diff --git a/main.tex b/main.tex
index 77c2cedef6e7ec8f85f4634143eae8afb53e304c..489bee52a9664a04f6238aa6d6fe51fcc600db58 100644
--- a/main.tex
+++ b/main.tex
@@ -1041,36 +1041,33 @@ inverse, for each $q$ considered.
 Consider the following tautology:
 
 \begin{align}
-	&\frac{ x }{ m }
+	\frac{ x }{ m }
 	- \frac{
 		(\aa r+2\bb)\aa
 	}{
 		2n^2
 	}
 	= \frac{ k }{ 2mn^2 }
-	\quad \text{for some } x \in \ZZ, k \in \ZZ_{>0}
+	\quad \text{for some } x \in \ZZ
+	\span \span \span \span \span
 	\label{eqn:finding_better_eps_problem}
-\\ &\iff
+\\ &\Longleftrightarrow& 
 	- (\aa r+2\bb)\aa m
-	\equiv k
-	\quad \mod 2n^2
-	\quad \text{for some } k \in \ZZ_{>0}
-\\ &\iff
+	&\equiv k &&
+	\mod 2n^2
+\\ &\Longleftrightarrow&
 	- \aa^2 m r - 2\aa\bb m
-	\equiv k
-	\quad \mod 2n^2
-	\quad \text{for some } k \in \ZZ_{>0}
-\\ &\Longrightarrow
+	&\equiv k &&
+	\mod 2n^2
+\\ &\Longrightarrow&
   \aa^2 \aa^{'}\bb m - 2\aa\bb m
-	\equiv k
-	\quad \mod \gcd(2n^2, \aa^2 mn)
-	\quad \text{for some } k \in \ZZ_{>0}
+	&\equiv k &&
+	\mod \gcd(2n^2, \aa^2 mn)
 	\label{eqn:better_eps_problem_k_mod_gcd2n2_a2mn}
-\\ &\Longrightarrow
+\\ &\Longrightarrow&
   -\aa\bb m
-	\equiv k
-	\quad \mod n
-	\quad \text{for some } k \in \ZZ_{>0}
+	&\equiv k &&
+	\mod n
 	\label{eqn:better_eps_problem_k_mod_n}
 \end{align}