
theorem Th39:
  for x,z being Element of F_Complex st x <> 0.F_Complex holds x *
  rpoly(1,z) is Hurwitz iff Re(z) < 0
proof
  let x,z be Element of F_Complex;
  set p = x * rpoly(1,z);
  assume
A1: x <> 0.F_Complex;
A2: now
    assume
A3: Re(z) < 0;
    now
      let y be Element of F_Complex;
      assume y is_a_root_of p;
      then 0.F_Complex = eval(p,y) by POLYNOM5:def 7
        .= x * eval(rpoly(1,z),y) by POLYNOM5:30
        .= x * (y - z) by Th29;
      then y - z = 0.F_Complex by A1,VECTSP_1:12;
      hence Re(y) < 0 by A3,RLVECT_1:21;
    end;
    hence p is Hurwitz;
  end;
  now
    eval(p,z) = x * eval(rpoly(1,z),z) by POLYNOM5:30
      .= x * (z - z) by Th29
      .= x * 0.F_Complex by RLVECT_1:15
      .= 0.F_Complex;
    then
A4: z is_a_root_of p by POLYNOM5:def 7;
    assume x * rpoly(1,z) is Hurwitz;
    hence Re(z) < 0 by A4;
  end;
  hence thesis by A2;
end;
