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.x*
  arctan.x+cos.x/(1+x^2)
proof
  assume that
A1: Z c= dom (cos(#)arctan) and
A2: Z c= ].-1,1.[;
A3: arctan is_differentiable_on Z by A2,SIN_COS9:81;
A4: for x st x in Z holds cos is_differentiable_in x by SIN_COS:63;
  Z c= dom cos /\ dom arctan by A1,VALUED_1:def 4;
  then Z c= dom cos by XBOOLE_1:18;
  then
A5: cos is_differentiable_on Z by A4,FDIFF_1:9;
  for x st x in Z holds ((cos(#)arctan)`|Z).x = -sin.x*arctan.x+cos.x/(1+x ^2)
  proof
    let x;
    assume
A6: x in Z;
    then
    ((cos(#)arctan)`|Z).x = (arctan.x)*diff(cos,x)+(cos.x)*diff(arctan,x)
    by A1,A5,A3,FDIFF_1:21
      .= (arctan.x)*(-sin.x)+(cos.x)*diff(arctan,x) by SIN_COS:63
      .= -sin.x*arctan.x+(cos.x)*((arctan)`|Z).x by A3,A6,FDIFF_1:def 7
      .= -sin.x*arctan.x+(cos.x)*(1/(1+x^2)) by A2,A6,SIN_COS9:81
      .= -sin.x*arctan.x+cos.x/(1+x^2);
    hence thesis;
  end;
  hence thesis by A1,A5,A3,FDIFF_1:21;
end;
