Lemma 2. Assuming that:

Then one of the following is true:

either $n$ is odd, say $n = 2 k + 1$ , and $B = X^{(\leq k)} - {k + 2, k + 3, \dots, 2 k + 1} \cup {1, 2, \dots, k + 1}$
or $n$ is even, say $n = 2 k$ , and $B = X^{(< k)} \cup {x \in X^{(k)} : 1 \in x} - {1, k + 2, k + 3, \dots, 2 k} \cup {2, 3, 4, \dots, k + 1}$ .