
theorem
  for L be Field for p be Polynomial of L st len p <> 0 holds Roots(p) =
  Roots(NormPolynomial p)
proof
  let L be Field;
  let p be Polynomial of L;
  assume
A1: len p <> 0;
  thus Roots(p) c= Roots(NormPolynomial p)
  proof
    let x be object;
    assume
A2: x in Roots(p);
    then reconsider x1=x as Element of L;
    x1 is_a_root_of p by A2,Def10;
    then x1 is_a_root_of NormPolynomial p by A1,Th59;
    hence thesis by Def10;
  end;
  thus Roots(NormPolynomial p) c= Roots(p)
  proof
    let x be object;
    assume
A3: x in Roots(NormPolynomial p);
    then reconsider x1=x as Element of L;
    x1 is_a_root_of NormPolynomial p by A3,Def10;
    then x1 is_a_root_of p by A1,Th59;
    hence thesis by Def10;
  end;
end;
