for n being Element of NAT holds
( (1 / (n + 1)) (#) ((#Z (n + 1)) * sin) is_differentiable_on REAL & ( for x being Real holds (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = ((sin . x) #Z n) * (cos . x) ) )