let A be non empty 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)))
( exp_R | A is bounded & [#] REAL is open Subset of REAL ) by Lm8, INTEGRA5:10;
hence integral ((r (#) exp_R),A) = (r * (exp_R . (upper_bound A))) - (r * (exp_R . (lower_bound A))) by Lm8, Th32, Th68, INTEGRA5:11, SIN_COS:66; :: thesis: verum