let A be 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