Sum 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 $u + v \in S$.

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.

Then:

As Integer Addition is Closed, both $x_1 + x_2$ and $y_1 + y_2$ are integers.

Hence the result.