/ ‘research/07 shelah-spencer QE/Shelah-Spencer QE.tex’

research/07 shelah-spencer QE/Shelah-Spencer QE.tex

@@ -126,7 +126,7 @@ \section{Quantifier Elimination}
 \end{align*}
 
 \begin{proof}
-	$(\Rightarrow)$ Fix $\B \subset \M$ witnessing existential statement. By remark 5.3 and lemma 3.8 in \cite{Laskowski}there is a unique $\B^* \in X_m$ maximally embeddable (with unique image) into $\M$ over $\B$ . By lemma \ref{C} $\B^* \in Y(\B, \Phi, \Gamma, m)$.
+	$(\Rightarrow)$ Fix $\B \subset \M$ witnessing existential statement. By remark 5.3 and lemma 3.8 in \cite{Laskowski} there is a unique $\B^* \in X_m$ maximally embeddable (with unique image) into $\M$ over $\B$ . By lemma \ref{C} $\B^* \in Y(\B, \Phi, \Gamma, m)$.
 
 	$(\Leftarrow)$ Take the embedding $g\colon B^* \to \M$ and restrict it to $\B \subseteq \B^*$ i.e. $f = g \mid \B$. As $\B^* \in Y(\B, \Phi, \Gamma, m)$ by lemma \ref{C} $f$ omits $\Phi$ and admits $\Gamma$. Thus is is a witness to $\exists y \theta(x, y)$.
 \end{proof}