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
  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=(sin.x)^2/(cos.x)^2
proof
A1: for x st x in Z holds (id Z).x = 1*x+0 by FUNCT_1:18;
  assume
A2: Z c= dom (tan-id Z);
  then
A3: Z c= dom tan /\ dom (id Z) by VALUED_1:12;
  then
A4: Z c= dom tan by XBOOLE_1:18;
A5: Z c= dom (id Z) by A3,XBOOLE_1:18;
  then
A6: id Z is_differentiable_on Z by A1,FDIFF_1:23;
  for x st x in Z holds tan is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then cos.x<>0 by A4,Th1;
    hence thesis by FDIFF_7:46;
  end;
  then
A7: tan is_differentiable_on Z by A4,FDIFF_1:9;
  for x st x in Z holds ((tan-id Z)`|Z).x=(sin.x)^2/(cos.x)^2
  proof
    let x;
    assume
A8: x in Z;
    then
A9: cos.x<>0 by A4,Th1;
    then
A10: (cos.x)^2 >0 by SQUARE_1:12;
    ((tan-id Z)`|Z).x= diff(tan,x) - diff(id Z,x) by A2,A6,A7,A8,FDIFF_1:19
      .=1/(cos.x)^2-diff(id Z,x) by A9,FDIFF_7:46
      .=1/(cos.x)^2-((id Z)`|Z).x by A6,A8,FDIFF_1:def 7
      .=1/(cos.x)^2-1 by A5,A1,A8,FDIFF_1:23
      .=1/(cos.x)^2-(cos.x)^2/(cos.x)^2 by A10,XCMPLX_1:60
      .=(1-(cos.x)^2)/(cos.x)^2
      .=((sin.x)^2+(cos.x)^2-(cos.x)^2)/(cos.x)^2 by SIN_COS:28
      .=(sin.x)^2/(cos.x)^2;
    hence thesis;
  end;
  hence thesis by A2,A6,A7,FDIFF_1:19;
end;
