reserve x for Real,

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

     g for PartFunc of REAL,REAL;

theorem
  Z c= dom (tan(#)arctan) & Z c= ].-1,1.[ implies (tan(#)arctan)
is_differentiable_on Z & for x st x in Z holds ((tan(#)arctan)`|Z).x = arctan.x
  /(cos.x)^2+tan.x/(1+x^2)
proof
  assume that
A1: Z c= dom (tan(#)arctan) and
A2: Z c= ].-1,1.[;
A3: arctan is_differentiable_on Z by A2,SIN_COS9:81;
  Z c= dom tan /\ dom arctan by A1,VALUED_1:def 4;
  then
A4: Z c= dom tan by XBOOLE_1:18;
  for x st x in Z holds tan is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then cos.x <> 0 by A4,FDIFF_8:1;
    hence thesis by FDIFF_7:46;
  end;
  then
A5: tan is_differentiable_on Z by A4,FDIFF_1:9;
  for x st x in Z holds ((tan(#)arctan)`|Z).x = arctan.x/(cos.x)^2+tan.x/
  (1+x^2)
  proof
    let x;
    assume
A6: x in Z;
    then
A7: cos.x <> 0 by A4,FDIFF_8:1;
    ((tan(#)arctan)`|Z).x = (arctan.x)*diff(tan,x)+(tan.x)*diff(arctan,x)
    by A1,A5,A3,A6,FDIFF_1:21
      .= (arctan.x)*(1/(cos.x)^2)+(tan.x)*diff(arctan,x) by A7,FDIFF_7:46
      .= arctan.x/(cos.x)^2+(tan.x)*((arctan)`|Z).x by A3,A6,FDIFF_1:def 7
      .= arctan.x/(cos.x)^2+(tan.x)*(1/(1+x^2)) by A2,A6,SIN_COS9:81
      .= arctan.x/(cos.x)^2+tan.x/(1+x^2);
    hence thesis;
  end;
  hence thesis by A1,A5,A3,FDIFF_1:21;
end;
