let A be non empty 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 1;
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 2;
dom (((sin * exp_R) ^2) ^) c= dom ((sin * exp_R) ^2) by RFUNCT_1:1;
then Z c= dom ((sin * exp_R) ^2) by A5;
then A6: Z c= dom (sin * exp_R) by VALUED_1:11;
A7: 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 A8: 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 A4, A8, RFUNCT_1:def 1
.= - ((exp_R . x) / (((sin * exp_R) . x) ^2)) by VALUED_1:11
.= - ((exp_R . x) / ((sin . (exp_R . x)) ^2)) by A6, A8, FUNCT_1:12 ;
hence f . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2)) by A1; :: thesis: verum

end;

A9: for x being Element of REAL st x in dom ((cot * exp_R) `| Z) holds
((cot * exp_R) `| Z) . x = f . x

proof

let x be Element of 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 A10: x in Z by A3, FDIFF_1:def 7;
then ((cot * exp_R) `| Z) . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2)) by A1, FDIFF_8:13
.= f . x by A7, A10 ;
hence ((cot * exp_R) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((cot * exp_R) `| Z) = dom f by A1, A3, FDIFF_1:def 7;
then (cot * exp_R) `| Z = f by A9, PARTFUN1:5;
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