reserve x,x0, r, s, h for Real,

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

  f, f1, f2 for PartFunc of REAL,REAL;

theorem
  Z c= dom (ln*(arctan)) & Z c= ].-1,1.[ & (for x st x in Z holds arctan
.x>0) implies ln*(arctan) is_differentiable_on Z & for x st x in Z holds ((ln*(
  arctan))`|Z).x = 1/((1+x^2)*arctan.x)
proof
  assume that
A1: Z c= dom (ln*arctan) and
A2: Z c= ].-1,1.[ and
A3: for x st x in Z holds arctan.x > 0;
A4: for x st x in Z holds ln*arctan is_differentiable_in x
  proof
    let x;
    assume
A5: x in Z;
    arctan is_differentiable_on Z by A2,Th81;
    then
A6: arctan is_differentiable_in x by A5,FDIFF_1:9;
    arctan.x >0 by A3,A5;
    hence thesis by A6,TAYLOR_1:20;
  end;
  then
A7: ln*arctan is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((ln*(arctan))`|Z).x = 1/((1+x^2)*arctan.x)
  proof
    let x;
    assume
A8: x in Z;
    then
A9: -1 < x by A2,XXREAL_1:4;
    arctan is_differentiable_on Z by A2,Th81;
    then
A10: arctan is_differentiable_in x by A8,FDIFF_1:9;
A11: x < 1 by A2,A8,XXREAL_1:4;
    arctan.x >0 by A3,A8;
    then diff(ln*arctan,x) = diff(arctan,x)/arctan.x by A10,TAYLOR_1:20
      .= (1/(1+x^2))/arctan.x by A9,A11,Th75
      .= 1/((1+x^2)*arctan.x) by XCMPLX_1:78;
    hence thesis by A7,A8,FDIFF_1:def 7;
  end;
  hence thesis by A1,A4,FDIFF_1:9;
end;
