let n be Element of NAT ; :: thesis: ( n <> 0 implies ( (- (1 / n)) (#) (cos * (AffineMap n,0 )) is_differentiable_on REAL & ( for x being Real holds (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = sin (n * x) ) ) )
assume A1: n <> 0 ; :: thesis: ( (- (1 / n)) (#) (cos * (AffineMap n,0 )) is_differentiable_on REAL & ( for x being Real holds (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = sin (n * x) ) )
A: [#] REAL = dom ((- (1 / n)) (#) (cos * (AffineMap n,0 ))) by FUNCT_2:def 1;
A2: [#] REAL = dom (cos * (AffineMap n,0 )) by FUNCT_2:def 1;
A3: for x being Real st x in REAL holds
(AffineMap n,0 ) . x = (n * x) + 0 by JORDAN16:def 3;
then A4: ( cos * (AffineMap n,0 ) is_differentiable_on REAL & ( for x being Real st x in REAL holds
((cos * (AffineMap n,0 )) `| REAL ) . x = - (n * (sin . ((n * x) + 0 ))) ) ) by A2, FDIFF_4:38;
for x being Real st x in REAL holds
(((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = sin (n * x)

proof

let x be Real; :: thesis: ( x in REAL implies (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = sin (n * x) )
assume x in REAL ; :: thesis: (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = sin (n * x)
(((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = (- (1 / n)) * (diff (cos * (AffineMap n,0 )),x) by A, A4, FDIFF_1:28
.= (- (1 / n)) * (((cos * (AffineMap n,0 )) `| REAL ) . x) by A4, FDIFF_1:def 8
.= (- (1 / n)) * (- (n * (sin . ((n * x) + 0 )))) by A2, A3, FDIFF_4:38
.= ((1 / n) * n) * (sin . ((n * x) + 0 ))
.= (n / n) * (sin . ((n * x) + 0 )) by XCMPLX_1:100
.= 1 * (sin . ((n * x) + 0 )) by A1, XCMPLX_1:60
.= sin (n * x) ;
hence (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = sin (n * x) ; :: thesis: verum

end;

hence ( (- (1 / n)) (#) (cos * (AffineMap n,0 )) is_differentiable_on REAL & ( for x being Real holds (((- (1 / n)) (#) (cos * (AffineMap n,0 ))) `| REAL ) . x = sin (n * x) ) ) by A, A4, FDIFF_1:28; :: thesis: verum