  Let $P$ and $Q$ be bounded posets. In this note, a lemma is introduced that
provides a set of sufficient conditions for the proper part of $P$ being
homotopy equivalent to the suspension of the proper part of~$Q$. An application
of this lemma is a unified proof of the sphericity of the higher Bruhat orders
under both inclusion order (a known proved earlier by Ziegler) and single step
inclusion order (which was not previously known).
