let A be non empty closed_interval Subset of REAL; :: thesis: ( exp_R (#) (cos - sin) is_integrable_on A & (exp_R (#) (cos - sin)) | A is bounded )
( dom exp_R = REAL & dom (cos - sin) = REAL ) by FUNCT_2:def 1;
then A c= (dom exp_R) /\ (dom (cos - sin)) ;
then A1: A c= dom (exp_R (#) (cos - sin)) by VALUED_1:def 4;
(exp_R (#) (cos - sin)) | A is continuous ;
hence ( exp_R (#) (cos - sin) is_integrable_on A & (exp_R (#) (cos - sin)) | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11; :: thesis: verum