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
  for E being Subset-Family of Omega st E is intersection_stable for D
  being Dynkin_System of Omega st E c= D holds sigma(E) c= D
proof
  let E be Subset-Family of Omega such that
A1: E is intersection_stable;
  reconsider G=generated_Dynkin_System(E) as Dynkin_System of Omega;
  G is intersection_stable by A1,Th22;
  then
A2: G is SigmaField of Omega by Th19;
  let D be Dynkin_System of Omega;
  assume E c= D;
  then
A3: G c= D by Def6;
  E c= G by Def6;
  then sigma(E) c= G by A2,PROB_1:def 9;
  hence thesis by A3;
end;
