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 implies cosec is_differentiable_on Z & for x st x in Z
  holds ( (cosec)`|Z).x = -cos.x/(sin.x)^2
proof
  assume Z c= dom cosec;
  then for x st x in Z holds sin.x<>0 by RFUNCT_1:3;
  hence thesis by FDIFF_4:40;
end;
