theorem :: FDIFF_4:30

for a being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (2 (#) ((#R (1 / 2)) * f)) & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) holds
( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) )