theorem :: INTEGRA9:52

for a being Real
for A being non empty closed_interval Subset of REAL
for f, f2 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
( f . x = a - x & f . x > 0 ) ) & dom (- (ln * f)) = Z & dom (- (ln * f)) = dom f2 & ( for x being Real st x in Z holds
f2 . x = 1 / (a - x) ) & f2 | A is continuous holds
integral (f2,A) = ((- (ln * f)) . (upper_bound A)) - ((- (ln * f)) . (lower_bound A))