let n be Element of NAT ; :: thesis: for A being closed-interval Subset of REAL st n <> 0 holds
integral (sin * (AffineMap n,0 )),A = (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) . (upper_bound A)) - (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) . (lower_bound A))
let A be closed-interval Subset of REAL ; :: thesis: ( n <> 0 implies integral (sin * (AffineMap n,0 )),A = (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) . (upper_bound A)) - (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) . (lower_bound A)) )
assume A1: n <> 0 ; :: thesis: integral (sin * (AffineMap n,0 )),A = (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) . (upper_bound A)) - (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) . (lower_bound A))
A2: [#] REAL = dom (sin * (AffineMap n,0 )) by FUNCT_2:def 1;
A3: for x being Real st x in REAL holds
(AffineMap n,0 ) . x = n * x

proof

let x be Real; :: thesis: ( x in REAL implies (AffineMap n,0 ) . x = n * x )
assume x in REAL ; :: thesis: (AffineMap n,0 ) . x = n * x
(AffineMap n,0 ) . x = (n * x) + 0 by JORDAN16:def 3
.= n * x ;
hence (AffineMap n,0 ) . x = n * x ; :: thesis: verum

end;

A4: for x being Real st x in dom (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) holds
(((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = (sin * (AffineMap n,0 )) . x

proof

let x be Real; :: thesis: ( x in dom (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) implies (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = (sin * (AffineMap n,0 )) . x )
assume x in dom (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) ; :: thesis: (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = (sin * (AffineMap n,0 )) . x
(((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = sin (n * x) by A1, Th8
.= sin . ((AffineMap n,0 ) . x) by A3
.= (sin * (AffineMap n,0 )) . x by A2, FUNCT_1:22 ;
hence (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = (sin * (AffineMap n,0 )) . x ; :: thesis: verum

end;

(- (1 / n)) (#) (cos * (AffineMap n,0 )) is_differentiable_on REAL by A1, Th8;
then dom (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) = dom (sin * (AffineMap n,0 )) by A2, FDIFF_1:def 8;
then A5: ((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL = sin * (AffineMap n,0 ) by A4, PARTFUN1:34;
( [#] REAL = dom (AffineMap n,0 ) & ( for x being Real st x in REAL holds
(AffineMap n,0 ) . x = (n * x) + 0 ) ) by FUNCT_2:def 1, JORDAN16:def 3;
then (AffineMap n,0 ) | REAL is continuous by FDIFF_1:31, FDIFF_1:33;
then A6: (AffineMap n,0 ) | A is continuous by FCONT_1:17;
sin | REAL is continuous by FDIFF_1:33, SIN_COS:73;
then A7: sin | ((AffineMap n,0 ) .: A) is continuous by FCONT_1:17;
then (sin * (AffineMap n,0 )) | A is continuous by A6, FCONT_1:26;
then A8: sin * (AffineMap n,0 ) is_integrable_on A by A2, INTEGRA5:11;
(sin * (AffineMap n,0 )) | A is bounded by A2, A6, A7, FCONT_1:26, INTEGRA5:10;
hence integral (sin * (AffineMap n,0 )),A = (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) . (upper_bound A)) - (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) . (lower_bound A)) by A1, A8, A5, Th8, INTEGRA5:13; :: thesis: verum