Proposition
13
[27P](Replaces 27Y) Addition is commutative.
Proof
By the lemma we can write
\begin{equation} S(h) + n = S(h+n)= h + S(n)\label{eq:Shn_ hSN} \end{equation}
14
intuitively the formula is symmetric and therefore also the definition of addition must have a symmetry. Precisely, let \(\tilde f_ n(h){\stackrel{.}{=}}f_ h(n)\) then \(\tilde f_ n(0)=n\) (as already noted) and for the lemma [27N] \(\tilde f_ n(S(h))=S\Big(\tilde f_ n(h)\Big)\) but then \(\tilde f\) satisfies the same recursive relation as \(f\) and therefore they are identical, so \(f_ h(n)=f_ n(h)\). (The idea is that if we had defined addition. recursively starting from left instead of right, we would have achieved the same result).