  The main purpose of the paper is to establish a closedness theorem over
Henselian valued fields $K$ of equicharacteristic zero (not necessarily
algebraically closed) with separated analytic structure. It says that every
projection with a projective fiber is a definably closed map. This remains
valid also for valued fields with analytic structure induced by a strictly
convergent Weierstrass systems, including the classical, complete rank one
valued fields with the Tate algebra of strictly convergent power series. As
application, we prove two theorems on existence of the limit and on piecewise
continuity.
