let A be non empty closed_interval Subset of REAL; :: thesis: for r being Real holds integral ((r (#) cos),A) = (r * (sin . (upper_bound A))) - (r * (sin . (lower_bound A)))
let r be Real; :: thesis: integral ((r (#) cos),A) = (r * (sin . (upper_bound A))) - (r * (sin . (lower_bound A)))
A1: [#] REAL is open Subset of REAL ;
( cos is_integrable_on A & cos | A is bounded ) by Lm11;
hence integral ((r (#) cos),A) = (r * (sin . (upper_bound A))) - (r * (sin . (lower_bound A))) by A1, Th27, Th68, SIN_COS:68; :: thesis: verum