Complement of Complement in Uniquely Complemented Lattice

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

For each $x \in S$, let $\neg x$ be the complement of $x$.

Then for each $x \in S$:


 * $\neg \neg x = x$

Proof
By the definition of a complement of $x$:


 * $\neg x \vee x = \top$
 * $\neg x \wedge x = \bot$

Since $\vee$ and $\wedge$ are commutative:


 * $x \vee \neg x = \top$
 * $x \wedge \neg x = \bot$

Thus by the definition of complement, $x$ is a complement of $\neg x$.

By the definition of a uniquely complemented lattice, $x = \neg \neg x$.