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 = (arctan / (#Z 2)) - (((id Z) (#) (f1 + (#Z 2))) ^) & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (((id Z) ^) (#) arctan)) . (upper_bound A)) - ((- (((id Z) ^) (#) arctan)) . (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 = (arctan / (#Z 2)) - (((id Z) (#) (f1 + (#Z 2))) ^) & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (((id Z) ^) (#) arctan)) . (upper_bound A)) - ((- (((id Z) ^) (#) arctan)) . (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 = (arctan / (#Z 2)) - (((id Z) (#) (f1 + (#Z 2))) ^) & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous implies integral (f,A) = ((- (((id Z) ^) (#) arctan)) . (upper_bound A)) - ((- (((id Z) ^) (#) arctan)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = (arctan / (#Z 2)) - (((id Z) (#) (f1 + (#Z 2))) ^) & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((- (((id Z) ^) (#) arctan)) . (upper_bound A)) - ((- (((id Z) ^) (#) arctan)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: - (((id Z) ^) (#) arctan) is_differentiable_on Z by A1, Th10;
AB: Z = (dom (arctan / (#Z 2))) /\ (dom (((id Z) (#) (f1 + (#Z 2))) ^)) by A1, VALUED_1:12;
then A4: Z c= dom (arctan / (#Z 2)) by XBOOLE_1:18;
A5: Z c= dom (((id Z) (#) (f1 + (#Z 2))) ^) by AB, XBOOLE_1:18;
dom (((id Z) (#) (f1 + (#Z 2))) ^) c= dom ((id Z) (#) (f1 + (#Z 2))) by RFUNCT_1:11;
then Z c= dom ((id Z) (#) (f1 + (#Z 2))) by XBOOLE_1:1, A5;
then Z c= (dom (id Z)) /\ (dom (f1 + (#Z 2))) by VALUED_1:def 4;
then A7: Z c= dom (f1 + (#Z 2)) by XBOOLE_1:18;
B: for x being Real st x in Z holds
f . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2))))

proof

let x be Real; :: thesis: ( x in Z implies f . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2)))) )
assume A8: x in Z ; :: thesis: f . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2))))
then A9: x in dom (((id Z) (#) (f1 + (#Z 2))) ^) by A5;
((arctan / (#Z 2)) - (((id Z) (#) (f1 + (#Z 2))) ^)) . x = ((arctan / (#Z 2)) . x) - ((((id Z) (#) (f1 + (#Z 2))) ^) . x) by VALUED_1:13, A1, A8
.= ((arctan / (#Z 2)) . x) - (1 / (((id Z) (#) (f1 + (#Z 2))) . x)) by RFUNCT_1:def 8, A9
.= ((arctan / (#Z 2)) . x) - (1 / (((id Z) . x) * ((f1 + (#Z 2)) . x))) by VALUED_1:5
.= ((arctan / (#Z 2)) . x) - (1 / (((id Z) . x) * ((f1 . x) + ((#Z 2) . x)))) by VALUED_1:def 1, A7, A8
.= ((arctan / (#Z 2)) . x) - (1 / (x * ((f1 . x) + ((#Z 2) . x)))) by FUNCT_1:35, A8
.= ((arctan / (#Z 2)) . x) - (1 / (x * (1 + ((#Z 2) . x)))) by A1, A8
.= ((arctan . x) / ((#Z 2) . x)) - (1 / (x * (1 + ((#Z 2) . x)))) by RFUNCT_1:def 4, A8, A4
.= ((arctan . x) / (x #Z 2)) - (1 / (x * (1 + ((#Z 2) . x)))) by TAYLOR_1:def 1
.= ((arctan . x) / (x #Z 2)) - (1 / (x * (1 + (x #Z 2)))) by TAYLOR_1:def 1
.= ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x #Z 2)))) by FDIFF_7:1
.= ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2)))) by FDIFF_7:1 ;
hence f . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2)))) by A1; :: thesis: verum

end;

A10: for x being Real st x in dom ((- (((id Z) ^) (#) arctan)) `| Z) holds
((- (((id Z) ^) (#) arctan)) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((- (((id Z) ^) (#) arctan)) `| Z) implies ((- (((id Z) ^) (#) arctan)) `| Z) . x = f . x )
assume x in dom ((- (((id Z) ^) (#) arctan)) `| Z) ; :: thesis: ((- (((id Z) ^) (#) arctan)) `| Z) . x = f . x
then A11: x in Z by A3, FDIFF_1:def 8;
then ((- (((id Z) ^) (#) arctan)) `| Z) . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2)))) by A1, Th10
.= f . x by B, A11 ;
hence ((- (((id Z) ^) (#) arctan)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((- (((id Z) ^) (#) arctan)) `| Z) = dom f by A1, A3, FDIFF_1:def 8;
then (- (((id Z) ^) (#) arctan)) `| Z = f by A10, PARTFUN1:34;
hence integral (f,A) = ((- (((id Z) ^) (#) arctan)) . (upper_bound A)) - ((- (((id Z) ^) (#) arctan)) . (lower_bound A)) by A1, A2, A3, INTEGRA5:13; :: thesis: verum