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% AUTHOR: Amlal El Mahrouss
% PURPOSE: Lemmas for Integrals.
\documentclass[11pt, a4paper]{article}
\usepackage{graphicx}
\usepackage{listings}
\usepackage{xcolor}
\usepackage{hyperref}
\usepackage{amsfonts}
\usepackage[margin=0.5in,top=1in,bottom=1in]{geometry}
\title{Lemmas for Integrals.}
\author{Amlal El Mahrouss\\amlal@nekernel.org}
\date{February 2026}
\begin{document}
\bf
\maketitle
\begin{center}
\rule[0.01cm]{17cm}{0.01cm}
\end{center}
\section{Lemmas}
\subsection{Definition}
Let two lemmas for the integrals $\alpha$ and $\beta$. $\beta$ and $\alpha$ shall be true given the following:
\subsubsection{Conditions of $\alpha$}
Let $\alpha$ be an integral of $t \in \mathbb{Z}$ which consists of a sum $S(t) = \sum_{k=1}^{n} (t!), \quad \forall k, n \in \mathbb{Z}$.
Consider the following integral:
\begin{equation}
\alpha(u) = \int S(t) du
\end{equation}
For all $\alpha$ in $0 < \alpha < e$, or $\alpha > e$.
For $\alpha \in \mathbb{R}, \quad u \in \mathbb{R}, \quad u > 0$.
\subsubsection{Conditions of $\beta$}
Let $\beta$ be an integral of $t \in \mathbb{Z}$ which consists of $\alpha \in \mathbb{R}$.
$\beta > \alpha$. Such that $\lim_{t \to e} (\alpha \to \infty$+).
\subsubsection{Conclusion}
We see that $\beta, \quad \alpha$ is defined if and only if $t > e$. We can conclude that the lemma holds and points toward $\infty+$.
\end{document}
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