let Z be open Subset of REAL ; :: thesis: ( 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 )) ; :: thesis: ( (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 4;
then A4: Z c= dom (sin - cos ) by XBOOLE_1:18;
then A5: ( 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;
A6: (sin - cos ) / exp_R is_differentiable_on Z by A2, FDIFF_7:43;
then A7: (1 / 2) (#) ((sin - cos ) / exp_R ) is_differentiable_on Z by FDIFF_2:19, A;
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; :: thesis: ( x in Z implies (((1 / 2) (#) ((sin - cos ) / exp_R )) `| Z) . x = (cos . x) / (exp_R . x) )
assume A8: x in Z ; :: thesis: (((1 / 2) (#) ((sin - cos ) / exp_R )) `| Z) . x = (cos . x) / (exp_R . x)
A9: exp_R is_differentiable_in x by SIN_COS:70;
A10: sin - cos is_differentiable_in x by A5, A8, FDIFF_1:16;
A11: (sin - cos ) . x = (sin . x) - (cos . x) by A4, A8, VALUED_1:13;
A12: exp_R . x <> 0 by SIN_COS:59;
(((1 / 2) (#) ((sin - cos ) / exp_R )) `| Z) . x = (1 / 2) * (diff ((sin - cos ) / exp_R ),x) by A, A1, A6, A8, FDIFF_1:28
.= (1 / 2) * ((((diff (sin - cos ),x) * (exp_R . x)) - ((diff exp_R ,x) * ((sin - cos ) . x))) / ((exp_R . x) ^2 )) by A9, A10, A12, FDIFF_2:14
.= (1 / 2) * ((((((sin - cos ) `| Z) . x) * (exp_R . x)) - ((diff exp_R ,x) * ((sin - cos ) . x))) / ((exp_R . x) ^2 )) by A5, A8, FDIFF_1:def 8
.= (1 / 2) * (((((cos . x) + (sin . x)) * (exp_R . x)) - ((diff exp_R ,x) * ((sin - cos ) . x))) / ((exp_R . x) ^2 )) by A4, A8, FDIFF_7:39
.= (1 / 2) * (((((cos . x) + (sin . x)) * (exp_R . x)) - ((exp_R . x) * ((sin . x) - (cos . x)))) / ((exp_R . x) ^2 )) by A11, SIN_COS:70
.= (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_1:79
.= (1 / 2) * ((2 * (cos . x)) * (1 / (exp_R . x))) by A12, XCMPLX_1:60
.= (cos . x) / (exp_R . x) ;
hence (((1 / 2) (#) ((sin - cos ) / exp_R )) `| Z) . x = (cos . x) / (exp_R . x) ; :: thesis: 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 A7; :: thesis: verum