reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
          right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
             right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;

theorem
  for n be Nat for L be non empty ZeroStr
  for p be Series of n,L
    holds (p extended_by_0) removed_last = p
proof
  let n be Nat;
  let L be non empty ZeroStr;
  let p be Series of n,L;
  set e0 =p extended_by_0;
  for a being Element of Bags n holds p.a = (e0 removed_last).a
  proof
    let b be Element of Bags n;
    thus (e0 removed_last).b = e0. (b bag_extend 0) by Def6
    .= p.b by HILB10_2:6;
  end;
  hence thesis;
end;
