Cassini's Identity/Negative Indices

Theorem
Let $n \in \Z_{<0}$ be a negative integer.

Let $F_n$ be the $n$th Fibonacci number (as extended to negative integers).

Then Cassini's Identity:


 * $F_{n + 1} F_{n - 1} - F_n^2 = \paren {-1}^n$

continues to hold.

Proof
Let $n \in \Z_{> 0}$.

Then: