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 Th2:
  sin.x<>0 implies cosec is_differentiable_in x & diff(cosec, x) =
  -cos.x/(sin.x)^2
proof
A1: sin is_differentiable_in x by SIN_COS:64;
  assume
A2: sin.x<>0;
  then diff(sin^,x)= - diff(sin,x)/(sin.x)^2 by A1,FDIFF_2:15
    .=-cos.x/(sin.x)^2 by SIN_COS:64;
  hence thesis by A2,A1,FDIFF_2:15;
end;
