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 & 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)) . (upper_bound A)) - ((f1 / (f2 + f1)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: for f, f1, f2 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)) . (upper_bound A)) - ((f1 / (f2 + f1)) . (lower_bound A))
let f, f1, f2 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)) . (upper_bound A)) - ((f1 / (f2 + f1)) . (lower_bound A)) )
assume that
A1: A c= Z and
A2: f1 = #Z 2 and
A3: for x being Real st x in Z holds
( f2 . x = 1 & x <> 0 & f . x = (2 * x) / ((1 + (x ^2)) ^2) ) and
A4: dom (f1 / (f2 + f1)) = Z and
A5: Z = dom f and
A6: f | A is continuous ; :: thesis: integral (f,A) = ((f1 / (f2 + f1)) . (upper_bound A)) - ((f1 / (f2 + f1)) . (lower_bound A))
A7: f is_integrable_on A by A1, A5, A6, INTEGRA5:11;
A8: for x being Real st x in Z holds
( f2 . x = 1 & x <> 0 ) by A3;
then A9: f1 / (f2 + f1) is_differentiable_on Z by A2, A4, FDIFF_6:7;
A10: for x being Element of REAL st x in dom ((f1 / (f2 + f1)) `| Z) holds
((f1 / (f2 + f1)) `| Z) . x = f . x

proof

let x be Element of 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 A11: x in Z by A9, FDIFF_1:def 7;
then ((f1 / (f2 + f1)) `| Z) . x = (2 * x) / ((1 + (x ^2)) ^2) by A2, A4, A8, FDIFF_6:7
.= f . x by A3, A11 ;
hence ((f1 / (f2 + f1)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((f1 / (f2 + f1)) `| Z) = dom f by A5, A9, FDIFF_1:def 7;
then (f1 / (f2 + f1)) `| Z = f by A10, PARTFUN1:5;
hence integral (f,A) = ((f1 / (f2 + f1)) . (upper_bound A)) - ((f1 / (f2 + f1)) . (lower_bound A)) by A1, A5, A6, A7, A9, INTEGRA5:10, INTEGRA5:13; :: thesis: verum