Theorem and proof blocks

Mathematical writing runs on numbered statements: theorems, lemmas, and definitions, each set off from the text and numbered so later proofs can cite them. The amsthm package supplies the machinery. You declare the kinds of blocks your document needs once in the preamble, then use them as ordinary environments.

\usepackage{amsthm}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]

\begin{theorem}[Optional name]
 Statement.
\end{theorem}
\begin{proof}
 Argument. \qedhere
\end{proof}

Declaring the environments

\newtheorem{theorem}{Theorem}[section] creates a theorem environment. The first argument is the environment name you will type, the second is the heading text that prints, and the trailing [section] ties the numbering to sections, so the third theorem of section 2 is Theorem 2.3 and the counter resets at each new section.

\newtheorem{lemma}[theorem]{Lemma} places the optional argument in the other position, where it means something different: [theorem] says lemmas share the theorem counter instead of keeping their own. A shared counter produces a single sequence, Theorem 2.1 followed by Lemma 2.2, which is easier to navigate than parallel sequences where Lemma 2.1 and Theorem 2.1 both exist.

\theoremstyle sets the look of every declaration that follows it. The plain style prints the body in italics, the traditional treatment for theorems, lemmas, and propositions. The definition style keeps the body upright, which suits definitions and examples. A third style, remark, is lighter still and suits notes and observations. Group your declarations under the style each one should use.

Using the blocks

The optional argument of the environment prints a name after the heading, so \begin{theorem}[Optional name] yields something like Theorem 2.4 (Fubini). The proof environment prints an italic Proof heading and closes with the end-of-proof mark, the small square set flush right.

\qedhere repositions that square when the proof does not end in ordinary text. If the last line of a proof is a displayed equation, the automatic mark would fall on an empty line below it; writing \qedhere inside the display puts the square on the equation’s own line. When a proof ends in plain text, as in the example, the command is harmless but unnecessary.

Theorem blocks are numbered objects like any other, so \label{thm:main} inside the environment and Theorem~\ref{thm:main} in the text work with the usual two-pass rules described in numbers and pointers for equations. The common declaration mistake is putting every environment under \theoremstyle{plain}, which sets multi-paragraph definitions in italics; keep definitions and examples under the definition style.