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 )) . (upper_bound A)) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) . (lower_bound A))
let A be closed-interval Subset of REAL ; :: thesis: integral (((#Z n) * sin ) (#) cos ),A = (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) . (upper_bound A)) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) . (lower_bound A))
A1: [#] REAL = dom (((#Z n) * sin ) (#) cos ) by FUNCT_2:def 1;
A2: 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 FUNCT_1:23, SIN_COS:27
.= (((#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;

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;

then ( dom ((#Z n) * sin ) = REAL & ( for x0 being Real st x0 in REAL holds
(#Z n) * sin is_differentiable_in x0 ) ) by FUNCT_2:def 1;
then (#Z n) * sin is_differentiable_on REAL by A1, FDIFF_1:16;
then A3: (((#Z n) * sin ) (#) cos ) | REAL is continuous by A1, FDIFF_1:29, FDIFF_1:33, SIN_COS:72;
then (((#Z n) * sin ) (#) cos ) | A is continuous by FCONT_1:17;
then A4: ((#Z n) * sin ) (#) cos is_integrable_on A by A1, INTEGRA5:11;
(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 A5: ((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) `| REAL = ((#Z n) * sin ) (#) cos by A2, PARTFUN1:34;
(((#Z n) * sin ) (#) cos ) | A is bounded by A1, A3, FCONT_1:17, INTEGRA5:10;
hence integral (((#Z n) * sin ) (#) cos ),A = (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) . (upper_bound A)) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) . (lower_bound A)) by A4, A5, Th3, INTEGRA5:13; :: thesis: verum