Product of Sequence of Fermat Numbers plus 2/Corollary

Corollary to Product of Sequence of Fermat Numbers plus 2
Let $F_n$ denote the $n$th Fermat number.

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Then:
 * $F_n \divides F_{n + m} - 2$

where $\divides$ denotes divisibility.

Proof
From Product of Sequence of Fermat Numbers plus 2:

and so all Fermat numbers of index less than $n + m$ are divisors of $F_{n + m} - 2$.

This of course includes $F_n$.