  The present paper is a continuation of author's paper arXiv:1909.00940
[math.AT] devoted to the lemmas of Alexander and Sperner, but is independent
from it. We begin by a step back from Alexander and Sperner to Lebesgue work on
the invariance of the dimension. In contrast with almost everybody else,
Lebesgue worked with cubes rather than with simplices. His methods were
developed by Hurewicz and Lusternik-Schnirelmann and then forgotten. In the
present paper these methods are recast in the language of cubical chains and
cochains.
  After this, we present a new approach to Lebesgue and Lusternik-Schnirelmann
theorems which is both conceptual and elementary. It is based on adaptation of
Serre's definition of products of singular cubical cochains to discrete
setting. The main results are new purely combinatorial "cubical lemmas". This
approach also clarifies the cubical versions of Sperner lemma of Kuhn and Ky
Fan. In particular, Ky Fan's lemma can be understood as a natural strengthening
of Lebesgue or Kuhn's results under a transversality assumption.
  The exposition does not assume any knowledge of algebraic topology.
