
theorem Th22:
  for z being Complex st Arg z <= PI holds Im z >= 0
proof
  let z be Complex;
  assume
A1: Arg z <= PI;
  per cases by A1,COMPTRIG:34,XXREAL_0:1;
  suppose
    Arg z = PI or Arg z = 0;
    hence thesis by Th21;
  end;
  suppose
    0 < Arg z & Arg z < PI;
    then Arg z in ].0,PI.[ by XXREAL_1:4;
    hence thesis by Th16;
  end;
end;
