theorem :: INTEGR12:45

for A being non empty closed_interval Subset of REAL
for f, f1 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 = 1 & arccos . x > 0 ) ) & Z c= dom (ln * arccos) & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) ^ holds
integral (f,A) = ((- (ln * arccos)) . (upper_bound A)) - ((- (ln * arccos)) . (lower_bound A))