An elementary proof that upper and lower powerdomain constructions commute

It was proved in [1] that lower and upper powerdomain constructions commute on all domains. In that proof, domains were represented as information systems. In [2] a rather complicated algebraic proof was given which relied on universality properties of powerdomains proved in the previous works of the author of [2]. Here we give an elementary algebraic proof that upper and lower powerdomain constructions commute. The proof is essentially a reduction of the problem to establishing a 1-1 correspondence between certain disjunctive and conjunctive normal forms.