  We give a new proof of quantifier elimination in the theory of all ordered
abelian groups in a suitable language. More precisely, this is only "quantifier
elimination relative to ordered sets" in the following sense. Each definable
set in the group is a union of a family of quantifier free definable sets,
where the parameter of the family runs over a set definable (with quantifiers)
in a sort which carries the structure of an ordered set with some additional
unary predicates.
  As a corollary, we find that all definable functions in ordered abelian
groups are piecewise affine linear on finitely many definable pieces.
