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 Th63:
  for n being Ordinal, b being bag of n holds (divisors b)/.1 =
  EmptyBag n & (divisors b)/.len divisors b = b
proof
  let n be Ordinal, b be bag of n;
  consider S being non empty finite Subset of Bags n such that
A1: divisors b = SgmX(BagOrder n, S) and
A2: for p being bag of n holds p in S iff p divides b by Def15;
A3: now
    let y be Element of Bags n;
    assume y in S;
    then y divides b by A2;
    then y <=' b by Th48;
    hence [y,b] in BagOrder n by Def13;
  end;
A4: now
    let y be Element of Bags n;
    assume y in S;
    EmptyBag n <=' y by Th48,Th57;
    hence [EmptyBag n, y] in BagOrder n by Def13;
  end;
A5: BagOrder n linearly_orders S by Lm4,ORDERS_1:38;
  EmptyBag n divides b;
  then EmptyBag n in S by A2;
  hence (divisors b)/.1 = EmptyBag n by A1,A5,A4,Th19;
  b in S by A2;
  hence thesis by A1,A5,A3,Th20;
end;
