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

theorem
  [:A,A:] \/ [:X \ A,X \ A:] c= [:X \ A,X:] \/ [:X,A:]
  proof
    let x be object;
    assume x in [:A,A:] \/ [:X \ A,X \ A:];
    then per cases by XBOOLE_0:def 3;
    suppose x in [:A,A:];
      then consider y,z be object such that
A1:   y in A and
A2:   z in A and
A3:   x = [y,z] by ZFMISC_1:def 2;
      x in [:X \ A,X:] or x in [:X,A:] by A1,A2,A3,ZFMISC_1:87;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose x in [:X \ A, X \ A:];
      then consider y,z be object such that
A4:   y in X \ A and
A5:   z in X \ A and
A6:   x = [y,z] by ZFMISC_1:def 2;
      x in [:X \ A,X:] or x in [:X,A:] by A4,A5,A6,ZFMISC_1:87;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
