  Let $K$ be an algebraically closed field endowed with a complete
non-archimedean norm with valuation ring $R$. Let $f\colon Y\to X$ be a map of
$K$-affinoid varieties. In this paper we study the analytic structure of the
image $f(Y)\subset X$; such an image is a typical example of a subanalytic set.
Using Embedded Resolution of Singularities, we derive in the zero
characteristic case a Uniformization Theorem for subanalytic sets: after
finitely many local blowing ups with smooth centres, a subanalytic set becomes
semi-analytic. To prove this we establish a Flattening Theorem for affinoid
varieties in the style of Hironaka, which allows a reduction to the study of
subanalytic sets arising from flat maps. Specifically we show that a map of
affinoid varieties can be rendered flat by using only finitely many local
blowing ups. The case of an image under a flat map is then dealt with by a
small extension of a result of Raynaud.
  Our result can be conveniently stated as a Quantifier Elimination theorem for
the valuation ring $R$ in an analytic expansion of the language of valued
fields. This formulation is in the style of Denef and van den Dries.
