theorem :: INTEGR12:15

for r being Real
for A being non empty closed_interval Subset of REAL
for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arctan * g holds
integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))