let n be Element of NAT ; :: thesis: for A being closed-interval Subset of REAL holds integral (((#Z n) * cos ) (#) sin ),A = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) . (sup A)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) . (inf A))
let A be closed-interval Subset of REAL ; :: thesis: integral (((#Z n) * cos ) (#) sin ),A = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) . (sup A)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) . (inf A))
A1: ( [#] REAL = dom (((#Z n) * cos ) (#) sin ) & dom ((#Z n) * cos ) = REAL & [#] REAL = dom ((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) & [#] REAL = dom ((#Z (n + 1)) * cos ) ) by FUNCT_2:def 1;
A: for x0 being Real holds (#Z n) * cos is_differentiable_in x0

proof

let x0 be Real; :: thesis: (#Z n) * cos is_differentiable_in x0
cos is_differentiable_in x0 by SIN_COS:68;
hence (#Z n) * cos is_differentiable_in x0 by TAYLOR_1:3; :: thesis: verum

end;

A2: (#Z n) * cos is_differentiable_on REAL

proof

for x0 being Real st x0 in REAL holds
(#Z n) * cos is_differentiable_in x0 by A;
hence (#Z n) * cos is_differentiable_on REAL by A1, FDIFF_1:16; :: thesis: verum

end;

(((#Z n) * cos ) (#) sin ) | REAL is continuous by A1, A2, SIN_COS:73, FDIFF_1:29, FDIFF_1:33;
then (((#Z n) * cos ) (#) sin ) | A is continuous by FCONT_1:17;
then A3: ( ((#Z n) * cos ) (#) sin is_integrable_on A & (((#Z n) * cos ) (#) sin ) | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A4: for x being Real st x in dom (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) `| REAL ) holds
(((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) `| REAL ) . x = (((#Z n) * cos ) (#) sin ) . x

proof

let x be Real; :: thesis: ( x in dom (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) `| REAL ) implies (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) `| REAL ) . x = (((#Z n) * cos ) (#) sin ) . x )
assume x in dom (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) `| REAL ) ; :: thesis: (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) `| REAL ) . x = (((#Z n) * cos ) (#) sin ) . x
(((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) `| REAL ) . x = ((cos . x) #Z n) * (sin . x) by Th4
.= ((#Z n) . (cos . x)) * (sin . x) by TAYLOR_1:def 1
.= (((#Z n) * cos ) . x) * (sin . x) by SIN_COS:27, FUNCT_1:23
.= (((#Z n) * cos ) (#) sin ) . x by A1, VALUED_1:def 4 ;
hence (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) `| REAL ) . x = (((#Z n) * cos ) (#) sin ) . x ; :: thesis: verum

end;

(- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos ) is_differentiable_on REAL by Th4;
then dom (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) `| REAL ) = dom (((#Z n) * cos ) (#) sin ) by A1, FDIFF_1:def 8;
then ((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) `| REAL = ((#Z n) * cos ) (#) sin by A4, PARTFUN1:34;
hence integral (((#Z n) * cos ) (#) sin ),A = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) . (sup A)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos )) . (inf A)) by A3, Th4, INTEGRA5:13; :: thesis: verum