let A be closed-interval Subset of REAL ; :: thesis: for r being Real holds integral (r (#) sinh ),A = (r * (cosh . (upper_bound A))) - (r * (cosh . (lower_bound A)))
let r be Real; :: thesis: integral (r (#) sinh ),A = (r * (cosh . (upper_bound A))) - (r * (cosh . (lower_bound A)))
X: A c= dom sinh by D5;
sinh | A is continuous by Lm10;
then A1: ( sinh is_integrable_on A & sinh | A is bounded ) by X, INTEGRA5:10, INTEGRA5:11;
[#] REAL is open Subset of REAL ;
hence integral (r (#) sinh ),A = (r * (cosh . (upper_bound A))) - (r * (cosh . (lower_bound A))) by A1, Th31, Th68, SIN_COS2:35; :: thesis: verum