
theorem Th13:
  for z being Complex holds Arg z = 0 iff z = |.z.|
proof
  let z be Complex;
  hereby
    assume Arg z=0;
    then z= |.z.|*cos 0 + |.z.|*sin 0 *<i> by COMPTRIG:62;
    hence z= |.z.| by SIN_COS:31;
  end;
  assume z= |.z.|;
  hence thesis by COMPTRIG:35,COMPLEX1:46;
end;
