 We have handled the case in which at least one of the slicings of the 00-point set is of the form 00=00+00+00. The only remaining case is when all slicings of the 00-point set are of the form 00+00+00 (or a permutation thereof). By the symmetries of the situation, we may assume that the 0*** slice has 00 points, and thus by Lemma 0 takes the form <math>\{0\} \times D_0</math>. Inspecting the *0**, *0**, *0** slices, we then see (from Lemma 0) that only the *0** slice can have 00 points; since we are assuming that this slicing is some permutation of 00=00+00+00, we conclude that the *0** slice must have exactly 00 points, and is thus described precisely by Lemma 0. Similarly for the **0* and ***0 slices. Indeed, by Lemma 0, we see that the 00-point set must agree exactly with <math>D_{0,0}</math> on any of these slices. In particular, on the remaining portion <math>\{0,0\}^0</math> of the cube, there are exactly 0 points of the 00-point set in <math>\{0,0\}^0</math>. 