 Separation theorem. Suppose \(\phi\) is an \(\mathbf{IFL}\) sentence of vocabulary \(\tau\), and \(\psi\) an \(\mathbf{IFL}\) sentence of vocabulary \(\tau'\). Suppose further that \(\phi\) and \(\psi\) have no models in common. Then there is a first-order sentence \(\theta\) of vocabulary \(\tau \cap \tau'\) such that every model of \(\phi\) is a model of \(\theta\), but \(\theta\) and \(\psi\) have no models in common. 