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

theorem Th33:
  for A be non degenerated Ring
  for B be non degenerated comRing
  for w be Element of B st A is Subring of B
  holds Ann_Poly(w,A) is proper Ideal of Polynom-Ring A
proof
  let A be non degenerated Ring;
  let B be non degenerated comRing;
  let w be Element of B;
  assume
A0: A is Subring of B;
A1: Ann_Poly(w,A) is add-closed by A0,Lm30;
A2: Ann_Poly(w,A) is left-ideal by A0,Th31;
A3: Ann_Poly(w,A) is right-ideal by A0,Lm32;
    Ann_Poly(w,A) is proper
    proof
      assume not Ann_Poly(w,A) is proper; then
A5:   1.Polynom-RingA in Ann_Poly(w,A);
A6:   1_.A in Ann_Poly(w,A) by A5,POLYNOM3:37;
A7:   Ext_eval(1_.A,w)= 1.B by A0,Th18;
      ex p be Polynomial of A st p = 1_.A & Ext_eval(p,w)= 0.B by A6;
      hence contradiction by A7;
    end;
  hence thesis by A1,A2,A3;
end;
