let A be non empty closed_interval Subset of REAL; :: thesis: integral ((exp_R (#) (cos - sin)),A) = ((exp_R (#) cos) . (upper_bound A)) - ((exp_R (#) cos) . (lower_bound A))
A1: ( dom (exp_R (#) cos) = REAL & [#] REAL is open Subset of REAL ) by FUNCT_2:def 1;
A2: dom (cos - sin) = REAL by FUNCT_2:def 1;
A3: for x being Element of REAL st x in dom ((exp_R (#) cos) `| REAL) holds
((exp_R (#) cos) `| REAL) . x = (exp_R (#) (cos - sin)) . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((exp_R (#) cos) `| REAL) implies ((exp_R (#) cos) `| REAL) . x = (exp_R (#) (cos - sin)) . x )
assume x in dom ((exp_R (#) cos) `| REAL) ; :: thesis: ((exp_R (#) cos) `| REAL) . x = (exp_R (#) (cos - sin)) . x
(exp_R (#) (cos - sin)) . x = (exp_R . x) * ((cos - sin) . x) by VALUED_1:5
.= (exp_R . x) * ((cos . x) - (sin . x)) by A2, VALUED_1:13 ;
hence ((exp_R (#) cos) `| REAL) . x = (exp_R (#) (cos - sin)) . x by A1, FDIFF_7:45; :: thesis: verum

end;

A4: ( exp_R (#) (cos - sin) is_integrable_on A & (exp_R (#) (cos - sin)) | A is bounded ) by Lm22;
A5: dom (exp_R (#) (cos - sin)) = (dom exp_R) /\ (dom (cos - sin)) by VALUED_1:def 4
.= REAL /\ (dom (cos - sin)) by SIN_COS:47
.= REAL by A2 ;
exp_R (#) cos is_differentiable_on REAL by A1, FDIFF_7:45;
then dom ((exp_R (#) cos) `| REAL) = dom (exp_R (#) (cos - sin)) by A5, FDIFF_1:def 7;
then (exp_R (#) cos) `| REAL = exp_R (#) (cos - sin) by A3, PARTFUN1:5;
hence integral ((exp_R (#) (cos - sin)),A) = ((exp_R (#) cos) . (upper_bound A)) - ((exp_R (#) cos) . (lower_bound A)) by A4, A1, FDIFF_7:45, INTEGRA5:13; :: thesis: verum