reserve x,a,b,c for Real,
  n for Nat,
  Z for open Subset of REAL,
  f, f1,f2 for PartFunc of REAL,REAL;

theorem
  not 0 in Z & Z c= dom ((id Z)^(#)cot) implies ((id Z)^(#)cot)
is_differentiable_on Z & for x st x in Z holds (((id Z)^(#)cot)`|Z).x = -cos.x/
  sin.x/x^2-1/x/(sin.x)^2
proof
  set f = id Z;
  assume that
A1: not 0 in Z and
A2: Z c= dom (f^(#)cot);
A3: f^ is_differentiable_on Z by A1,FDIFF_5:4;
A4: Z c= dom (f^) /\ dom cot by A2,VALUED_1:def 4;
  then
A5: Z c= dom cot by XBOOLE_1:18;
A6: for x st x in Z holds cot is_differentiable_in x & diff(cot, x)=-1/(sin.
  x)^2
  proof
    let x;
    assume x in Z;
    then sin.x<>0 by A5,Th2;
    hence thesis by FDIFF_7:47;
  end;
  then for x st x in Z holds cot is_differentiable_in x;
  then
A7: cot is_differentiable_on Z by A5,FDIFF_1:9;
A8: Z c= dom (f^) by A4,XBOOLE_1:18;
  for x st x in Z holds ((f^(#)cot)`|Z).x =-cos.x/sin.x/x^2-1/x/(sin.x)^2
  proof
    let x;
    assume
A9: x in Z;
    then ((f^(#)cot)`|Z).x= (cot.x)*diff(f^,x)+((f^).x)*diff(cot,x) by A2,A3,A7
,FDIFF_1:21
      .=(cot.x)*((f^)`|Z).x+((f^).x)*diff(cot,x) by A3,A9,FDIFF_1:def 7
      .=(cot.x)*(-1/x^2)+((f^).x)*diff(cot,x) by A1,A9,FDIFF_5:4
      .=-(cot.x)*(1/x^2)+((f^).x)*(-1/(sin.x)^2) by A6,A9
      .=-(cos.x/sin.x)*(1/x^2)-((f^).x)/(sin.x)^2 by A5,A9,RFUNCT_1:def 1
      .=-cos.x/sin.x/x^2-(f.x)"/(sin.x)^2 by A8,A9,RFUNCT_1:def 2
      .=-cos.x/sin.x/x^2-1/x/(sin.x)^2 by A9,FUNCT_1:18;
    hence thesis;
  end;
  hence thesis by A2,A3,A7,FDIFF_1:21;
end;
