The Poincare-Miranda Theorem

equivalent result was established earlier by Bohl [2] in 1904. It was Hadamard [4] who in 1910 gave (using the Kroneckex index) the first proof fox arbitrary n. In 1912 Brouwer gave another proof using the simplicial approximation technique, and notions of degree. A short and simple proof of the BohlwBrouwer theorem was given in 1929 by Knaster-Kuratowski-Mazurkiewicz [7]; the proof was based on the lemma of Sperner [12]. In this paper, we also apply combinatorial methods of proof taken from the Sperner lemma, but our proof seems to be simpler because it does not require the notion of barycentric coordinates and barycentric subdivision. Let us begin by discussing the Poincare-Miranda Theorem Let k > 1 be a given natural number and let Zk := {i/k:i c Z}, whexe Z denotes the set of integers. Let Zkn denote the Cartesian product of n copies of the set Zk Zkn {z {l, * * *, n} Zk 0 z is a map}