Combinatorial Lemma for Lebesgue–Brouwer Partition of Cube on Euclidean Space

The purpose of this article is to present a new combinatorial lemma that can be used in fixed point theorem, for example, to prove Brouwer theorem. Partition of a three-dimensional cube used by Lebesgue and Brouwer in dimension theory is taken into consideration. Explanation is given for generalization of the lemma on Euclidean $n$-space, $n>3$.