  We develop an extension theory for analytic valuation rings in order to
establish Ax-Kochen-Ersov type results for these structures. New is that we can
add in salient cases lifts of the residue field and the value group and show
that the induced structure on the lifted residue field is just its field
structure, and on the lifted value group is just its ordered abelian group
structure. This restores an analogy with the non-analytic AKE-setting that was
missing in earlier treatments of analytic AKE-theory.
