let Z be open Subset of REAL; :: thesis: ( 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 * (cos . x)) / (exp_R . x) ) ) )
assume Z c= dom ((sin - cos) / exp_R) ; :: thesis: ( (sin - cos) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((sin - cos) / exp_R) `| Z) . x = (2 * (cos . 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 Th39;
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 * (cos . x)) / (exp_R . x)

proof

let x be Real; :: thesis: ( x in Z implies (((sin - cos) / exp_R) `| Z) . x = (2 * (cos . x)) / (exp_R . x) )
A5: exp_R . x <> 0 by SIN_COS:54;
assume A6: x in Z ; :: thesis: (((sin - cos) / exp_R) `| Z) . x = (2 * (cos . x)) / (exp_R . x)
then A7: (sin - cos) . x = (sin . x) - (cos . x) by A1, VALUED_1:13;
( exp_R is_differentiable_in x & sin - cos is_differentiable_in x ) by A2, A6, 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 A5, FDIFF_2:14
.= (((((sin - cos) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((sin - cos) . x))) / ((exp_R . x) ^2) by A2, A6, FDIFF_1:def 7
.= ((((cos . x) + (sin . x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((sin - cos) . x))) / ((exp_R . x) ^2) by A1, A6, Th39
.= ((((cos . x) + (sin . x)) * (exp_R . x)) - ((exp_R . x) * ((sin . x) - (cos . x)))) / ((exp_R . x) ^2) by A7, SIN_COS:65
.= ((2 * (cos . x)) * (exp_R . x)) / ((exp_R . x) * (exp_R . x))
.= (2 * (cos . x)) * ((exp_R . x) / ((exp_R . x) * (exp_R . x))) by XCMPLX_1:74
.= (2 * (cos . x)) * (((exp_R . x) / (exp_R . x)) / (exp_R . x)) by XCMPLX_1:78
.= (2 * (cos . x)) * (1 / (exp_R . x)) by A5, XCMPLX_1:60
.= (2 * (cos . x)) / (exp_R . x) by XCMPLX_1:99 ;
hence (((sin - cos) / exp_R) `| Z) . x = (2 * (cos . x)) / (exp_R . x) by A4, A6, FDIFF_1:def 7; :: thesis: 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 * (cos . x)) / (exp_R . x) ) ) by A2, A3, FDIFF_2:21; :: thesis: verum