theorem :: INTEGR11:54

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