reserve X for set,
        D for a_partition of X,
        TG for non empty TopologicalGroup;
reserve A for Subset of X;
reserve US for UniformSpace;

theorem
  for Omega being non empty set, D being a_partition of Omega holds
    UniCl D is Dynkin_System of Omega
  proof
    let Omega be non empty set, D be a_partition of Omega;
    now
      hereby
        let f be SetSequence of Omega;
        assume that
A1:     rng f c= UniCl D and
        f is disjoint_valued;
        UniCl D c= bool Omega;
        then rng f c= bool Omega by A1;
        then reconsider a = rng f as Subset-Family of Omega;
        union a in UniCl D by A1,ROUGHS_4:def 3;
        hence Union f in UniCl D by CARD_3:def 4;
      end;
      thus for X be Subset of Omega st X in UniCl D holds X` in UniCl D
        by PROB_1:def 1;
      UniCl D is with_empty_element;
      hence {} in UniCl D;
    end;
    hence thesis by DYNKIN:def 5;
  end;
