Complement in Distributive Lattice is Unique

Theorem
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a bounded distributive lattice.

Then every $a \in S$ admits at most one complement.

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

Then every $a \in S$ has a unique complement $\neg a$.

Proof
Let $a \in S$, and suppose that $b, c \in S$ are complements for $a$.

Then:

Interchanging $c$ and $b$ in the above gives that $c = c \wedge b$ as well.

Hence $b = c$, as desired.