Just grammar

pdf/grandpa.tex

@@ -364,8 +364,8 @@ Then if $V_{r,v,t} \subseteq V_{r,v',t'}$ and $C_{r,v,t} \subseteq C_{r,v',t'}$,
 \begin{proof} 
 Since $v$ sees that $r$ is completable at time $t$, both $V_{r,v,t}$ and $C_{r,v,t}$ contain votes from $n-f$ voters and so the same holds for $V_{r,v',t'}$ and $C_{r,v',t'}$. 
 By Lemma \ref{lem:ghost-monotonicity} (ii), $g(V_{r,v',t'}) \geq g(V_{r,v,t})$.
-Using Lemma \ref{lem:impossible}, since it is impossible for $C_{r,v,t}$ to have a supermajority for any children of $g(V_{r,v,t}$, it is impossible for $C_{r,v',t'})$ as well and so $E_{r,v',t'} \leq g(V_{r,v,t})$.
-But now $E_{r,v,t}$,$E_{r,v',t'}$ are the last blocks on $\textrm{chain}(g(V_{r,v,t}))$ that it is possible for $C_{r,v,t},C_{r,v',t'}$ respectively to have a supermajority for.
+As it is impossible for $C_{r,v,t}$ to have a supermajority for any children of $g(V_{r,v,t}$, it follows from Lemma \ref{lem:impossible} (i \& ii) that it is impossible for $C_{r,v',t'})$ as well, and so $E_{r,v',t'} \leq g(V_{r,v,t})$.
+But now $E_{r,v,t}$ and $E_{r,v',t'}$ are the last blocks on $\textrm{chain}(g(V_{r,v,t}))$ for which it is possible for $C_{r,v,t}$ and $C_{r,v',t'}$ respectively to have a supermajority.
 Thus by Lemma \ref{lem:impossible} (ii), $E_{r,v',t'} \leq E_{r,v,t}$.
 \end{proof}