let r be Real; :: thesis: for A being closed-interval Subset of REAL st r <> 0 holds
integral (exp_R * (AffineMap r,0 )),A = (((1 / r) (#) (exp_R * (AffineMap r,0 ))) . (upper_bound A)) - (((1 / r) (#) (exp_R * (AffineMap r,0 ))) . (lower_bound A))
let A be closed-interval Subset of REAL ; :: thesis: ( r <> 0 implies integral (exp_R * (AffineMap r,0 )),A = (((1 / r) (#) (exp_R * (AffineMap r,0 ))) . (upper_bound A)) - (((1 / r) (#) (exp_R * (AffineMap r,0 ))) . (lower_bound A)) )
assume B: r <> 0 ; :: thesis: integral (exp_R * (AffineMap r,0 )),A = (((1 / r) (#) (exp_R * (AffineMap r,0 ))) . (upper_bound A)) - (((1 / r) (#) (exp_R * (AffineMap r,0 ))) . (lower_bound A))
A1: dom (exp_R * (AffineMap r,0 )) = REAL by FUNCT_2:def 1;
A2: [#] REAL = dom (AffineMap r,0 ) by FUNCT_2:def 1;
A3: for x being Real st x in REAL holds
(AffineMap r,0 ) . x = (r * x) + 0 by JORDAN16:def 3;
(AffineMap r,0 ) | REAL is continuous by A2, A3, FDIFF_1:31, FDIFF_1:33;
then A4: (AffineMap r,0 ) | A is continuous by FCONT_1:17;
B1: exp_R | REAL is continuous by FDIFF_1:33, SIN_COS:71;
exp_R | ((AffineMap r,0 ) .: A) is continuous by B1, FCONT_1:17;
then (exp_R * (AffineMap r,0 )) | A is continuous by A4, FCONT_1:26;
then A5: ( exp_R * (AffineMap r,0 ) is_integrable_on A & (exp_R * (AffineMap r,0 )) | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A6: for x being Real st x in dom (((1 / r) (#) (exp_R * (AffineMap r,0 ))) `| REAL ) holds
(((1 / r) (#) (exp_R * (AffineMap r,0 ))) `| REAL ) . x = (exp_R * (AffineMap r,0 )) . x by B, Th5;
(1 / r) (#) (exp_R * (AffineMap r,0 )) is_differentiable_on REAL by B, Th5;
then dom (((1 / r) (#) (exp_R * (AffineMap r,0 ))) `| REAL ) = dom (exp_R * (AffineMap r,0 )) by A1, FDIFF_1:def 8;
then ((1 / r) (#) (exp_R * (AffineMap r,0 ))) `| REAL = exp_R * (AffineMap r,0 ) by A6, PARTFUN1:34;
hence integral (exp_R * (AffineMap r,0 )),A = (((1 / r) (#) (exp_R * (AffineMap r,0 ))) . (upper_bound A)) - (((1 / r) (#) (exp_R * (AffineMap r,0 ))) . (lower_bound A)) by B, A5, Th5, INTEGRA5:13; :: thesis: verum