Associatehood is Equivalence Relation

Theorem
Let $$\left({D, +, \circ}\right)$$ be an integral domain whose zero is $$0_D$$ and whose unity is $$1_D$$.

Let $$\sim$$ be the relation defined on $$D$$ as:

$$\forall x, y \in D: x \sim y$$ iff $$x$$ is an associate of $$y$$

Then $$\sim$$ is an equivalence relation.

Proof
Checking in turn each of the critera for equivalence:

Reflexive
Clearly $$x \backslash x$$ as $$x = 1_D \circ x$$, so $$x \sim x$$.

Symmetric
By the definition, $$x \sim y \iff x \backslash y \and y \backslash x \iff y \sim x$$.

Transitive
$$ $$ $$ $$ $$