theorem Th7: :: FDIFF_9:7

for a, b being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (cosec * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( cosec * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * f) `| Z) . x = - ((a * (cos . ((a * x) + b))) / ((sin . ((a * x) + b)) ^2)) ) )