theorem
  dom arctan is open
proof
  for x0 st x0 in ].-PI/2,PI/2.[ holds 0 < diff(tan,x0)
  proof
    let x0;
    assume
A1: x0 in ].-PI/2,PI/2.[;
    then 0 < cos.x0 by COMPTRIG:11;
    then (cos.x0)^2 > 0;
    then 1/(cos.x0)^2 > 0 /(cos.x0)^2;
    hence thesis by A1,Lm3;
  end;
  then rng (tan|].-PI/2,PI/2.[) is open by Lm1,Th1,FDIFF_2:41;
  hence thesis by FUNCT_1:33;
end;
