A combinatorial identity arising from cobordism theory

Let $\underline\alpha=(\alpha_1,\alpha_2,\dots,\alpha_m)\in{\Bbb R}_{>0}^m$. Let $\mathop{\underline\alpha\,}_{i,j}$ be the vector obtained from $\underline\alpha$ by deleting the entries $\alpha_i$ and $\alpha_j$. A. Besser and P. Moree [Arch. Math. (Basel) 79 (2002), no. 6, 463--471; MR1967264 (2004a:11014)] introduced some invariants and near invariants related to the solutions $\underline\epsilon\in\{\pm1}^{m-2}$ of the linear inequality ${|\alpha_i-\alpha_j|}<\langle\underline\epsilon, \mathop{\underline\alpha\,}_{i,j}\rangle<\alpha_i+\alpha_j$, where $\langle·,·\rangle$ denotes the usual inner product and $\mathop{\underline\alpha\,}_{i,j}$ the vector obtained from $\underline\alpha$ by deleting $\alpha_i$ and $\alpha_j$. The main result of [op. cit.] is extended here to a much more general setting, namely that of certain maps from finite sets to ${-1,1}$.