CARD_4:84
theorem Th85: :: CARD_4:85
theorem Th86: :: CARD_4:86
theorem Th87: :: CARD_4:87