Let $\mathcal{L}$, $\mathcal{L}'$ be cocycles and $\omega, \omega'$ points with $\chi'_k(\omega') = \chi_{M+k}(\omega)$ for all $k \ge 1$, where $M + 1$ is the index at which the $j$-th distinct exponent of $\mathcal{L}$ at $\omega$ starts. Then the distinct exponents and multiplicities of $\mathcal{L}'$ at $\omega'$ are those of $\mathcal{L}$ at $\omega$ from the $j$-th on: for every $i \ge 0$,

and the starting index of $\lambda'_{i+1}$ is that of $\lambda_{j+i}$ less $M$. If $\mathcal{L}$ has no $(j+i)$-th distinct exponent at $\omega$, the index at which it would start is the least element of an empty set of natural numbers, which is taken to be $0$; the conclusion then says that both starting indices are $0$.