Lemma 8.2.4 and Corollary 9.1.2 tell us that the rules CrL and ClR and each application of ClL with principal formula other than x:□A can be eliminated in S(T). It is however easy to show that we cannot eliminate left contractions of lwffs of the form x:□A and retain completeness of S(T). As an example, consider the S(T)-theorem x: ~ □ ~ (B ⊃ □B) and the proof (6.1) we gave in §6.1 (and see also the discussion at the end of §6.3).
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