Simple Proof Methods

The following $\LaTeX$ source code shows how to create definitions, theorems, proofs, and corollaries:

\documentclass[12pt]{article}
 
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm} % The Theorem library.
 
\setlength{\parskip}{10pt} % Space between paragraphs.
\setlength\parindent{0pt} % Eliminating paragraph indentation.

% Definitions of the definition, theorem, and corollary environments:
\newtheorem{definition}{Definition}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}

\begin{document}
 
\begin{definition}
{\normalfont [Even integers]} An integer $n$ is called \textbf{even} if there exists an integer $k$ such that $n = 2k$.
\end{definition}

\begin{theorem}
{\normalfont [Sum of even integers]} If $m$ and $n$ are both even integers, then their sum, $m + n$, is also even.
\end{theorem}

\begin{proof}
{[Sum of even integers]} Let $m$ and $n$ be two even integers. According to the definition of even integers, there exists $k_1 \in \mathbb{Z}$ such that $m = 2k_1$ and $k_2 \in \mathbb{Z}$ such that $n = 2k_2$.

Notice that
$$m + n = 2k_1 + 2k_2 = 2(k_1 + k_2),$$
which is, by definition, an even integer because $(k_1 + k_2)$ is an integer.
\end{proof}

\begin{corollary}
{\normalfont [Square of an odd integer]} If $n$ is an odd integer, then $n^2$ is also odd.
\end{corollary}

This is a corollary of a probably more general theorem:
\begin{theorem}
{\normalfont [Product of odd integers]} If $n_1, n_2, n_3, \dots, n_m$ are odd integers, then their product, $n_1\cdot n_2\cdot n_3\cdot\dots\cdot n_m$, is also odd.
\end{theorem}
 
\end{document}