
theorem Th41:
  for f,g being Polynomial of F_Complex holds f *' g is Hurwitz
  iff f is Hurwitz & g is Hurwitz
proof
  let f,g be Polynomial of F_Complex;
A1: now
    assume that
A2: f is Hurwitz and
A3: g is Hurwitz;
    now
      let z be Element of F_Complex;
      assume z is_a_root_of f*'g;
      then
A4:   0.F_Complex = eval(f*'g,z) by POLYNOM5:def 7
        .= eval(f,z) * eval(g,z) by POLYNOM4:24;
      per cases by A4,VECTSP_1:12;
      suppose
        eval(f,z)=0.F_Complex;
        then z is_a_root_of f by POLYNOM5:def 7;
        hence Re(z) < 0 by A2;
      end;
      suppose
        eval(g,z)=0.F_Complex;
        then z is_a_root_of g by POLYNOM5:def 7;
        hence Re(z) < 0 by A3;
      end;
    end;
    hence f*'g is Hurwitz;
  end;
  now
    assume
A5: f*'g is Hurwitz;
    now
      let z be Element of F_Complex;
      assume z is_a_root_of f;
      then z is_a_root_of (f*'g) by Lm12;
      hence Re(z) < 0 by A5;
    end;
    hence f is Hurwitz;
    now
      let z be Element of F_Complex;
      assume z is_a_root_of g;
      then z is_a_root_of (f*'g) by Lm12;
      hence Re(z) < 0 by A5;
    end;
    hence g is Hurwitz;
  end;
  hence thesis by A1;
end;
