Fermat Number whose Index is Sum of Integers

Theorem
Let $F_n = 2^{\left({2^n}\right)} + 1$ be the $n$th Fermat number.

Let $k \in \Z_{>0}$.

Then:
 * $F_{n+k} - 1 = \left({F_n - 1}\right)^{2 k}$

Proof
By the definition of Fermat number