:: FDIFF_2 semantic presentation
theorem Th1: :: FDIFF_2:1
theorem Th2: :: FDIFF_2:2
theorem Th3: :: FDIFF_2:3
theorem Th4: :: FDIFF_2:4
theorem Th5: :: FDIFF_2:5
theorem Th6: :: FDIFF_2:6
theorem Th7: :: FDIFF_2:7
theorem Th8: :: FDIFF_2:8
theorem Th9: :: FDIFF_2:9
theorem Th10: :: FDIFF_2:10
Lemma97:
for x0 being Real
for f being PartFunc of REAL , REAL st ex N being Neighbourhood of x0 st N c= dom f & ( for h being convergent_to_0 Real_Sequence
for c being constant Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h " ) (#) ((f * (h + c)) - (f * c)) is convergent ) holds
( f is_differentiable_in x0 & ( for h being convergent_to_0 Real_Sequence
for c being constant Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
diff f,x0 = lim ((h " ) (#) ((f * (h + c)) - (f * c))) ) )
theorem Th11: :: FDIFF_2:11
theorem Th12: :: FDIFF_2:12
Lemma128:
for x0 being Real
for f2, f1 being PartFunc of REAL , REAL st ex N being Neighbourhood of x0 st N c= dom (f2 * f1) & f1 is_differentiable_in x0 & f2 is_differentiable_in f1 . x0 holds
( f2 * f1 is_differentiable_in x0 & diff (f2 * f1),x0 = (diff f2,(f1 . x0)) * (diff f1,x0) )
theorem Th13: :: FDIFF_2:13
theorem Th14: :: FDIFF_2:14
theorem Th15: :: FDIFF_2:15
theorem Th16: :: FDIFF_2:16
theorem Th17: :: FDIFF_2:17
theorem Th18: :: FDIFF_2:18
theorem Th19: :: FDIFF_2:19
theorem Th20: :: FDIFF_2:20
Lemma173:
for f being PartFunc of REAL , REAL holds (f (#) f) " {0} = f " {0}
theorem Th21: :: FDIFF_2:21
theorem Th22: :: FDIFF_2:22
theorem Th23: :: FDIFF_2:23
theorem Th24: :: FDIFF_2:24
theorem Th25: :: FDIFF_2:25
theorem Th26: :: FDIFF_2:26
theorem Th27: :: FDIFF_2:27
theorem Th28: :: FDIFF_2:28
theorem Th29: :: FDIFF_2:29
theorem Th30: :: FDIFF_2:30
theorem Th31: :: FDIFF_2:31
theorem Th32: :: FDIFF_2:32
theorem Th33: :: FDIFF_2:33
theorem Th34: :: FDIFF_2:34
theorem Th35: :: FDIFF_2:35
theorem Th36: :: FDIFF_2:36
theorem Th37: :: FDIFF_2:37
theorem Th38: :: FDIFF_2:38
theorem Th39: :: FDIFF_2:39
theorem Th40: :: FDIFF_2:40
theorem Th41: :: FDIFF_2:41
theorem Th42: :: FDIFF_2:42
theorem Th43: :: FDIFF_2:43
theorem Th44: :: FDIFF_2:44
theorem Th45: :: FDIFF_2:45
theorem Th46: :: FDIFF_2:46
theorem Th47: :: FDIFF_2:47
theorem Th48: :: FDIFF_2:48
for
p,
g being
Real for
f being
one-to-one PartFunc of
REAL ,
REAL st
f is_differentiable_on ].p,g.[ & ( for
x0 being
Real st
x0 in ].p,g.[ holds
0
< diff f,
x0 or for
x0 being
Real st
x0 in ].p,g.[ holds
diff f,
x0 < 0 ) holds
(
f | ].p,g.[ is
one-to-one &
(f | ].p,g.[) " is_differentiable_on dom ((f | ].p,g.[) " ) & ( for
x0 being
Real st
x0 in dom ((f | ].p,g.[) " ) holds
diff ((f | ].p,g.[) " ),
x0 = 1
/ (diff f,(((f | ].p,g.[) " ) . x0)) ) )
theorem Th49: :: FDIFF_2:49