theorem :: INTEGR11:36

for A being non empty closed_interval Subset of REAL
for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st A c= Z & f1 = #Z 2 & ( for x being Real st x in Z holds
( f2 . x = 1 & x <> 0 & f . x = (2 * x) / ((1 + (x ^2)) ^2) ) ) & dom (f1 / (f2 + f1)) = Z & Z = dom f & f | A is continuous holds
integral (f,A) = ((f1 / (f2 + f1)) . (upper_bound A)) - ((f1 / (f2 + f1)) . (lower_bound A))