Oriented matroids and Ky Fan’s theorem

L. Lovász (Matroids and Sperner’s Lemma, Europ. J. Comb.1 (1980), 65–66) has shown that Sperner’s combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. We prove that Ky Fan’s theorem admits an oriented matroid generalization of similar nature. Classical Ky Fan’s theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating matroid Cm,r.