let A be closed-interval Subset of REAL ; :: thesis: integral (sin * (AffineMap 2,0 )),A = (((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) . (sup A)) - (((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) . (inf A))
A1: [#] REAL = dom (sin * (AffineMap 2,0 )) by FUNCT_2:def 1;
A2: [#] REAL = dom (AffineMap 2,0 ) by FUNCT_2:def 1;
for x being Real st x in REAL holds
(AffineMap 2,0 ) . x = (2 * x) + 0 by JORDAN16:def 3;
then (AffineMap 2,0 ) | REAL is continuous by A2, FDIFF_1:31, FDIFF_1:33;
then A3: (AffineMap 2,0 ) | A is continuous by FCONT_1:17;
B1: sin | REAL is continuous by SIN_COS:73, FDIFF_1:33;
sin | ((AffineMap 2,0 ) .: A) is continuous by B1, FCONT_1:17;
then (sin * (AffineMap 2,0 )) | A is continuous by A3, FCONT_1:26;
then A4: ( sin * (AffineMap 2,0 ) is_integrable_on A & (sin * (AffineMap 2,0 )) | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A5: for x being Real st x in REAL holds
(AffineMap 2,0 ) . x = 2 * x

proof

let x be Real; :: thesis: ( x in REAL implies (AffineMap 2,0 ) . x = 2 * x )
assume x in REAL ; :: thesis: (AffineMap 2,0 ) . x = 2 * x
(AffineMap 2,0 ) . x = (2 * x) + 0 by JORDAN16:def 3
.= 2 * x ;
hence (AffineMap 2,0 ) . x = 2 * x ; :: thesis: verum

end;

A6: (- (1 / 2)) (#) (cos * (AffineMap 2,0 )) is_differentiable_on REAL by Lm3, A5, FDIFF_6:38;
A7: for x being Real st x in dom (((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) `| REAL ) holds
(((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) `| REAL ) . x = (sin * (AffineMap 2,0 )) . x

proof

let x be Real; :: thesis: ( x in dom (((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) `| REAL ) implies (((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) `| REAL ) . x = (sin * (AffineMap 2,0 )) . x )
assume x in dom (((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) `| REAL ) ; :: thesis: (((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) `| REAL ) . x = (sin * (AffineMap 2,0 )) . x
(((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) `| REAL ) . x = sin (2 * x) by Lm3, A5, FDIFF_6:38
.= sin . ((AffineMap 2,0 ) . x) by A5
.= (sin * (AffineMap 2,0 )) . x by A1, FUNCT_1:22 ;
hence (((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) `| REAL ) . x = (sin * (AffineMap 2,0 )) . x ; :: thesis: verum

end;

dom (((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) `| REAL ) = dom (sin * (AffineMap 2,0 )) by A1, A6, FDIFF_1:def 8;
then ((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) `| REAL = sin * (AffineMap 2,0 ) by A7, PARTFUN1:34;
hence integral (sin * (AffineMap 2,0 )),A = (((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) . (sup A)) - (((- (1 / 2)) (#) (cos * (AffineMap 2,0 ))) . (inf A)) by Lm3, A4, A5, FDIFF_6:38, INTEGRA5:13; :: thesis: verum