it at most one time. V. N. Latyshev applied Dilworth theorem for estimation of the number of not $m$-divisible polylinear words of degree $n$ over the alphabet $\{a_1,\dots,a_n\}$. The estimate is ${(m - 1)}^{2n}$ and is rather sharp. Let us recall this theorem.

: [*Let $n$ be the maximal number of elements of an antichain in a given finite partially ordered set $M$. Then $M$ can be divided into $n$ disjoint chains.*]{}

Consider a polylinear word $W$ consisting of $n$ letters. Put $a_i\succ
a_j$ if $i>j$ and the letter $a_i$ is located in $W$ to the right from $a_j$. The condition of not $k$-divisibility means absence of an antichain consisting of $n$ elements. Then by Dilworth theorem all positions (and the letters $a_i$ as well) split into $(n-1)$ chains. Attach a specific color to each chain. Then the coloring of positions and of letters uniquely determines the word $W$. Furthermore the number of these colorings does not exceed $(n-1)^k\times (n-1)^k=(n-1)^{2k}$.

The above estimate implies validity of polylinear identities corresponding to an irreducible module whose Young diagram includes the square of size $n^4$. This in turn enables, firstly, to obtain a transparent proof for Regev theorem which asserts that a tensor 