“…Before the present work, density elimination proofs were known for some (classes of) hypersequent calculi, though for logics different from Ł∀ and RPL∀; see [3], [1], [22], [11], [23], [8], [4], [5], [6], [24], [7], [28], and [29] (in chronological order). In all these works except [1], such proofs use the cut rule even if no application of it is in an initial formal proof.…”