
theorem Th14:
  for z being Complex st z <> 0 holds Arg(z) < PI iff Arg - z >= PI
proof
  let z be Complex;
  assume
A1: z<>0;
  thus Arg(z)<PI implies Arg(-z)>=PI
  proof
    Arg(z)>=0 by COMPTRIG:34;
    then
A2: PI+0<=PI+Arg(z) by XREAL_1:7;
    assume Arg(z)<PI;
    hence thesis by A1,A2,Th12;
  end;
  thus Arg(-z)>=PI implies Arg(z)<PI
  proof
    2*PI>Arg(-z) by COMPTRIG:34;
    then
A3: PI+PI-PI>Arg(-z)-PI by XREAL_1:9;
    assume Arg(-z)>=PI;
    then Arg(--z)=Arg(-z)-PI by A1,Th12;
    hence thesis by A3;
  end;
end;
