 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem
  for K,L be Field, w be Element of L st K is Subring of L &
    w is_integral_over K holds Ann_Poly(w,K) is maximal
   proof
     let K,L be Field;
     let w be Element of L;
     assume
A1:  K is Subring of L & w is_integral_over K; then
     consider g be Element of Polynom-Ring K such that
A2:  g <> 0_.K & {g}-Ideal = Ann_Poly(w,K) & g = NormPolynomial(g)
       by ALGNUM_1:33;
A3:  g <> 0.Polynom-Ring K by A2,POLYNOM3:def 10;
reconsider g1 = g as non zero Element of Polynom-Ring K by A3,STRUCT_0:def 12;
     Ann_Poly(w,K) is prime by A1,ALGNUM_1:32; then
     g1 is prime by A2,RING_2:24;
     hence thesis by A2,RING_2:26;
   end;
