Maximal Element of Nest is Greatest Element

Theorem
Let $A$ be a nest.

Let $x$ be a maximal element of $A$.

Then $x$ is the greatest element of $A$.

Proof
Let $x$ be a maximal element of $A$.

Let $y \in A$ be arbitrary such that $x \ne y$.

We have by definition of maximal element that:
 * $x \not \subset y$

and as $x \ne y$:
 * $x \nsubseteq y$

But because $A$ is a nest:
 * $x \subseteq y$ or $y \subseteq x$

from which it follows that it must be the case that:
 * $y \subseteq x$

As $y$ is arbitrary, the result follows by definition of greatest element.