reserve i,j for Nat;
reserve A,B for Ring;
reserve K, L for Field;

theorem Th41:
  for K,L be Field, z be Element of L,f,g be Element of Polynom-Ring K st
  z is_integral_over K &
  {f}-Ideal = Ann_Poly(z,K) & f = NormPolynomial(f) &
  {g}-Ideal = Ann_Poly(z,K) & g = NormPolynomial(g)
  holds f = g
  proof
    let K,L be Field;
    let z be Element of L;
    let f,g be Element of Polynom-Ring K;
    assume that
A1:  z is_integral_over K and
A2: {f}-Ideal = Ann_Poly(z,K) and
A3:  f = NormPolynomial(f) and
A4: {g}-Ideal = Ann_Poly(z,K) and
A5:  g = NormPolynomial(g);
    reconsider f as Element of the carrier of Polynom-Ring K;
     NormPolynomial(f) <> 0.(Polynom-Ring K) by A3,A2,A1,Th35,IDEAL_1:47; then
     f <> 0_.K by A3,POLYNOM3:def 10; then
A6: f is non zero Element of the carrier of Polynom-Ring K by Lm37;
    reconsider g as Element of the carrier of Polynom-Ring K;
     NormPolynomial(g) <> 0.(Polynom-Ring K) by A5,A4,A1,Th35,IDEAL_1:47; then
     g <> 0_.K by A5,POLYNOM3:def 10; then
     g is non zero Element of the carrier of Polynom-Ring K by Lm37;
     hence thesis by A3,A6,A5,RING_2:21,RING_4:30,A4,A2;
    end;
