Integer Combination of Coprime Integers

Theorem
Let $a, b \in \Z$ be integers, not both zero.

Then:
 * $a$ and $b$ are coprime


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

In such an integer combination $m a + n b = 1$, the integers $m$ and $n$ are also coprime.

Proof
The proof can conveniently be divided into two parts: