diff --git a/research/07 shelah-spencer QE/Shelah-Spencer QE.tex b/research/07 shelah-spencer QE/Shelah-Spencer QE.tex index 1dbec96c..3880bef0 100644 --- a/research/07 shelah-spencer QE/Shelah-Spencer QE.tex +++ b/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}