# Help:Proofs

Proofs make up the main part of $\mathsf{Pr} \infty \mathsf{fWiki}$. This page lists information regarding their page structure.

## Introduction

### One theorem per page

The rule is to present only one theorem per page. Theorems with more than one statement have to be refactored. Reasons not to allow multiple results are:

- If there are Multiple Proofs of one or all of the statements, organizing the proofs becomes impractical.
- The page title becomes less meaningful.

In order to maintain an overview of related theorems, a page called *Properties of (Something)* can be created, with multiple theorems (but not the proofs) transcluded.

For an example of how this is done, see Trigonometric Identities.

Equivalence proofs are an exception to this rule: they may show the equivalence between more than two statements. For the precise policies on it, see Help:Equivalence Proofs.

### Multiple Proofs

*See also: Help:Multiple Proofs*

$\mathsf{Pr} \infty \mathsf{fWiki}$ strives to present as many proofs as possible, including multiple proofs of the same theorem. Each proof is placed on a subpage and transcluded as a separate section.

## Page structure

Proofs follow the standard structure of pages, as explained at Help:Page Structure. We list here additional elements that are specific to proofs.

### Lemmas

Lemmas that do not otherwise merit their own page, are put on a subpage of the proof page. They are:

- transcluded below the "Proof" section heading
- followed by a horizontal line.

In order to keep pages concise, it is usually a good idea not to transclude the proof of the lemma:

... == Proof == === First transcluded lemma === === Second transcluded lemma === ---- The rest of the proof.

### Corollaries

Corollaries are treated like lemmas, but are transcluded right before the "Proof" section heading.

### Q.E.D

*For the mathematical context, see Definition:QED*

Each proof, whether of a theorem or a lemma, is concluded using a Halmos Symbol, which can either be $\blacksquare$ or $\square$. It has multiple advantages:

- It gives a clear indication of the end of the proof.
- The
`{{qed}}`

template adds the page to the Proven Results category. This allows to count the number of proofs.

For instructions, see the `{{qed}}`

template.

### Outline of Proof

## Types of proofs

### Equivalence Proofs

Proofs that show the equivalence of two or more statements are subject to a particular structure. See Help:Equivalence Proofs.