let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * a)) (#) (sin * f)) & ( for x being Real st x in Z holds
f . x = (2 * a) * x ) & a <> 0 holds
( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) )
let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * a)) (#) (sin * f)) & ( for x being Real st x in Z holds
f . x = (2 * a) * x ) & a <> 0 holds
( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) )
let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((1 / (4 * a)) (#) (sin * f)) & ( for x being Real st x in Z holds
f . x = (2 * a) * x ) & a <> 0 implies ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) ) )
assume that
A1: Z c= dom ((1 / (4 * a)) (#) (sin * f)) and
A2: for x being Real st x in Z holds
f . x = (2 * a) * x and
A3: a <> 0 ; :: thesis: ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) )
A4: ( Z c= dom (sin * f) & ( for x being Real st x in Z holds
f . x = ((2 * a) * x) + 0 ) ) by A1, A2, VALUED_1:def 5;
then A5: sin * f is_differentiable_on Z by FDIFF_4:37;
for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x))

proof

let x be Real; :: thesis: ( x in Z implies (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) )
assume A6: x in Z ; :: thesis: (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x))
then (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / (4 * a)) * (diff ((sin * f),x)) by A1, A5, FDIFF_1:20
.= (1 / (4 * a)) * (((sin * f) `| Z) . x) by A5, A6, FDIFF_1:def 7
.= (1 / (4 * a)) * ((2 * a) * (cos . (((2 * a) * x) + 0))) by A4, A6, FDIFF_4:37
.= ((1 / (4 * a)) * (2 * a)) * (cos . (((2 * a) * x) + 0))
.= (((1 / 4) * (1 / a)) * (2 * a)) * (cos . ((2 * a) * x)) by XCMPLX_1:102
.= (((1 / 4) * 2) * ((1 / a) * a)) * (cos . ((2 * a) * x))
.= ((1 / 2) * 1) * (cos . ((2 * a) * x)) by A3, XCMPLX_1:106
.= (1 / 2) * (cos ((2 * a) * x)) by SIN_COS:def 19 ;
hence (((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ; :: thesis: verum

end;

hence ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) ) by A1, A5, FDIFF_1:20; :: thesis: verum