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 (cos*arctan) & Z c= ].-1,1.[ implies cos*arctan
  is_differentiable_on Z & for x st x in Z holds ((cos*arctan)`|Z).x = -sin.(
  arctan.x)/(1+x^2)
proof
  assume that
A1: Z c= dom (cos*arctan) and
A2: Z c= ].-1,1.[;
A3: for x st x in Z holds cos*arctan is_differentiable_in x
  proof
    let x;
    assume
A4: x in Z;
A5: cos is_differentiable_in arctan.x by SIN_COS:63;
    arctan is_differentiable_on Z by A2,SIN_COS9:81;
    then arctan is_differentiable_in x by A4,FDIFF_1:9;
    hence thesis by A5,FDIFF_2:13;
  end;
  then
A6: cos*arctan is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((cos*arctan)`|Z).x = -sin.(arctan.x)/(1+x^2)
  proof
    let x;
    assume
A7: x in Z;
A8: arctan is_differentiable_on Z by A2,SIN_COS9:81;
    then
A9: arctan is_differentiable_in x by A7,FDIFF_1:9;
A10: cos is_differentiable_in arctan.x by SIN_COS:63;
    ((cos*arctan)`|Z).x = diff(cos*arctan,x) by A6,A7,FDIFF_1:def 7
      .= diff(cos,arctan.x)*diff(arctan,x) by A9,A10,FDIFF_2:13
      .= (-sin.(arctan.x))*diff(arctan,x) by SIN_COS:63
      .= -sin.(arctan.x)*diff(arctan,x)
      .= -sin.(arctan.x)*((arctan)`|Z).x by A7,A8,FDIFF_1:def 7
      .= -sin.(arctan.x)*(1/(1+x^2)) by A2,A7,SIN_COS9:81
      .= -sin.(arctan.x)/(1+x^2);
    hence thesis;
  end;
  hence thesis by A1,A3,FDIFF_1:9;
end;
