Complement in Distributive Lattice is Unique/Corollary

Corollary to Complement in Distributive Lattice is Unique
Let $\struct {S, \vee, \wedge, \preceq}$ be a Boolean lattice.

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

Proof
By definition, a Boolean lattice is a complemented distributive lattice.

A complemented lattice is bounded by definition.

By Complement in Distributive Lattice is Unique, it follows that every $a \in S$ has at most one complement.

Since $\struct {S, \vee, \wedge, \preceq}$ is complemented, the result follows.