theorem :: SIN_COS9:109

for r being Real
for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) )