# Completeness Theorem

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Lemma 4.3 and Theorem 4.5 provide a method to construct a proof for a given tautology in terms of the axioms of AXSP3B, as we show next with an example.

Example. The formula $p \vee (\neg (p \wedge \neg p))$ is a tautology as shown by its true table. The formula is a disjunction, we put $\eta = (\neg (p \wedge \neg p)$ . We prove that this formula is a Theorem by using Lemma 4.3.
Let $\nu$ be a Valuation such that $\nu (p)=0$ , then $\nu (\eta )=2$. According to case 2 (e) of Lemma 4.3 $\neg (p \wedge p) \vdash \neg (p \wedge p)$ and $\neg (p \wedge p) \vdash \neg (\neg \eta \wedge \neg \eta )$
According to axiom PB10 $\neg (\neg \eta \wedge \neg \eta )) \wedge \neg (p \wedge p) \rightarrow \neg (\neg (\eta \vee p) \wedge \neg (\eta \vee p))$. According to axiom PB1