Partial Difference of Integer Combinations is Integer Combination

Lemma
Let $a, b \in \Z$ be integers.

Let $S = \left\{{ax + by: x, y \in \Z}\right\}$ be the set of integer combinations of $a$ and $b$.

Let $u \in S$ and $v \in S$.

Then:
 * $u \mathop {\dot -} v \in S$

where $\dot -$ denotes the extension of the partial subtraction operator to the integers.

Proof
As both $u, v \in S$, $u$ and $v$ can be expressed as:

where $x_1, x_2, y_1, y_2$ are integers.

Let $u \ge v$.

Then:

As Integer Subtraction is Closed, both $x_1 - x_2$ and $y_1 - y_2$ are integers.

Thus $u \mathop {\dot -} v \in S$.

Let $u < v$.

Then by definition of the partial subtraction operator:
 * $u \mathop {\dot -} v = 0$

From Set of Integer Combinations includes Zero, $0 \in S$.

Hence the result.