Complement of Complement in Uniquely Complemented Lattice

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Theorem

Let $\struct {S, \wedge, \vee, \preceq}$ 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, $x = \neg \neg x$.

$\blacksquare$