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 Th14:
  the set of all union P where P is Subset of D = UniCl D
  proof
    set F = the set of all union P where P is Subset of D;
    thus F c= UniCl D
    proof
      let x be object;
      assume x in F;
      then consider P be Subset of D such that
A2:   x = union P;
      P c= D & D c= bool X;
      then P c= bool X;
      then reconsider Y = P as Subset-Family of X;
      Y c= D & union Y = union P;
      hence thesis by A2,CANTOR_1:def 1;
    end;
    let x be object;
    assume x in UniCl D;
    then consider Y be Subset-Family of X such that
A3: Y c= D and
A4: x = union Y by CANTOR_1:def 1;
    reconsider P = Y as Subset of D by A3;
    x = union P by A4;
    hence thesis;
  end;
