Non-Zero Integer has Unique Positive Integer Associate

Theorem
Let $a \in \Z$ be an integer such that $a \ne 0$.

Then $a$ has a unique associate $b \in \Z_{>0}$.

Proof
Let $a, b, c \in \Z_{\ne 0}$ such that $b > 0$ and $c > 0$.

Let $a \sim b$ and $a \sim c$ where $\sim$ denotes the relation of associatehood.

By definition of associatehood:
 * $a \mathop \backslash b$ and $b \mathop \backslash a$

and:
 * $a \mathop \backslash c$ and $c \mathop \backslash a$

From Divides is Antisymmetric/Corollary/Proof 2:
 * $a = \pm b$

and
 * $a = \pm c$

That is:
 * $\pm b = \pm c$

which means:
 * $b = c$ or $b = -c$

But as both $b > 0$ and $c > 0$:
 * $b = c$

Hence the result.