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

theorem Th10:
  for L being non empty ZeroStr,
    p being Series of n,L holds
       {0.L} \/ rng p = rng (p extended_by_0)
proof
  let L be non empty ZeroStr,
  p be Series of n,L;
A1:dom p = Bags n & dom (p extended_by_0) = Bags (n+1) by FUNCT_2:def 1;
A2:rng p c= rng (p extended_by_0)
  proof
    let y be object;
    assume y in rng p;
    then consider x be object such that
A3:   x in dom p & p.x=y by FUNCT_1:def 3;
    reconsider x as Element of Bags n by A3;
    (p extended_by_0).(x bag_extend 0) = p.x by Th6;
    hence thesis by A1,FUNCT_1:def 3,A3;
  end;
  set b0= (the bag of n) bag_extend 1;
  b0.n = 1 by HILBASIS:def 1;
  then
A4: (p extended_by_0).b0 =0.L by Def3;
  0.L in rng (p extended_by_0) by A1,A4,FUNCT_1:def 3;
  hence {0.L} \/rng p c= rng (p extended_by_0) by A2,ZFMISC_1:137;
  let y be object;
  assume y in rng (p extended_by_0);
  then consider x be object such that
A5:x in dom (p extended_by_0) & (p extended_by_0).x=y by FUNCT_1:def 3;
  reconsider x as Element of Bags (n+1) by A5;
  per cases;
  suppose x.n<>0;
    then y = 0.L by A5,Def3;
    hence thesis by ZFMISC_1:136;
  end;
  suppose x.n = 0; then
A6: y = p.(0,n)-cut x by Def3,A5;
    n-'0=n by NAT_D:40;
    then ((0,n)-cut x) in Bags n by PRE_POLY:def 12;
    then p.(0,n)-cut x in rng p by A1,FUNCT_1:def 3;
    hence thesis by A6,ZFMISC_1:136;
  end;
end;
