Product of nth Lucas and Fibonacci Numbers

Theorem
Let $L_k$ be the $k$th Lucas number.

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

Then:
 * $\forall n \in \N_{>0}: F_n L_n = F_{2 n}$

Proof
By definition of Lucas numbers:
 * $L_n = F_{n - 1} + F_{n + 1}$

Hence:
 * $F_n L_n = F_n \paren {F_{n - 1} + F_{n + 1} }$

From Honsberger's Identity:
 * $\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$

The result follows by setting $m = n$.