Addition of Natural Numbers is Provable/General Form

Theorem
Let $y \in \N$ be a natural number.

Let $s^a$ denote the application of the successor mapping $a$ times.

Let $\sqbrk a$ denote $\map {s^a} 0$.

Then there exists a formal proof of:
 * $\forall x: x + \sqbrk y = \map {s^y} x$

in minimal arithmetic.

Proof
Proceed by induction on $y$.

Basis for the Induction
Let $y = 0$.

Then $\sqbrk y = 0$ by definition.

Then the theorem to prove is:
 * $\forall x: x + 0 = x$

But that is Axiom $\text M 3$ of minimal arithmetic.

Induction Hypothesis
Fix some $y \in \N$.

Suppose that there is a formal proof of:
 * $\forall x: x + \sqbrk y = \map {s^y} x$

Induction Step
The following is a formal proof:

By expanding the definitions of $\sqbrk y$ and $\map {s^y} x$, the proven result is syntactically identical to:
 * $\forall x: x + \sqbrk {\map s y} = \map {s^{\map s y} } x$

which was the theorem to be proven.

Thus, by the Principle of Mathematical Induction, there is such a formal proof for every $y \in \N$.