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

theorem Lm30:
  for A,B be Ring, w be Element of B, x, y being Element of Polynom-Ring A
  st A is Subring of B & x in Ann_Poly(w,A) & y in Ann_Poly(w,A)
  holds x + y in Ann_Poly(w,A)
proof
  let A,B;
  let w be Element of B;
  let x,y be Element of Polynom-Ring A;
  assume that
A0: A is Subring of B and
A1: x in Ann_Poly(w,A) and
A2: y in Ann_Poly(w,A);
    reconsider x1=x, y1=y as Polynomial of A by POLYNOM3:def 10;
    set M ={p where p is Polynomial of A:Ext_eval(p,w)=0.B};
    consider x2 be Polynomial of A such that
A3: x2 = x1 and
A4: Ext_eval(x2,w)=0.B by A1;
    consider y2 be Polynomial of A such that
A5: y2 = y1 and
A6: Ext_eval(y2,w)=0.B by A2;
A7: Ext_eval(x2 + y2,w) = Ext_eval(x1,w) + 0.B by A0,Th19,A6,A3
      .= 0.B by A3,A4;
    consider t be Polynomial of A such that
A8: t = x1+y1 and
A9: Ext_eval(t,w) = 0.B by A3,A5,A7;
    x1+ y1 in M by A8,A9;
    hence thesis by POLYNOM3:def 10;
  end;
