let Z be open Subset of REAL; ( Z c= dom ((id Z) (#) sin) implies ( (id Z) (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) sin) `| Z) . x = (sin . x) + (x * (cos . x)) ) ) )
A1:
for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0
by FUNCT_1:18;
assume A2:
Z c= dom ((id Z) (#) sin)
; ( (id Z) (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) sin) `| Z) . x = (sin . x) + (x * (cos . x)) ) )
then
Z c= (dom (id Z)) /\ (dom sin)
by VALUED_1:def 4;
then A3:
Z c= dom (id Z)
by XBOOLE_1:18;
then A4:
id Z is_differentiable_on Z
by A1, FDIFF_1:23;
A5:
sin is_differentiable_on Z
by FDIFF_1:26, SIN_COS:68;
now for x being Real st x in Z holds
(((id Z) (#) sin) `| Z) . x = (sin . x) + (x * (cos . x))let x be
Real;
( x in Z implies (((id Z) (#) sin) `| Z) . x = (sin . x) + (x * (cos . x)) )assume A6:
x in Z
;
(((id Z) (#) sin) `| Z) . x = (sin . x) + (x * (cos . x))hence (((id Z) (#) sin) `| Z) . x =
((sin . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff (sin,x)))
by A2, A4, A5, FDIFF_1:21
.=
((sin . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff (sin,x)))
by A4, A6, FDIFF_1:def 7
.=
((sin . x) * 1) + (((id Z) . x) * (diff (sin,x)))
by A3, A1, A6, FDIFF_1:23
.=
((sin . x) * 1) + (((id Z) . x) * (cos . x))
by SIN_COS:64
.=
(sin . x) + (x * (cos . x))
by A6, FUNCT_1:18
;
verum end;
hence
( (id Z) (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) sin) `| Z) . x = (sin . x) + (x * (cos . x)) ) )
by A2, A4, A5, FDIFF_1:21; verum