  We prove the closedness theorem over Henselian valued fields, which was
established over rank one valued fields in one of our recent papers. In the
proof, as before, we use the local behaviour of definable functions of one
variable and the so-called fiber shrinking, which is a relaxed version of curve
selection. Now our approach applies also relative quantifier elimination for
ordered abelian groups due to Cluckers--Halupczok. Afterwards the closedness
theorem will allow us to achieve i.a. the \L{}ojasiewicz inequality, curve
selection and extending hereditarily rational functions as well as to develop
the theory of regulous functions and sheaves.
