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 (exp_R*arctan) & Z c= ].-1,1.[ implies exp_R*arctan
is_differentiable_on Z & for x st x in Z holds ((exp_R*arctan)`|Z).x = exp_R.(
  arctan.x)/(1+x^2)
proof
  assume that
A1: Z c= dom (exp_R*arctan) and
A2: Z c= ].-1,1.[;
A3: for x st x in Z holds exp_R*arctan is_differentiable_in x
  proof
    let x;
    assume
A4: x in Z;
    arctan is_differentiable_on Z by A2,Th81;
    then
A5: arctan is_differentiable_in x by A4,FDIFF_1:9;
    exp_R is_differentiable_in arctan.x by SIN_COS:65;
    hence thesis by A5,FDIFF_2:13;
  end;
  then
A6: exp_R*arctan is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((exp_R*arctan)`|Z).x = exp_R.(arctan.x)/(1+x^2)
  proof
    let x;
    assume
A7: x in Z;
A8: exp_R is_differentiable_in arctan.x by SIN_COS:65;
A9: arctan is_differentiable_on Z by A2,Th81;
    then arctan is_differentiable_in x by A7,FDIFF_1:9;
    then diff(exp_R*arctan,x) = diff(exp_R,arctan.x)*diff(arctan,x) by A8,
FDIFF_2:13
      .= diff(exp_R,arctan.x)*((arctan)`|Z).x by A7,A9,FDIFF_1:def 7
      .= diff(exp_R,arctan.x)*(1/(1+x^2)) by A2,A7,Th81
      .= exp_R.(arctan.x)/(1+x^2) by SIN_COS:65;
    hence thesis by A6,A7,FDIFF_1:def 7;
  end;
  hence thesis by A1,A3,FDIFF_1:9;
end;
