Integer Combination of Coprime Integers

Theorem
Two integers are coprime there exists an integer combination of them equal to $1$:
 * $\forall a, b \in \Z: a \perp b \iff \exists m, n \in \Z: m a + n b = 1$

Proof 1
Note that in the integer combination $m a + n b = 1$, the integers $m$ and $n$ are also coprime.

Also known as
This result is sometimes known as Bézout's Identity, as is the more general Bézout's Lemma.

Some sources refer to this result as the Euclidean Algorithm, but the latter as generally understood is the procedure that can be used to establish the values of $m$ and $n$, and for any pair of integers, not necessarily coprime.