theorem :: SIN_COS9:97

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