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 Th50:
  for n be Nat
    for L be non empty ZeroStr
      for p be Series of n+1,L st not n in vars p
  holds (p removed_last) extended_by_0 = p
proof
  let n be Nat;
  let L be non empty ZeroStr;
  let p be Series of n+1,L such that
A1: not n in vars p;
  set r=p removed_last;
  for a being Element of Bags (n+1) holds p.a = (r extended_by_0).a
  proof
    let b be Element of Bags (n+1);
    per cases;
    suppose
A2:   b.n<>0;
      then
A3:   (r extended_by_0).b = 0.L by HILB10_2:def 3;
A4:   Bags(n+1)=dom p by PARTFUN1:def 2;
      not b in Support p by A1,A2,Def5;
      hence thesis by A3,A4,POLYNOM1:def 3;
    end;
    suppose
A5:   b.n=0;
      then
A6:   (r extended_by_0).b = r.(0,n)-cut b by HILB10_2:def 3;
      set c = (0,n)-cut b;
      n-'0 = n-0;
      then r.c = p. ( c bag_extend 0) by Def6;
      hence thesis by A6,A5,HILB10_2:4;
    end;
  end;
  hence thesis;
end;
