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: Z c= dom ((1 / 2) (#) ((#Z 2) * arccot)) by VALUED_1:def 5;
A4: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A5: (1 / 2) (#) ((#Z 2) * arccot) is_differentiable_on Z by A1, A2, SIN_COS9:94;
A3: Z = dom (arccot / (f1 + (#Z 2))) by A1, VALUED_1:8;
then Z c= (dom arccot) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0})) by RFUNCT_1:def 4;
then Z c= (dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) by XBOOLE_1:18;
then A7: 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 A8: Z c= dom (f1 + (#Z 2)) by XBOOLE_1:1, A7;
A9: 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 A10: x in Z ; :: thesis: f . x = - ((arccot . x) / (1 + (x ^2)))
(- (arccot / (f1 + (#Z 2)))) . x = - ((arccot / (f1 + (#Z 2))) . x) by VALUED_1:8
.= - ((arccot . x) / ((f1 + (#Z 2)) . x)) by A3, A10, RFUNCT_1:def 4
.= - ((arccot . x) / ((f1 . x) + ((#Z 2) . x))) by VALUED_1:def 1, A8, A10
.= - ((arccot . x) / ((f1 . x) + (x #Z 2))) by TAYLOR_1:def 1
.= - ((arccot . x) / (1 + (x #Z 2))) by A1, A10
.= - ((arccot . x) / (1 + (x ^2))) by FDIFF_7:1 ;
hence f . x = - ((arccot . x) / (1 + (x ^2))) by A1; :: thesis: verum

end;

A11: 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 A12: x in Z by A5, FDIFF_1:def 8;
then (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) by A1, A2, SIN_COS9:94
.= f . x by A9, A12 ;
hence (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = f . x ; :: thesis: verum

end;

dom (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) = dom f by A1, A5, FDIFF_1:def 8;
then ((1 / 2) (#) ((#Z 2) * arccot)) `| Z = f by A11, 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, A4, SIN_COS9:94, INTEGRA5:13; :: thesis: verum