A Suspension Lemma for Bounded Posets

LetPandQbe bounded posets. In this note, a lemma is introduced that provides a set of sufficient conditions for the proper part ofPbeing homotopy equivalent to the suspension of the proper part ofQ. An application of this lemma is a unified proof of the sphericity of the higher Bruhat orders under both inclusion order (which is a known result) and single step inclusion order (which was not known so far).