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 Th62:
  for n being Ordinal,i being Element of NAT, b being bag of n st
  i in dom divisors b holds ((divisors b)/.i) qua Element of Bags n divides b
proof
  let n be Ordinal,i be Element of NAT, b be bag of n;
  assume i in dom divisors b;
  then
A1: (divisors b)/.i = (divisors b).i & (divisors b).i in rng divisors b by
FUNCT_1:def 3,PARTFUN1:def 6;
  reconsider pid = (divisors b)/.i as bag of n;
  consider S being non empty finite Subset of Bags n such that
A2: divisors b = SgmX(BagOrder n, S) and
A3: for p being bag of n holds p in S iff p divides b by Def15;
  BagOrder n linearly_orders S by Lm4,ORDERS_1:38;
  then pid in S by A2,A1,Def2;
  hence thesis by A3;
end;
