theorem :: FDIFF_4:31

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)) ) )