let A be 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)) . (sup A)) - ((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) . (inf 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)) . (sup A)) - ((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) . (inf 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)) . (sup A)) - ((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) . (inf A)) )
assume A1: ( 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)) ) ; :: thesis: integral arccos ,A = ((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) . (sup A)) - ((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) . (inf A))
arccos | A is continuous ;
then A2: ( arccos is_integrable_on A & arccos | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A3: ((id Z) (#) arccos ) - ((#R (1 / 2)) * f) is_differentiable_on Z by A1, FDIFF_7:24;
A4: for x being 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 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 A3, FDIFF_1:def 8;
hence ((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x by A1, FDIFF_7:24; :: thesis: verum

end;

dom ((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) `| Z) = dom arccos by A1, A3, FDIFF_1:def 8;
then (((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) `| Z = arccos by A4, PARTFUN1:34;
hence integral arccos ,A = ((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) . (sup A)) - ((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) . (inf A)) by A1, A2, A3, INTEGRA5:13; :: thesis: verum