
theorem Th40:
  for f being Polynomial of F_Complex for z being Element of
  F_Complex st z <> 0.F_Complex holds f is Hurwitz iff z * f is Hurwitz
proof
  let f be Polynomial of F_Complex;
  let z be Element of F_Complex;
  assume
A1: z <> 0.F_Complex;
A2: now
    assume
A3: f is Hurwitz;
    now
      let x be Element of F_Complex;
      assume x is_a_root_of (z*f);
      then 0.F_Complex = eval(z*f,x) by POLYNOM5:def 7
        .= z * eval(f,x) by POLYNOM5:30;
      then eval(f,x) = 0.F_Complex by A1,VECTSP_1:12;
      then x is_a_root_of f by POLYNOM5:def 7;
      hence Re(x) < 0 by A3;
    end;
    hence z*f is Hurwitz;
  end;
  now
    assume
A4: z*f is Hurwitz;
    now
      let x be Element of F_Complex;
      assume
A5:   x is_a_root_of f;
      eval(z*f,x) = z * eval(f,x) by POLYNOM5:30
        .= z * 0.F_Complex by A5,POLYNOM5:def 7
        .= 0.F_Complex;
      then x is_a_root_of (z*f) by POLYNOM5:def 7;
      hence Re(x) < 0 by A4;
    end;
    hence f is Hurwitz;
  end;
  hence thesis by A2;
end;
