This paper of tutorial nature gives some further details of proofs of some theorems related to the quantum dynamical Yang-Baxter equation. This mainly expands proofs given in “Lectures on the dynamical Yang-Baxter equation” by Etingof and Schiffmann, math.QA/9908064. This concerns the intertwining operator, the fusion matrix, the exchange matrix and the difference operators. The last part expands proofs given in “Traces of intertwiners for quantum groups and difference equations, I” by Etingof and Varchenko, math.QA/9907181. This concerns the dual Macdonald-Ruijsenaars equations.