let Z be open Subset of REAL; ( Z c= dom ((sin + cos) / exp_R) implies ( (sin + cos) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((sin + cos) / exp_R) `| Z) . x = - ((2 * (sin . x)) / (exp_R . x)) ) ) )
assume
Z c= dom ((sin + cos) / exp_R)
; ( (sin + cos) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((sin + cos) / exp_R) `| Z) . x = - ((2 * (sin . x)) / (exp_R . x)) ) )
then
Z c= (dom (sin + cos)) /\ ((dom exp_R) \ (exp_R " {0}))
by RFUNCT_1:def 1;
then A1:
Z c= dom (sin + cos)
by XBOOLE_1:18;
then A2:
sin + cos is_differentiable_on Z
by Th38;
A3:
( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
exp_R . x <> 0 ) )
by FDIFF_1:26, SIN_COS:54, TAYLOR_1:16;
then A4:
(sin + cos) / exp_R is_differentiable_on Z
by A2, FDIFF_2:21;
for x being Real st x in Z holds
(((sin + cos) / exp_R) `| Z) . x = - ((2 * (sin . x)) / (exp_R . x))
proof
let x be
Real;
( x in Z implies (((sin + cos) / exp_R) `| Z) . x = - ((2 * (sin . x)) / (exp_R . x)) )
reconsider xx =
x as
Element of
REAL by XREAL_0:def 1;
A5:
(sin + cos) . xx = (sin . xx) + (cos . xx)
by VALUED_1:1;
A6:
exp_R . x <> 0
by SIN_COS:54;
assume A7:
x in Z
;
(((sin + cos) / exp_R) `| Z) . x = - ((2 * (sin . x)) / (exp_R . x))
then
(
exp_R is_differentiable_in x &
sin + cos is_differentiable_in x )
by A2, FDIFF_1:9, SIN_COS:65;
then diff (
((sin + cos) / exp_R),
x) =
(((diff ((sin + cos),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((sin + cos) . x))) / ((exp_R . x) ^2)
by A6, FDIFF_2:14
.=
(((((sin + cos) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((sin + cos) . x))) / ((exp_R . x) ^2)
by A2, A7, FDIFF_1:def 7
.=
((((cos . x) - (sin . x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((sin + cos) . x))) / ((exp_R . x) ^2)
by A1, A7, Th38
.=
((((cos . x) - (sin . x)) * (exp_R . x)) - ((exp_R . x) * ((sin . x) + (cos . x)))) / ((exp_R . x) ^2)
by A5, SIN_COS:65
.=
((- (2 * (sin . x))) * (exp_R . x)) / ((exp_R . x) * (exp_R . x))
.=
(- (2 * (sin . x))) * ((exp_R . x) / ((exp_R . x) * (exp_R . x)))
by XCMPLX_1:74
.=
(- (2 * (sin . x))) * (((exp_R . x) / (exp_R . x)) / (exp_R . x))
by XCMPLX_1:78
.=
(- (2 * (sin . x))) * (1 / (exp_R . x))
by A6, XCMPLX_1:60
.=
(- (2 * (sin . x))) / (exp_R . x)
by XCMPLX_1:99
.=
- ((2 * (sin . x)) / (exp_R . x))
by XCMPLX_1:187
;
hence
(((sin + cos) / exp_R) `| Z) . x = - ((2 * (sin . x)) / (exp_R . x))
by A4, A7, FDIFF_1:def 7;
verum
end;
hence
( (sin + cos) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((sin + cos) / exp_R) `| Z) . x = - ((2 * (sin . x)) / (exp_R . x)) ) )
by A2, A3, FDIFF_2:21; verum