let A be non empty closed_interval Subset of REAL; :: thesis: for f, f1 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 f, f1 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;
A3: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A4: (1 / 2) (#) ((#Z 2) * arccot) is_differentiable_on Z by A1, A2, SIN_COS9:94;
A5: 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 1;
then Z c= (dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) by XBOOLE_1:18;
then A6: Z c= dom ((f1 + (#Z 2)) ^) by RFUNCT_1:def 2;
dom ((f1 + (#Z 2)) ^) c= dom (f1 + (#Z 2)) by RFUNCT_1:1;
then A7: Z c= dom (f1 + (#Z 2)) by A6;
A8: 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 A9: 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 A5, A9, RFUNCT_1:def 1
.= - ((arccot . x) / ((f1 . x) + ((#Z 2) . x))) by A7, A9, VALUED_1:def 1
.= - ((arccot . x) / ((f1 . x) + (x #Z 2))) by TAYLOR_1:def 1
.= - ((arccot . x) / (1 + (x #Z 2))) by A1, A9
.= - ((arccot . x) / (1 + (x ^2))) by FDIFF_7:1 ;
hence f . x = - ((arccot . x) / (1 + (x ^2))) by A1; :: thesis: verum

end;

A10: for x being Element of 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 Element of 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 A11: x in Z by A4, FDIFF_1:def 7;
then (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) by A1, A2, SIN_COS9:94
.= f . x by A8, A11 ;
hence (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = f . x ; :: thesis: verum

end;

dom (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) = dom f by A1, A4, FDIFF_1:def 7;
then ((1 / 2) (#) ((#Z 2) * arccot)) `| Z = f by A10, PARTFUN1:5;
hence integral (f,A) = (((1 / 2) (#) ((#Z 2) * arccot)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arccot)) . (lower_bound A)) by A1, A2, A3, INTEGRA5:13, SIN_COS9:94; :: thesis: verum