theorem :: INTEGRA9:64

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 ((2 / 3) (#) ((#R (3 / 2)) * f)) = Z & dom ((2 / 3) (#) ((#R (3 / 2)) * f)) = dom f2 & ( for x being Real st x in Z holds
f2 . x = (a + x) #R (1 / 2) ) & f2 | A is continuous holds
integral (f2,A) = (((2 / 3) (#) ((#R (3 / 2)) * f)) . (upper_bound A)) - (((2 / 3) (#) ((#R (3 / 2)) * f)) . (lower_bound A))