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 & ( 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;
A3: dom sin = REAL by SIN_COS:24;
( rng sin c= dom sin & dom sin = dom cos ) by SIN_COS:24;
then dom (cos * sin) = REAL by A3, RELAT_1:27;
then A4: dom (- (cos * sin)) = REAL by VALUED_1:8;
A5: cos * sin is_differentiable_on Z by FDIFF_10:9;
then A6: (- 1) (#) (cos * sin) is_differentiable_on Z by A4, FDIFF_1:20;
A7: 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 A8: x in Z ; :: thesis: ((- (cos * sin)) `| Z) . x = (sin . (sin . x)) * (cos . x)
((- (cos * sin)) `| Z) . x = ((- 1) (#) ((cos * sin) `| Z)) . x by A5, FDIFF_2:19
.= (- 1) * (((cos * sin) `| Z) . x) by VALUED_1:6
.= (- 1) * (- ((sin . (sin . x)) * (cos . x))) by A8, FDIFF_10:9
.= (sin . (sin . x)) * (cos . x) ;
hence ((- (cos * sin)) `| Z) . x = (sin . (sin . x)) * (cos . x) ; :: thesis: verum

end;

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

proof

let x be Element of 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 A10: x in Z by A6, FDIFF_1:def 7;
then ((- (cos * sin)) `| Z) . x = (sin . (sin . x)) * (cos . x) by A7
.= f . x by A1, A10 ;
hence ((- (cos * sin)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((- (cos * sin)) `| Z) = dom f by A1, A6, FDIFF_1:def 7;
then (- (cos * sin)) `| Z = f by A9, PARTFUN1:5;
hence integral (f,A) = ((- (cos * sin)) . (upper_bound A)) - ((- (cos * sin)) . (lower_bound A)) by A1, A2, A6, INTEGRA5:13; :: thesis: verum