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*arccot) & Z c= ].-1,1.[ & (for x st x in Z holds
  arccot.x > -1 & arccot.x < 1) implies arctan*arccot is_differentiable_on Z &
  for x st x in Z holds ((arctan*arccot)`|Z).x = -1/((1+x^2)*(1+(arccot.x)^2))
proof
  assume that
A1: Z c= dom (arctan*arccot) and
A2: Z c= ].-1,1.[ and
A3: for x st x in Z holds arccot.x > -1 & arccot.x < 1;
A4: for x st x in Z holds arctan*arccot is_differentiable_in x
  proof
    let x;
    assume
A5: x in Z;
    then
A6: arccot.x > -1 & arccot.x < 1 by A3;
    arccot is_differentiable_on Z by A2,SIN_COS9:82;
    then arccot is_differentiable_in x by A5,FDIFF_1:9;
    hence thesis by A6,SIN_COS9:85;
  end;
  then
A7: arctan*arccot is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((arctan*arccot)`|Z).x = -1/((1+x^2)*(1+(arccot.x )^2))
  proof
    let x;
    assume
A8: x in Z;
    then
A9: arccot.x > -1 & arccot.x < 1 by A3;
A10: arccot is_differentiable_on Z by A2,SIN_COS9:82;
    then
A11: arccot is_differentiable_in x by A8,FDIFF_1:9;
    ((arctan*arccot)`|Z).x = diff(arctan*arccot,x) by A7,A8,FDIFF_1:def 7
      .= diff(arccot,x)/(1+(arccot.x)^2) by A11,A9,SIN_COS9:85
      .= ((arccot)`|Z).x/(1+(arccot.x)^2) by A8,A10,FDIFF_1:def 7
      .= (-1/(1+x^2))/(1+(arccot.x)^2) by A2,A8,SIN_COS9:82
      .= -1/(1+x^2)/(1+(arccot.x)^2)
      .= -1/((1+x^2)*(1+(arccot.x)^2)) by XCMPLX_1:78;
    hence thesis;
  end;
  hence thesis by A1,A4,FDIFF_1:9;
end;
