theorem :: INTEGR11:61

for A being non empty closed_interval Subset of REAL
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds
( f . x = 1 / ((cos . x) ^2) & cos . x <> 0 ) ) & dom tan = Z & Z = dom f & f | A is continuous holds
integral (f,A) = (tan . (upper_bound A)) - (tan . (lower_bound A))