theorem :: INTEGR13:14

for A being non empty closed_interval Subset of REAL
for f, f1 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = (- ((cos / sin) / f1)) - (((id Z) ^) / (sin ^2)) & f1 = #Z 2 & Z c= dom (((id Z) ^) (#) cot) & Z = dom f & f | A is continuous holds
integral (f,A) = ((((id Z) ^) (#) cot) . (upper_bound A)) - ((((id Z) ^) (#) cot) . (lower_bound A))