Lower Closure is Increasing

Theorem
Let $\struct {S, \preceq}$ be an ordered set.

Let $x, y$ be elements of $S$ such that
 * $x \preceq y$

Then:
 * $x^\preceq \subseteq y^\preceq$

where $y^\preceq$ denotes the lower closure of $y$.

Proof
Let $z \in x^\preceq$.

By definition of lower closure of element:
 * $z \preceq x$

By definition of transitivity:
 * $z \preceq y$

Thus again by definition of lower closure of element:
 * $z \in y^\preceq$