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 (cosec*tan) implies cosec*tan is_differentiable_on Z & for x
  st x in Z holds ((cosec*tan)`|Z).x = -cos.(tan.x)/(cos.x)^2/(sin.(tan.x))^2
proof
  assume
A1: Z c= dom (cosec*tan);
A2: for x st x in Z holds sin.(tan.x)<>0
  proof
    let x;
    assume x in Z;
    then tan.x in dom cosec by A1,FUNCT_1:11;
    hence thesis by RFUNCT_1:3;
  end;
  dom (cosec*tan) c= dom tan by RELAT_1:25;
  then
A3: Z c= dom tan by A1,XBOOLE_1:1;
A4: for x st x in Z holds cosec*tan is_differentiable_in x
  proof
    let x;
    assume
A5: x in Z;
    then cos.x<>0 by A3,FDIFF_8:1;
    then
A6: tan is_differentiable_in x by FDIFF_7:46;
    sin.(tan.x)<>0 by A2,A5;
    then cosec is_differentiable_in tan.x by Th2;
    hence thesis by A6,FDIFF_2:13;
  end;
  then
A7: cosec*tan is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((cosec*tan)`|Z).x = -cos.(tan.x)/(cos.x)^2/(sin.
  (tan.x))^2
  proof
    let x;
    assume
A8: x in Z;
    then
A9: cos.x<>0 by A3,FDIFF_8:1;
    then
A10: tan is_differentiable_in x by FDIFF_7:46;
A11: sin.(tan.x)<>0 by A2,A8;
    then cosec is_differentiable_in tan.x by Th2;
    then diff(cosec*tan,x) = diff(cosec, tan.x)*diff(tan,x) by A10,FDIFF_2:13
      .=(-cos.(tan.x)/(sin.(tan.x))^2) * diff(tan,x) by A11,Th2
      .=(1/(cos.x)^2)*(-cos.(tan.x)/(sin.(tan.x))^2) by A9,FDIFF_7:46
      .=-cos.(tan.x)/(cos.x)^2/(sin.(tan.x))^2;
    hence thesis by A7,A8,FDIFF_1:def 7;
  end;
  hence thesis by A1,A4,FDIFF_1:9;
end;
