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