theorem :: INTEGR11:38

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
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = 1 / (1 - (sin . x)) ) ) & dom (tan + sec) = Z & Z = dom f & f | A is continuous holds
integral (f,A) = ((tan + sec) . (upper_bound A)) - ((tan + sec) . (lower_bound A))