diff --git a/notes.tex b/notes.tex
index 29755a2..c47e720 100644
--- a/notes.tex
+++ b/notes.tex
@@ -1,10 +1,11 @@
-\documentclass[a4paper]{article}
+\documentclass[a4paper,11pt]{article}
 \usepackage{chao}
+\DeclareMathOperator{\supp}{supp}
 \title{Cogirth of Perturbed Graphic Matroids}
 \author{}
 \date{}
 \begin{document}
-% \maketitle
+\maketitle
 
 We want to use basepacking on perturbed graphic matroid. Basepacking works for deletion closed matroid classes with constant cogirth-packing gap. 
 PGMs are closed under deletion. It remains to show PGMs have constant gap.
@@ -85,8 +86,8 @@ Let $A$ be the binary representation of $M$. Note that the cocircuit space of $M
 
 Now we consider the hitting sets. It follows from definitions that the minimum hitting set of $\{B-e |\forall B ,\forall e\in B\cap C(B,f)\}$ is exactly the minimum cocircuit of $(M+f)/f$, which is the minimum set in the cocircuit space with $yv=0$. So the minimum hitting set of $B-e$ is always an even cocycle.
 
-Consider a set $X$ which fails to hit all $\{C(B,f)|\forall B\}$. Then there must be a set $F\subseteq E\setminus X$ that span the new element $f$. On matrix, this is equivalent to the existence of $\chi_F$ that $A\chi_F=v$. Using Farkas' lemma on $\mathbb F_2$, this implies that there dose not exists $y$ that ${\rm supp}(yA)\in X\land yv=1$. (chatGPT says one can use Farkas lemma on finite field like this, need to verify).
-So $X$ hits all $\{C(B,f)|\forall B\}$ iff there is such a $y$. Minimizing $X$ pushes ${\rm supp}(yA)$ which is an odd cocycle. Hence, the minimum hitting set for $C(B,f)$ is always an odd cocycle.
+Consider a set $X$ which fails to hit all $\{C(B,f)|\forall B\}$. Then there must be a set $F\subseteq E\setminus X$ that span the new element $f$. On matrix, this is equivalent to the existence of $\chi_F$ that $A\chi_F=v$. Using Farkas' lemma on $\mathbb F_2$, this implies that there dose not exists $y$ that $\supp(yA)\in X\land yv=1$. (chatGPT says one can use Farkas lemma on finite field like this, need to verify).
+So $X$ hits all $\{C(B,f)|\forall B\}$ iff there is such a $y$. Minimizing $X$ pushes $\supp(yA)$ which is an odd cocycle. Hence, the minimum hitting set for $C(B,f)$ is always an odd cocycle.
 
 The theorem then follows directly from the fact that any cocycle is either odd or even.