Please check my below proof. Thank your for your help!
Theorem:
For Fibonacci sequence, $f_{m+n+1}=f_m f_n+ f_{m+1}f_{n+1}$
Proof:
Let $P(n)$ is the statement $\forall m \in \mathbb{N}(f_{m+n+1}=f_m f_n+ f_{m+1}f_{n+1})$.
It is clear that $P(0)$ is true.
Assuming that $P(k)$ is true i.e. $\forall m \in \mathbb{N}(f_{m+k+1}=f_m f_k+ f_{m+1}f_{k+1})$.
Since $P(k)$ is true for all $m$, then $P(k)$ is true for $(m+1)$ too.
Substitute $(m+1)$ for $m$, we have $f_{(m+1)+k+1}=f_{m+1} f_k+ f_{(m+1)+1}f_{k+1}=f_{m+1} f_k+ f_{m+2}f_{k+1}$.
$\iff f_{(m+1)+k+1}=f_{m+1} f_k+ f_{m+2}f_{k+1}$
We now prove $P(k+1)$ is true.
$f_{m+(k+1)+1}=f_{(m+1)+k+1}=f_{m+1} f_k+ f_{m+2}f_{k+1}$
$=f_{m+1} f_k+(f_{m+1}+f_m)f_{k+1}$
$=f_{m+1} f_k+f_{m+1} f_{k+1}+f_m f_{k+1}$
$=f_{m+1}(f_k+f_{k+1})+f_m f_{k+1}$
$=f_{m+1} f_{k+2} + f_m f_{k+1}$
$=f_m f_{k+1}+f_{m+1} f_{k+2}$
$=f_m f_{k+1}+f_{m+1} f_{(k+1)+1}$.
To sum up, $f_{m+(k+1)+1}=f_m f_{k+1}+f_{m+1} f_{(k+1)+1}$. This implies $P(k+1)$ is true.
By principle of induction, $P(n)$ is true for all $n \in \mathbb{N}$.
Your proof is correct. You may fix $m$ and just focus on $n$ to save some effort.