Figure 3

The idea of the proof of Sperner’s lemma (Theorem 13 ) for \(\pmb{n=3}\) . Then simplex \(\langle {p^{1},p^{2}} \rangle\) is embedded in \(\langle{e^{1},e^{2},e^{3}} \rangle\) as \(\langle{v^{1},v^{2}} \rangle\). The triangulation T of that simplex consists of small sectors in \(\langle{v^{1},v^{2}} \rangle\) whose vertices are labeled by the function l. The labels at the other vertices are produced by \(l'\) (see the proof). The simplices \(\sigma_{1},\ldots,\sigma_{5}\) come from combinatorial Lemma 1. The simplices \(\sigma'_{1},\ldots,\sigma'_{5}\) come from the pairing of simplices by the procedure described in the proof of Sperner’s lemma (see also endnote j). Finally, observe that the simplex \(\langle{e^{1},e^{2},e^{3}} \rangle\) is triangulated as in the claim (and proof) of Lemma 12.