
theorem Th21:
  for z being Complex holds Im z = 0 iff Arg z = 0 or Arg z = PI
proof
  let z be Complex;
  hereby
A1: Arg(Re z+0*<i>) = 0 or Arg(Re z+0*<i>) = PI by COMPTRIG:35,36;
    assume Im z = 0;
    hence Arg z = 0 or Arg z = PI by A1,COMPLEX1:13;
  end;
  assume Arg z = 0 or Arg z = PI;
  hence thesis by Th19,Th20;
end;
