Suppose an non-empty $n$ -place relation $R$ is definable by an $L_{0}$ -formula $\phi$ in $\mathfrak{N}_{0}$ (for the empty relation the conclusion is obvious). Consider $\pi$ from the set of parameters of $L_{0}$ into the set of formulas of $L_{1}$ such that $\pi_{\forall}=v_{1}=v_{1}$ , $\pi_{+}=+v_{1}v_{2}v_{3}$ and $\pi_{\cdot}=\cdot v_{1}v_{2}v_{3}$ . Since $\exists v_{1}v_{1}=v_{1}$ , $\forall v_{1}\forall v_{2}\exists x\forall v_{3}+v_{1}v_{2}v_{3}\leftrightarrow v_{3}=x$ , and $\forall v_{1}\forall v_{2}\exists x\forall v_{3}\cdot v_{1}v_{2}v_{3}\leftrightarrow v_{3}=x$ are true in $\mathfrak{N}_{1}$ , $\pi$ is an interpretation of $L_{0}$ into $\mbox{Th}\mathfrak{N}_{1}$ , and $\mathfrak{N}_{1}$ is a model of $\mbox{Th}\mathfrak{N}_{1}$ . Then, $|{}^{\pi}\mathfrak{N}_{1}|=\mathbb{N}$ , $a+^{^{\pi}\mathfrak{N}_{1}}b=c$ iff $+^{\mathfrak{N}_{1}}abc$ iff $a+b=c$ , and $a\cdot^{^{\pi}\mathfrak{N}_{1}}b=c$ iff $\cdot^{\mathfrak{N}_{1}}abc$ iff $a\cdot b=c$ , implying that $^{\pi}\mathfrak{N}_{1}=\mathfrak{N}_{0}$ . But then, for every $s:V\rightarrow\mathbb{N}$ , $\vDash_{\mathfrak{N}_{0}}\phi[s]$ iff $\vDash_{^{\pi}\mathfrak{N}_{1}}\phi[s]$ iff $\vDash_{\mathfrak{N}_{1}}\phi^{n}[s]$ (Lemma 27B), implying that $R$ is definable in $\mathfrak{N}_{1}$ by $\phi^{n}$ .