\right]
\implies
\left[
- \bigforall_{D \text{ s.t. } \patchof{D} = \bot}
+ \bigforall_{D \text{ s.t. } \isforeign{D}}
D \isin C \equiv D \le C
\right]
$$
\proof{
-Consider some $D$ s.t. $\patchof{D} = \bot$.
+Consider some $D$ s.t. $\isforeign{D}$.
If $D = C$, trivially true. For $D \neq C$,
by Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
And by Exact Ancestors $D \le L \equiv D \le C$.
$$
\left[
C \hasparents \set A \land
- \patchof{C} = \bot \land
- \bigforall_{A \in \set A} \patchof{A} = \bot
+ \isforeign{C} \land
+ \bigforall_{A \in \set A} \isforeign{A}
\right]
\implies
\left[
\bigforall_{D}
D \le C
\implies
- \patchof{D} = \bot
+ \isforeign{D}
\right]
$$
\proof{
-Consider some $D \le C$. If $D = C$, $\patchof{D} = \bot$ trivially.
+Consider some $D \le C$. If $D = C$, $\isforeign{D}$ trivially.
If $D \neq C$ then $D \le A$ where $A \in \set A$. By Foreign
-Contents of $A$, $\patchof{D} = \bot$.
+Contents of $A$, $\isforeign{D}$.
}