let A be non empty closed_interval Subset of REAL; :: thesis: for Z being open Subset of REAL st A c= Z holds
integral (((- (id Z)) (#) sin),A) = (((- sin) + ((id Z) (#) cos)) . (upper_bound A)) - (((- sin) + ((id Z) (#) cos)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z implies integral (((- (id Z)) (#) sin),A) = (((- sin) + ((id Z) (#) cos)) . (upper_bound A)) - (((- sin) + ((id Z) (#) cos)) . (lower_bound A)) )
assume A1: A c= Z ; :: thesis: integral (((- (id Z)) (#) sin),A) = (((- sin) + ((id Z) (#) cos)) . (upper_bound A)) - (((- sin) + ((id Z) (#) cos)) . (lower_bound A))
A2: (- sin) + ((id Z) (#) cos) is_differentiable_on Z by Th11;
A3: for x being Real st x in Z holds
(- (id Z)) . x = ((- 1) * x) + 0

proof

let x be Real; :: thesis: ( x in Z implies (- (id Z)) . x = ((- 1) * x) + 0 )
assume A4: x in Z ; :: thesis: (- (id Z)) . x = ((- 1) * x) + 0
(- (id Z)) . x = - ((id Z) . x) by VALUED_1:8
.= - x by A4, FUNCT_1:18
.= ((- 1) * x) + 0 ;
hence (- (id Z)) . x = ((- 1) * x) + 0 ; :: thesis: verum

end;

A5: for x being Real st x in Z holds
((- (id Z)) (#) sin) . x = - (x * (sin . x))

proof

let x be Real; :: thesis: ( x in Z implies ((- (id Z)) (#) sin) . x = - (x * (sin . x)) )
assume A6: x in Z ; :: thesis: ((- (id Z)) (#) sin) . x = - (x * (sin . x))
((- (id Z)) (#) sin) . x = ((- (id Z)) . x) * (sin . x) by VALUED_1:5
.= (((- 1) * x) + 0) * (sin . x) by A3, A6
.= - (x * (sin . x)) ;
hence ((- (id Z)) (#) sin) . x = - (x * (sin . x)) ; :: thesis: verum

end;

A7: for x being Element of REAL st x in dom (((- sin) + ((id Z) (#) cos)) `| Z) holds
(((- sin) + ((id Z) (#) cos)) `| Z) . x = ((- (id Z)) (#) sin) . x

proof

let x be Element of REAL ; :: thesis: ( x in dom (((- sin) + ((id Z) (#) cos)) `| Z) implies (((- sin) + ((id Z) (#) cos)) `| Z) . x = ((- (id Z)) (#) sin) . x )
assume x in dom (((- sin) + ((id Z) (#) cos)) `| Z) ; :: thesis: (((- sin) + ((id Z) (#) cos)) `| Z) . x = ((- (id Z)) (#) sin) . x
then A8: x in Z by A2, FDIFF_1:def 7;
then (((- sin) + ((id Z) (#) cos)) `| Z) . x = - (x * (sin . x)) by Th11
.= ((- (id Z)) (#) sin) . x by A5, A8 ;
hence (((- sin) + ((id Z) (#) cos)) `| Z) . x = ((- (id Z)) (#) sin) . x ; :: thesis: verum

end;

dom ((- (id Z)) (#) sin) = (dom (- (id Z))) /\ REAL by SIN_COS:24, VALUED_1:def 4
.= dom (- (id Z)) by XBOOLE_1:28
.= dom (id Z) by VALUED_1:8 ;
then A9: Z = dom ((- (id Z)) (#) sin) by RELAT_1:45;
then dom (((- sin) + ((id Z) (#) cos)) `| Z) = dom ((- (id Z)) (#) sin) by A2, FDIFF_1:def 7;
then A10: ((- sin) + ((id Z) (#) cos)) `| Z = (- (id Z)) (#) sin by A7, PARTFUN1:5;
((- (id Z)) (#) sin) | A is continuous ;
then A11: (- (id Z)) (#) sin is_integrable_on A by A1, A9, INTEGRA5:11;
((- (id Z)) (#) sin) | A is bounded by A1, A9, INTEGRA5:10;
hence integral (((- (id Z)) (#) sin),A) = (((- sin) + ((id Z) (#) cos)) . (upper_bound A)) - (((- sin) + ((id Z) (#) cos)) . (lower_bound A)) by A1, A11, A10, Th11, INTEGRA5:13; :: thesis: verum