 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 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>. Suppose that 0000 was in the set; then since all permutations of 0000, 0000 are known to lie in the set, then 0000, 0000 must lie outside the set. Also, as 0000 lies in the set, at least one of 0000, 0000 lie outside the set. This leaves only 0 points in <math>\{0,0\}^0</math>, a contradiction. Thus 0000 lies outside the set; similarly 0000 lies outside the set. Suppose that 0000 was in the set; then since all permutations of 0000, 0000 are known to lie in the set, then 0000, 0000 must lie outside the set. Also, as 0000 lies in the set, at least one of 0000, 0000 lie outside the set. This leaves only 0 points in <math>\{0,0\}^0</math>, a contradiction. Thus 0000 lies outside the set; similarly 0000 lies outside the set. L