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:
 * $(1): \quad \forall x \in S: x \vee \top = \top$
 * $(2): \quad \forall x \in S: x \wedge \bot = \bot$

That is, $\bot$ is a zero element for $\wedge$, and $\top$ 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.