the following definitions.

a\) If a word $v$ includes a subword of the form $u^t$ then we say that $v$ includes a [*period of length $t.$*]{}

b\) If a word $u$ is the beginning of a word $v$ then these words are called [*incomparable*]{}.

c\) A word $v$ is [*a tail*]{} of a word $u$ if there exists a word $w$ such that $u=wv$.

d\) A word $v$ is [*a $k$-tail*]{} of a word $u$ if $v$ consists of $k$ first letters of some tail $u$.

d\*) [*A $k$-beginning*]{} is the same as $k$-tail.

e\) A word $u$ [*is to the left from*]{} a word $v$ if $u$ begins to the left from the beginning of $v$.

a\) For a real number $x$ put $\ulcorner x\urcorner := -[-x].$

b\) Let $|u|$ denote the length of a word $u$.

The proof uses the following sufficient conditions for presence of a period:

\[Lm0.1\] In a word $W$ of length $x$ either first $[x/d]$ tails are pairwise comparable or $W$ includes a period of length $d$.

Suppose $W$ includes no word of the form $u^{d}$. Consider first $[x/{d}]$ tails. Suppose some two of them, say $v_1$ and $v_2$, are incomparable and $v_1=u\cdot v_2$. 