theorem :: INTEGRA9:47

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 ) ) & Z = dom f & dom f = dom f2 & ( for x being Real st x in Z holds
f2 . x = - (1 / ((a + x) ^2)) ) & f2 | A is continuous holds
integral (f2,A) = ((f ^) . (upper_bound A)) - ((f ^) . (lower_bound A))