  We give a procedure for translating geometric Kripke frame axioms into
structural hypersequent rules for the corresponding intermediate logics in
Int^*/Geo that admit weakening, contraction and in some cases, cut. We give a
procedure for translating labelled sequents in the corresponding logic to
hypersequents that share the same linear models (which correspond to
G\"odel-Dummett logic). We prove that labelled proofs Int^*/Geo can be
translated into hypersequent proofs that may use the linearity rule, which
corresponds to the well-known communication rule for G\"odel-Dummett logic.
