theorem :: FDIFF_4:51

for Z being open Subset of REAL st Z c= dom (((1 / 2) (#) ((#Z 2) * sin)) - cos) & ( for x being Real st x in Z holds
( sin . x > 0 & cos . x < 1 ) ) holds
( ((1 / 2) (#) ((#Z 2) * sin)) - cos is_differentiable_on Z & ( for x being Real st x in Z holds
((((1 / 2) (#) ((#Z 2) * sin)) - cos) `| Z) . x = ((sin . x) |^ 3) / (1 - (cos . x)) ) )