fibonacci numbers - Proof with mathematical induction: $F_{m+n+1}=F_{m+1}F_{n+1}+F_{m}F_{n} quad text{for} m,n geqslant 0$

Today when I was proving the identity with Fibonacci sequence, namely: $$F_{m+n+1}=F_{m+1}F_{n+1}+F_{m}F_{n} \quad \text{for} \ m,n \geqslant 0$$
I have used mathematical induction. But we know that if we have some property which depends on natural numbers, namely $P(n)$ and we have needed properties on $P(n)$ we can prove that $P(n)$ holds for any $n\in \mathbb{N}$ -- this is the principle of math induction.

However in that post I have applied mathematical induction on $n$ and proved that it holds for any $n\in \mathbb{N}\cup \{0\}$ (the case $n=0$ also true).

However, we should prove that above identity is true for any $m,n\geqslant 0$, but in my proof i have done it only for natural parameter $n$. What about $m$?

Can anyone explain this subtle moment, please?