let A be closed-interval Subset of REAL ; :: thesis: for Z being open Subset of REAL
for f1, f2, f being PartFunc of REAL ,REAL st A c= Z & f1 = #Z 2 & ( for x being Real st x in Z holds
( f2 . x = 1 & x <> 0 & f . x = (2 * x) / ((1 + (x ^2 )) ^2 ) ) ) & dom (f1 / (f2 + f1)) = Z & Z = dom f & f | A is continuous holds
integral f,A = ((f1 / (f2 + f1)) . (sup A)) - ((f1 / (f2 + f1)) . (inf A))
let Z be open Subset of REAL ; :: thesis: for f1, f2, f being PartFunc of REAL ,REAL st A c= Z & f1 = #Z 2 & ( for x being Real st x in Z holds
( f2 . x = 1 & x <> 0 & f . x = (2 * x) / ((1 + (x ^2 )) ^2 ) ) ) & dom (f1 / (f2 + f1)) = Z & Z = dom f & f | A is continuous holds
integral f,A = ((f1 / (f2 + f1)) . (sup A)) - ((f1 / (f2 + f1)) . (inf A))
let f1, f2, f be PartFunc of REAL ,REAL ; :: thesis: ( A c= Z & f1 = #Z 2 & ( for x being Real st x in Z holds
( f2 . x = 1 & x <> 0 & f . x = (2 * x) / ((1 + (x ^2 )) ^2 ) ) ) & dom (f1 / (f2 + f1)) = Z & Z = dom f & f | A is continuous implies integral f,A = ((f1 / (f2 + f1)) . (sup A)) - ((f1 / (f2 + f1)) . (inf A)) )
assume A1: ( A c= Z & f1 = #Z 2 & ( for x being Real st x in Z holds
( f2 . x = 1 & x <> 0 & f . x = (2 * x) / ((1 + (x ^2 )) ^2 ) ) ) & dom (f1 / (f2 + f1)) = Z & Z = dom f & f | A is continuous ) ; :: thesis: integral f,A = ((f1 / (f2 + f1)) . (sup A)) - ((f1 / (f2 + f1)) . (inf A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
B: ( Z c= dom (f1 / (f2 + f1)) & f1 = #Z 2 & ( for x being Real st x in Z holds
( f2 . x = 1 & x <> 0 ) ) ) by A1;
then A3: f1 / (f2 + f1) is_differentiable_on Z by FDIFF_6:7;
A4: for x being Real st x in dom ((f1 / (f2 + f1)) `| Z) holds
((f1 / (f2 + f1)) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((f1 / (f2 + f1)) `| Z) implies ((f1 / (f2 + f1)) `| Z) . x = f . x )
assume x in dom ((f1 / (f2 + f1)) `| Z) ; :: thesis: ((f1 / (f2 + f1)) `| Z) . x = f . x
then A5: x in Z by A3, FDIFF_1:def 8;
then ((f1 / (f2 + f1)) `| Z) . x = (2 * x) / ((1 + (x ^2 )) ^2 ) by B, FDIFF_6:7
.= f . x by A1, A5 ;
hence ((f1 / (f2 + f1)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((f1 / (f2 + f1)) `| Z) = dom f by A1, A3, FDIFF_1:def 8;
then (f1 / (f2 + f1)) `| Z = f by A4, PARTFUN1:34;
hence integral f,A = ((f1 / (f2 + f1)) . (sup A)) - ((f1 / (f2 + f1)) . (inf A)) by A1, A2, A3, INTEGRA5:13; :: thesis: verum