let a be Real; :: thesis: for A being non empty closed_interval Subset of REAL
for f, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) ) & dom ((- 1) (#) (f ^)) = Z & dom ((- 1) (#) (f ^)) = dom f2 & ( for x being Real st x in Z holds
f2 . x = 1 / ((a + x) ^2) ) & f2 | A is continuous holds
integral (f2,A) = (((- 1) (#) (f ^)) . (upper_bound A)) - (((- 1) (#) (f ^)) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; :: thesis: for f, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) ) & dom ((- 1) (#) (f ^)) = Z & dom ((- 1) (#) (f ^)) = dom f2 & ( for x being Real st x in Z holds
f2 . x = 1 / ((a + x) ^2) ) & f2 | A is continuous holds
integral (f2,A) = (((- 1) (#) (f ^)) . (upper_bound A)) - (((- 1) (#) (f ^)) . (lower_bound A))
let f, f2 be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) ) & dom ((- 1) (#) (f ^)) = Z & dom ((- 1) (#) (f ^)) = dom f2 & ( for x being Real st x in Z holds
f2 . x = 1 / ((a + x) ^2) ) & f2 | A is continuous holds
integral (f2,A) = (((- 1) (#) (f ^)) . (upper_bound A)) - (((- 1) (#) (f ^)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) ) & dom ((- 1) (#) (f ^)) = Z & dom ((- 1) (#) (f ^)) = dom f2 & ( for x being Real st x in Z holds
f2 . x = 1 / ((a + x) ^2) ) & f2 | A is continuous implies integral (f2,A) = (((- 1) (#) (f ^)) . (upper_bound A)) - (((- 1) (#) (f ^)) . (lower_bound A)) )
assume that
A1: A c= Z and
A2: for x being Real st x in Z holds
( f . x = a + x & f . x <> 0 ) and
A3: dom ((- 1) (#) (f ^)) = Z and
A4: dom ((- 1) (#) (f ^)) = dom f2 and
A5: for x being Real st x in Z holds
f2 . x = 1 / ((a + x) ^2) and
A6: f2 | A is continuous ; :: thesis: integral (f2,A) = (((- 1) (#) (f ^)) . (upper_bound A)) - (((- 1) (#) (f ^)) . (lower_bound A))
A7: f2 is_integrable_on A by A1, A3, A4, A6, INTEGRA5:11;
A8: (- 1) (#) (f ^) is_differentiable_on Z by A2, A3, FDIFF_4:15;
A9: for x being Element of REAL st x in dom (((- 1) (#) (f ^)) `| Z) holds
(((- 1) (#) (f ^)) `| Z) . x = f2 . x

proof

let x be Element of REAL ; :: thesis: ( x in dom (((- 1) (#) (f ^)) `| Z) implies (((- 1) (#) (f ^)) `| Z) . x = f2 . x )
assume x in dom (((- 1) (#) (f ^)) `| Z) ; :: thesis: (((- 1) (#) (f ^)) `| Z) . x = f2 . x
then A10: x in Z by A8, FDIFF_1:def 7;
then (((- 1) (#) (f ^)) `| Z) . x = 1 / ((a + x) ^2) by A2, A3, FDIFF_4:15
.= f2 . x by A5, A10 ;
hence (((- 1) (#) (f ^)) `| Z) . x = f2 . x ; :: thesis: verum

end;

dom (((- 1) (#) (f ^)) `| Z) = dom f2 by A3, A4, A8, FDIFF_1:def 7;
then ((- 1) (#) (f ^)) `| Z = f2 by A9, PARTFUN1:5;
hence integral (f2,A) = (((- 1) (#) (f ^)) . (upper_bound A)) - (((- 1) (#) (f ^)) . (lower_bound A)) by A1, A3, A4, A6, A7, A8, INTEGRA5:10, INTEGRA5:13; :: thesis: verum