let A be non empty closed_interval Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = exp_R (#) (cos - sin) & Z c= dom (exp_R (#) cos) & Z = dom f & f | A is continuous holds
integral (f,A) = ((exp_R (#) cos) . (upper_bound A)) - ((exp_R (#) cos) . (lower_bound A))
let f be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & f = exp_R (#) (cos - sin) & Z c= dom (exp_R (#) cos) & Z = dom f & f | A is continuous holds
integral (f,A) = ((exp_R (#) cos) . (upper_bound A)) - ((exp_R (#) cos) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & f = exp_R (#) (cos - sin) & Z c= dom (exp_R (#) cos) & Z = dom f & f | A is continuous implies integral (f,A) = ((exp_R (#) cos) . (upper_bound A)) - ((exp_R (#) cos) . (lower_bound A)) )
assume A1: ( A c= Z & f = exp_R (#) (cos - sin) & Z c= dom (exp_R (#) cos) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((exp_R (#) cos) . (upper_bound A)) - ((exp_R (#) cos) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: exp_R (#) cos is_differentiable_on Z by A1, FDIFF_7:45;
dom f = (dom exp_R) /\ (dom (cos - sin)) by A1, VALUED_1:def 4;
then A4: Z c= dom (cos - sin) by A1, XBOOLE_1:18;
A5: for x being Real st x in Z holds
f . x = (exp_R . x) * ((cos . x) - (sin . x))

proof

let x be Real; :: thesis: ( x in Z implies f . x = (exp_R . x) * ((cos . x) - (sin . x)) )
assume A6: x in Z ; :: thesis: f . x = (exp_R . x) * ((cos . x) - (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 A4, A6, VALUED_1:13 ;
hence f . x = (exp_R . x) * ((cos . x) - (sin . x)) by A1; :: thesis: verum

end;

A7: for x being Element of REAL st x in dom ((exp_R (#) cos) `| Z) holds
((exp_R (#) cos) `| Z) . x = f . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((exp_R (#) cos) `| Z) implies ((exp_R (#) cos) `| Z) . x = f . x )
assume x in dom ((exp_R (#) cos) `| Z) ; :: thesis: ((exp_R (#) cos) `| Z) . x = f . x
then A8: x in Z by A3, FDIFF_1:def 7;
then ((exp_R (#) cos) `| Z) . x = (exp_R . x) * ((cos . x) - (sin . x)) by A1, FDIFF_7:45
.= f . x by A5, A8 ;
hence ((exp_R (#) cos) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((exp_R (#) cos) `| Z) = dom f by A1, A3, FDIFF_1:def 7;
then (exp_R (#) cos) `| Z = f by A7, PARTFUN1:5;
hence integral (f,A) = ((exp_R (#) cos) . (upper_bound A)) - ((exp_R (#) cos) . (lower_bound A)) by A1, A2, FDIFF_7:45, INTEGRA5:13; :: thesis: verum