  We prove Riemann's theorems on extensions of functions over certain mixed
characteristic analytic adic spaces, first introduced by Johansson and Newton.
We use these results to reprove a theorem of de Jong identifying global
sections of an $\mathcal{O}_K$-flat normal formal scheme, locally formally of
finite type over $\mathcal{O}_K$, with locally powerbounded sections over the
generic fibre.
