Complement in Distributive Lattice is Unique/Corollary

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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.

$\blacksquare$