Exercise 11, § 1

Let $(x_i)$ be a family of n elements of a lattice ordered group G; for each integer k (0 ≤ k ≤ n), let d_k (resp. m_k) denote the inf (resp. sup) of the $\binom{n}{k}$ sums of k terms $x_i$ with distinct indices. Show that $d_k + m_{n-k} = x_1 + x_2 + \ldots + x_n$.