theorem Th20:
  for E being Subset-Family of Omega for X,Y being Subset of Omega
  holds X in E & Y in generated_Dynkin_System(E) & E is intersection_stable
  implies X/\ Y in generated_Dynkin_System(E)
proof
  let E be Subset-Family of Omega;
  let X,Y be Subset of Omega;
  assume that
A1: X in E and
A2: Y in generated_Dynkin_System(E) and
A3: E is intersection_stable;
  reconsider G=generated_Dynkin_System(E) as Dynkin_System of Omega;
  E c= generated_Dynkin_System(E) by Def6;
  then reconsider X as Element of G by A1;
  for x being object holds x in E implies x in DynSys(X,G)
  proof
    let x be object;
    assume
A4: x in E;
    then reconsider x as Subset of Omega;
A5: E c= G by Def6;
    x /\ X in E by A1,A3,A4,FINSUB_1:def 2;
    hence thesis by A5,Def7;
  end;
  then E c= DynSys(X,G);
  then generated_Dynkin_System(E) c= DynSys(X,G) by Def6;
  hence thesis by A2,Def7;
end;
