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 (arctan*arctan) & Z c= ].-1,1.[ & (for x st x in Z holds
  arctan.x > -1 & arctan.x < 1) implies arctan*arctan is_differentiable_on Z &
  for x st x in Z holds ((arctan*arctan)`|Z).x = 1/((1+x^2)*(1+(arctan.x)^2))
proof
  assume that
A1: Z c= dom (arctan*arctan) and
A2: Z c= ].-1,1.[ and
A3: for x st x in Z holds arctan.x > -1 & arctan.x < 1;
A4: for x st x in Z holds arctan*arctan is_differentiable_in x
  proof
    let x;
    assume
A5: x in Z;
    then
A6: arctan.x > -1 & arctan.x < 1 by A3;
    arctan is_differentiable_on Z by A2,SIN_COS9:81;
    then arctan is_differentiable_in x by A5,FDIFF_1:9;
    hence thesis by A6,SIN_COS9:85;
  end;
  then
A7: arctan*arctan is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((arctan*arctan)`|Z).x = 1/((1+x^2)*(1+(arctan.x) ^2))
  proof
    let x;
    assume
A8: x in Z;
    then
A9: arctan.x > -1 & arctan.x < 1 by A3;
A10: arctan is_differentiable_on Z by A2,SIN_COS9:81;
    then
A11: arctan is_differentiable_in x by A8,FDIFF_1:9;
    ((arctan*arctan)`|Z).x = diff(arctan*arctan,x) by A7,A8,FDIFF_1:def 7
      .= diff(arctan,x)/(1+(arctan.x)^2) by A11,A9,SIN_COS9:85
      .= ((arctan)`|Z).x/(1+(arctan.x)^2) by A8,A10,FDIFF_1:def 7
      .= (1/(1+x^2))/(1+(arctan.x)^2) by A2,A8,SIN_COS9:81
      .= 1/((1+x^2)*(1+(arctan.x)^2)) by XCMPLX_1:78;
    hence thesis;
  end;
  hence thesis by A1,A4,FDIFF_1:9;
end;
