theorem :: INTEGR13:56

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
f1 . x = 1 ) & f = (arctan / (#Z 2)) - (((id Z) (#) (f1 + (#Z 2))) ^) & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (((id Z) ^) (#) arctan)) . (upper_bound A)) - ((- (((id Z) ^) (#) arctan)) . (lower_bound A))