reserve i,j for Nat;
reserve A,B for Ring;

theorem Th31:
  for B be comRing, z be Element of B, p, x being Element of Polynom-Ring A
  st A is Subring of B & x in Ann_Poly(z,A) holds p * x in Ann_Poly(z,A)
proof
  let B be comRing;
  let w be Element of B;
  let p,x be Element of Polynom-Ring A;
  assume that
A0: A is Subring of B and
A1: x in Ann_Poly(w,A);
    set M ={p where p is Polynomial of A:Ext_eval(p,w)=0.B};
    reconsider p1=p, x1=x as Polynomial of A by POLYNOM3:def 10;
    consider x2 be Polynomial of A such that
A2: x2 = x1 and
A3: Ext_eval(x2,w)=0.B by A1;
    Ext_eval(p1 *'x1,w) = Ext_eval(p1,w) * 0.B by A0,A2,A3,Th24.= 0.B;
    then
    consider t be Polynomial of A such that
A4: t = p1 *'x1 and
A5: Ext_eval(t,w) = 0.B;
    p1 *'x1 in M by A4,A5;
     hence thesis by POLYNOM3:def 10;
end;
