:: FDIFF_5 semantic presentation
Lemma34:
for a being Real
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL , REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = a ) & f2 = #Z 2 holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) )
Lemma43:
for a being Real
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL , REAL st Z c= dom (f1 - f2) & ( for x being Real st x in Z holds
f1 . x = a ) & f2 = #Z 2 holds
( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - f2) `| Z) . x = - (2 * x) ) )
Lemma44:
for Z being open Subset of REAL
for f being PartFunc of REAL , REAL st Z c= dom ((#R (1 / 2)) * f) & ( for x being Real st x in Z holds
( f . x = x & f . x > 0 ) ) holds
( (#R (1 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = (1 / 2) * (x #R (- (1 / 2))) ) )
theorem Th1: :: FDIFF_5:1
theorem Th2: :: FDIFF_5:2
theorem Th3: :: FDIFF_5:3
theorem Th4: :: FDIFF_5:4
theorem Th5: :: FDIFF_5:5
theorem Th6: :: FDIFF_5:6
theorem Th7: :: FDIFF_5:7
theorem Th8: :: FDIFF_5:8
theorem Th9: :: FDIFF_5:9
theorem Th10: :: FDIFF_5:10
theorem Th11: :: FDIFF_5:11
theorem Th12: :: FDIFF_5:12
theorem Th13: :: FDIFF_5:13
theorem Th14: :: FDIFF_5:14
theorem Th15: :: FDIFF_5:15
theorem Th16: :: FDIFF_5:16
theorem Th17: :: FDIFF_5:17
theorem Th18: :: FDIFF_5:18
theorem Th19: :: FDIFF_5:19
theorem Th20: :: FDIFF_5:20
theorem Th21: :: FDIFF_5:21
theorem Th22: :: FDIFF_5:22
theorem Th23: :: FDIFF_5:23
theorem Th24: :: FDIFF_5:24
theorem Th25: :: FDIFF_5:25