reserve Omega, F for non empty set,
  f for SetSequence of Omega,
  X,A,B for Subset of Omega,
  D for non empty Subset-Family of Omega,
  n,m for Element of NAT,
  h,x,y,z,u,v,Y,I for set;

theorem Th16:
  D is Dynkin_System of Omega & D is intersection_stable implies
  for f being SetSequence of Omega holds rng f c= D implies union rng f in D
proof
  assume that
A1: D is Dynkin_System of Omega and
A2: D is intersection_stable;
  let f be SetSequence of Omega;
  assume rng f c= D;
  then
A3: rng disjointify(f) c= D by A1,A2,Th15;
  disjointify(f) is disjoint_valued by Th5;
  then Union disjointify(f) in D by A1,A3,Def5;
  hence thesis by Th6;
end;
