If D is a set of subsets of a finite set such that aϵD,b ⊃ a ⇒ bϵD, then D is called a down-set. Berge proved that any down-set D is the disjoint union of pairs {a, b} such that a ∩ b = /b/ together with the singleton {⊘} if |D| is odd. A proof is given of the corresponding result for finite distributive lattices, together with a number of generalizations and analogues of it; when specialized to the lattice of all subsets of a finite set, this proof is rather simpler than was Berge's.