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

theorem Lm32:
  for B be comRing
  for w be Element of B, p, x being Element of Polynom-Ring A
  st A is Subring of B & x in Ann_Poly(w,A) holds x * p in Ann_Poly(w,A)
proof
  let B be comRing;
  let w be Element of B;
  let p,x be Element of Polynom-Ring 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;
  assume that
A0:  A is Subring of B and
A1:  x in Ann_Poly(w,A);
     consider x2 be Polynomial of A such that
A2:  x2 = x1 and
A3:  Ext_eval(x2,w)=0.B by A1;
     Ext_eval(x1*'p1,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 = x1 *'p1 and
A5: Ext_eval(t,w) = 0.B;
     x1 *'p1 in M by A4,A5;
  hence thesis by POLYNOM3:def 10;
end;
