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

theorem Th38:
  for K,L be Field, w be Element of L st K is Subring of L
  holds Ann_Poly(w,K) is quasi-prime
  proof
    let K,L;
    let w be Element of L;
    assume
A0:  K is Subring of L;
    set M = {p where p is Polynomial of K:Ext_eval(p,w)=0.L};
    for p, q being Element of Polynom-Ring K st p*q in Ann_Poly(w,K) holds
    p in Ann_Poly(w,K) or q in Ann_Poly(w,K)
    proof
      let p, q be Element of Polynom-Ring K;
      assume
A1:   p*q in Ann_Poly(w,K);
      reconsider p1=p, q1=q as Polynomial of K by POLYNOM3:def 10;
      p1*'q1 in Ann_Poly(w,K) by A1,POLYNOM3:def 10; then
      consider t be Polynomial of K such that
A5:   t = p1*'q1 and
A6:   Ext_eval(t,w)=0.L;
      Ext_eval(p1,w) * Ext_eval(q1,w) = 0.L by A0,Th24,A6,A5;
      then per cases by VECTSP_2:def 1;
      suppose Ext_eval(p1,w)=0.L;
        hence thesis;
      end;
      suppose Ext_eval(q1,w)=0.L;
        hence thesis;
      end;
     end;
     hence thesis by RING_1:def 1;
  end;
