 We now define the interpretation function \(I\). Let \(a\) be any constant in the expanded language. By \((=\)I) and \((\exists\)I), \(\Gamma''\vdash \exists x x=a\), and so \(\exists x x=a \in \Gamma''\). By the construction of \(\Gamma'\), there is a sentence in the form \((\exists x x=a \rightarrow c_i =a)\) in \(\Gamma''\). We have that \(c_i =a\) is in \(\Gamma''\). As above, there is exactly one \(c_j\) in \(d\) such that \(c_{i}=c_{j}\) is in \(\Gamma''\). Let \(I(a)=c_j\). Notice that if \(c_i\) is a constant in the domain \(d\), then \(I\)(c\(_i)=c_i\). That is each \(c_i\) in \(d\) denotes itself.