theorem :: FDIFF_9:32

for Z being open Subset of REAL st not 0 in Z & Z c= dom (((id Z) ^) (#) sec) holds
( ((id Z) ^) (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) sec) `| Z) . x = (- ((1 / (cos . x)) / (x ^2))) + (((sin . x) / x) / ((cos . x) ^2)) ) )