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 = 1/((cos.(
  arctan.x))^2*(1+x^2))
proof
  assume that
A1: Z c= dom (tan*arctan) and
A2: Z c= ].-1,1.[;
A3: for x st x in Z holds tan*arctan is_differentiable_in x
  proof
    let x;
    assume
A4: x in Z;
    then arctan.x in dom tan by A1,FUNCT_1:11;
    then cos.(arctan.x) <> 0 by FDIFF_8:1;
    then
A5: tan is_differentiable_in arctan.x by FDIFF_7:46;
    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: tan*arctan is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((tan*arctan)`|Z).x = 1/((cos.(arctan.x))^2*(1+x ^2))
  proof
    let x;
    assume
A7: x in Z;
    then arctan.x in dom tan by A1,FUNCT_1:11;
    then
A8: cos.(arctan.x) <> 0 by FDIFF_8:1;
    then
A9: tan is_differentiable_in arctan.x by FDIFF_7:46;
A10: arctan is_differentiable_on Z by A2,SIN_COS9:81;
    then
A11: arctan is_differentiable_in x by A7,FDIFF_1:9;
    ((tan*arctan)`|Z).x = diff(tan*arctan,x) by A6,A7,FDIFF_1:def 7
      .= diff(tan,arctan.x)*diff(arctan,x) by A11,A9,FDIFF_2:13
      .= (1/(cos.(arctan.x))^2)*diff(arctan,x) by A8,FDIFF_7:46
      .= (1/(cos.(arctan.x))^2)*((arctan)`|Z).x by A7,A10,FDIFF_1:def 7
      .= (1/(cos.(arctan.x))^2)*(1/(1+x^2)) by A2,A7,SIN_COS9:81
      .= 1/((cos.(arctan.x))^2*(1+x^2)) by XCMPLX_1:102;
    hence thesis;
  end;
  hence thesis by A1,A3,FDIFF_1:9;
end;
