reserve x,x0, r, s, h for Real,

  n for Element of NAT,
  rr, y for set,
  Z for open Subset of REAL,

  f, f1, f2 for PartFunc of REAL,REAL;

theorem
  dom arccot is open
proof
  for x0 st x0 in ].0,PI.[ holds diff(cot,x0) < 0
  proof
    let x0;
    assume
A1: x0 in ].0,PI.[;
    then 0 < sin.x0 by COMPTRIG:7;
    then (sin.x0)^2 > 0;
    then 1/(sin.x0)^2 > 0 /(sin.x0)^2;
    then -1/(sin.x0)^2 < -0;
    hence thesis by A1,Lm4;
  end;
  then rng (cot|].0,PI.[) is open by Lm2,Th2,FDIFF_2:41;
  hence thesis by FUNCT_1:33;
end;
