
theorem
  for f being Polynomial of F_Complex st deg(f) >= 1 for rho being
Element of F_Complex st Re(rho) < 0 & |.eval(f,rho).| < |.eval(f*',rho).| holds
  f is Hurwitz iff F*(f,rho) div rpoly(1,rho) is Hurwitz
proof
  let f be Polynomial of F_Complex;
  assume
A1: deg(f) >= 1;
  let rho be Element of F_Complex;
  assume that
A2: Re(rho) < 0 and
A3: |.eval(f,rho).| < |.eval(f*',rho).|;
  reconsider ef = eval(f,rho), ef1 = eval(f*',rho) as Element of F_Complex;
  now
    -1 < deg rpoly(1,rho) by Th27;
    then
A4: deg 0_.(F_Complex) < deg rpoly(1,rho) by Th20;
    eval(ef1 * f - ef * (f*'),rho) = eval(ef1*f,rho) - eval(ef*(f*'),rho)
    by POLYNOM4:21
      .= ef1 * eval(f,rho) - eval(ef*(f*'),rho) by POLYNOM5:30
      .= ef1 * eval(f,rho) - ef * eval((f*'),rho) by POLYNOM5:30
      .= 0.F_Complex by RLVECT_1:15;
    then rho is_a_root_of (ef1 * f - ef * (f*')) by POLYNOM5:def 7;
    then consider t being Polynomial of F_Complex such that
A5: F*(f,rho) = rpoly(1,rho) *' t by Th33;
    F*(f,rho) = rpoly(1,rho) *' t + 0_.(F_Complex) by A5,POLYNOM3:28;
    then
A6: F*(f,rho) = (F*(f,rho) div rpoly(1,rho)) *' rpoly(1,rho) by A5,A4,Def5;
    (1_F_Complex) * rpoly(1,rho) is Hurwitz by A2,Th39;
    then
A7: rpoly(1,rho) is Hurwitz by POLYNOM5:27;
    assume F*(f,rho) div rpoly(1,rho) is Hurwitz;
    then F*(f,rho) is Hurwitz by A6,A7,Th41;
    hence f is Hurwitz by A1,A3,Th50;
  end;
  hence thesis by A1,A2,Th51;
end;
