Prove Theorem 33C, stating that true (in $\displaystyle N$) quantifier-free sentences are theorems of $\displaystyle A_E.$

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Theorem 33C. For any quantifier-free sentence $\displaystyle \tau$ true in $\displaystyle N$, $\displaystyle A_E \vdash \tau$.

Proof in the textbook (incomplete)

Start with the atomic sentences; these will be of the form $\displaystyle t_1 = t_2$ or $\displaystyle t_1 < t_2$ for variable-free terms $\displaystyle t_1$ and $\displaystyle t_2$. Show that $\displaystyle A_E$ proves $\displaystyle \tau$ if $\displaystyle \tau$ is true in $\displaystyle N$, and refutes $\displaystyle \tau$ (i.e., proves $\displaystyle \neg \tau$) if $\displaystyle \tau$ is false in $\displaystyle N$.

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Lemma. For any variable-free term t, there is a unique number n such that $\displaystyle A_E \vdash t = S^n0$.

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The necessary definitions can be found in here. (slide 4 and 5)

Any hint that I can start with will be appreciated.