reserve X for set;
reserve S for Subset-Family of X;

theorem
  for S be cap-finite-partition-closed diff-finite-partition-closed
  Subset-Family of X holds
  {union x where x is finite Subset of S:x is mutually-disjoint} is
  Ring_of_sets
  proof
    let S be cap-finite-partition-closed diff-finite-partition-closed
    Subset-Family of X;
    {union x where x is finite Subset of S:x is mutually-disjoint}
    is cap-closed cup-closed by Thm86,thmCup;
    hence thesis by thmIL;
  end;
