reserve y for set,
  x,r,a,b for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2,f3 for PartFunc of REAL,REAL;

theorem
  Z c= dom ((sin-cos)/exp_R) implies (sin-cos)/exp_R
is_differentiable_on Z & for x st x in Z holds (((sin-cos)/exp_R)`|Z).x =2*cos.
  x/exp_R.x
proof
  assume Z c= dom ((sin-cos)/exp_R);
  then Z c= dom (sin-cos) /\ (dom exp_R \ (exp_R)"{0}) by RFUNCT_1:def 1;
  then
A1: Z c= dom (sin-cos) by XBOOLE_1:18;
  then
A2: sin-cos is_differentiable_on Z by Th39;
A3: exp_R is_differentiable_on Z & for x st x in Z holds exp_R.x<>0 by
FDIFF_1:26,SIN_COS:54,TAYLOR_1:16;
  then
A4: (sin-cos)/exp_R is_differentiable_on Z by A2,FDIFF_2:21;
  for x st x in Z holds (((sin-cos)/exp_R)`|Z).x =2*cos.x/exp_R.x
  proof
    let x;
A5: exp_R.x <>0 by SIN_COS:54;
    assume
A6: x in Z;
    then
A7: (sin-cos).x=sin.x-cos.x by A1,VALUED_1:13;
    exp_R is_differentiable_in x & sin-cos is_differentiable_in x by A2,A6,
FDIFF_1:9,SIN_COS:65;
    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 A5,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 7
      .=((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 A7,
SIN_COS:65
      .=((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_1:74
      .=(2*cos.x)*((exp_R.x)/(exp_R.x)/(exp_R.x)) by XCMPLX_1:78
      .=(2*cos.x)*(1/exp_R.x) by A5,XCMPLX_1:60
      .=2*cos.x/exp_R.x by XCMPLX_1:99;
    hence thesis by A4,A6,FDIFF_1:def 7;
  end;
  hence thesis by A2,A3,FDIFF_2:21;
end;
