reserve i,j,k,n,m for Nat,
        b,b1,b2 for bag of n;

theorem Th8:
  for L being non empty ZeroStr, p being Series of n,L holds
      b bag_extend 0 in Support (p extended_by_0) iff
    b in Support p
proof
  let L be non empty ZeroStr,
  p be Series of n,L;
  set B= b bag_extend 0,P=p extended_by_0;
  B.n=0 by HILBASIS:def 1;
  then
A1: P.B = p.(0,n)-cut B by Def3
       .= p.b by Th5;
  thus B in Support P implies b in Support p
  proof
    assume B in Support P;
    then p.b<>0.L & b in Bags n & dom p = Bags n
      by POLYNOM1:def 3,A1,FUNCT_2:def 1,PRE_POLY:def 12;
    hence thesis by POLYNOM1:def 3;
  end;
  assume b in Support p;
  then
A2: p.b <> 0.L by POLYNOM1:def 3;
  B in Bags (n+1) & dom P = Bags (n+1) by FUNCT_2:def 1;
  hence thesis by POLYNOM1:def 3,A2,A1;
end;
