the $(a_e-a_d)$-tail $u_d$. Compare $a_{i+2d} - a_i$ and $a_{i+4d} - a_{i+2d}$. We may assume that $a_{i+4d} - a_{i+2d}\geqslant a_{i+2d} - a_i$. Let $a_{j+1} - a_j = \inf\limits_k {(a_{k+1} - a_k)}, 0\leqslant j<2d.$ We may assume that $j<d.$ By assumption the $(a_{2d}-a_j)$-tail $u_j$ and the $(a_{2d} - a_{j+1})$-tail $u_{j+1}$ are incomparable with the $(a_{4d}-a_{2d})$-tail $u_{2d}$. Since $a_{4d}-a_{2d}\geqslant a_{2d}-a_j>a_{2d}-a_{j+1}$, the $(a_2d-a_j)$-tail $u_j$ and the $(a_{2d}-a_{j+1})$-tail $u_{j+1}$ are mutually incomparable. Since ${{a_{2d}-a_j}\over {a_{2d}-a_{j+1}}}\leqslant {{d+1}\over {d}},$ the $(a_{j+1}-a_j)$-tail $u_j$ in degree $d$ is included into the $(a_2d-a_j)$-tail $u_j$. A contradiction.

If a word $W$ is not $n$-divisible in ordinary sense then $W$ is not $4nd$-divisible (in tail sense).

Set $p_{n, d}:=4nd-1$.

Let $W$ be a not $n$-cancellable word. Consider $U$, the $[\left|
W\right|/d]$-tail of $W$. Then $W$ is not $(p_{n, d}+1)$-divisible. Let $\Omega$ be the set of tails of $W$, which begin in $U$. Then by lemma \[Lm0.1\] any two elements of $\Omega$ are comparable. There is a natural bijection between $\Omega$, the letters of $U$ and positive integers from $1$ to $\left|\Omega\right|=\left|U\right|$.

Let us introduce a word $\theta$ which is lexicographically less than any other word.

In the current proof of theorem \[c:main1\] all tails are assumed to belong to $\Omega$.

Estimates on 