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 & ( for x being Real st x in Z holds
f . x = (sin . (sin . x)) * (cos . x) ) & Z = dom f & f | A is continuous holds
integral f,A = ((- (cos * sin )) . (upper_bound A)) - ((- (cos * sin )) . (lower_bound A))
let 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
f . x = (sin . (sin . x)) * (cos . x) ) & Z = dom f & f | A is continuous holds
integral f,A = ((- (cos * sin )) . (upper_bound A)) - ((- (cos * sin )) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
f . x = (sin . (sin . x)) * (cos . x) ) & Z = dom f & f | A is continuous implies integral f,A = ((- (cos * sin )) . (upper_bound A)) - ((- (cos * sin )) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f . x = (sin . (sin . x)) * (cos . x) ) & Z = dom f & f | A is continuous ) ; :: thesis: integral f,A = ((- (cos * sin )) . (upper_bound A)) - ((- (cos * sin )) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
B1: dom sin = REAL by SIN_COS:27;
( rng sin c= dom sin & dom sin = dom cos ) by SIN_COS:27;
then dom (cos * sin ) = REAL by B1, RELAT_1:46;
then B2: dom (- (cos * sin )) = REAL by VALUED_1:8;
A4: cos * sin is_differentiable_on Z by FDIFF_10:9;
then A5: (- 1) (#) (cos * sin ) is_differentiable_on Z by B2, FDIFF_1:28, A;
A6: for x being Real st x in Z holds
((- (cos * sin )) `| Z) . x = (sin . (sin . x)) * (cos . x)

proof

let x be Real; :: thesis: ( x in Z implies ((- (cos * sin )) `| Z) . x = (sin . (sin . x)) * (cos . x) )
assume A7: x in Z ; :: thesis: ((- (cos * sin )) `| Z) . x = (sin . (sin . x)) * (cos . x)
((- (cos * sin )) `| Z) . x = ((- 1) (#) ((cos * sin ) `| Z)) . x by A4, FDIFF_2:19, A
.= (- 1) * (((cos * sin ) `| Z) . x) by VALUED_1:6
.= (- 1) * (- ((sin . (sin . x)) * (cos . x))) by A7, FDIFF_10:9
.= (sin . (sin . x)) * (cos . x) ;
hence ((- (cos * sin )) `| Z) . x = (sin . (sin . x)) * (cos . x) ; :: thesis: verum

end;

A8: for x being Real st x in dom ((- (cos * sin )) `| Z) holds
((- (cos * sin )) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((- (cos * sin )) `| Z) implies ((- (cos * sin )) `| Z) . x = f . x )
assume x in dom ((- (cos * sin )) `| Z) ; :: thesis: ((- (cos * sin )) `| Z) . x = f . x
then A9: x in Z by A5, FDIFF_1:def 8;
then ((- (cos * sin )) `| Z) . x = (sin . (sin . x)) * (cos . x) by A6
.= f . x by A1, A9 ;
hence ((- (cos * sin )) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((- (cos * sin )) `| Z) = dom f by A1, A5, FDIFF_1:def 8;
then (- (cos * sin )) `| Z = f by A8, PARTFUN1:34;
hence integral f,A = ((- (cos * sin )) . (upper_bound A)) - ((- (cos * sin )) . (lower_bound A)) by A1, A2, A5, INTEGRA5:13; :: thesis: verum