Join with Complement is Top

Theorem
Let $\left({S, \vee, \wedge, \neg}\right)$ be a Boolean algebra, defined as in Definition 2.

Then:
 * $\exists \top \in S: \forall a \in S: a \vee \neg a = \top$

where $\wedge$ denotes the meet operation in $S$.

This element $\top$ is unique for any given $S$, and is named top.

Proof
Let $\exists r, s \in S: r \vee \neg r = a, \ s \vee \neg s = b$

Then:

Thus, whatever $r$ and $s$ may be:
 * $r \vee \neg r = s \vee \neg s$

This unique element can be assigned the symbol $\top$ and named top as required.