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 Th21:
  for E being Subset-Family of Omega for X,Y being Subset of Omega
holds X in generated_Dynkin_System(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 generated_Dynkin_System(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;
  defpred P[set] means ex X being Element of G st $1=DynSys(X,G);
  consider h such that
A4: for x holds x in h iff x in bool bool Omega & P[x] from XFAMILY:sch
  1;
A5: for Y st Y in h holds Y is Dynkin_System of Omega
  proof
    let Y;
    assume Y in h;
    then ex X being Element of G st Y=DynSys(X,G) by A4;
    hence thesis;
  end;
  h is non empty
  proof
    set X = the Element of G;
    DynSys(X,G) in h by A4;
    hence thesis;
  end;
  then reconsider h as non empty set;
  DynSys(X,G)in h by A1,A4;
  then
A6: meet h c= DynSys(X,G) by SETFAM_1:3;
  for x being object holds x in E implies x in meet h
  proof
    let x be object;
    reconsider xx=x as set by TARSKI:1;
    assume
A7: x in E;
    for Y st Y in h holds x in Y
    proof
      let Y;
      assume Y in h;
      then consider X being Element of G such that
A8:   Y=DynSys(X,G) by A4;
      xx/\ X in G by A3,A7,Th20;
      hence thesis by A7,A8,Def7;
    end;
    hence thesis by SETFAM_1:def 1;
  end;
  then
A9: E c= meet h;
  meet h is Dynkin_System of Omega by A5,Th11;
  then G c= meet h by A9,Def6;
  then G c= DynSys(X,G) by A6;
  hence thesis by A2,Def7;
end;
