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, b being bag of n holds (decomp b)/.1 = <*
  EmptyBag n, b*> & (decomp b)/.len decomp b = <*b, EmptyBag n*>
proof
  let n be Ordinal, b be bag of n;
  reconsider p = (divisors b)/.1 as bag of n;
  p = EmptyBag n & 1 in dom decomp b by Th63,FINSEQ_5:6;
  hence (decomp b)/.1 = <*EmptyBag n, b-'EmptyBag n*> by Def16
    .= <*EmptyBag n, b*> by Th52;
  reconsider p = (divisors b)/.len decomp b as bag of n;
  dom decomp b = dom divisors b by Def16;
  then len decomp b = len divisors b by FINSEQ_3:29;
  then
A1: p = b by Th63;
  len decomp b in dom decomp b by FINSEQ_5:6;
  hence (decomp b)/.len decomp b = <*b,b-'b*> by A1,Def16
    .= <*b, EmptyBag n*> by Th54;
end;
