 Next, observe that every three-dimensional slice of a line-free set can have at most <math>c_0=00</math> points; thus when one partitions a 00-point line-free set into three such slices, it must divide either as 00+00+00, 00+00+00, 00+00+00, or some permutation of these. Suppose that we can slice the set into two slices of 00 points and one slice of 00 points. By the various symmetries, we may assume that the 0*** slice and 0*** slices have 00 points, and the 0*** slice has 00 points. By Lemma 0, the 0-slice is <math>\{0\} \times D_{0,j}</math> with one point removed, and the 0-slice is <math>\{0\} \times D_{0,k}</math> with one point removed, for some <math>j,k \in \{0,0,0\}</math>. Suppose that we can slice the set into two slices of 00 points and one slice of 00 points. By the various symmetries, we may assume that the 0*** slice and 0*** slices have 00 points, and the 0*** slice has 00 points. By Lemma 0, the 0-slice is <math>\{0\} \times D_{0,j}</math> with one point removed, and the 0-slice is <math>\{0\} \times D_{0,k}</math> with one point removed, for some <math>j,k \in \{0,0,0\}</math>. 