:: LIMFUNC4 semantic presentation
Lemma1:
for b1, b2 being Real holds
( 0 < b1 implies ( b2 - b1 < b2 & b2 < b2 + b1 ) )
Lemma2:
for b1, b2 being PartFunc of REAL , REAL
for b3 being Real_Sequence holds
( rng b3 c= dom (b1 * b2) implies ( rng b3 c= dom b2 & rng (b2 * b3) c= dom b1 ) )
theorem Th1: :: LIMFUNC4:1
theorem Th2: :: LIMFUNC4:2
theorem Th3: :: LIMFUNC4:3
theorem Th4: :: LIMFUNC4:4
theorem Th5: :: LIMFUNC4:5
theorem Th6: :: LIMFUNC4:6
theorem Th7: :: LIMFUNC4:7
theorem Th8: :: LIMFUNC4:8
theorem Th9: :: LIMFUNC4:9
theorem Th10: :: LIMFUNC4:10
theorem Th11: :: LIMFUNC4:11
theorem Th12: :: LIMFUNC4:12
theorem Th13: :: LIMFUNC4:13
theorem Th14: :: LIMFUNC4:14
theorem Th15: :: LIMFUNC4:15
theorem Th16: :: LIMFUNC4:16
theorem Th17: :: LIMFUNC4:17
theorem Th18: :: LIMFUNC4:18
theorem Th19: :: LIMFUNC4:19
theorem Th20: :: LIMFUNC4:20
theorem Th21: :: LIMFUNC4:21
theorem Th22: :: LIMFUNC4:22
theorem Th23: :: LIMFUNC4:23
theorem Th24: :: LIMFUNC4:24
theorem Th25: :: LIMFUNC4:25
theorem Th26: :: LIMFUNC4:26
theorem Th27: :: LIMFUNC4:27
theorem Th28: :: LIMFUNC4:28
theorem Th29: :: LIMFUNC4:29
theorem Th30: :: LIMFUNC4:30
theorem Th31: :: LIMFUNC4:31
theorem Th32: :: LIMFUNC4:32
theorem Th33: :: LIMFUNC4:33
theorem Th34: :: LIMFUNC4:34
theorem Th35: :: LIMFUNC4:35
theorem Th36: :: LIMFUNC4:36
theorem Th37: :: LIMFUNC4:37
theorem Th38: :: LIMFUNC4:38
theorem Th39: :: LIMFUNC4:39
theorem Th40: :: LIMFUNC4:40
theorem Th41: :: LIMFUNC4:41
theorem Th42: :: LIMFUNC4:42
theorem Th43: :: LIMFUNC4:43
theorem Th44: :: LIMFUNC4:44
theorem Th45: :: LIMFUNC4:45
theorem Th46: :: LIMFUNC4:46
theorem Th47: :: LIMFUNC4:47
theorem Th48: :: LIMFUNC4:48
theorem Th49: :: LIMFUNC4:49
theorem Th50: :: LIMFUNC4:50
theorem Th51: :: LIMFUNC4:51
theorem Th52: :: LIMFUNC4:52
theorem Th53: :: LIMFUNC4:53
theorem Th54: :: LIMFUNC4:54
theorem Th55: :: LIMFUNC4:55
theorem Th56: :: LIMFUNC4:56
theorem Th57: :: LIMFUNC4:57
theorem Th58: :: LIMFUNC4:58
theorem Th59: :: LIMFUNC4:59
theorem Th60: :: LIMFUNC4:60
theorem Th61: :: LIMFUNC4:61
theorem Th62: :: LIMFUNC4:62
theorem Th63: :: LIMFUNC4:63
theorem Th64: :: LIMFUNC4:64
theorem Th65: :: LIMFUNC4:65
theorem Th66: :: LIMFUNC4:66
theorem Th67: :: LIMFUNC4:67
theorem Th68: :: LIMFUNC4:68
theorem Th69: :: LIMFUNC4:69
theorem Th70: :: LIMFUNC4:70
theorem Th71: :: LIMFUNC4:71
theorem Th72: :: LIMFUNC4:72
theorem Th73: :: LIMFUNC4:73
for b
1 being
Realfor b
2, b
3 being
PartFunc of
REAL ,
REAL holds
( b
2 is_convergent_in b
1 & b
3 is_left_convergent_in lim b
2,b
1 & ( for b
4, b
5 being
Real holds
not ( b
4 < b
1 & b
1 < b
5 & ( for b
6, b
7 being
Real holds
not ( b
4 < b
6 & b
6 < b
1 & b
6 in dom (b3 * b2) & b
7 < b
5 & b
1 < b
7 & b
7 in dom (b3 * b2) ) ) ) ) & ex b
4 being
Real st
( 0
< b
4 & ( for b
5 being
Real holds
not ( b
5 in (dom b2) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) & not b
2 . b
5 < lim b
2,b
1 ) ) ) implies ( b
3 * b
2 is_convergent_in b
1 &
lim (b3 * b2),b
1 = lim_left b
3,
(lim b2,b1) ) )
theorem Th74: :: LIMFUNC4:74
theorem Th75: :: LIMFUNC4:75
for b
1 being
Realfor b
2, b
3 being
PartFunc of
REAL ,
REAL holds
( b
2 is_convergent_in b
1 & b
3 is_right_convergent_in lim b
2,b
1 & ( for b
4, b
5 being
Real holds
not ( b
4 < b
1 & b
1 < b
5 & ( for b
6, b
7 being
Real holds
not ( b
4 < b
6 & b
6 < b
1 & b
6 in dom (b3 * b2) & b
7 < b
5 & b
1 < b
7 & b
7 in dom (b3 * b2) ) ) ) ) & ex b
4 being
Real st
( 0
< b
4 & ( for b
5 being
Real holds
not ( b
5 in (dom b2) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) & not
lim b
2,b
1 < b
2 . b
5 ) ) ) implies ( b
3 * b
2 is_convergent_in b
1 &
lim (b3 * b2),b
1 = lim_right b
3,
(lim b2,b1) ) )
theorem Th76: :: LIMFUNC4:76
theorem Th77: :: LIMFUNC4:77
for b
1 being
Realfor b
2, b
3 being
PartFunc of
REAL ,
REAL holds
( b
2 is_convergent_in b
1 & b
3 is_convergent_in lim b
2,b
1 & ( for b
4, b
5 being
Real holds
not ( b
4 < b
1 & b
1 < b
5 & ( for b
6, b
7 being
Real holds
not ( b
4 < b
6 & b
6 < b
1 & b
6 in dom (b3 * b2) & b
7 < b
5 & b
1 < b
7 & b
7 in dom (b3 * b2) ) ) ) ) & ex b
4 being
Real st
( 0
< b
4 & ( for b
5 being
Real holds
not ( b
5 in (dom b2) /\ (].(b1 - b4),b1.[ \/ ].b1,(b1 + b4).[) & not b
2 . b
5 <> lim b
2,b
1 ) ) ) implies ( b
3 * b
2 is_convergent_in b
1 &
lim (b3 * b2),b
1 = lim b
3,
(lim b2,b1) ) )