Complement in Boolean Algebra is Unique

Theorem
Let $\left({S, \vee, \wedge}\right)$ be a Boolean algebra.

Then for all $a \in S$, there is a unique $b \in S$ such that:


 * $a \wedge b = \bot, a \vee b = \top$

i.e., a valid choice for $\neg a$ as in axiom $(BA \ 4)$ for Boolean algebras.

Proof
Suppose $b, c \in S$ both satisfy the identities.

Then:

That is, $b = c$.

The result follows.