Lemma doGenericServer_key_in_output_list : forall net h os st' ms, raft_intermediate_reachable net -> doGenericServer h (nwState net h) = (os, st', ms) -> key_in_output_list client id os -> exists e : entry, eClient e = client /\ eId e = id /\ In e (applied_entries (update name_eq_dec (nwState net) h st')).