Euler-Binet Formula/Negative Index

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 the Euler-Binet Formula:


 * $F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5} = \dfrac {\phi^n - \hat \phi^n} {\phi - \hat \phi}$

continues to hold.

In the above:
 * $\phi$ is the golden mean
 * $\hat \phi = 1 - \phi = -\dfrac 1 \phi$

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

Then: