reserve x for Real,

  Z for open Subset of REAL;

theorem
  Z c= dom (exp_R*exp_R) implies exp_R*exp_R is_differentiable_on Z &
  for x st x in Z holds ((exp_R*exp_R)`|Z).x = exp_R.(exp_R.x)*exp_R.x
proof
A1: for x st x in Z holds exp_R*exp_R is_differentiable_in x
  proof
    let x;
    assume x in Z;
A2: exp_R is_differentiable_in exp_R.x by SIN_COS:65;
    exp_R is_differentiable_in x by SIN_COS:65;
    hence thesis by A2,FDIFF_2:13;
  end;
  assume
A3: Z c= dom (exp_R*exp_R);
  then
A4: exp_R*exp_R is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((exp_R*exp_R)`|Z).x = exp_R.(exp_R.x)*exp_R.x
  proof
    let x;
    assume
A5: x in Z;
A6: exp_R is_differentiable_in exp_R.x by SIN_COS:65;
    exp_R is_differentiable_in x by SIN_COS:65;
    then diff(exp_R*exp_R,x) = diff(exp_R,exp_R.x)*diff(exp_R,x) by A6,
FDIFF_2:13
      .=exp_R.(exp_R.x)*diff(exp_R,x) by SIN_COS:65
      .=exp_R.(exp_R.x)*exp_R.x by SIN_COS:65;
    hence thesis by A4,A5,FDIFF_1:def 7;
  end;
  hence thesis by A3,A1,FDIFF_1:9;
end;
