\section{Some lemmas}
-\subsection{Alternative (overlapping) formulations of $\mergeof{C}{L}{M}{R}$}
+\subsection{Alternative (overlapping) formulations of $\commitmergeof{C}{L}{M}{R}$}
$$
D \isin C \equiv
\begin{cases}
\text{as above with L and R exchanged}
\end{cases}
$$
-\proof{ ~ Truth table (ordered by original definition): \\
+\proof{ ~ Truth table (ordered by original definitions): \\
\begin{tabular}{cccc|c|cc}
$D = C$ &
$\isin L$ &
\right]
\implies
\left[
- \bigforall_{D \text{ s.t. } \isforeign{D}}
+ \bigforall_{D \in \foreign}
D \isin C \equiv D \le C
\right]
$$
\proof{
-Consider some $D$ s.t. $\isforeign{D}$.
+Consider some $D \in \foreign$.
If $D = C$, trivially true. For $D \neq C$,
by Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
And by Exact Ancestors $D \le L \equiv D \le C$.