proof of uniqueness (hint) $f(m,n)=n+1$ only if $n \ge m$

Question

I'm a bit confused about this proof.

let define on $\Bbb N$ a binary fucntion $f$ that satisfies

(1) $f(m,n)=n+1$ if $n \ge m$

and

(2) $f$ is commutative

if I write the values of $f$ in a 2x2 table using (1) I see that the commutativity makes the solution $f(m,n)=max(m,n)+1$ unique... but from where should I start to make that a formal proof?

Answer 1

Just separate two cases, if $n\geq m$, then you already know that $f(m,n)=n+1$. If $n<m$, then prove that $f(m,n)=m+1$ and you are done.