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*exp_R) implies sin*exp_R is_differentiable_on Z & for x
  st x in Z holds ((sin*exp_R)`|Z).x = exp_R.x * cos.(exp_R.x)
proof
A1: for x st x in Z holds sin*exp_R is_differentiable_in x
  proof
    let x;
    assume x in Z;
    exp_R is_differentiable_in x & sin is_differentiable_in exp_R.x by
SIN_COS:64,65;
    hence thesis by FDIFF_2:13;
  end;
  assume
A2: Z c= dom (sin*exp_R);
  then
A3: sin*exp_R is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((sin*exp_R)`|Z).x =exp_R.x * cos.(exp_R.x)
  proof
    let x;
    exp_R is_differentiable_in x & sin is_differentiable_in exp_R.x by
SIN_COS:64,65;
    then
A4: diff(sin*exp_R,x) = diff(sin,exp_R.x)*diff(exp_R,x) by FDIFF_2:13
      .=cos.(exp_R.x)*diff(exp_R,x) by SIN_COS:64
      .=exp_R.x*cos.(exp_R.x) by SIN_COS:65;
    assume x in Z;
    hence thesis by A3,A4,FDIFF_1:def 7;
  end;
  hence thesis by A2,A1,FDIFF_1:9;
end;
