An independence theorem and its consequences

The following theorem is proved: Let $A_1,\dots,A_d$ be linear operators in a vector space $V$, $v\in V$, and let the word $C=A_{k_1}A_{k_2}\dots A_{k_n}$ be maximal in the right lexicographical order among all words of length $n$ satisfying the condition $Cv\ne0$. If all the operators corresponding to the subwords of $C$ are nilpotent, then the vectors $v$, $A_{k_n}v$, $A_{k_{n-1}}A_{k_n}v,\dots,A_{k_1}A_{k_2}\cdots A_{k_n}v$ are independent.
As a corollary, a proof is presented of Shestakov's conjecture about the number of nil-conditions necessary for a subalgebra of a matrix algebra to be nilpotent.
Bibliography: 5 titles.