  A theorem of McCord of 1966 and Quillen's Theorem A of 1973 provide
sufficient conditions for a map between two posets to be a homotopy equivalence
at the level of complexes. We give an alternative elementary proof of this
result and we deduce also a stronger statement: under the hypotheses of the
theorem, the map is not only a homotopy equivalence but a simple homotopy
equivalence. This leads then to stronger formulations of the simplicial version
of Quillen's Theorem A, the Nerve lemma and other known results.
