open neighborhood $U$ of $\operatorname{{tp}}_{\Delta}(ab/M)/E_\Delta$. Then $[\operatorname{{tp}}_{\Delta}(ab/M)]_{E_\Delta}$ is contained in the preimage $U'$ of $U$ under the quotient map. Since $[\operatorname{{tp}}_{\Delta}(ab/M)]_{E_\Delta}$ is closed and $U'$ is open, by the compactness of $S_{G,\Delta}(M)$, we can find a $\Delta$-formula (over $M$) $\varphi(x)$ such that $[\operatorname{{tp}}_{\Delta}(ab/M)]_{E_\Delta} \subseteq [\varphi(x)] \subseteq U'$, and so $U_\varphi$ is an open neighborhood of $\operatorname{{tp}}_{\Delta}(ab/M)/E_\Delta$ contained in $U$.

Consider any $\varphi$ as above. Then $\operatorname{{tp}}_{\Delta}(ab/M) \vdash E_\Delta'(ab,x) \vdash \varphi(x)$. So there is $\psi(x) \in \operatorname{{tp}}_{\Delta}(ab/M)$ such that

$$(\exists y)(\psi(y) \wedge E_\Delta'(x,y)) \vdash \varphi(x).\label{eq: 1}$$

Clearly $\models \psi(ab)$.

Consider any $\delta(w) \in p$. As $\models \delta(a) \wedge \psi(ab)$, we get $\delta(w) \wedge \psi(wb) \in \operatorname{{tp}}_\Delta(a/M,b)$. (Note that in order to have that $\psi(wb)$ is a $\Delta$-formula over $M,b$, we use our convention which allows Boolean combinations of instances of formulas from $\Delta$ with parameters from $\operatorname{{dcl}}(M,b)$; namely, we have to use parameters which are products of the form $b\cdot g \in \operatorname{{dcl}}(M,b)$ where $g \in G$.) By the assumption that $\operatorname{{tp}}_{\Delta}(a/M,b)$ is finitely satisfiable in $M$, there exists $g_{\delta,\varphi} \in G$ such that $\models \delta(g_{\delta,\varphi}) \wedge \psi(g_{\delta,\varphi} \cdot b)$. By (\[eq: 1\]), we conclude that $E_\Delta'(g_{\delta,\varphi} \cdot b, x) \vdash \varphi(x)$, and so $$g_{\delta,\varphi} (q/E_\Delta)=\operatorname{{tp}}_\Delta(g_{\delta,\varphi} \cdot b/M)/E_\Delta \in U_\varphi.\label{eq: 2}$$

