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

theorem Th35:
  for K,L be Field, z be Element of L st z is_integral_over K
  holds Ann_Poly(z,K) <> {0.Polynom-Ring K}
proof
  let K,L;
  let z be Element of L;
  assume
A1:  z is_integral_over K;
set M = {p where p is Polynomial of K:Ext_eval(p,z)=0.L};
  consider f be Polynomial of K such that
A2:  LC f = 1.K and
A3:  Ext_eval(f,z) = 0.L by A1;
     not f in {0.Polynom-Ring K}
     proof
     assume
A5:  f in {0.Polynom-Ring K};
     reconsider f as Element of Polynom-Ring K by POLYNOM3:def 10;
     f in {0.Polynom-Ring K}-Ideal by A5,IDEAL_1:47; then
     f in the set of all 0.Polynom-Ring K*g
     where g is Element of Polynom-Ring K by IDEAL_1:64; then
     consider g1 being Element of Polynom-Ring K such that
A6: f = 0.Polynom-Ring K * g1;
    reconsider g2 = g1 as Polynomial of K by POLYNOM3:def 10;
    reconsider h2 = 0.Polynom-Ring K as Polynomial of K by POLYNOM3:def 10;
    f = 0_.K by POLYNOM3:def 10,A6;
    hence contradiction by FUNCOP_1:7,A2;
    end;
    hence thesis by A3;
end;
