theorem :: FDIFF_4:35

for a being Real
for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (- ((#R (1 / 2)) * f)) & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & 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 = x * (((a ^2) - (x |^ 2)) #R (- (1 / 2))) ) )