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 ) & dom arccot = Z & Z = dom (((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) holds
integral arccot ,A = ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) . (sup A)) - ((((id Z) (#) arccot ) + ((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 ) & dom arccot = Z & Z = dom (((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) holds
integral arccot ,A = ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) . (sup A)) - ((((id Z) (#) arccot ) + ((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 ) & dom arccot = Z & Z = dom (((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) implies integral arccot ,A = ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) . (sup A)) - ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) . (inf A)) )
assume A1: ( A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) & dom arccot = Z & Z = dom (((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) ) ; :: thesis: integral arccot ,A = ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) . (sup A)) - ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) . (inf A))
Ab: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
A c= ].(- 1),1.[ by A1, XBOOLE_1:1;
then arccot | A is continuous by SIN_COS9:54, FCONT_1:17, Ab, XBOOLE_1:1;
then A2: ( arccot is_integrable_on A & arccot | A is bounded ) by INTEGRA5:10, INTEGRA5:11, A1;
A3: ((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z by A1, SIN_COS9:104;
A4: for x being Real st x in dom ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) holds
((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x

proof

let x be Real; :: thesis: ( x in dom ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) implies ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x )
assume x in dom ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) ; :: thesis: ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x
then x in Z by A3, FDIFF_1:def 8;
hence ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x by A1, SIN_COS9:104; :: thesis: verum

end;

dom ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) = dom arccot by A1, A3, FDIFF_1:def 8;
then (((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z = arccot by A4, PARTFUN1:34;
hence integral arccot ,A = ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) . (sup A)) - ((((id Z) (#) arccot ) + ((1 / 2) (#) (ln * (f1 + f2)))) . (inf A)) by A1, A2, SIN_COS9:104, INTEGRA5:13; :: thesis: verum