In mathematics, informal logic and argument mapping, a **lemma** (plural **lemmas** or **lemmata**) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".^{[1]}^{[2]}^{[3]} In many cases, a lemma derives its importance from the theorem it aims to prove, however, a lemma can also turn out to be more important than originally thought.^{[4]} The word "lemma" derives from the Ancient Greek λῆμμα ("anything which is received",^{[3]} such as a gift, profit, or a bribe).

## Comparison with theorem

There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof.^{[4]}

## Well-known lemmas

A good stepping stone can lead to many others. Some powerful results in mathematics are known as lemmas, first named for their originally minor purpose.^{[a]} These include, among others:

While these results originally seemed too simple or too technical to warrant independent interest, they have eventually turned out to be central to the theories in which they occur.

## See also

Look up in Wiktionary, the free dictionary.lemma |

- Corollary
- Co-premise
- Fundamental lemma
- Inference objection
- List of lemmas
- Objection
- Porism
- Theorem
- Theorem terminology

## Notes

**^**Conscientious mathematicians occasionally are given to the impulse of elevating the names to "theorem" rather than "lemma", but due to long use, the familiar, more humble name "lemma" invariably prevails.

## References

**^**"The Definitive Glossary of Higher Mathematical Jargon — Lemma".*Math Vault*. 2019-08-01. Retrieved 2019-11-28.**^**Higham, Nicholas J. (1998).*Handbook of Writing for the Mathematical Sciences*. Society for Industrial and Applied Mathematics. pp. 16. ISBN 0-89871-420-6.- ^
^{a}^{b}"Definition of lemma | Dictionary.com".*www.dictionary.com*. Retrieved 2019-11-28. - ^
^{a}^{b}Richeson, Dave (2008-09-23). "What is the difference between a theorem, a lemma, and a corollary?".*David Richeson: Division by Zero*. Retrieved 2019-11-28.

## External links

*This article incorporates material from Lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*