Way Above Closure is Subset of Upper Closure of Element

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

Let $x \in S$.

Then:
 * $x^\gg \subseteq x^\succeq$

where
 * $x^\gg$ denotes the way above closure of $x$
 * $x^\succeq$ denotes the upper closure of $x$.

Proof
Let $y \in x^\gg$.

By definition of way above closure:
 * $x \ll y$

where $\ll$ denotes the way below relation.

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

Thus by definition of upper closure of element:
 * $y \in x^\succeq$