let A be closed-interval Subset of REAL ; :: thesis: for r being Real holds integral (r (#) exp_R ),A = (r * (exp_R . (upper_bound A))) - (r * (exp_R . (lower_bound A)))
let r be Real; :: thesis: integral (r (#) exp_R ),A = (r * (exp_R . (upper_bound A))) - (r * (exp_R . (lower_bound A)))
X: A c= dom exp_R by D4;
exp_R | A is continuous ;
then A1: ( exp_R is_integrable_on A & exp_R | A is bounded ) by X, INTEGRA5:10, INTEGRA5:11;
[#] REAL is open Subset of REAL ;
hence integral (r (#) exp_R ),A = (r * (exp_R . (upper_bound A))) - (r * (exp_R . (lower_bound A))) by A1, Th32, Th68, SIN_COS:71; :: thesis: verum