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 4;
then A1: Z c= dom (sin - cos ) by XBOOLE_1:18;
then A2: ( sin - cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin - cos ) `| Z) . x = (cos . x) + (sin . x) ) ) by Th39;
A3: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;
A4: for x being Real st x in Z holds
exp_R . x <> 0 by SIN_COS:59;
then A5: (sin - cos ) / exp_R is_differentiable_on Z by A2, A3, 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) )
assume A6: x in Z ; :: thesis: (((sin - cos ) / exp_R ) `| Z) . x = (2 * (cos . x)) / (exp_R . x)
A7: exp_R is_differentiable_in x by SIN_COS:70;
A8: sin - cos is_differentiable_in x by A2, A6, FDIFF_1:16;
A9: (sin - cos ) . x = (sin . x) - (cos . x) by A1, A6, VALUED_1:13;
A10: exp_R . x <> 0 by SIN_COS:59;
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 A7, A8, 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 8
.= ((((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 A9, SIN_COS:70
.= ((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_1:75
.= (2 * (cos . x)) * (((exp_R . x) / (exp_R . x)) / (exp_R . x)) by XCMPLX_1:79
.= (2 * (cos . x)) * (1 / (exp_R . x)) by A10, XCMPLX_1:60
.= (2 * (cos . x)) / (exp_R . x) by XCMPLX_1:100 ;
hence (((sin - cos ) / exp_R ) `| Z) . x = (2 * (cos . x)) / (exp_R . x) by A5, A6, FDIFF_1:def 8; :: 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, A4, FDIFF_2:21; :: thesis: verum