let A be closed-interval Subset of REAL ; :: thesis: integral (exp_R * (AffineMap 2,0 )),A = (((1 / 2) (#) (exp_R * (AffineMap 2,0 ))) . (sup A)) - (((1 / 2) (#) (exp_R * (AffineMap 2,0 ))) . (inf A))
A1: dom (exp_R * (AffineMap 2,0 )) = REAL by FUNCT_2:def 1;
A2: [#] REAL = dom (AffineMap 2,0 ) by FUNCT_2:def 1;
A3: for x being Real st x in REAL holds
(AffineMap 2,0 ) . x = (2 * x) + 0 by JORDAN16:def 3;
(AffineMap 2,0 ) | REAL is continuous by A2, A3, FDIFF_1:31, FDIFF_1:33;
then A4: (AffineMap 2,0 ) | A is continuous by FCONT_1:17;
B1: exp_R | REAL is continuous by SIN_COS:71, FDIFF_1:33;
exp_R | ((AffineMap 2,0 ) .: A) is continuous by B1, FCONT_1:17;
then (exp_R * (AffineMap 2,0 )) | A is continuous by A4, FCONT_1:26;
then A5: ( exp_R * (AffineMap 2,0 ) is_integrable_on A & (exp_R * (AffineMap 2,0 )) | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A6: for x being Real st x in dom (((1 / 2) (#) (exp_R * (AffineMap 2,0 ))) `| REAL ) holds
(((1 / 2) (#) (exp_R * (AffineMap 2,0 ))) `| REAL ) . x = (exp_R * (AffineMap 2,0 )) . x by Th3;
dom (((1 / 2) (#) (exp_R * (AffineMap 2,0 ))) `| REAL ) = dom (exp_R * (AffineMap 2,0 )) by A1, Th3, FDIFF_1:def 8;
then ((1 / 2) (#) (exp_R * (AffineMap 2,0 ))) `| REAL = exp_R * (AffineMap 2,0 ) by A6, PARTFUN1:34;
hence integral (exp_R * (AffineMap 2,0 )),A = (((1 / 2) (#) (exp_R * (AffineMap 2,0 ))) . (sup A)) - (((1 / 2) (#) (exp_R * (AffineMap 2,0 ))) . (inf A)) by A5, Th3, INTEGRA5:13; :: thesis: verum