theorem :: INTEGR13:53

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