:: 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
Lemma10:
for b1 being Real
for b2 being PartFunc of REAL , REAL holds
( ex b3 being Neighbourhood of b1 st b3 c= dom b2 & ( for b3 being convergent_to_0 Real_Sequence
for b4 being constant Real_Sequence holds
( rng b4 = {b1} & rng (b3 + b4) c= dom b2 implies (b3 " ) (#) ((b2 * (b3 + b4)) - (b2 * b4)) is convergent ) ) implies ( b2 is_differentiable_in b1 & ( for b3 being convergent_to_0 Real_Sequence
for b4 being constant Real_Sequence holds
( rng b4 = {b1} & rng (b3 + b4) c= dom b2 implies diff b2,b1 = lim ((b3 " ) (#) ((b2 * (b3 + b4)) - (b2 * b4))) ) ) ) )
theorem Th11: :: FDIFF_2:11
theorem Th12: :: FDIFF_2:12
Lemma13:
for b1 being Real
for b2, b3 being PartFunc of REAL , REAL holds
( ex b4 being Neighbourhood of b1 st b4 c= dom (b2 * b3) & b3 is_differentiable_in b1 & b2 is_differentiable_in b3 . b1 implies ( b2 * b3 is_differentiable_in b1 & diff (b2 * b3),b1 = (diff b2,(b3 . b1)) * (diff b3,b1) ) )
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
Lemma17:
for b1 being PartFunc of REAL , REAL holds (b1 (#) b1) " {0} = b1 " {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 b
1, b
2 being
Realfor b
3 being
one-to-one PartFunc of
REAL ,
REAL holds
( b
3 is_differentiable_on ].b1,b2.[ & ( ( for b
4 being
Real holds
not ( b
4 in ].b1,b2.[ & not 0
< diff b
3,b
4 ) ) or for b
4 being
Real holds
not ( b
4 in ].b1,b2.[ & not
diff b
3,b
4 < 0 ) ) implies ( b
3 | ].b1,b2.[ is
one-to-one &
(b3 | ].b1,b2.[) " is_differentiable_on dom ((b3 | ].b1,b2.[) " ) & ( for b
4 being
Real holds
( b
4 in dom ((b3 | ].b1,b2.[) " ) implies
diff ((b3 | ].b1,b2.[) " ),b
4 = 1
/ (diff b3,(((b3 | ].b1,b2.[) " ) . b4)) ) ) ) )
theorem Th49: :: FDIFF_2:49