Identities of Boolean Algebra are also Zeroes

Theorem
Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra, defined as in Definition 1.

Let the identity for $\vee$ be $\bot$ and the identity for $\wedge$ be $\top$.

Then:

That is:
 * $\bot$ is a zero element for $\wedge$
 * $\top$ is a zero element for $\vee$.

Proof
Let $x \in S$.

Then:

So $x \vee \top = \top$.

The result $x \wedge \bot = \bot$ follows from the Duality Principle.