reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem
  for n being Ordinal holds decomp EmptyBag n = <* <*EmptyBag n,
  EmptyBag n*> *>
proof
  let n be Ordinal;
  len<*EmptyBag n, EmptyBag n*> = 2 by FINSEQ_1:44;
  then reconsider
  E = <*EmptyBag n, EmptyBag n*> as Element of 2-tuples_on Bags n
  by FINSEQ_2:92;
  reconsider e = <* E *> as FinSequence of 2-tuples_on Bags n;
A1: dom e = Seg 1 by FINSEQ_1:38;
A2: <* EmptyBag n *> = divisors EmptyBag n by Th65;
A3: for i being Element of NAT, p being bag of n st i in dom e & p = (
  divisors EmptyBag n)/.i holds e/.i = <*p, (EmptyBag n)-'p*>
  proof
    let i be Element of NAT, p be bag of n such that
A4: i in dom e and
A5: p = (divisors EmptyBag n)/.i;
A6: i = 1 by A1,A4,FINSEQ_1:2,TARSKI:def 1;
    then
A7: (divisors EmptyBag n)/.i = EmptyBag n by A2,FINSEQ_4:16;
    thus e/.i = E by A6,FINSEQ_4:16
      .= <*p, (EmptyBag n)-'p*> by A5,A7,Th52;
  end;
  dom e = dom divisors EmptyBag n by A2,A1,FINSEQ_1:38;
  hence thesis by A3,Def16;
end;
