 reserve a,x for Real;
 reserve n for Element of NAT;
 reserve A for non empty closed_interval Subset of REAL;
 reserve f,h,f1,f2 for PartFunc of REAL,REAL;
 reserve Z for open Subset of REAL;

theorem Th9:
  Z c= dom ((1/2)(#)((sin-cos)/exp_R)) implies
(1/2)(#)((sin-cos)/exp_R) is_differentiable_on Z &
  for x st x in Z holds (((1/2)(#)((sin-cos)/exp_R))`|Z).x =cos.x/exp_R.x
proof
   assume
A1:Z c= dom ((1/2)(#)((sin-cos)/exp_R));
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 1;
then A3:Z c= dom (sin-cos) by XBOOLE_1:18; then
A4:sin-cos is_differentiable_on Z &
  for x st x in Z holds ((sin-cos)`|Z).x =cos.x+sin.x by FDIFF_7:39;
A5:(sin-cos)/exp_R is_differentiable_on Z by A2,FDIFF_7:43;then
A6:(1/2)(#)((sin-cos)/exp_R) is_differentiable_on Z by FDIFF_2:19;
 for x st x in Z holds (((1/2)(#)((sin-cos)/exp_R))`|Z).x =cos.x/exp_R.x
   proof
     let x;
     assume
A7:x in Z;
A8:exp_R is_differentiable_in x by SIN_COS:65;
A9:sin-cos is_differentiable_in x by A4,A7,FDIFF_1:9;
A10:(sin-cos).x=sin.x-cos.x by A3,A7,VALUED_1:13;
A11:exp_R.x <>0 by SIN_COS:54;
   (((1/2)(#)((sin-cos)/exp_R))`|Z).x=(1/2)*(diff((sin-cos)/exp_R,x))
    by A1,A5,A7,FDIFF_1:20
 .=(1/2)*((diff(sin-cos,x) * exp_R.x
    - diff(exp_R,x) *(sin-cos).x)/(exp_R.x)^2) by A8,A9,A11,FDIFF_2:14
 .=(1/2)*((((sin-cos)`|Z).x* exp_R.x
    - diff(exp_R,x) *(sin-cos).x)/(exp_R.x)^2) by A4,A7,FDIFF_1:def 7
 .=(1/2)*(((cos.x+sin.x)* exp_R.x
    - diff(exp_R,x) *(sin-cos).x)/(exp_R.x)^2) by A3,A7,FDIFF_7:39
 .=(1/2)*(((cos.x+sin.x)* exp_R.x
    - exp_R.x*(sin.x-cos.x))/(exp_R.x)^2) by A10,SIN_COS:65
 .=(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_1:78
 .=(1/2)*((2*cos.x)*(1/exp_R.x)) by A11,XCMPLX_1:60
 .=cos.x/exp_R.x;
   hence thesis;
  end;
 hence thesis by A6;
end;
