\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[margin=0.5in, top=1in]{geometry}
\usepackage{amsmath,amssymb}
\usepackage{tikz}
\newcommand{\N}{\mathbb{N}}
\input{../.headers/additions.tex}
\begin{document}

\begin{center}
	Soit $E, F$ des ensembles.
\end{center}
	\begin{center}
	\section{Lemme des bergers}
		
	\end{center}

	\begin{align*}
		\left( |E|, |F| \in \N \land \exists p\in \N, \begin{cases}
			\exists I\subset \N, \begin{cases}
				E = \coprod_{i\in I}E_i \\
				\forall i\in I,\ |E_i| = p
			\end{cases} \\
			\text{ou} \\
			\exists f\in \surjections(E, F),\ \forall x\in F,\ |f^{\leftarrow}(\{x\} )| = p
		\end{cases} \right) \implies |E| &= \frac{|F|}{p} \\
	\end{align*}

	\begin{center}
		\section{Principe des tirroirs}
	\end{center}

	\begin{align*}
		|E| < |F| < +\infty \implies \begin{cases}
			\surjections(E, F) &= \emptyset \\
			\injections(F, E) &= \emptyset \\
		\end{cases}
	\end{align*}
\end{document}