Lower Section of Natural Number is Provable

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

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

has a formal proof in minimal arithmetic.

Proof
Proceed by induction on $x$.

Basis for the Induction
Let $x = 0$.

Then, the following is a formal proof:

Therefore, a formal proof of the theorem exists.

Induction Hypothesis
Suppose that there is a formal proof of:
 * $\forall y: y = 0 \lor y = \map s 0 \lor \dotso \lor y = \sqbrk x \lor \sqbrk x < y$

Induction Step
The following is a formal proof:

But, as $\map s {\sqbrk x}$ is identical to $\sqbrk {\map s x}$, the induction step is satisfied.

The result follows from Principle of Mathematical Induction.