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 * (sin . 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 * (sin . 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 by Th38;
A3: ( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
exp_R . x <> 0 ) ) by FDIFF_1:34, SIN_COS:59, 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; :: thesis: ( x in Z implies (((sin + cos ) / exp_R ) `| Z) . x = - ((2 * (sin . x)) / (exp_R . x)) )
A5: (sin + cos ) . x = (sin . x) + (cos . x) by VALUED_1:1;
A6: exp_R . x <> 0 by SIN_COS:59;
assume A7: x in Z ; :: thesis: (((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_1:16, SIN_COS:70;
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 8
.= ((((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:70
.= ((- (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_1:75
.= (- (2 * (sin . x))) * (((exp_R . x) / (exp_R . x)) / (exp_R . x)) by XCMPLX_1:79
.= (- (2 * (sin . x))) * (1 / (exp_R . x)) by A6, XCMPLX_1:60
.= (- (2 * (sin . x))) / (exp_R . x) by XCMPLX_1:100
.= - ((2 * (sin . x)) / (exp_R . x)) by XCMPLX_1:188 ;
hence (((sin + cos ) / exp_R ) `| Z) . x = - ((2 * (sin . x)) / (exp_R . x)) by A4, A7, 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 * (sin . x)) / (exp_R . x)) ) ) by A2, A3, FDIFF_2:21; :: thesis: verum