3 files changed, 41 insertions, 10 deletions
diff --git a/source/dn001.07/paper.tex b/source/dn001.07/paper.tex
index d2caef5..252d730 100644
--- a/source/dn001.07/paper.tex
+++ b/source/dn001.07/paper.tex
@@ -10,6 +10,11 @@
 \usepackage{amsfonts}
 \usepackage[margin=0.5in,top=1in,bottom=1in]{geometry}
 
+\newenvironment{proof}{\paragraph{PROOF:}}{\hfill$\square$}
+
+\newtheorem{thm}{LEMMA}[section]
+\newtheorem{defn}[thm]{DEFINITION}
+
 \title{Lemmas for Integrals.}
 \author{Amlal El Mahrouss\\amlal@nekernel.org}
 \date{February 2026}
@@ -21,7 +26,11 @@
 	\rule[0.01cm]{17cm}{0.01cm}
 \end{center}
 
-\section{Definitions}
+\begin{abstract}
+    The paper covers several lemmas and definitions that we will define to help us connect the dots for series in analysis.
+\end{abstract}
+
+\section{DEFINITIONS}
 
 We assume that $u \in \mathbb{R}, \quad \forall u \in \mathbb{R} : u < \pi, \quad 0 < u$
 
@@ -30,6 +39,19 @@ We assume that $u \in \mathbb{R}, \quad \forall u \in \mathbb{R} : u < \pi, \qua
 \end{equation}
 Where $u, u_{0} > 0, \quad u, u_{0} \in \mathbb{R}$, such that $S : (u, \pi) \to \mathbb{R}$.
 
-\section{Lemmas}
+\section{LEMMAS}
+
+\subsection{FIRST LEMMA}
+
+\begin{defn}We assume the value $B_{0} \in \mathbb{R}, \quad \pi > B_{0} > 0$. \end{defn}
+\begin{thm}For all variables of $B \to B_{n}$ such that $0 < B_{n} < \pi$, the variable $B_{n}$ is defined in $I_{1}$.\end{thm}
+
+\begin{proof}
+We assume that:
+\begin{equation}
+    \lim_{B_{n} \to \infty} = B_{n} \to I_{1}, \quad
+\end{equation}
+If and only if: $B_{0} > 0, \quad B_{n} \in \mathbb{R}$.
+\end{proof}
 
 \end{document}
diff --git a/source/wg04/paper.2.tex b/source/wg04/paper.2.tex
index 0cbfd64..93c3a59 100644
--- a/source/wg04/paper.2.tex
+++ b/source/wg04/paper.2.tex
@@ -22,15 +22,19 @@
 	\rule[0.01cm]{17cm}{0.01cm}
 \end{center}
 
-\section{Definitions}
+\begin{abstract}
+    The paper covers several definitions that are made to be used for sum analysis; their purpose has been kept abstract for flexibility reasons.
+\end{abstract}
+
+\section{DEFINITIONS}
 
 We assume $x \in \mathbb{R}, \quad a, b \in \mathbb{N}, \quad 0 < a < b < \cos(0)$ for:
 
-\subsection{The S-Integral:}
+\subsection{DEFINITION OF THE S-INTEGRAL:}
 \begin{equation}
     I_{1} \coloneqq \int_{a}^{b} S(x)dx
 \end{equation}
-\subsection{The S-Function:}
+\subsection{DEFINITION OF THE S-FUNCTION:}
 \begin{equation}
     S(x) = \sum_{k=1}^{t} (\frac{dx}{du}) \cdot x, \quad x > 0, \quad t > k
 \end{equation}
diff --git a/source/wg04/paper.tex b/source/wg04/paper.tex
index d205919..aa0cc59 100644
--- a/source/wg04/paper.tex
+++ b/source/wg04/paper.tex
@@ -22,23 +22,28 @@
 	\rule[0.01cm]{17cm}{0.01cm}
 \end{center}
 
-\section{Definitions}
+\begin{abstract}
+    The paper covers several definitions that are made to be used in analysis; their purpose has been kept abstract for flexibility reasons.
+\end{abstract}
 
+\section{DEFINITIONS}
+
+This section will cover the definitions and properties of the formulas we will use in this paper.
 We assume $x, y, z \in \mathbb{R}, \quad n \in \mathbb{N}, \quad \cos(\pi) < a < b, \quad t = a$
-\subsection{The A-Integral:}
+\subsection{DEFINITION OF THE A-INTEGRAL:}
 \begin{equation}
     I_{1} \coloneqq \sum_{n}^{t} \int_{a}^{b} S(x) d \cdot x
 \end{equation}
-\subsection{The A-Function:}
+\subsection{DEFINITION OF THE A-FUNCTION:}
 \begin{equation}
 	S(x) : (x, y) \to \mathbb{R}
 \end{equation}
-\subsection{The A-Equation:}
+\subsection{DEFINITION OF THE A-EQUATION:}
 \begin{equation}
 	I_{2} \coloneqq \int_{a}^{b} E_{1}d \cdot x, \quad E_{1} = Z(x) := \frac{(S(x) \cdot y)}{\Delta t \cdot z}
 \end{equation}
 Where $\Delta t \in \mathbb{R}$ such that $0 < \Delta t$.
-\subsection{The K-Equation:}
+\subsection{DEFINITION OF THE K-EQUATION:}
 \begin{equation}
 	K_{n} \coloneqq \int_{a}^{b} S_{n}(u_{n}) \quad d \cdot u_{n}, \quad \forall u_{n} \in \mathbb{R}
 \end{equation}