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 Th19:
  D is Dynkin_System of Omega & D is intersection_stable implies D
  is SigmaField of Omega
proof
  assume that
A1: D is Dynkin_System of Omega and
A2: D is intersection_stable;
A3: for f st rng f c= D holds Intersection f in D
  proof
    let f such that
A4: rng f c= D;
A5: for n holds (f.n)`in D
    proof
      let n;
      f.n in rng f by NAT_1:51;
      hence thesis by A1,A4,Def5;
    end;
A6: for n holds (Complement f).n in D
    proof
      let n;
      (Complement f).n=(f.n)` by PROB_1:def 2;
      hence thesis by A5;
    end;
    for x being object st x in rng Complement f holds x in D
    proof
      let x be object;
      assume x in rng Complement f;
      then consider z being object such that
A7:   z in dom Complement f and
A8:   x=(Complement f).z by FUNCT_1:def 3;
      reconsider n=z as Element of NAT by A7,FUNCT_2:def 1;
      x=(Complement f).n by A8;
      hence thesis by A6;
    end;
    then rng Complement f c= D;
    then Union Complement f in D by A1,A2,Th16;
    then (Union Complement f)` in D by A1,Def5;
    hence thesis by PROB_1:def 3;
  end;
  for X st X in D holds X`in D by A1,Def5;
  hence thesis by A3,PROB_1:15;
end;
