Mastering Theorem Environments — theorem, definition, proof, and Friends

A complete guide to Folio's 18+ built-in theorem environments. Covers theorem, definition, proof, lemma, corollary, example, remark, and more — usage, numbering, named arguments, and italic/roman styles explained.

Folio Official

March 3, 2026

1. Theorem Environment Basics

Folio comes with amsthm-equivalent theorem environments built in. No \newtheorem declarations needed.

Basic syntax:

\begin{theorem}[Fermat's Little Theorem]
Let $p$ be a prime and $\gcd(a, p) = 1$.
Then $a^{p-1} \equiv 1 \pmod{p}$.
\end{theorem}

Result:

Theorem 1 (Fermat's Little Theorem).

Let

p

be a prime and

g cd (a, p) = 1

. Then

a^{p - 1} \equiv 1 (mod p)

2. Complete List of Theorem Environments

Folio provides the following environments. All are automatically numbered (except proof).

2.1. Theorem-Type (Italic Body)

Environments for stating theorem-like assertions. The body text is displayed in italic:

Theorem 2 (Sample Theorem).

theorem environment. For any prime

p

and integer

a

with

g cd (a, p) = 1

, we have

a^{p - 1} \equiv 1 (mod p)

Lemma 3 (Sample Lemma).

lemma environment. If

n

is composite, then

n

has a prime factor at most

n

Proposition 4 (Sample Proposition).

proposition environment. The composition of continuous functions is continuous.

Corollary 5 (Sample Corollary).

corollary environment. The order of a finite group is divisible by the order of each element.

Claim 6 (Sample Claim).

claim environment. This construction is unique.

Fact 7 (Sample Fact).

fact environment.

π

is a transcendental number.

Hypothesis 8 (Sample Hypothesis).

hypothesis environment.

P \neq = NP

2.2. Definition-Type (Roman Body)

Environments for definitions and examples. The body text is displayed in roman (upright) style:

Definition 9 (Sample Definition).

definition environment. A subset

H

of a group

G

is a subgroup if

H

itself forms a group under the operation of

G

Example 10 (Sample Example).

example environment. In

Z /6 Z

, the order of

\overset{ˉ}{2}

3

Remark 11 (Sample Remark).

remark environment. The converse of this theorem does not hold in general.

Note 12 (Sample Note).

note environment. Unless otherwise stated, all rings are assumed to be commutative.

Exercise 13 (Sample Exercise).

exercise environment. Find all Sylow

2

-subgroups of

S_{4}

Problem 14 (Sample Problem).

problem environment. Prove that

\sum_{n = 1}^{\infty} \frac{1}{n ^{2}} = \frac{π ^{2}}{6}

Assumption 15 (Sample Assumption).

assumption environment. Assume

f

is continuous on

[a, b]

Observation 16 (Sample Observation).

observation environment. The determinant changes sign when two rows are swapped.

Question 17 (Sample Question).

question environment. Can every even integer be expressed as the sum of two primes?

Solution 18 (Sample Solution).

solution environment. Substituting

x = 2

gives

f (2) = 4 + 2 + 1 = 7

Answer.

answer environment. The number of ways is

(3 10) = 120

. □

2.3. Proof Environment

The proof environment is unnumbered and automatically appends a QED symbol ( $□$ ):

Proof.

We prove that

2

is irrational by contradiction. Suppose

2 = p / q

where

p, q

are coprime positive integers. Then

2 q^{2} = p^{2}

, so

p

is even. Let

p = 2 m

; then

q^{2} = 2 m^{2}

, so

q

is also even. This contradicts the assumption that

p, q

are coprime. □

Named proofs are also supported:

Proof.

Using \begin{proof}[Proof of Theorem 1.1] you can specify a custom name. □

3. Environment Reference Table

Environment	Display Label	Body Style	Numbered
`theorem`	Theorem	Italic	Yes
`lemma`	Lemma	Italic	Yes
`proposition`	Proposition	Italic	Yes
`corollary`	Corollary	Italic	Yes
`claim`	Claim	Italic	Yes
`fact`	Fact	Italic	Yes
`hypothesis`	Hypothesis	Italic	Yes
`definition`	Definition	Roman	Yes
`example`	Example	Roman	Yes
`remark`	Remark	Roman	Yes
`note`	Note	Roman	Yes
`exercise`	Exercise	Roman	Yes
`problem`	Problem	Roman	Yes
`assumption`	Assumption	Roman	Yes
`observation`	Observation	Roman	Yes
`question`	Question	Roman	Yes
`solution`	Solution	Roman	Yes
`answer`	Answer	Roman	Yes
`proof`	Proof	Roman	No

Note 19.

The environment labels (e.g., "Theorem," "Definition") automatically switch based on the user's language setting. In Japanese mode, they display as their Japanese equivalents.

4. Combining with Cross-References

You can label theorem environments with \label and reference them with \ref:

\begin{theorem}[Lagrange's Theorem]
\label{thm:lagrange}
For a subgroup $H$ of a finite group $G$, $|H|$ divides $|G|$.
\end{theorem}

By Theorem \ref{thm:lagrange}...

Theorem 21 (Lagrange's Theorem).

For a subgroup

H

of a finite group

G

∣ H ∣

divides

∣ G ∣

By Theorem 21, the possible orders of subgroups of a group of order $12$ are $1, 2, 3, 4, 6, 12$ .

5. Summary

Folio's theorem environments provide amsthm-equivalent functionality with zero configuration. Using 19 different environments, you can clearly express the structure of mathematical arguments. Next, we'll look at code blocks and algorithms.

LaTeX Folio Tutorial Theorem Environments theorem proof

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 followers·107 articles

Mastering Theorem Environments — theorem, definition, proof, and Friends

Folio Official

March 3, 2026

1. Theorem Environment Basics

Folio comes with amsthm-equivalent theorem environments built in. No \newtheorem declarations needed.

Basic syntax:

\begin{theorem}[Fermat's Little Theorem]
Let $p$ be a prime and $\gcd(a, p) = 1$.
Then $a^{p-1} \equiv 1 \pmod{p}$.
\end{theorem}

Result:

Theorem 1 (Fermat's Little Theorem).

Let

p

be a prime and

g cd (a, p) = 1

. Then

a^{p - 1} \equiv 1 (mod p)

2. Complete List of Theorem Environments

Folio provides the following environments. All are automatically numbered (except proof).

2.1. Theorem-Type (Italic Body)

Environments for stating theorem-like assertions. The body text is displayed in italic:

Theorem 2 (Sample Theorem).

theorem environment. For any prime

p

and integer

a

with

g cd (a, p) = 1

, we have

a^{p - 1} \equiv 1 (mod p)

Lemma 3 (Sample Lemma).

lemma environment. If

n

is composite, then

n

has a prime factor at most

n

Proposition 4 (Sample Proposition).

proposition environment. The composition of continuous functions is continuous.

Corollary 5 (Sample Corollary).

corollary environment. The order of a finite group is divisible by the order of each element.

Claim 6 (Sample Claim).

claim environment. This construction is unique.

Fact 7 (Sample Fact).

fact environment.

π

is a transcendental number.

Hypothesis 8 (Sample Hypothesis).

hypothesis environment.

P \neq = NP

2.2. Definition-Type (Roman Body)

Environments for definitions and examples. The body text is displayed in roman (upright) style:

Definition 9 (Sample Definition).

definition environment. A subset

H

of a group

G

is a subgroup if

H

itself forms a group under the operation of

G

Example 10 (Sample Example).

example environment. In

Z /6 Z

, the order of

\overset{ˉ}{2}

3

Remark 11 (Sample Remark).

remark environment. The converse of this theorem does not hold in general.

Note 12 (Sample Note).

note environment. Unless otherwise stated, all rings are assumed to be commutative.

Exercise 13 (Sample Exercise).

exercise environment. Find all Sylow

2

-subgroups of

S_{4}

Problem 14 (Sample Problem).

problem environment. Prove that

\sum_{n = 1}^{\infty} \frac{1}{n ^{2}} = \frac{π ^{2}}{6}

Assumption 15 (Sample Assumption).

assumption environment. Assume

f

is continuous on

[a, b]

Observation 16 (Sample Observation).

observation environment. The determinant changes sign when two rows are swapped.

Question 17 (Sample Question).

question environment. Can every even integer be expressed as the sum of two primes?

Solution 18 (Sample Solution).

solution environment. Substituting

x = 2

gives

f (2) = 4 + 2 + 1 = 7

Answer.

answer environment. The number of ways is

(3 10) = 120

. □

2.3. Proof Environment

The proof environment is unnumbered and automatically appends a QED symbol ( $□$ ):

Proof.

We prove that

2

is irrational by contradiction. Suppose

2 = p / q

where

p, q

are coprime positive integers. Then

2 q^{2} = p^{2}

, so

p

is even. Let

p = 2 m

; then

q^{2} = 2 m^{2}

, so

q

is also even. This contradicts the assumption that

p, q

are coprime. □

Named proofs are also supported:

Proof.

Using \begin{proof}[Proof of Theorem 1.1] you can specify a custom name. □

3. Environment Reference Table

Environment	Display Label	Body Style	Numbered
`theorem`	Theorem	Italic	Yes
`lemma`	Lemma	Italic	Yes
`proposition`	Proposition	Italic	Yes
`corollary`	Corollary	Italic	Yes
`claim`	Claim	Italic	Yes
`fact`	Fact	Italic	Yes
`hypothesis`	Hypothesis	Italic	Yes
`definition`	Definition	Roman	Yes
`example`	Example	Roman	Yes
`remark`	Remark	Roman	Yes
`note`	Note	Roman	Yes
`exercise`	Exercise	Roman	Yes
`problem`	Problem	Roman	Yes
`assumption`	Assumption	Roman	Yes
`observation`	Observation	Roman	Yes
`question`	Question	Roman	Yes
`solution`	Solution	Roman	Yes
`answer`	Answer	Roman	Yes
`proof`	Proof	Roman	No

Note 19.

The environment labels (e.g., "Theorem," "Definition") automatically switch based on the user's language setting. In Japanese mode, they display as their Japanese equivalents.

4. Combining with Cross-References

You can label theorem environments with \label and reference them with \ref:

\begin{theorem}[Lagrange's Theorem]
\label{thm:lagrange}
For a subgroup $H$ of a finite group $G$, $|H|$ divides $|G|$.
\end{theorem}

By Theorem \ref{thm:lagrange}...

Theorem 21 (Lagrange's Theorem).

For a subgroup

H

of a finite group

G

∣ H ∣

divides

∣ G ∣

By Theorem 21, the possible orders of subgroups of a group of order $12$ are $1, 2, 3, 4, 6, 12$ .

5. Summary

LaTeX Folio Tutorial Theorem Environments theorem proof

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 followers·107 articles

Mastering Theorem Environments — theorem, definition, proof, and Friends

1. Theorem Environment Basics

2. Complete List of Theorem Environments

2.1. Theorem-Type (Italic Body)

2.2. Definition-Type (Roman Body)

2.3. Proof Environment

3. Environment Reference Table

4. Combining with Cross-References

5. Summary

Share your expertise with the world

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Mastering Theorem Environments — theorem, definition, proof, and Friends

1. Theorem Environment Basics

2. Complete List of Theorem Environments

2.1. Theorem-Type (Italic Body)

2.2. Definition-Type (Roman Body)

2.3. Proof Environment

3. Environment Reference Table

4. Combining with Cross-References

5. Summary

Share your expertise with the world

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Folio LaTeX Reference: A Complete Guide to Commands, Environments, and Syntax

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