let A be closed-interval Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL
for Z being open Subset of REAL st A c= Z & f = - (((id Z) (#) ((sin * ln ) ^2 )) ^ ) & Z c= dom (cot * ln ) & Z = dom f & f | A is continuous holds
integral f,A = ((cot * ln ) . (upper_bound A)) - ((cot * ln ) . (lower_bound A))
let f be PartFunc of REAL ,REAL ; :: thesis: for Z being open Subset of REAL st A c= Z & f = - (((id Z) (#) ((sin * ln ) ^2 )) ^ ) & Z c= dom (cot * ln ) & Z = dom f & f | A is continuous holds
integral f,A = ((cot * ln ) . (upper_bound A)) - ((cot * ln ) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z & f = - (((id Z) (#) ((sin * ln ) ^2 )) ^ ) & Z c= dom (cot * ln ) & Z = dom f & f | A is continuous implies integral f,A = ((cot * ln ) . (upper_bound A)) - ((cot * ln ) . (lower_bound A)) )
assume A1: ( A c= Z & f = - (((id Z) (#) ((sin * ln ) ^2 )) ^ ) & Z c= dom (cot * ln ) & Z = dom f & f | A is continuous ) ; :: thesis: integral f,A = ((cot * ln ) . (upper_bound A)) - ((cot * ln ) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: cot * ln is_differentiable_on Z by A1, FDIFF_8:15;
AB: Z = dom (((id Z) (#) ((sin * ln ) ^2 )) ^ ) by VALUED_1:8, A1;
then Z c= dom ((id Z) (#) ((sin * ln ) ^2 )) by RFUNCT_1:11;
then Z c= (dom (id Z)) /\ (dom ((sin * ln ) ^2 )) by VALUED_1:def 4;
then Z c= dom ((sin * ln ) ^2 ) by XBOOLE_1:18;
then A6: Z c= dom (sin * ln ) by VALUED_1:11;
B: for x being Real st x in Z holds
f . x = - (1 / (x * ((sin . (ln . x)) ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies f . x = - (1 / (x * ((sin . (ln . x)) ^2 ))) )
assume C: x in Z ; :: thesis: f . x = - (1 / (x * ((sin . (ln . x)) ^2 )))
(- (((id Z) (#) ((sin * ln ) ^2 )) ^ )) . x = - ((((id Z) (#) ((sin * ln ) ^2 )) ^ ) . x) by VALUED_1:8
.= - (1 / (((id Z) (#) ((sin * ln ) ^2 )) . x)) by RFUNCT_1:def 8, AB, C
.= - (1 / (((id Z) . x) * (((sin * ln ) ^2 ) . x))) by VALUED_1:5
.= - (1 / (x * (((sin * ln ) ^2 ) . x))) by FUNCT_1:35, C
.= - (1 / (x * (((sin * ln ) . x) ^2 ))) by VALUED_1:11
.= - (1 / (x * ((sin . (ln . x)) ^2 ))) by A6, FUNCT_1:22, C ;
hence f . x = - (1 / (x * ((sin . (ln . x)) ^2 ))) by A1; :: thesis: verum

end;

A7: for x being Real st x in dom ((cot * ln ) `| Z) holds
((cot * ln ) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((cot * ln ) `| Z) implies ((cot * ln ) `| Z) . x = f . x )
assume x in dom ((cot * ln ) `| Z) ; :: thesis: ((cot * ln ) `| Z) . x = f . x
then A8: x in Z by A3, FDIFF_1:def 8;
then ((cot * ln ) `| Z) . x = - (1 / (x * ((sin . (ln . x)) ^2 ))) by A1, FDIFF_8:15
.= f . x by B, A8 ;
hence ((cot * ln ) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((cot * ln ) `| Z) = dom f by A1, A3, FDIFF_1:def 8;
then (cot * ln ) `| Z = f by A7, PARTFUN1:34;
hence integral f,A = ((cot * ln ) . (upper_bound A)) - ((cot * ln ) . (lower_bound A)) by A1, A2, FDIFF_8:15, INTEGRA5:13; :: thesis: verum