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 (cot(#)arccot) & Z c= ].-1,1.[ implies (cot(#)arccot)
is_differentiable_on Z & for x st x in Z holds ((cot(#)arccot)`|Z).x = -arccot.
  x/(sin.x)^2-cot.x/(1+x^2)
proof
  assume that
A1: Z c= dom (cot(#)arccot) and
A2: Z c= ].-1,1.[;
A3: arccot is_differentiable_on Z by A2,SIN_COS9:82;
  Z c= dom cot /\ dom arccot by A1,VALUED_1:def 4;
  then
A4: Z c= dom cot by XBOOLE_1:18;
  for x st x in Z holds cot is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then sin.x <> 0 by A4,FDIFF_8:2;
    hence thesis by FDIFF_7:47;
  end;
  then
A5: cot is_differentiable_on Z by A4,FDIFF_1:9;
  for x st x in Z holds ((cot(#)arccot)`|Z).x = -arccot.x/(sin.x)^2-cot.x
  /(1+x^2)
  proof
    let x;
    assume
A6: x in Z;
    then
A7: sin.x <> 0 by A4,FDIFF_8:2;
    ((cot(#)arccot)`|Z).x = (arccot.x)*diff(cot,x)+(cot.x)*diff(arccot,x)
    by A1,A5,A3,A6,FDIFF_1:21
      .= (arccot.x)*(-1/(sin.x)^2)+(cot.x)*diff(arccot,x) by A7,FDIFF_7:47
      .= -arccot.x/(sin.x)^2+(cot.x)*((arccot)`|Z).x by A3,A6,FDIFF_1:def 7
      .= -arccot.x/(sin.x)^2+(cot.x)*(-1/(1+x^2)) by A2,A6,SIN_COS9:82
      .= -arccot.x/(sin.x)^2-cot.x/(1+x^2);
    hence thesis;
  end;
  hence thesis by A1,A5,A3,FDIFF_1:21;
end;
