let r be Real; :: thesis: for A being non empty closed_interval Subset of REAL
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds
integral (f,A) = (() . ()) - (() . ())

let A be non empty closed_interval Subset of REAL; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds
integral (f,A) = (() . ()) - (() . ())

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds
integral (f,A) = (() . ()) - (() . ())

let f be PartFunc of REAL,REAL; :: thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous implies integral (f,A) = (() . ()) - (() . ()) )

assume that
A1: A c= Z and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
f . x = - (r / (1 + (x ^2))) and
A4: Z = dom f and
A5: f | A is continuous ; :: thesis: integral (f,A) = (() . ()) - (() . ())
A6: r (#) arccot is_differentiable_on Z by ;
A7: for x being Element of REAL st x in dom (() `| Z) holds
(() `| Z) . x = f . x
proof
let x be Element of REAL ; :: thesis: ( x in dom (() `| Z) implies (() `| Z) . x = f . x )
assume x in dom (() `| Z) ; :: thesis: (() `| Z) . x = f . x
then A8: x in Z by ;
then (() `| Z) . x = - (r / (1 + (x ^2))) by
.= f . x by A3, A8 ;
hence ((r (#) arccot) `| Z) . x = f . x ; :: thesis: verum
end;
dom (() `| Z) = dom f by ;
then A9: (r (#) arccot) `| Z = f by ;
( f is_integrable_on A & f | A is bounded ) by ;
hence integral (f,A) = (() . ()) - (() . ()) by ; :: thesis: verum