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) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound 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) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound 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) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound 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) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound 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) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound A))
A6: r (#) arctan is_differentiable_on Z by A2, SIN_COS9:83;
A7: for x being Element of REAL st x in dom ((r (#) arctan) `| Z) holds
((r (#) arctan) `| Z) . x = f . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((r (#) arctan) `| Z) implies ((r (#) arctan) `| Z) . x = f . x )
assume x in dom ((r (#) arctan) `| Z) ; :: thesis: ((r (#) arctan) `| Z) . x = f . x
then A8: x in Z by A6, FDIFF_1:def 7;
then ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) by A2, SIN_COS9:83
.= f . x by A3, A8 ;
hence ((r (#) arctan) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((r (#) arctan) `| Z) = dom f by A4, A6, FDIFF_1:def 7;
then A9: (r (#) arctan) `| Z = f by A7, PARTFUN1:5;
( f is_integrable_on A & f | A is bounded ) by A1, A4, A5, INTEGRA5:10, INTEGRA5:11;
hence integral (f,A) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound A)) by A1, A2, A9, INTEGRA5:13, SIN_COS9:83; :: thesis: verum