diff --git a/out/AStudyOnDynamicEquations.pdf b/out/AStudyOnDynamicEquations.pdf
index 4a29e08..b310feb 100644
Binary files a/out/AStudyOnDynamicEquations.pdf and b/out/AStudyOnDynamicEquations.pdf differ
diff --git a/src/sections/proof-of-main-theorem.tex b/src/sections/proof-of-main-theorem.tex
index 49d45b6..695e5c7 100644
--- a/src/sections/proof-of-main-theorem.tex
+++ b/src/sections/proof-of-main-theorem.tex
@@ -46,13 +46,10 @@ \subsection*{Proof of theorem ~\ref{main_theorem}}
 \]
 Now, let's express the nominator of ~\eqref{eq:proof1} as follows
 \begin{align*}
-    \\
     \polynomialP{m}{\sigma(b)}{\sigma(x)} - \polynomialP{m}{b}{x}
     &= \polynomialP{m}{b}{x}^{\Delta}_{x} (x, \sigma(b)) \cdot \Delta x + \polynomialP{m}{b}{x}^{\Delta}_{b} (x,b) \cdot \Delta b \\
-    \\
     \polynomialP{m}{\sigma(b)}{\sigma(x)} - \polynomialP{m}{b}{x}
     &= \polynomialP{m}{b}{x}^{\Delta}_{x} (x, \sigma(b)) \cdot (\sigma(x) - x) + \polynomialP{m}{b}{x}^{\Delta}_{b} (x,b) \cdot (\sigma(b) - b)
-    \\
 \end{align*}
 We can collapse the terms $(\sigma(x) - x), \; (\sigma(b) - b)$ in above expressions, as $b\to x$.
 Therefore,
@@ -70,4 +67,4 @@ \subsection*{Proof of theorem ~\ref{main_theorem}}
     + \pTsDerivative{\polynomialP{m}{b}{x}}{b} \bigg |_{x = t, \; b = t}
 \end{equation*}
 
-This completes the proof. \qed
\ No newline at end of file
+This completes the proof. \qed