from Lemma 10.2.2 instantiated to 2 x = [+] a , we obtain that {? | [+] a ? ? [?]} ? {¬? } is consistent. But this is impossible because, from the hypothesis (2) implies (1) Suppose that Now (AB2) guarantees that ?? a ? ? ?. In any normal modal logic we have that (2? ? 3) ? 3?. We use this with the previous formula and we obtain ([?] a ? ? ?? a ) ? ?. Note that ?? a ? ? because otherwise [?] a ? ? ? and, from the hypothesis we obtain that ? ? [?] -impossible. Now, if we prove that, Suppose that Now note that [?] a ? ? [? ] implies, using Lemma 10.2.2 instantiated with 2 x = [?] a , that {? | 10.2. Soundness and Completeness We now need to establish the analogous results for the starred modalities . In what follows ,
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