Way Below Relation is Transitive

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $x, y, z \in S$ such that
 * $x \ll y \ll z$

Then
 * $x \ll z$

Proof
By Way Below implies Preceding:
 * $x \preceq y$

By definition of reflexivity:
 * $z \preceq z$

Thus by Preceding and Way Below implies Way Below:
 * $x \ll z$