Axiom: The empty set exists, which is to say that there is a set \( \emptyset \) containing no elements, \[ \forall x, \left( x \notin \emptyset \right) \]

Subset of Naturals with a Max Element has Bounded Difference

Suppose \( S \subseteq \mathbb{ N } _ 1 \) has a max element \( n \) and min element \( n \), and \( D := \left\{ x - y: x \neq y \in S \right\} \)
\[
\max \left( D \right) \le m - n
\]

Suppose \( x - y \in D \) since \( x , y\in S \) then \( x \le m \) and \( y \ge n \) therefore:
\[
x - y \le m - y \le m - n
\]
thus \( \max \left( D \right) \le m - n \) as needed.