let n be Element of NAT ; :: thesis: for Z being open Subset of REAL st Z c= dom ((1 / n) (#) ((#Z n) * sin)) & n > 0 holds
( (1 / n) (#) ((#Z n) * sin) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * sin)) `| Z) . x = ((sin . x) #Z (n - 1)) * (cos . x) ) )
let Z be open Subset of REAL; :: thesis: ( Z c= dom ((1 / n) (#) ((#Z n) * sin)) & n > 0 implies ( (1 / n) (#) ((#Z n) * sin) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * sin)) `| Z) . x = ((sin . x) #Z (n - 1)) * (cos . x) ) ) )
assume that
A1: Z c= dom ((1 / n) (#) ((#Z n) * sin)) and
A2: n > 0 ; :: thesis: ( (1 / n) (#) ((#Z n) * sin) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * sin)) `| Z) . x = ((sin . x) #Z (n - 1)) * (cos . x) ) )

A3: now :: thesis: for x being Real st x in Z holds
(#Z n) * sin is_differentiable_in x

let x be Real; :: thesis: ( x in Z implies (#Z n) * sin is_differentiable_in x )
assume x in Z ; :: thesis: (#Z n) * sin is_differentiable_in x
sin is_differentiable_in x by SIN_COS:64;
hence (#Z n) * sin is_differentiable_in x by TAYLOR_1:3; :: thesis: verum

end;

Z c= dom ((#Z n) * sin) by A1, VALUED_1:def 5;
then A4: (#Z n) * sin is_differentiable_on Z by A3, FDIFF_1:9;
for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * sin)) `| Z) . x = ((sin . x) #Z (n - 1)) * (cos . x)

proof

let x be Real; :: thesis: ( x in Z implies (((1 / n) (#) ((#Z n) * sin)) `| Z) . x = ((sin . x) #Z (n - 1)) * (cos . x) )
A5: sin is_differentiable_in x by SIN_COS:64;
assume x in Z ; :: thesis: (((1 / n) (#) ((#Z n) * sin)) `| Z) . x = ((sin . x) #Z (n - 1)) * (cos . x)
then (((1 / n) (#) ((#Z n) * sin)) `| Z) . x = (1 / n) * (diff (((#Z n) * sin),x)) by A1, A4, FDIFF_1:20
.= (1 / n) * ((n * ((sin . x) #Z (n - 1))) * (diff (sin,x))) by A5, TAYLOR_1:3
.= (1 / n) * ((n * ((sin . x) #Z (n - 1))) * (cos . x)) by SIN_COS:64
.= (((1 / n) * n) * ((sin . x) #Z (n - 1))) * (cos . x)
.= (((n ") * n) * ((sin . x) #Z (n - 1))) * (cos . x) by XCMPLX_1:215
.= (1 * ((sin . x) #Z (n - 1))) * (cos . x) by A2, XCMPLX_0:def 7
.= ((sin . x) #Z (n - 1)) * (cos . x) ;
hence (((1 / n) (#) ((#Z n) * sin)) `| Z) . x = ((sin . x) #Z (n - 1)) * (cos . x) ; :: thesis: verum

end;

hence ( (1 / n) (#) ((#Z n) * sin) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * sin)) `| Z) . x = ((sin . x) #Z (n - 1)) * (cos . x) ) ) by A1, A4, FDIFF_1:20; :: thesis: verum