Adjust first direction of proof for new statement of main lemma

main.tex

@@ -311,24 +311,30 @@ are equivalent to the following more numerical conditions:
 \end{lemma}
 
 \begin{proof}
-Let $u,v$ be Chern characters with positive ranks and
-$\Delta(u),\Delta(v) \geq 0$.
+Let $u,v$ be Chern characters with
+$\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.
 
 
 For the forwards implication, assume that the suppositions of the lemma are
-satisfied. The pseudo-wall intersects $\Theta_v^-$, at some point $Q$ further up
-the hyperbola branch than $P$ (to satisfy supposition b). At $Q$, we have
-$\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too. This means that $\Theta_u$ must
+satisfied. Let $Q$ be the point on $\Theta_v^-$ (above $P$) where $u$ is a
+pseudo-semistabilizer of $v$.
+Firstly, consequence 3 is part of the definition for $u$ being a
+pseudo-semistabilizer at a point with same $\beta$ value of $P$ (since the
+pseudo-wall surrounds $P$).
+If $u$ were to have 0 rank, it's tilt slope would be decreasing as $\beta$
+increases, contradicting supposition b. So $u$ must have strictly non-zero rank,
+and we can consider it's characteristic curves (or that of $-u$ in case of
+negative rank).
+$\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too. This means that $\Theta_{\pm u}$ must
 intersect $\Theta_v^-$ at $Q$. Considering the shapes of the hyperbolae alone,
 there are 3 distinct ways that they can intersect, as illustrated in Fig
 \ref{fig:hyperbol-intersection}. These cases are distinguished by whether it is
 the left, or the right branch of $\Theta_u$ involved, as well as the positions
 of the base. However, considering supposition b, only case 3 (green in
-figure) is valid. This is because we need $\nu_{P}(u)>0$ (or $\nu_{P}(-u)>0$ in
-case 1 involving $\Theta_u^+$).
+figure) is possible. This is because we need $\nu_{P}(u)>0$ (or $\nu_{P}(-u)>0$ in
+case 1 involving $\Theta_u^+$), to satisfy supposition b.
 Recalling how the sign of $\nu_{\alpha,\beta}(\pm u)$ changes (illustrated in
 Fig \ref{fig:charact_curves_vis}), we can eliminate cases 1 and 2.
-In passing, note that this implies consequence 3.
 
 \begin{sagesilent}
 def hyperbola_intersection_plot():
@@ -436,15 +442,21 @@ def correct_hyperbola_intersection_plot():
 \end{subfigure}
 \end{figure}
 
-Fixing attention on the only valid case (2), illustrated in Fig
-\ref{fig:correct-hyperbol-intersection}. We must have $\Theta_u^-$ taking a
+Fixing attention on the only possible case (2), illustrated in Fig
+\ref{fig:correct-hyperbol-intersection}.
+$P$ is on the left of $V_{\pm u}$ (first part of consequence 2), so $u$ must have positive rank (consequence 1)
+to ensure that $\chern_1^{\beta{P}} \geq 0$ (since the pseudo-wall passed over
+$P$).
+Furthermore, $P$ being on the left of $V_u$ implies
+$\chern_1^{\beta{P}}(u) \geq 0$,
+and therefore $\chern_2^{P}(u) > 0$ (consequence 4) to satisfy supposition b.
+Next considering the way the hyperbolae intersect, we must have $\Theta_u^-$ taking a
 base-point to the right $\Theta_v$, but then, further up, crossing over to the
 left side. The latter fact implies that the assymptote for $\Theta_u^-$ must be
 to the left of the one for $\Theta_v^-$. Given that they are parallel and
 intersect the $\beta$-axis at $\beta=\mu(u)$ and $\beta=\mu(v)$ respectively. We
-must have $\mu(u)<\mu(v)$, that is, $V_u$ is strictly to the left of $V_v$.
-Finally, the fact that it is the left branch of the hyperbola for $u$ implies
-consequence 1 and $\beta(P)<\mu(u)$ (consequence 2).
+must have $\mu(u)<\mu(v)$ (second part of consequence 2),
+that is, $V_u$ is strictly to the left of $V_v$.
 
 
 Conversely, suppose that the consequences 2 and 3 are satisfied. Consequence 2