theorem :: SIN_COS9:111

for Z being open Subset of REAL st not 0 in Z & Z c= dom (arctan * ((id Z) ^)) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds
( arctan * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) ) )