The Fibonacci sequence $(F_n)_{n\ge 0}$ is defined by the recurrence relation $F_{n+2}=F_{n+1}+F_{n}$ with $F_{0}=0$ and $F_{1}=1$. Prove that for any $m$, $n \in \mathbb{N}$, we have $$F_{m+n+1}=F_{m+1}F_{n+1}+F_{m}F_{n}.$$ Deduce from here that $F_{2n+1}=F^2_{n+1}+F^2_{n}$ for any $n \in \mathbb{N}$