Vajda's Identity/Formulation 2

Theorem
Let $F_n$ be the $n$th Fibonacci number.

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

Proof
We have:

Now substitute:

where:
 * $\phi$ denotes the golden mean
 * $\hat \phi = 1 - \phi$

first into $(2)$:

and then into $(1)$:

$(1) = (2)$ and hence the result.