Divisor Relation in Integral Domain is Transitive

Theorem
Let $$\left({D, +, \circ}\right)$$ be an integral domain.

Let $$x, y, z \in D$$. Then:

$$x \backslash y \land y \backslash z \Longrightarrow x \backslash z$$.

Proof
Let $$x \backslash y \land y \backslash z$$.

Then from the definition of divisor, we have:


 * $$x \backslash y \iff \exists s \in D: y = s \circ x$$.
 * $$y \backslash z \iff \exists t \in D: z = t \circ y$$.

Then $$z = t \circ \left({s \circ x}\right) = \left({t \circ s}\right) \circ x$$.

Thus $$\exists \left({t \circ s}\right) \in D: z = \left({t \circ s}\right) \circ x$$ and the result follows.