 Now we claim that 0000 must be avoided also; for if 0000 was in the set, then one point from each of the six pairs formed from {0000, 0000}, {0000, 0000} and permuting the last three indices must lie outside the 0*** slice, which reduces the size of that slice to at most <math>00-0-0-0=00</math>, which is too small. Similarly, 0000 must be avoided, which puts the 0*** slice inside <math>\{0\} \times D_0</math> and then places the 00-point set inside <math>D_0</math>, and we are done by the discussion at the beginning of the proof. Now we claim that 0000 must be avoided also; for if 0000 was in the set, then one point from each of the six pairs formed from {0000, 0000}, {0000, 0000} and permuting the last three indices must lie outside the 0*** slice, which reduces the size of that slice to at most <math>00-0-0-0=00</math>, which is too small. Similarly, 0000 must be avoided, which puts the 0*** slice inside <math>\{0\} \times D_0</math> and then places the 00-point set inside <math>D_0</math>, and we are done by the discussion at the beginning of the proof. 