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 / (id Z)) + (ln / (f1 + (#Z 2))) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous holds
integral f,A = ((ln (#) arctan ) . (sup A)) - ((ln (#) arctan ) . (inf 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 / (id Z)) + (ln / (f1 + (#Z 2))) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous holds
integral f,A = ((ln (#) arctan ) . (sup A)) - ((ln (#) arctan ) . (inf 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 / (id Z)) + (ln / (f1 + (#Z 2))) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous implies integral f,A = ((ln (#) arctan ) . (sup A)) - ((ln (#) arctan ) . (inf A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = (arctan / (id Z)) + (ln / (f1 + (#Z 2))) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous ) ; :: thesis: integral f,A = ((ln (#) arctan ) . (sup A)) - ((ln (#) arctan ) . (inf A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
Z = (dom (arctan / (id Z))) /\ (dom (ln / (f1 + (#Z 2)))) by VALUED_1:def 1, A1;
then A4: ( Z c= dom (arctan / (id Z)) & Z c= dom (ln / (f1 + (#Z 2))) ) by XBOOLE_1:18;
Z c= (dom arctan ) /\ ((dom (id Z)) \ ((id Z) " {0 })) by A4, RFUNCT_1:def 4;
then A5: Z c= dom arctan by XBOOLE_1:18;
Z c= (dom ln ) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0 })) by A4, RFUNCT_1:def 4;
then A6: ( Z c= dom ln & Z c= (dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0 }) ) by XBOOLE_1:18;
Z c= (dom arctan ) /\ (dom ln ) by XBOOLE_1:19, A5, A6;
then A7: Z c= dom (ln (#) arctan ) by VALUED_1:def 4;
A3: ln (#) arctan is_differentiable_on Z by A1, A7, SIN_COS9:127;
A8: Z c= dom ((f1 + (#Z 2)) ^ ) by RFUNCT_1:def 8, A6;
dom ((f1 + (#Z 2)) ^ ) c= dom (f1 + (#Z 2)) by RFUNCT_1:11;
then A9: Z c= dom (f1 + (#Z 2)) by XBOOLE_1:1, A8;
A10: for x being Real st x in Z holds
f . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies f . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2 ))) )
assume A11: x in Z ; :: thesis: f . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2 )))
((arctan / (id Z)) + (ln / (f1 + (#Z 2)))) . x = ((arctan / (id Z)) . x) + ((ln / (f1 + (#Z 2))) . x) by VALUED_1:def 1, A1, A11
.= ((arctan . x) * (((id Z) . x) " )) + ((ln / (f1 + (#Z 2))) . x) by RFUNCT_1:def 4, A4, A11
.= ((arctan . x) * (((id Z) . x) " )) + ((ln . x) * (((f1 + (#Z 2)) . x) " )) by RFUNCT_1:def 4, A4, A11
.= ((arctan . x) / x) + ((ln . x) / ((f1 + (#Z 2)) . x)) by FUNCT_1:35, A11
.= ((arctan . x) / x) + ((ln . x) / ((f1 . x) + ((#Z 2) . x))) by VALUED_1:def 1, A9, A11
.= ((arctan . x) / x) + ((ln . x) / (1 + ((#Z 2) . x))) by A1, A11
.= ((arctan . x) / x) + ((ln . x) / (1 + (x #Z 2))) by TAYLOR_1:def 1
.= ((arctan . x) / x) + ((ln . x) / (1 + (x ^2 ))) by FDIFF_7:1 ;
hence f . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2 ))) by A1; :: thesis: verum

end;

A12: for x being Real st x in dom ((ln (#) arctan ) `| Z) holds
((ln (#) arctan ) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((ln (#) arctan ) `| Z) implies ((ln (#) arctan ) `| Z) . x = f . x )
assume x in dom ((ln (#) arctan ) `| Z) ; :: thesis: ((ln (#) arctan ) `| Z) . x = f . x
then A13: x in Z by A3, FDIFF_1:def 8;
then ((ln (#) arctan ) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2 ))) by A1, A7, SIN_COS9:127
.= f . x by A13, A10 ;
hence ((ln (#) arctan ) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((ln (#) arctan ) `| Z) = dom f by A1, A3, FDIFF_1:def 8;
then (ln (#) arctan ) `| Z = f by A12, PARTFUN1:34;
hence integral f,A = ((ln (#) arctan ) . (sup A)) - ((ln (#) arctan ) . (inf A)) by A1, A2, A7, SIN_COS9:127, INTEGRA5:13; :: thesis: verum