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. Hence and . Ep, ??) M iff for all epistemic formulas ? , x ? r ep

. Remark, ) and (3) follows from the definition of truth-sets and the equivalence between (3) and (4) follows from the definition of r ep(·, ·) Note that in all other cases, the equivalence between (1) and (2) follows from the definition of r ep(·, ·), the equivalence between (2) and (3) follows from the definition of truth-sets, the equivalence between (3) and (4) follows from (H ), the equivalence between (4) and (5) follows from logical reasoning, the equivalence between (5) and (6) follows from the definition of truth-sets and the equivalence between, ) and (7) follows from the definition of r ep(·, ·)

?. 1. Hence, ? m ¬? ? x iff ? 1 . . . ? m ? ? x and ? 1

?. 1. Hence, ? m (? ? ?) ? x iff ? 1 . . . ? m ? ? x and ? 1

?. 1. Hence, ? m K a ? ? x iff ? 1 . . . ? m ? ? x and for all maximal consistent theories y containing ? 1 . . . ? m ?, if K a x ?

?. 1. Hence, ? m [?]? ? x iff ? 1 . . . ? m ? ? x and if ? 1 . . . ? m ? ? x

?. 1. Hence, ? m ?? iff ? 1 . . . ? m ? and for all epistemic formulas ?, if ? 1 . . . ? m ? ? x

. Remark, ) and (2) follows from Lemma (26), the equivalence between (2) and (3) follows from (H) and the equivalence between (3) and (4) follows from Lemmas (26) and (28) Note that in all the other cases, the equivalence between (1) and (2) follows from Lemma, the equivalence between (2) and (3) follows from (H) and the equivalence between (3) and (4) follows from Lemma