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 (exp_R(#)(sin-cos)) implies exp_R(#)(sin-cos)
is_differentiable_on Z & for x st x in Z holds ((exp_R(#)(sin-cos))`|Z).x =2 *
  exp_R.x * sin.x
proof
  assume
A1: Z c= dom (exp_R(#)(sin-cos));
  then Z c= dom (sin-cos) /\ dom exp_R by VALUED_1:def 4;
  then
A2: Z c= dom (sin-cos) by XBOOLE_1:18;
  then
A3: sin-cos is_differentiable_on Z by Th39;
A4: exp_R is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:16;
  for x st x in Z holds ((exp_R(#)(sin-cos))`|Z).x =2 * exp_R.x * sin.x
  proof
    let x;
    assume
A5: x in Z;
    then
    ((exp_R(#)(sin-cos))`|Z).x = ((sin-cos).x)*diff(exp_R,x) + (exp_R.x)*
    diff((sin-cos),x) by A1,A3,A4,FDIFF_1:21
      .=(sin.x-cos.x)*diff(exp_R,x)+ (exp_R.x)*diff((sin-cos),x) by A2,A5,
VALUED_1:13
      .=(sin.x-cos.x)*(exp_R.x)+ (exp_R.x)*diff((sin-cos),x) by TAYLOR_1:16
      .=(sin.x-cos.x)*(exp_R.x)+(exp_R.x)*((sin-cos)`|Z).x by A3,A5,
FDIFF_1:def 7
      .=(sin.x-cos.x)*(exp_R.x)+(exp_R.x)*(cos.x+sin.x) by A2,A5,Th39
      .=2 * exp_R.x * sin.x;
    hence thesis;
  end;
  hence thesis by A1,A3,A4,FDIFF_1:21;
end;
