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 = - (exp_R / ((sin * exp_R ) ^2 )) & Z c= dom (cot * exp_R ) & Z = dom f & f | A is continuous holds
integral f,A = ((cot * exp_R ) . (upper_bound A)) - ((cot * exp_R ) . (lower_bound A))
let f be PartFunc of REAL ,REAL ; :: thesis: for Z being open Subset of REAL st A c= Z & f = - (exp_R / ((sin * exp_R ) ^2 )) & Z c= dom (cot * exp_R ) & Z = dom f & f | A is continuous holds
integral f,A = ((cot * exp_R ) . (upper_bound A)) - ((cot * exp_R ) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z & f = - (exp_R / ((sin * exp_R ) ^2 )) & Z c= dom (cot * exp_R ) & Z = dom f & f | A is continuous implies integral f,A = ((cot * exp_R ) . (upper_bound A)) - ((cot * exp_R ) . (lower_bound A)) )
assume A1: ( A c= Z & f = - (exp_R / ((sin * exp_R ) ^2 )) & Z c= dom (cot * exp_R ) & Z = dom f & f | A is continuous ) ; :: thesis: integral f,A = ((cot * exp_R ) . (upper_bound A)) - ((cot * exp_R ) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: cot * exp_R is_differentiable_on Z by A1, FDIFF_8:13;
A4: Z = dom (exp_R / ((sin * exp_R ) ^2 )) by A1, VALUED_1:8;
then Z c= (dom exp_R ) /\ ((dom ((sin * exp_R ) ^2 )) \ (((sin * exp_R ) ^2 ) " {0 })) by RFUNCT_1:def 4;
then Z c= (dom ((sin * exp_R ) ^2 )) \ (((sin * exp_R ) ^2 ) " {0 }) by XBOOLE_1:18;
then A5: Z c= dom (((sin * exp_R ) ^2 ) ^ ) by RFUNCT_1:def 8;
dom (((sin * exp_R ) ^2 ) ^ ) c= dom ((sin * exp_R ) ^2 ) by RFUNCT_1:11;
then Z c= dom ((sin * exp_R ) ^2 ) by XBOOLE_1:1, A5;
then A7: Z c= dom (sin * exp_R ) by VALUED_1:11;
A8: for x being Real st x in Z holds
f . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies f . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2 )) )
assume A9: x in Z ; :: thesis: f . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2 ))
(- (exp_R / ((sin * exp_R ) ^2 ))) . x = - ((exp_R / ((sin * exp_R ) ^2 )) . x) by VALUED_1:8
.= - ((exp_R . x) / (((sin * exp_R ) ^2 ) . x)) by RFUNCT_1:def 4, A4, A9
.= - ((exp_R . x) / (((sin * exp_R ) . x) ^2 )) by VALUED_1:11
.= - ((exp_R . x) / ((sin . (exp_R . x)) ^2 )) by FUNCT_1:22, A7, A9 ;
hence f . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2 )) by A1; :: thesis: verum

end;

A10: for x being Real st x in dom ((cot * exp_R ) `| Z) holds
((cot * exp_R ) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((cot * exp_R ) `| Z) implies ((cot * exp_R ) `| Z) . x = f . x )
assume x in dom ((cot * exp_R ) `| Z) ; :: thesis: ((cot * exp_R ) `| Z) . x = f . x
then A11: x in Z by A3, FDIFF_1:def 8;
then ((cot * exp_R ) `| Z) . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2 )) by A1, FDIFF_8:13
.= f . x by A8, A11 ;
hence ((cot * exp_R ) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((cot * exp_R ) `| Z) = dom f by A1, A3, FDIFF_1:def 8;
then (cot * exp_R ) `| Z = f by A10, PARTFUN1:34;
hence integral f,A = ((cot * exp_R ) . (upper_bound A)) - ((cot * exp_R ) . (lower_bound A)) by A1, A2, FDIFF_8:13, INTEGRA5:13; :: thesis: verum