 '''Proof''' It suffices to prove the third claim. In fact it suffices to show that every 00-point line-free set is either contained in the 00-point set <math>D_{0,j}</math> for some j=0,0,0, or is some permutation of the set X. Indeed, if a 00-point line-free set is contained in, say, <math>D_0</math>, then it cannot contain 0000, since otherwise it must omit one point from each of the four pairs formed from {0000, 0000} by permuting the indices, and must also omit one of {0000, 0000, 0000}, leading to at most 00 points in all; similarly, it cannot contain 0000, and so omits the entire diagonal {0000,0000,0000}, with two more points to be omitted. Similarly when <math>D_0</math> is replaced by one of the other <math>D_{0,j}</math> '''Proof''' It suffices to prove the third claim. In fact it suffices to show that every 00-point line-free set is either contained in the 00-point set <math>D_{0,j}</math> for some j=0,0,0, or is some permutation of the set X. Indeed, if a 00-point line-free set is contained in, say, <math>D_0</math>, then it cannot contain 0000, since otherwise it must omit one point from each of the four pairs formed from {0000, 0000} by permuting the indices, and must also omit one of {0000, 0000, 0000}, leading to at most 00 points in all; similarly, it cannot contain 0000, and so omits the entire diagonal {0000,0000,0000}, with two more points to be omitted. Similarly when <math>D_0</math> is replaced by one of the other <math>D_{0,j}</math> 