theorem :: INTEGR11:68

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 & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 & x <> 0 ) ) & dom arcsin = Z & Z c= dom (((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) holds
integral (arcsin,A) = ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) . (lower_bound A))