
theorem Th59:
  for L be Field for p be Polynomial of L st len p <> 0 for x be
  Element of L holds x is_a_root_of p iff x is_a_root_of NormPolynomial(p)
proof
  let L be Field;
  let p be Polynomial of L;
  assume
A1: len p <> 0;
  then len p >= 0+1 by NAT_1:13;
  then len p = len p-'1+1 by XREAL_1:235;
  then p.(len p-'1) <> 0.L by ALGSEQ_1:10;
  then
A2: (p.(len p-'1))" <> 0.L by VECTSP_1:25;
  let x be Element of L;
  thus x is_a_root_of p implies x is_a_root_of NormPolynomial(p)
  proof
    assume x is_a_root_of p;
    then eval(p,x) = 0.L;
    then eval(NormPolynomial(p),x) = 0.L/p.(len p-'1) by A1,Th58
      .= 0.L * (p.(len p-'1))"
      .= 0.L;
    hence thesis;
  end;
  assume x is_a_root_of NormPolynomial(p);
  then 0.L = eval(NormPolynomial(p),x)
    .= eval(p,x)/p.(len p-'1) by A1,Th58
    .= eval(p,x) * (p.(len p-'1))";
  then eval(p,x) = 0.L by A2,VECTSP_1:12;
  hence thesis;
end;
