
theorem Th19:
  for z being Complex holds Arg z = 0 iff Re z >= 0 & Im z = 0
proof
  let z being Complex;
A1: |.z.| = |.z.|+0*<i>;
  hereby
    assume Arg z = 0;
    then
A2: z = |.z.| by Th13;
    then Re z = |.z.| by A1,COMPLEX1:12;
    hence Re z >= 0 & Im z = 0 by A1,A2,COMPLEX1:12,46;
  end;
  assume that
A3: Re z >= 0 and
A4: Im z = 0;
  z = Re z +0*<i> by A4,COMPLEX1:13;
  hence thesis by A3,COMPTRIG:35;
end;
