let A be closed-interval Subset of REAL; :: thesis: for f1, f 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
f1 . x = 1 ) & f = arccot / (f1 + (#Z 2)) & Z c= ].(- 1),1.[ & Z c= dom ((#Z 2) * arccot) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((1 / 2) (#) ((#Z 2) * arccot))) . (upper_bound A)) - ((- ((1 / 2) (#) ((#Z 2) * arccot))) . (lower_bound A))
let f1, f 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
f1 . x = 1 ) & f = arccot / (f1 + (#Z 2)) & Z c= ].(- 1),1.[ & Z c= dom ((#Z 2) * arccot) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((1 / 2) (#) ((#Z 2) * arccot))) . (upper_bound A)) - ((- ((1 / 2) (#) ((#Z 2) * arccot))) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = arccot / (f1 + (#Z 2)) & Z c= ].(- 1),1.[ & Z c= dom ((#Z 2) * arccot) & Z = dom f & f | A is continuous implies integral (f,A) = ((- ((1 / 2) (#) ((#Z 2) * arccot))) . (upper_bound A)) - ((- ((1 / 2) (#) ((#Z 2) * arccot))) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = arccot / (f1 + (#Z 2)) & Z c= ].(- 1),1.[ & Z c= dom ((#Z 2) * arccot) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((- ((1 / 2) (#) ((#Z 2) * arccot))) . (upper_bound A)) - ((- ((1 / 2) (#) ((#Z 2) * arccot))) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: - ((1 / 2) (#) ((#Z 2) * arccot)) is_differentiable_on Z by A1, Th8;
Z c= (dom arccot) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0})) by RFUNCT_1:def 4, A1;
then Z c= (dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) by XBOOLE_1:18;
then A5: Z c= dom ((f1 + (#Z 2)) ^) by RFUNCT_1:def 8;
dom ((f1 + (#Z 2)) ^) c= dom (f1 + (#Z 2)) by RFUNCT_1:11;
then A6: Z c= dom (f1 + (#Z 2)) by XBOOLE_1:1, A5;
A7: for x being Real st x in Z holds
f . x = (arccot . x) / (1 + (x ^2))

proof

let x be Real; :: thesis: ( x in Z implies f . x = (arccot . x) / (1 + (x ^2)) )
assume A8: x in Z ; :: thesis: f . x = (arccot . x) / (1 + (x ^2))
then (arccot / (f1 + (#Z 2))) . x = (arccot . x) / ((f1 + (#Z 2)) . x) by A1, RFUNCT_1:def 4
.= (arccot . x) / ((f1 . x) + ((#Z 2) . x)) by VALUED_1:def 1, A6, A8
.= (arccot . x) / ((f1 . x) + (x #Z 2)) by TAYLOR_1:def 1
.= (arccot . x) / (1 + (x #Z 2)) by A1, A8
.= (arccot . x) / (1 + (x ^2)) by FDIFF_7:1 ;
hence f . x = (arccot . x) / (1 + (x ^2)) by A1; :: thesis: verum

end;

A9: for x being Real st x in dom ((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) holds
((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) implies ((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = f . x )
assume x in dom ((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) ; :: thesis: ((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = f . x
then A10: x in Z by A3, FDIFF_1:def 8;
then ((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = (arccot . x) / (1 + (x ^2)) by A1, Th8
.= f . x by A7, A10 ;
hence ((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) = dom f by A1, A3, FDIFF_1:def 8;
then (- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z = f by A9, PARTFUN1:34;
hence integral (f,A) = ((- ((1 / 2) (#) ((#Z 2) * arccot))) . (upper_bound A)) - ((- ((1 / 2) (#) ((#Z 2) * arccot))) . (lower_bound A)) by A1, A2, Th8, INTEGRA5:13; :: thesis: verum