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

proof

let x0 be Real; :: thesis: (#Z n) * sin is_differentiable_in x0
sin is_differentiable_in x0 by SIN_COS:69;
hence (#Z n) * sin is_differentiable_in x0 by TAYLOR_1:3; :: thesis: verum

end;

A2: (#Z n) * sin is_differentiable_on REAL

proof

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

end;

(((#Z n) * sin ) (#) cos ) | REAL is continuous by A1, A2, SIN_COS:72, FDIFF_1:29, FDIFF_1:33;
then (((#Z n) * sin ) (#) cos ) | A is continuous by FCONT_1:17;
then A3: ( ((#Z n) * sin ) (#) cos is_integrable_on A & (((#Z n) * sin ) (#) cos ) | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A4: for x being Real st x in dom (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) `| REAL ) holds
(((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) `| REAL ) . x = (((#Z n) * sin ) (#) cos ) . x

proof

let x be Real; :: thesis: ( x in dom (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) `| REAL ) implies (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) `| REAL ) . x = (((#Z n) * sin ) (#) cos ) . x )
assume x in dom (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) `| REAL ) ; :: thesis: (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) `| REAL ) . x = (((#Z n) * sin ) (#) cos ) . x
(((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) `| REAL ) . x = ((sin . x) #Z n) * (cos . x) by Th3
.= ((#Z n) . (sin . x)) * (cos . x) by TAYLOR_1:def 1
.= (((#Z n) * sin ) . x) * (cos . x) by SIN_COS:27, FUNCT_1:23
.= (((#Z n) * sin ) (#) cos ) . x by A1, VALUED_1:def 4 ;
hence (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) `| REAL ) . x = (((#Z n) * sin ) (#) cos ) . x ; :: thesis: verum

end;

(1 / (n + 1)) (#) ((#Z (n + 1)) * sin ) is_differentiable_on REAL by Th3;
then dom (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) `| REAL ) = dom (((#Z n) * sin ) (#) cos ) by A1, FDIFF_1:def 8;
then ((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) `| REAL = ((#Z n) * sin ) (#) cos by A4, PARTFUN1:34;
hence integral (((#Z n) * sin ) (#) cos ),A = (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) . (sup A)) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) . (inf A)) by A3, Th3, INTEGRA5:13; :: thesis: verum