Proving Hilbert’s Axioms in Tarski Geometry

Hilbert’s geometry and Tarski’s geometry have the same models, since both are able to define addition and multiplication. But has anyone given explicit proofs of Hilbert’s axioms in Tarski geometry? The purpose of this note is to show that this has been done by Wanda Szmielew, by citing the specific theorems of her development that are needed. We define an explicit translation of Hilbert’s language into Tarski’s language and use Szmielew’s theorems to verify that this translation is an interpretation, i.e. takes theorems into theorems. Of course this is not an original result, but it doesn’t seem to appear explicitly in the literature and one geometer wanted to see it written out.