Lower Section of Natural Number is Provable

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

Then the following WFF:
 * $\forall y: y = 0 \lor y = \map s 0 \lor \dotso \lor y = \sqbrk x \lor \exists z: \map s z + \sqbrk x = y$

has a formal proof from the axioms of Robinson arithmetic.

Proof
Proceed by induction on $x$.

Base Case
Let $x = 0$.

Then, the following is a formal proof:

Therefore, a formal proof of the theorem exists.