  Several years ago Th\'erien and Wilke exhibited a decidable characterization
of the languages of words that are definable in FO2(<,+1). Their proof relies
on three separate ingredients. The first one is the characterization of the
languages that are definable in FO2(<) as those whose syntactic semigroup
belongs to the variety DA. Then, this result is combined with a wreath product
argument showing that being definable in FO2(<,+1) corresponds to having a
syntactic semigroup in DA*D. Finally, proving that membership of a semigroup in
DA*D is decidable requires a third ingredient: the "locality" of DA, a result
proved by Almeida. In this note we present a new self-contained and simple
proof that definability in FO2(<,+1) is decidable. We obtain the locality of DA
as a corollary.
