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 & Z = dom f & f = (exp_R * cos) (#) sin 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 & Z = dom f & f = (exp_R * cos) (#) sin 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 & Z = dom f & f = (exp_R * cos) (#) sin implies integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A)) )
assume A1: ( A c= Z & Z = dom f & f = (exp_R * cos) (#) sin ) ; :: thesis: integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A))
then Z = (dom (exp_R * cos)) /\ (dom sin) by VALUED_1:def 4;
then A2: Z c= dom (exp_R * cos) by XBOOLE_1:18;
then A3: exp_R * cos is_differentiable_on Z by FDIFF_7:36;
sin is_differentiable_on Z by FDIFF_1:26, SIN_COS:68;
then f | Z is continuous by A1, A3, FDIFF_1:21, FDIFF_1:25;
then f | A is continuous by A1, FCONT_1:16;
then A4: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A5: Z c= dom (- (exp_R * cos)) by A2, VALUED_1:8;
then A6: (- 1) (#) (exp_R * cos) is_differentiable_on Z by A3, FDIFF_1:20;
A7: for x being Real st x in Z holds
((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x)

proof

let x be Real; :: thesis: ( x in Z implies ((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x) )
assume A8: x in Z ; :: thesis: ((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x)
A9: cos is_differentiable_in x by SIN_COS:63;
A10: exp_R is_differentiable_in cos . x by SIN_COS:65;
A11: exp_R * cos is_differentiable_in x by A3, A8, FDIFF_1:9;
((- (exp_R * cos)) `| Z) . x = diff ((- (exp_R * cos)),x) by A6, A8, FDIFF_1:def 7
.= (- 1) * (diff ((exp_R * cos),x)) by A11, FDIFF_1:15
.= (- 1) * ((diff (exp_R,(cos . x))) * (diff (cos,x))) by A9, A10, FDIFF_2:13
.= (- 1) * ((diff (exp_R,(cos . x))) * (- (sin . x))) by SIN_COS:63
.= (- 1) * ((exp_R . (cos . x)) * (- (sin . x))) by SIN_COS:65
.= (exp_R . (cos . x)) * (sin . x) ;
hence ((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x) ; :: thesis: verum

end;

A12: for x being Real st x in Z holds
f . x = (exp_R . (cos . x)) * (sin . x)

proof

let x be Real; :: thesis: ( x in Z implies f . x = (exp_R . (cos . x)) * (sin . x) )
assume A13: x in Z ; :: thesis: f . x = (exp_R . (cos . x)) * (sin . x)
then ((exp_R * cos) (#) sin) . x = ((exp_R * cos) . x) * (sin . x) by A1, VALUED_1:def 4
.= (exp_R . (cos . x)) * (sin . x) by A2, A13, FUNCT_1:12 ;
hence f . x = (exp_R . (cos . x)) * (sin . x) by A1; :: thesis: verum

end;

A14: 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 A15: x in Z by A6, FDIFF_1:def 7;
then ((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x) by A7
.= f . x by A15, A12 ;
hence ((- (exp_R * cos)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((- (exp_R * cos)) `| Z) = dom f by A1, A6, FDIFF_1:def 7;
then (- (exp_R * cos)) `| Z = f by A14, PARTFUN1:5;
hence integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A)) by A1, A4, A3, A5, FDIFF_1:20, INTEGRA5:13; :: thesis: verum