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