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 = - ((sin (#) cos) ^) & Z c= dom (ln * cot) & Z = dom f & f | A is continuous holds
integral (f,A) = ((ln * cot) . (upper_bound A)) - ((ln * cot) . (lower_bound A))
let f be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & f = - ((sin (#) cos) ^) & Z c= dom (ln * cot) & Z = dom f & f | A is continuous holds
integral (f,A) = ((ln * cot) . (upper_bound A)) - ((ln * cot) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & f = - ((sin (#) cos) ^) & Z c= dom (ln * cot) & Z = dom f & f | A is continuous implies integral (f,A) = ((ln * cot) . (upper_bound A)) - ((ln * cot) . (lower_bound A)) )
assume A1: ( A c= Z & f = - ((sin (#) cos) ^) & Z c= dom (ln * cot) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((ln * cot) . (upper_bound A)) - ((ln * cot) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: ln * cot is_differentiable_on Z by A1, FDIFF_8:19;
A4: Z = dom ((sin (#) cos) ^) by A1, VALUED_1:8;
A5: for x being Real st x in Z holds
f . x = - (1 / ((sin . x) * (cos . x)))

proof

let x be Real; :: thesis: ( x in Z implies f . x = - (1 / ((sin . x) * (cos . x))) )
assume A6: x in Z ; :: thesis: f . x = - (1 / ((sin . x) * (cos . x)))
(- ((sin (#) cos) ^)) . x = - (((sin (#) cos) ^) . x) by VALUED_1:8
.= - (1 / ((sin (#) cos) . x)) by A4, A6, RFUNCT_1:def 2
.= - (1 / ((sin . x) * (cos . x))) by VALUED_1:5 ;
hence f . x = - (1 / ((sin . x) * (cos . x))) by A1; :: thesis: verum

end;

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

proof

let x be Element of REAL ; :: thesis: ( x in dom ((ln * cot) `| Z) implies ((ln * cot) `| Z) . x = f . x )
assume x in dom ((ln * cot) `| Z) ; :: thesis: ((ln * cot) `| Z) . x = f . x
then A8: x in Z by A3, FDIFF_1:def 7;
then ((ln * cot) `| Z) . x = - (1 / ((sin . x) * (cos . x))) by A1, FDIFF_8:19
.= f . x by A5, A8 ;
hence ((ln * cot) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((ln * cot) `| Z) = dom f by A1, A3, FDIFF_1:def 7;
then (ln * cot) `| Z = f by A7, PARTFUN1:5;
hence integral (f,A) = ((ln * cot) . (upper_bound A)) - ((ln * cot) . (lower_bound A)) by A1, A2, FDIFF_8:19, INTEGRA5:13; :: thesis: verum