  A set S is independent if no two vertices from S are adjacent. In this paper
we prove that if F is a collection of maximum independent sets of a graph, then
there is a matching from S-{intersection of all members of F} into {union of
all members of F}-S, for every independent set S. Based on this finding we give
alternative proofs for a number of well-known lemmata, as the "Maximum Stable
Set Lemma" due to Claude Berge and the "Clique Collection Lemma" due to
Andr\'as Hajnal.
