Exercise 3, § 5

Let $R$ and $S$ be relations in $\mathscr{T}$, let $T$ be a term in $\mathscr{T}$, and let $x$ and $y$ be distinct letters. Suppose that $y$ is not a constant of $T$ and that $x$ does not appear in $T$. Let $\mathscr{T}'$ be the theory obtained by adjoining $S$ to the axioms of $\mathscr{T}$. If $R$ is functional in $x$ in $\mathscr{T}'$, and if $(T|y)S$ is a theorem in $\mathscr{T}$, then the relation $(T|y)R$ is functional in $X$ in $\mathscr{T}$.