reserve x,x0, r, s, h for Real,

  n for Element of NAT,
  rr, y for set,
  Z for open Subset of REAL,

  f, f1, f2 for PartFunc of REAL,REAL;

theorem
  Z c= dom (arctan*exp_R) & (for x st x in Z holds exp_R.x < 1) implies
arctan*exp_R is_differentiable_on Z & for x st x in Z holds ((arctan*exp_R)`|Z)
  .x = exp_R.x/(1+(exp_R.x)^2)
proof
  assume that
A1: Z c= dom (arctan*exp_R) and
A2: for x st x in Z holds exp_R.x < 1;
A3: for x st x in Z holds arctan*exp_R is_differentiable_in x
  proof
    let x;
A4: exp_R is_differentiable_in x by SIN_COS:65;
    assume x in Z;
    then
A5: exp_R.x < 1 by A2;
    exp_R.x + 0 > 0 + (-1) by SIN_COS:54;
    hence thesis by A5,A4,Th85;
  end;
  then
A6: arctan*exp_R is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((arctan*exp_R)`|Z).x = exp_R.x/(1+(exp_R.x)^2)
  proof
    let x;
A7: exp_R.x + 0 > 0 + (-1) by SIN_COS:54;
A8: exp_R is_differentiable_in x by SIN_COS:65;
    assume
A9: x in Z;
    then
A10: exp_R.x < 1 by A2;
    ((arctan*exp_R)`|Z).x = diff(arctan*exp_R,x) by A6,A9,FDIFF_1:def 7
      .= diff(exp_R,x)/(1+(exp_R.x)^2) by A7,A10,A8,Th85
      .= (exp_R.x)/(1+(exp_R.x)^2) by SIN_COS:65;
    hence thesis;
  end;
  hence thesis by A1,A3,FDIFF_1:9;
end;
