Prove Theorem 33C, stating that true (in ) quantifier-free sentences are theorems of

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Theorem 33C. For any quantifier-free sentence true in , .

Proof in the textbook (incomplete)

Start with the atomic sentences; these will be of the form or for variable-free terms and . Show that proves if is true in , and refutes (i.e., proves ) if is false in .

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Lemma. For any variable-free term t, there is a unique number n such that .

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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.