theorem Th33: :: INTEGR11:33

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