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, $x = \neg \neg x$.
$\blacksquare$