:: SETLIM_1 semantic presentation
Lm1:
for i, j being Element of NAT holds
( not i <= j or i = j or i + 1 <= j )
theorem Th1: :: SETLIM_1:1
theorem Th2: :: SETLIM_1:2
for
X,
y being
set st
X <> {} & ( for
x being
set st
x in X holds
x = y ) holds
meet X = y
theorem Th3: :: SETLIM_1:3
theorem Th4: :: SETLIM_1:4
theorem Th5: :: SETLIM_1:5
theorem Th6: :: SETLIM_1:6
theorem Th7: :: SETLIM_1:7
theorem Th8: :: SETLIM_1:8
theorem Th9: :: SETLIM_1:9
theorem Th10: :: SETLIM_1:10
theorem :: SETLIM_1:11
canceled;
theorem Th12: :: SETLIM_1:12
theorem Th13: :: SETLIM_1:13
theorem :: SETLIM_1:14
theorem Th15: :: SETLIM_1:15
theorem Th16: :: SETLIM_1:16
theorem :: SETLIM_1:17
Lm2:
for X being set
for A being Subset of X
for A1 being SetSequence of X st A1 is constant & the_value_of A1 = A holds
for n being Element of NAT holds
( A1 . n = A & Union A1 = A & Intersection A1 = A )
theorem Th18: :: SETLIM_1:18
theorem Th19: :: SETLIM_1:19
theorem Th20: :: SETLIM_1:20
theorem Th21: :: SETLIM_1:21
:: deftheorem Def1 defines monotone SETLIM_1:def 1 :
:: deftheorem Def2 defines inferior_setsequence SETLIM_1:def 2 :
:: deftheorem Def3 defines superior_setsequence SETLIM_1:def 3 :
theorem Th22: :: SETLIM_1:22
theorem Th23: :: SETLIM_1:23
theorem Th24: :: SETLIM_1:24
theorem Th25: :: SETLIM_1:25
theorem Th26: :: SETLIM_1:26
theorem Th27: :: SETLIM_1:27
theorem Th28: :: SETLIM_1:28
theorem Th29: :: SETLIM_1:29
theorem :: SETLIM_1:30
theorem :: SETLIM_1:31
theorem :: SETLIM_1:32
theorem Th33: :: SETLIM_1:33
theorem :: SETLIM_1:34
canceled;
theorem :: SETLIM_1:35
theorem Th36: :: SETLIM_1:36
theorem :: SETLIM_1:37
theorem Th38: :: SETLIM_1:38
theorem :: SETLIM_1:39
theorem :: SETLIM_1:40
theorem :: SETLIM_1:41
theorem :: SETLIM_1:42
theorem :: SETLIM_1:43
theorem Th44: :: SETLIM_1:44
theorem Th45: :: SETLIM_1:45
theorem Th46: :: SETLIM_1:46
theorem Th47: :: SETLIM_1:47
theorem Th48: :: SETLIM_1:48
theorem Th49: :: SETLIM_1:49
theorem Th50: :: SETLIM_1:50
theorem Th51: :: SETLIM_1:51
theorem Th52: :: SETLIM_1:52
theorem Th53: :: SETLIM_1:53
theorem Th54: :: SETLIM_1:54
theorem Th55: :: SETLIM_1:55
theorem Th56: :: SETLIM_1:56
theorem Th57: :: SETLIM_1:57
theorem Th58: :: SETLIM_1:58
theorem Th59: :: SETLIM_1:59
:: deftheorem defines lim_inf SETLIM_1:def 4 :
:: deftheorem defines lim_sup SETLIM_1:def 5 :
:: deftheorem Def6 defines lim SETLIM_1:def 6 :
theorem :: SETLIM_1:60
theorem :: SETLIM_1:61
theorem :: SETLIM_1:62
theorem :: SETLIM_1:63
theorem Th64: :: SETLIM_1:64
theorem :: SETLIM_1:65
theorem :: SETLIM_1:66
theorem :: SETLIM_1:67
theorem :: SETLIM_1:68
theorem Th69: :: SETLIM_1:69
theorem Th70: :: SETLIM_1:70
theorem :: SETLIM_1:71
:: deftheorem defines constant SETLIM_1:def 7 :
:: deftheorem defines @inferior_setsequence SETLIM_1:def 8 :
:: deftheorem defines @superior_setsequence SETLIM_1:def 9 :
theorem :: SETLIM_1:72
theorem :: SETLIM_1:73
:: deftheorem defines lim_inf SETLIM_1:def 10 :
:: deftheorem defines lim_sup SETLIM_1:def 11 :
:: deftheorem Def12 defines convergent SETLIM_1:def 12 :
:: deftheorem Def13 defines lim SETLIM_1:def 13 :
theorem Th74: :: SETLIM_1:74
theorem Th75: :: SETLIM_1:75
theorem :: SETLIM_1:76
theorem :: SETLIM_1:77
theorem :: SETLIM_1:78
:: deftheorem defines @Complement SETLIM_1:def 14 :
theorem Th79: :: SETLIM_1:79
theorem :: SETLIM_1:80
theorem :: SETLIM_1:81
theorem :: SETLIM_1:82
theorem :: SETLIM_1:83
theorem :: SETLIM_1:84
theorem :: SETLIM_1:85
theorem Th86: :: SETLIM_1:86
theorem Th87: :: SETLIM_1:87
theorem Th88: :: SETLIM_1:88
theorem Th89: :: SETLIM_1:89
theorem Th90: :: SETLIM_1:90
theorem Th91: :: SETLIM_1:91
theorem :: SETLIM_1:92