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