let A be closed-interval Subset of REAL ; :: thesis: for Z being open Subset of REAL
for g, f being PartFunc of REAL , REAL st A c= Z & ( for x being Real st x in Z holds
( g . x = x & g . x <> 0 & f . x = - (1 / (x ^2 )) ) ) & Z = dom g & dom f = Z & f | A is continuous holds
integral f,A = ((g ^ ) . (sup A)) - ((g ^ ) . (inf A))
let Z be open Subset of REAL ; :: thesis: for g, f being PartFunc of REAL , REAL st A c= Z & ( for x being Real st x in Z holds
( g . x = x & g . x <> 0 & f . x = - (1 / (x ^2 )) ) ) & Z = dom g & dom f = Z & f | A is continuous holds
integral f,A = ((g ^ ) . (sup A)) - ((g ^ ) . (inf A))
let g, f be PartFunc of REAL , REAL ; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
( g . x = x & g . x <> 0 & f . x = - (1 / (x ^2 )) ) ) & Z = dom g & dom f = Z & f | A is continuous implies integral f,A = ((g ^ ) . (sup A)) - ((g ^ ) . (inf A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( g . x = x & g . x <> 0 & f . x = - (1 / (x ^2 )) ) ) & Z = dom g & dom f = Z & f | A is continuous ) ; :: thesis: integral f,A = ((g ^ ) . (sup A)) - ((g ^ ) . (inf A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
B: ( Z c= dom g & ( for x being Real st x in Z holds
( g . x = x & g . x <> 0 ) ) ) by A1;
then A3: g ^ is_differentiable_on Z by FDIFF_5:4;
A4: for x being Real st x in dom ((g ^ ) `| Z) holds
((g ^ ) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((g ^ ) `| Z) implies ((g ^ ) `| Z) . x = f . x )
assume x in dom ((g ^ ) `| Z) ; :: thesis: ((g ^ ) `| Z) . x = f . x
then A5: x in Z by A3, FDIFF_1:def 8;
then ((g ^ ) `| Z) . x = - (1 / (x ^2 )) by B, FDIFF_5:4
.= f . x by A1, A5 ;
hence ((g ^ ) `| Z) . x = f . x ; :: thesis: verum

end;

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