Tweak e_{v,q} lemma/dfn

tex/bounds-on-semistabilisers.tex

@@ -925,21 +925,26 @@ for $d$.
 Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_v$ in the
 proof of Theorem \ref{thm:rmax_with_uniform_eps}:
 
-\begin{lemmadfn}[
+\begin{lemmadfn}[%
 	Finding a better alternative to $\epsilon_v$:
 	$\epsilon_{v,q}$
 	]
 	\label{lemdfn:epsilon_q}
 	Suppose $d \in \frac{1}{\lcm(m,2n^2)}\ZZ$ satisfies the condition in
-	eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
+	Equation \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
 	That is:
 
 	\begin{equation*}
 		\sage{positive_radius_condition_with_q.subs([q_value_expr,beta_value_expr]).factor()}
+		\qquad
+		\text{for some integers }
+		a_v, b_q, n
+		\:\text{with }(a_v, n)=1
 	\end{equation*}
 
 	\noindent
-	Then we have:
+	And $r$ satisfies $\aa r + \bb \equiv 0 \pmod{n}$,
+	then we have:
 
 	\begin{equation}
 		\label{eqn:epsilon_q_lemma_prop}
@@ -1021,13 +1026,13 @@ and the final logical equivalence is just a simplification of the expressions.
 	\nonumber
 \end{align}
 
-In our situation, we want to find the least $k>0$ satisfying 
-eqn \ref{eqn:finding_better_eps_problem}.
+In our situation, we want to find the least $k>0$ satisfying
+Equation \ref{eqn:finding_better_eps_problem}.
 Since such a $k$ must also satisfy eqn \ref{eqn:better_eps_problem_k_mod_n},
 we can pick the smallest $k_{q,v} \in \ZZ_{>0}$ which satisfies this new condition
 (a computation only depending on $q$ and $\beta$, but not $r$).
-We are then guaranteed that $k_{v,q}$ is less than any $k$ satisfying eqn
-\ref{eqn:finding_better_eps_problem}, giving the first inequality in eqn
+We are then guaranteed that $k_{v,q}$ is less than any $k$ satisfying Equation
+\ref{eqn:finding_better_eps_problem}, giving the first inequality in Equation
 \ref{eqn:epsilon_q_lemma_prop}.
 Furthermore, $k_{v,q}\geq 1$ gives the second part of the inequality:
 $\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.