theorem :: SIN_COS9:99

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