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 (tan*((id Z)^)) implies (tan*((id Z)^))
is_differentiable_on Z & for x st x in Z holds ((tan*((id Z)^))`|Z).x = -1/(x^2
  *(cos.(1/x))^2)
proof
  set f = id Z;
  assume that
A1: not 0 in Z and
A2: Z c= dom (tan*((id Z)^));
A3: f^ is_differentiable_on Z by A1,FDIFF_5:4;
  dom (tan*(f^)) c= dom (f^) by RELAT_1:25;
  then
A4: Z c= dom (f^) by A2,XBOOLE_1:1;
A5: for x st x in Z holds (cos.((f^).x))<>0
  proof
    let x;
    assume x in Z;
    then f^.x in dom (sin/cos) by A2,FUNCT_1:11;
    hence thesis by Th1;
  end;
A6: for x st x in Z holds tan*(f^) is_differentiable_in x
  proof
    let x;
    assume
A7: x in Z;
    then (cos.((f^).x))<>0 by A5;
    then
A8: tan is_differentiable_in (f^).x by FDIFF_7:46;
    f^ is_differentiable_in x by A3,A7,FDIFF_1:9;
    hence thesis by A8,FDIFF_2:13;
  end;
  then
A9: tan*(f^) is_differentiable_on Z by A2,FDIFF_1:9;
  for x st x in Z holds ((tan*(f^))`|Z).x = -1/(x^2*(cos.(1/x))^2)
  proof
    let x;
    assume
A10: x in Z;
    then
A11: f^ is_differentiable_in x by A3,FDIFF_1:9;
A12: cos.((f^).x)<>0 by A5,A10;
    then tan is_differentiable_in (f^).x by FDIFF_7:46;
    then diff(tan*(f^),x) = diff(tan,(f^).x)*diff((f^),x) by A11,FDIFF_2:13
      .=(1/(cos.((f^).x))^2) * diff((f^),x) by A12,FDIFF_7:46
      .=diff((f^),x)/(cos.((f.x)"))^2 by A4,A10,RFUNCT_1:def 2
      .=diff((f^),x)/(cos.(1*x"))^2 by A10,FUNCT_1:18
      .=((f^)`|Z).x/(cos.(1*x"))^2 by A3,A10,FDIFF_1:def 7
      .=(-1/x^2)/(cos.(1*x"))^2 by A1,A10,FDIFF_5:4
      .=(-1)/x^2/(cos.(1/x))^2
      .=(-1)/(x^2*(cos.(1/x))^2) by XCMPLX_1:78
      .=-1/(x^2*(cos.(1/x))^2);
    hence thesis by A9,A10,FDIFF_1:def 7;
  end;
  hence thesis by A2,A6,FDIFF_1:9;
end;
