theorem :: INTEGR14:30

for a being Real
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 & ( for x being Real st x in Z holds
f . x = ((sin . (a * x)) - ((cos . (a * x)) ^2)) / ((cos . (a * x)) ^2) ) & Z c= dom (((1 / a) (#) (sec * f1)) - (id Z)) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((((1 / a) (#) (sec * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (sec * f1)) - (id Z)) . (lower_bound A))