reserve y for set,
  x,a,b for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2,g for PartFunc of REAL,REAL;

theorem Th5:
  not 0 in Z implies sin*((id Z)^) is_differentiable_on Z & for x
  st x in Z holds ((sin*((id Z)^))`|Z).x = -(1/x^2)*cos.(1/x)
proof
  set f = id Z;
  assume
A1: not 0 in Z;
  then
A2: Z c= dom (sin*(f^)) by Lm4;
  then for y being object st y in Z holds y in dom (f^) by FUNCT_1:11;
  then
A3: Z c= dom (f^);
A4: f^ is_differentiable_on Z by A1,Th4;
A5: for x st x in Z holds sin*(f^) is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then
A6: f^ is_differentiable_in x by A4,FDIFF_1:9;
    sin is_differentiable_in (f^).x by SIN_COS:64;
    hence thesis by A6,FDIFF_2:13;
  end;
  then
A7: sin*(f^) is_differentiable_on Z by A2,FDIFF_1:9;
  for x st x in Z holds ((sin*(f^))`|Z).x = -(1/x^2)*cos.(1/x)
  proof
    let x;
A8: sin is_differentiable_in (f^).x by SIN_COS:64;
    assume
A9: x in Z;
    then f^ is_differentiable_in x by A4,FDIFF_1:9;
    then diff(sin*(f^),x) = diff(sin, (f^).x)*diff(f^,x) by A8,FDIFF_2:13
      .=cos.((f^).x)*diff(f^,x) by SIN_COS:64
      .=cos.((f.x)")*diff(f^,x) by A3,A9,RFUNCT_1:def 2
      .=cos.((f.x)")*((f^)`|Z).x by A4,A9,FDIFF_1:def 7
      .=cos.((f.x)")*(-1/x^2) by A1,A9,Th4
      .=cos.(1*x")*(-1/x^2) by A9,FUNCT_1:18
      .=cos.(1/x)*(-1/x^2) by XCMPLX_0:def 9;
    hence thesis by A7,A9,FDIFF_1:def 7;
  end;
  hence thesis by A2,A5,FDIFF_1:9;
end;
