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 * cos.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 Th38;
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 * cos.x
  proof
    let x;
     reconsider xx=x as Element of REAL by XREAL_0:def 1;
    assume
A5: x in Z;
    then
    ((exp_R(#)(sin+cos))`|Z).x = ((sin+cos).xx)*diff(exp_R,x) + (exp_R.x)*
    diff((sin+cos),x) by A1,A3,A4,FDIFF_1:21
      .=(sin.xx+cos.xx)*diff(exp_R,x)+ (exp_R.x)*diff((sin+cos),x)
               by VALUED_1:1
      .=(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,Th38
      .=2 * exp_R.x * cos.x;
    hence thesis;
  end;
  hence thesis by A1,A3,A4,FDIFF_1:21;
end;
