let A be non empty closed_interval Subset of REAL; :: thesis: 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 arccos = Z & Z c= dom (((id Z) (#) arccos) - ((#R (1 / 2)) * f)) holds
integral (arccos,A) = ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: 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 arccos = Z & Z c= dom (((id Z) (#) arccos) - ((#R (1 / 2)) * f)) holds
integral (arccos,A) = ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (lower_bound A))
let f, f1, f2 be PartFunc of REAL,REAL; :: thesis: ( 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 arccos = Z & Z c= dom (((id Z) (#) arccos) - ((#R (1 / 2)) * f)) implies integral (arccos,A) = ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (lower_bound A)) )
assume that
A1: A c= Z and
A2: ( 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 ) ) ) and
A3: dom arccos = Z and
A4: Z c= dom (((id Z) (#) arccos) - ((#R (1 / 2)) * f)) ; :: thesis: integral (arccos,A) = ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (lower_bound A))
A5: arccos | A is bounded by A1, A3, INTEGRA5:10;
A6: ((id Z) (#) arccos) - ((#R (1 / 2)) * f) is_differentiable_on Z by A2, A4, FDIFF_7:24;
A7: for x being Element of REAL st x in dom ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) holds
((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) implies ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x )
assume x in dom ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) ; :: thesis: ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x
then x in Z by A6, FDIFF_1:def 7;
hence ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x by A2, A4, FDIFF_7:24; :: thesis: verum

end;

dom ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z) = dom arccos by A3, A6, FDIFF_1:def 7;
then (((id Z) (#) arccos) - ((#R (1 / 2)) * f)) `| Z = arccos by A7, PARTFUN1:5;
hence integral (arccos,A) = ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) arccos) - ((#R (1 / 2)) * f)) . (lower_bound A)) by A1, A3, A5, A6, INTEGRA5:11, INTEGRA5:13; :: thesis: verum