Writing a mathematics course or paper

This guide writes one page of a course from start to finish — a definition, a theorem, its proof, an aligned calculation and a reference back to the theorem — and shows the habits that keep a long mathematical document consistent.

The page we are writing

Definition 1.1 (Group). A group is a set G with an associative operation, an identity element e, and an inverse for every element.
Theorem 1.2 (Uniqueness of the identity). A group has exactly one identity element.
Proof. Let e and e′ be two identity elements. Then e = e · e′ = e′. □
By Theorem 1.2, we may speak of the identity of a group.

Step 1 — A chapter

Insert a Heading, set its level to Chapter, and type "Groups". Statements will now be numbered 1.1, 1.2… If you prefer 1, 2, 3 throughout, do not use chapters: start with an H1 section instead.

Step 2 — The definition

Insert a Theorem block.
Click its heading and choose Definition.
In the name field after the heading, type "Group".
Type the statement. For the symbols, type $G$ , $e$ and so on: each becomes a formula as you close the $.

The heading reads Definition 1.1 (Group).

Step 3 — The theorem

Press Enter on the empty last line of the definition to leave it — or click below it — and insert another Theorem block.
Leave its kind as Theorem, and name it "Uniqueness of the identity".
Type the statement.

It is numbered 1.2: definitions and theorems share one counter, so the reader finds any statement by its number alone.

Step 4 — The proof

Hover the theorem and click Add a proof. Type the argument.

To set a calculation on its own lines inside the proof, press Enter for a new line, then insert an Equation from the left column: it lands inside the proof. Choose align to line the steps up:

e &= e \cdot e' \\
  &= e'

Choose Plain numbering for a calculation you will never refer to.

The □ that closes the proof is added by LaTeX.

Step 5 — Referring to the theorem

In the paragraph after the proof, type "By Theorem ", press Ctrl+Alt+R, and choose Theorem 1.2 in the list. Then finish the sentence.

If you later insert a lemma before the theorem, it becomes Theorem 1.3 — in its heading and in every sentence that refers to it.

Habits that pay off

Name the important results. A name is printed in the heading and makes the statement easy to find in the reference list.
Refer, never retype a number. A number typed by hand is wrong after the next insertion.
Number the equations you refer to, and only those. Mark the others Plain.
Put a sentence before a list in a statement. A statement that starts directly with a list prints its first item on the heading's line.
Keep one statement per block. A corollary after its theorem is a block of its own, not a paragraph inside the theorem.
Use remarks and examples for commentary. They are not numbered, so adding one never shifts your numbering.

Useful commands

You want	You type
ℝ, ℕ, ℤ, ℚ, ℂ	`\mathbb{R}`, `\mathbb{N}`…
A fraction	`\frac{a}{b}`
Indices and powers	`x_i`, `x^{2n}`
A sum with bounds	`\sum_{k=0}^{n} k`
An integral	`\int_a^b f(x)\,dx`
A limit	`\lim_{n \to \infty} u_n`
Words inside a formula	`\text{if } x > 0`
A definition by cases	`\begin{cases} x & \text{if } x \ge 0 \\ -x & \text{otherwise} \end{cases}`
A norm, an absolute value	`\lVert x \rVert`, `\lvert x \rvert`
Brackets that grow	`\left( \frac{a}{b} \right)`

The symbol list has the rest, searchable by name.

What to know before a long document

Statements hold paragraphs, lists and equations only — no figure or table inside a theorem. Place them just after it.
The kinds are fixed: theorem, lemma, proposition, corollary, definition, remark, example, proof. For an axiom, a conjecture or an exercise, use the closest kind and say so in its name.
All numbered statements share one counter, restarted at each chapter.
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