
theorem Th7:
  for n be Ordinal, b, b1 be bag of n holds b1 in rng divisors b
  iff b1 divides b
proof
  let n be Ordinal, b, b1 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 PRE_POLY:def 16;
  field (BagOrder n) = Bags n by ORDERS_1:15;
  then
A3: BagOrder n linearly_orders S by ORDERS_1:37,38;
  hereby
    assume b1 in rng divisors b;
    then b1 in S by A1,A3,PRE_POLY:def 2;
    hence b1 divides b by A2;
  end;
  assume b1 divides b;
  then b1 in S by A2;
  hence thesis by A1,A3,PRE_POLY:def 2;
end;
