for a, b being Real
for A being non empty closed_interval Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( f1 . x = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = (a (#) arccos) - (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) holds
integral (f,A) = ((f1 (#) arccos) . (upper_bound A)) - ((f1 (#) arccos) . (lower_bound A))