Redundancy and Helly

The classical Helly’s Theorem about finite sets of convex sets is given an unusually simple proof based on a ‘Redundancy Lemma’. Because the proof is topological it extends immediately to a Helly’s Theorem for the well-known combinatorial topology representation of oriented matroids which is reviewed. The same proof is then used to strengthen Helly’s Theorem in a useful way relative to the Farkas Lemma, both for linear inequality systems and for topologically represented oriented matroids.