theorem :: INTEGRA9:59

for a, b 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 = x + b & f1 . x > 0 ) ) & dom ((id Z) - ((a + b) (#) f)) = Z & Z = dom f2 & ( for x being Real st x in Z holds
f2 . x = (x - a) / (x + b) ) & f2 | A is continuous holds
integral (f2,A) = (((id Z) - ((a + b) (#) f)) . (upper_bound A)) - (((id Z) - ((a + b) (#) f)) . (lower_bound A))