Proof: We assume that there exists a protocol A that achieves consensus in λ rounds and arrive at a contradiction. The proof is based on the standard bivalency argument using forward induction. A particular configuration C of a synchronous system is univalent if there is only one value that the correct players can agree upon. C is said to be bivalent if it is not univalent (either 1-valent or 0-valent). In the following, a l-round partial run rl denotes the execution of A up to the end of round l. We prove two lemmas similar to [1]. The second one contradicts the first and completes the necessity proof of the theorem. Lemma: Any (λ− 1)-round run rλ−1 is univalent. Proof: Suppose rλ−1 is bivalent. w.l.g. assume that the λround run r obtained by extending rλ−1 by one round such that no player crashes in round λ is 0-valent. Let r be a 1valent extension of rλ−1 where some players crash in round λ. The only difference between r and r is that some messages {m1,m2, . . . ,ms} were sent in r but not in r. We define runs r for all 2 ≤ i ≤ s + 1, as follows: For every i, 1 ≤ i ≤ s, r is identical to r, except that the message mi was sent in round λ. If mi was sent along the k-cast ∆i then for every player other than the recipients of ∆i, r i+1 is indistinguishable from r. Note that, since n > t + k, this includes at least one correct player. This implies that each of these runs is 1-valent. However the view of any correct player c in r is the same as that in r, which means that c should decide 0 in r, giving the contradiction. Lemma: There is a bivalent (λ− 1)-round run rλ−1. Proof: We show by induction on l that for each l, 0 ≤ l ≤ λ− 1, there is a bivalent l-round partial run rl.