let A be closed-interval Subset of REAL ; :: thesis: for Z being open Subset of REAL
for f2, f1 being PartFunc of REAL ,REAL st A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom arctan & Z = dom (((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) holds
integral arctan ,A = ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) . (sup A)) - ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) . (inf A))
let Z be open Subset of REAL ; :: thesis: for f2, f1 being PartFunc of REAL ,REAL st A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom arctan & Z = dom (((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) holds
integral arctan ,A = ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) . (sup A)) - ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) . (inf A))
let f2, f1 be PartFunc of REAL ,REAL ; :: thesis: ( A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom arctan & Z = dom (((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) implies integral arctan ,A = ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) . (sup A)) - ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) . (inf A)) )
assume that
A1: A c= Z and
A2: Z c= ].(- 1),1.[ and
A3: ( f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) ) and
A4: Z = dom arctan and
A5: Z = dom (((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) ; :: thesis: integral arctan ,A = ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) . (sup A)) - ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) . (inf A))
( ].(- 1),1.[ c= [.(- 1),1.] & A c= ].(- 1),1.[ ) by A1, A2, XBOOLE_1:1, XXREAL_1:25;
then arctan | A is continuous by FCONT_1:17, SIN_COS9:53, XBOOLE_1:1;
then A6: ( arctan is_integrable_on A & arctan | A is bounded ) by A1, A4, INTEGRA5:10, INTEGRA5:11;
A7: ((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z by A2, A3, A5, SIN_COS9:103;
A8: for x being Real st x in dom ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) holds
((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x

proof

let x be Real; :: thesis: ( x in dom ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) implies ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x )
assume x in dom ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) ; :: thesis: ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x
then x in Z by A7, FDIFF_1:def 8;
hence ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x by A2, A3, A5, SIN_COS9:103; :: thesis: verum

end;

dom ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) = dom arctan by A4, A7, FDIFF_1:def 8;
then (((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z = arctan by A8, PARTFUN1:34;
hence integral arctan ,A = ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) . (sup A)) - ((((id Z) (#) arctan ) - ((1 / 2) (#) (ln * (f1 + f2)))) . (inf A)) by A1, A2, A3, A5, A6, INTEGRA5:13, SIN_COS9:103; :: thesis: verum