Fibonacci Number in terms of Larger Fibonacci Numbers

Theorem
Let $F_k$ be the $k$th Fibonacci number.

Then:
 * $\displaystyle \forall m,n : F_{m-n} = \left({-1}\right)^{n+1} F_{m-1} F_n + \left({-1}\right)^{n} F_m F_{n-1}$

Proof
The result follows from Fibonacci Numbers in terms of Smaller Fibonacci Numbers and Fibonacci Numbers with Negative Index: