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 Th67:
  Z c= ].-1,1.[ & (for x st x in Z holds arctan.x<>0) implies
arctan^ is_differentiable_on Z & for x st x in Z holds ((arctan^)`|Z).x = -1/((
  arctan.x)^2*(1+x^2))
proof
  assume that
A1: Z c= ].-1,1.[ and
A2: for x st x in Z holds arctan.x<>0;
A3: arctan is_differentiable_on Z by A1,SIN_COS9:81;
  then
A4: arctan^ is_differentiable_on Z by A2,FDIFF_2:22;
  for x st x in Z holds ((arctan^)`|Z).x = -1/((arctan.x)^2*(1+x^2))
  proof
    let x;
    assume
A5: x in Z;
    then
A6: arctan.x<>0 & arctan is_differentiable_in x by A2,A3,FDIFF_1:9;
    ((arctan^)`|Z).x = diff(arctan^,x) by A4,A5,FDIFF_1:def 7
      .= -diff(arctan,x)/(arctan.x)^2 by A6,FDIFF_2:15
      .= -((arctan)`|Z).x/(arctan.x)^2 by A3,A5,FDIFF_1:def 7
      .= -(1/(1+x^2))/(arctan.x)^2 by A1,A5,SIN_COS9:81
      .= -1/((arctan.x)^2*(1+x^2)) by XCMPLX_1:78;
    hence thesis;
  end;
  hence thesis by A2,A3,FDIFF_2:22;
end;
