 The final item in this proof is a lemma that for every sentence \(\theta\) in the expanded language, \(M\vDash \theta\) if and only if \(\theta\) is in \(\Gamma''\). This proceeds by induction on the complexity of \(\theta\). The case where \(\theta\) is atomic follows from the definitions of \(M\) (i.e., the domain \(d\) and the interpretation function \(I\)). The other cases follow from the various clauses in the definition of satisfaction.