Divisor Relation on Positive Integers is Partial Ordering

Theorem
The divisor relation is a partial ordering of $\Z_{>0}$.

Proof
Checking in turn each of the criteria for an ordering:

Divisor Relation is Transitive

 * $\forall x, y, z \in \Z: x \mathop \backslash y \land y \mathop \backslash z \implies x \mathop \backslash z$: