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 Th81:
  Z c= ].-1,1.[ implies arctan is_differentiable_on Z & for x st x
  in Z holds ((arctan)`|Z).x = 1/(1+x^2)
proof
  assume
A1: Z c= ].-1,1.[;
  then
A2: arctan is_differentiable_on Z by Th73,FDIFF_1:26;
  for x st x in Z holds ((arctan)`|Z).x = 1/(1+x^2)
  proof
    let x;
    assume
A3: x in Z;
    then
A4: -1 <= x by A1,XXREAL_1:4;
A5: x <= 1 by A1,A3,XXREAL_1:4;
    thus ((arctan)`|Z).x = diff(arctan,x) by A2,A3,FDIFF_1:def 7
      .= 1/(1+x^2) by A4,A5,Th75;
  end;
  hence thesis by A1,Th73,FDIFF_1:26;
end;
