theorem :: INTEGRA9:54

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