The main part of this paper begins with the deceptively simple Redundancy Lemma for Euclidean spaces. This states that if the intersection \(\bigcap R\) of a family of halfspaces does not meet a specified hyperplane \(H^=\), then each of the closed halfspaces \(H^\leq\) and \(H^\geq\) bounded by the plane either contains \(\bigcap R\) in its interior, or fails to intersect it. The reader will see that this can be proved easily using either connectedness or convexity. This leads to a very quick inductive proof of Helly's theorem for halfspaces. Moreover, because few properties of Euclidean spaces were assumed, the proofs carry over into piecewise-linear combinatorial topology, and the theory of oriented matroids. In this way, the author derives, among other results, a Helly-type theorem for oriented matroids and some results about systems of linear inequalities related to (and strengthening) Farkas' lemma.