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 ((1/2)(#)(( #Z 2)*(arctan))) & Z c=].-1,1.[ implies (1/2)(#)(
  ( #Z 2)*(arctan)) is_differentiable_on Z & for x st x in Z holds (((1/2)(#)((
  #Z 2)*(arctan)))`|Z).x = arctan.x / (1+x^2)
proof
  assume that
A1: Z c= dom ((1/2)(#)(( #Z 2)*(arctan))) and
A2: Z c=].-1,1.[;
A3: Z c= dom (( #Z 2)*(arctan)) by A1,VALUED_1:def 5;
  then
A4: ( #Z 2)*(arctan) is_differentiable_on Z by A2,Th91;
  for x st x in Z holds (((1/2)(#)(( #Z 2)*(arctan)))`|Z).x = arctan.x / (
  1+x^2)
  proof
    let x;
    assume
A5: x in Z;
    then (((1/2)(#)(( #Z 2)*(arctan)))`|Z).x = (1/2)*diff((( #Z 2)*arctan),x)
    by A1,A4,FDIFF_1:20
      .= (1/2)*((( #Z 2)*(arctan))`|Z).x by A4,A5,FDIFF_1:def 7
      .= (1/2)*((2*(arctan.x) #Z (2-1)) / (1+x^2)) by A2,A3,A5,Th91
      .= (1/2)*(2*arctan.x / (1+x^2)) by PREPOWER:35
      .= arctan.x / (1+x^2);
    hence thesis;
  end;
  hence thesis by A1,A4,FDIFF_1:20;
end;
