:: LIMFUNC4 semantic presentation
Lemma23:
for g, r being Real st 0 < g holds
( r - g < r & r < r + g )
Lemma25:
for f2, f1 being PartFunc of REAL , REAL
for s being Real_Sequence st rng s c= dom (f2 * f1) holds
( rng s c= dom f1 & rng (f1 * s) c= dom f2 )
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
x0 being
Real for
f1,
f2 being
PartFunc of
REAL ,
REAL st
f1 is_convergent_in x0 &
f2 is_left_convergent_in lim f1,
x0 & ( for
r1,
r2 being
Real st
r1 < x0 &
x0 < r2 holds
ex
g1,
g2 being
Real st
(
r1 < g1 &
g1 < x0 &
g1 in dom (f2 * f1) &
g2 < r2 &
x0 < g2 &
g2 in dom (f2 * f1) ) ) & ex
g being
Real st
( 0
< g & ( for
r being
Real st
r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds
f1 . r < lim f1,
x0 ) ) holds
(
f2 * f1 is_convergent_in x0 &
lim (f2 * f1),
x0 = lim_left f2,
(lim f1,x0) )
theorem Th74: :: LIMFUNC4:74
theorem Th75: :: LIMFUNC4:75
for
x0 being
Real for
f1,
f2 being
PartFunc of
REAL ,
REAL st
f1 is_convergent_in x0 &
f2 is_right_convergent_in lim f1,
x0 & ( for
r1,
r2 being
Real st
r1 < x0 &
x0 < r2 holds
ex
g1,
g2 being
Real st
(
r1 < g1 &
g1 < x0 &
g1 in dom (f2 * f1) &
g2 < r2 &
x0 < g2 &
g2 in dom (f2 * f1) ) ) & ex
g being
Real st
( 0
< g & ( for
r being
Real st
r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds
lim f1,
x0 < f1 . r ) ) holds
(
f2 * f1 is_convergent_in x0 &
lim (f2 * f1),
x0 = lim_right f2,
(lim f1,x0) )
theorem Th76: :: LIMFUNC4:76
theorem Th77: :: LIMFUNC4:77
for
x0 being
Real for
f1,
f2 being
PartFunc of
REAL ,
REAL st
f1 is_convergent_in x0 &
f2 is_convergent_in lim f1,
x0 & ( for
r1,
r2 being
Real st
r1 < x0 &
x0 < r2 holds
ex
g1,
g2 being
Real st
(
r1 < g1 &
g1 < x0 &
g1 in dom (f2 * f1) &
g2 < r2 &
x0 < g2 &
g2 in dom (f2 * f1) ) ) & ex
g being
Real st
( 0
< g & ( for
r being
Real st
r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds
f1 . r <> lim f1,
x0 ) ) holds
(
f2 * f1 is_convergent_in x0 &
lim (f2 * f1),
x0 = lim f2,
(lim f1,x0) )