apply Dilworth theorem. After that we color the tails and their first letters according to their location in chains obtained by application of Dilworth theorem.

We have to know the position in any chain where neighboring tails begin to differ. It is of interest what is the $``$frequency$"$ of this position in a $p$-tail for some $p\leqslant n$. Further we somewhat generalize our reasoning dividing tails into segments consisting of several letters each and determining the segment containg the position where neighboring tails begin to differ. Lemma \[c:lem2\] connects the $``$frequencies$"$ in question for $p$-tails and $kp$-tails for $k = 3$.

To complete the proof, we construct a hierarchical structure based on lemma \[c:lem2\], that is, we consecutively consider segments of $n$-tails, subsegments of these segments and so on. Furthermore we consider the greatest possible number of tails in the subset to which Dilworth theorem is applied, and then we estimate from above the total number of tails and hence of the letters in the word $W$.

Periodicity and $n$-divisibility properties
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Let $a_1, a_2,\ldots ,a_l$ be the alphabet used for constructing words. The ordering $a_1\prec a_2\prec\dots\prec a_l$ induces lexicographical ordering for words over the alphabet. For convenience, we introduce 