Integer Multiples Greater than Positive Integer Closed under Multiplication

Theorem
Let $n \Z$ be the set of integer multiples of $n$.

Let $p \in \Z: p \ge 0$ be a positive integer.

Let $S \subseteq n \Z$ be defined as:
 * $S := \set {x \in n \Z: x > p}$

that is, the set of integer multiples of $n$ greater than $p$.

Then the algebraic structure $\struct {S, \times}$ is closed under multiplication.

Proof
Let $x, y \in S$.

From Integer Multiples Closed under Multiplication, $x y \in n \Z$.

As $x, y > p$ we have that:
 * $\exists r \in \Z_{>0}: x = p + r$
 * $\exists s \in \Z_{>0}: y = p + s$

Thus it follows that:

So $x y > p$ and $x y \in n \Z$.

Hence by definition $x y \in S$, and so $S$ is closed under multiplication.