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)) )
A1: dom (exp_R * (AffineMap r,0 )) = REAL by FUNCT_2:def 1;
( [#] REAL = dom (AffineMap r,0 ) & ( for x being Real st x in REAL holds
(AffineMap r,0 ) . x = (r * x) + 0 ) ) by FUNCT_2:def 1, JORDAN16:def 3;
then (AffineMap r,0 ) | REAL is continuous by FDIFF_1:31, FDIFF_1:33;
then A2: (AffineMap r,0 ) | A is continuous by FCONT_1:17;
assume A3: 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))
then (1 / r) (#) (exp_R * (AffineMap r,0 )) is_differentiable_on REAL by Th5;
then A4: dom (((1 / r) (#) (exp_R * (AffineMap r,0 ))) `| REAL ) = dom (exp_R * (AffineMap r,0 )) by A1, FDIFF_1:def 8;
exp_R | REAL is continuous by FDIFF_1:33, SIN_COS:71;
then A5: exp_R | ((AffineMap r,0 ) .: A) is continuous by FCONT_1:17;
then (exp_R * (AffineMap r,0 )) | A is continuous by A2, FCONT_1:26;
then A6: exp_R * (AffineMap r,0 ) is_integrable_on A by A1, INTEGRA5:11;
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 A3, Th5;
then A7: ((1 / r) (#) (exp_R * (AffineMap r,0 ))) `| REAL = exp_R * (AffineMap r,0 ) by A4, PARTFUN1:34;
(exp_R * (AffineMap r,0 )) | A is bounded by A1, A2, A5, FCONT_1:26, INTEGRA5:10;
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 A3, A6, A7, Th5, INTEGRA5:13; :: thesis: verum