:: FDIFF_6 semantic presentation
Lemma31:
for x being Real holds 1 - (cos (2 * x)) = 2 * ((sin x) ^2 )
Lemma32:
for x being Real holds 1 + (cos (2 * x)) = 2 * ((cos x) ^2 )
Lemma33:
for x being Real st sin . x > 0 holds
sin . x = ((1 - (cos . x)) * (1 + (cos . x))) #R (1 / 2)
Lemma35:
for x being Real st sin . x > 0 & cos . x < 1 & cos . x > - 1 holds
(sin . x) / ((1 - (cos . x)) #R (1 / 2)) = (1 + (cos . x)) #R (1 / 2)
theorem Th1: :: FDIFF_6:1
theorem Th2: :: FDIFF_6:2
Lemma42:
for x being Real st sin . x > 0 & cos . x < 1 & cos . x > - 1 holds
(sin . x) / ((1 + (cos . x)) #R (1 / 2)) = (1 - (cos . x)) #R (1 / 2)
theorem Th3: :: FDIFF_6:3
theorem Th4: :: FDIFF_6:4
theorem Th5: :: FDIFF_6:5
theorem Th6: :: FDIFF_6:6
theorem Th7: :: FDIFF_6:7
theorem Th8: :: FDIFF_6:8
theorem Th9: :: FDIFF_6:9
theorem Th10: :: FDIFF_6:10
theorem Th11: :: FDIFF_6:11
theorem Th12: :: FDIFF_6:12
theorem Th13: :: FDIFF_6:13
theorem Th14: :: FDIFF_6:14
theorem Th15: :: FDIFF_6:15
theorem Th16: :: FDIFF_6:16
theorem Th17: :: FDIFF_6:17
theorem Th18: :: FDIFF_6:18
theorem Th19: :: FDIFF_6:19
theorem Th20: :: FDIFF_6:20
theorem Th21: :: FDIFF_6:21
theorem Th22: :: FDIFF_6:22
theorem Th23: :: FDIFF_6:23
theorem Th24: :: FDIFF_6:24
theorem Th25: :: FDIFF_6:25
theorem Th26: :: FDIFF_6:26
theorem Th27: :: FDIFF_6:27
theorem Th28: :: FDIFF_6:28
theorem Th29: :: FDIFF_6:29
theorem Th30: :: FDIFF_6:30
theorem Th31: :: FDIFF_6:31
theorem Th32: :: FDIFF_6:32
theorem Th33: :: FDIFF_6:33
theorem Th34: :: FDIFF_6:34
theorem Th35: :: FDIFF_6:35
theorem Th36: :: FDIFF_6:36
theorem Th37: :: FDIFF_6:37
theorem Th38: :: FDIFF_6:38
theorem Th39: :: FDIFF_6:39
theorem Th40: :: FDIFF_6:40
Lemma85:
for Z being open Subset of REAL
for f1 being PartFunc of REAL , REAL st Z c= dom (f1 + (2 (#) sin )) & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( f1 + (2 (#) sin ) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (2 (#) sin )) `| Z) . x = 2 * (cos . x) ) )
theorem Th41: :: FDIFF_6:41
Lemma86:
for Z being open Subset of REAL
for f1 being PartFunc of REAL , REAL st Z c= dom (f1 + (2 (#) cos )) & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( f1 + (2 (#) cos ) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (2 (#) cos )) `| Z) . x = - (2 * (sin . x)) ) )
theorem Th42: :: FDIFF_6:42
theorem Th43: :: FDIFF_6:43
theorem Th44: :: FDIFF_6:44
theorem Th45: :: FDIFF_6:45
theorem Th46: :: FDIFF_6:46
theorem Th47: :: FDIFF_6:47
theorem Th48: :: FDIFF_6:48
theorem Th49: :: FDIFF_6:49
theorem Th50: :: FDIFF_6:50