let Z be open Subset of REAL; ( Z c= dom ((1 / 2) (#) ((sin - cos) / exp_R)) implies ( (1 / 2) (#) ((sin - cos) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((sin - cos) / exp_R)) `| Z) . x = (cos . x) / (exp_R . x) ) ) )
assume A1:
Z c= dom ((1 / 2) (#) ((sin - cos) / exp_R))
; ( (1 / 2) (#) ((sin - cos) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((sin - cos) / exp_R)) `| Z) . x = (cos . x) / (exp_R . x) ) )
then A2:
Z c= dom ((sin - cos) / exp_R)
by VALUED_1:def 5;
then
Z c= (dom (sin - cos)) /\ ((dom exp_R) \ (exp_R " {0}))
by RFUNCT_1:def 1;
then A3:
Z c= dom (sin - cos)
by XBOOLE_1:18;
then A4:
( sin - cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin - cos) `| Z) . x = (cos . x) + (sin . x) ) )
by FDIFF_7:39;
A5:
(sin - cos) / exp_R is_differentiable_on Z
by A2, FDIFF_7:43;
then A6:
(1 / 2) (#) ((sin - cos) / exp_R) is_differentiable_on Z
by FDIFF_2:19;
for x being Real st x in Z holds
(((1 / 2) (#) ((sin - cos) / exp_R)) `| Z) . x = (cos . x) / (exp_R . x)
proof
let x be
Real;
( x in Z implies (((1 / 2) (#) ((sin - cos) / exp_R)) `| Z) . x = (cos . x) / (exp_R . x) )
assume A7:
x in Z
;
(((1 / 2) (#) ((sin - cos) / exp_R)) `| Z) . x = (cos . x) / (exp_R . x)
A8:
exp_R is_differentiable_in x
by SIN_COS:65;
A9:
sin - cos is_differentiable_in x
by A4, A7, FDIFF_1:9;
A10:
(sin - cos) . x = (sin . x) - (cos . x)
by A3, A7, VALUED_1:13;
A11:
exp_R . x <> 0
by SIN_COS:54;
(((1 / 2) (#) ((sin - cos) / exp_R)) `| Z) . x =
(1 / 2) * (diff (((sin - cos) / exp_R),x))
by A1, A5, A7, FDIFF_1:20
.=
(1 / 2) * ((((diff ((sin - cos),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((sin - cos) . x))) / ((exp_R . x) ^2))
by A8, A9, A11, FDIFF_2:14
.=
(1 / 2) * ((((((sin - cos) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((sin - cos) . x))) / ((exp_R . x) ^2))
by A4, A7, FDIFF_1:def 7
.=
(1 / 2) * (((((cos . x) + (sin . x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((sin - cos) . x))) / ((exp_R . x) ^2))
by A3, A7, FDIFF_7:39
.=
(1 / 2) * (((((cos . x) + (sin . x)) * (exp_R . x)) - ((exp_R . x) * ((sin . x) - (cos . x)))) / ((exp_R . x) ^2))
by A10, SIN_COS:65
.=
(1 / 2) * ((2 * (cos . x)) * ((exp_R . x) / ((exp_R . x) * (exp_R . x))))
.=
(1 / 2) * ((2 * (cos . x)) * (((exp_R . x) / (exp_R . x)) / (exp_R . x)))
by XCMPLX_1:78
.=
(1 / 2) * ((2 * (cos . x)) * (1 / (exp_R . x)))
by A11, XCMPLX_1:60
.=
(cos . x) / (exp_R . x)
;
hence
(((1 / 2) (#) ((sin - cos) / exp_R)) `| Z) . x = (cos . x) / (exp_R . x)
;
verum
end;
hence
( (1 / 2) (#) ((sin - cos) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((sin - cos) / exp_R)) `| Z) . x = (cos . x) / (exp_R . x) ) )
by A6; verum