let n be Element of NAT ; :: thesis: ( n <> 0 implies ( (1 / n) (#) (sin * (AffineMap (n,0))) is_differentiable_on REAL & ( for x being Real holds (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = cos (n * x) ) ) )
assume A1: n <> 0 ; :: thesis: ( (1 / n) (#) (sin * (AffineMap (n,0))) is_differentiable_on REAL & ( for x being Real holds (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = cos (n * x) ) )
A2: [#] REAL = dom ((1 / n) (#) (sin * (AffineMap (n,0)))) by FUNCT_2:def 1;
A3: ( [#] REAL = dom (sin * (AffineMap (n,0))) & ( for x being Real st x in REAL holds
(AffineMap (n,0)) . x = (n * x) + 0 ) ) by FCONT_1:def 4, FUNCT_2:def 1;
then A4: sin * (AffineMap (n,0)) is_differentiable_on REAL by FDIFF_4:37;
hence (1 / n) (#) (sin * (AffineMap (n,0))) is_differentiable_on REAL by A2, FDIFF_1:20; :: thesis: for x being Real holds (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = cos (n * x)
A5: for x being Real st x in REAL holds
(((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = cos (n * x)

proof

let x be Real; :: thesis: ( x in REAL implies (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = cos (n * x) )
assume A6: x in REAL ; :: thesis: (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = cos (n * x)
(((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = (1 / n) * (diff ((sin * (AffineMap (n,0))),x)) by A2, A4, FDIFF_1:20, A6
.= (1 / n) * (((sin * (AffineMap (n,0))) `| REAL) . x) by A4, FDIFF_1:def 7, A6
.= (1 / n) * (n * (cos . ((n * x) + 0))) by A3, FDIFF_4:37, A6
.= (n * (1 / n)) * (cos . ((n * x) + 0))
.= (n / n) * (cos . ((n * x) + 0)) by XCMPLX_1:99
.= 1 * (cos . ((n * x) + 0)) by A1, XCMPLX_1:60
.= cos (n * x) ;
hence (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = cos (n * x) ; :: thesis: verum

end;

let x be Real; :: thesis: (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = cos (n * x)
x in REAL by XREAL_0:def 1;
hence (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = cos (n * x) by A5; :: thesis: verum