Lemma (mathematics)

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In mathematics, a "helping theorem" or lemma (plural lemmas or lemmata) is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself.[1] The word derives from the Ancient Greek λῆμμα ("anything which is received, such as a gift, profit, or a bribe").

Comparison with theorem[edit]

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 theorem  – a step in the direction of proof[2] – or a short theorem appearing at an intermediate stage in a proof.[3]

Well-known lemmas[edit]

A good stepping stone can lead to many others; some powerful results in mathematics are known as lemmas, such as Bézout's lemma, Dehn's lemma, Euclid's lemma, Farkas' lemma, Fatou's lemma, Gauss's lemma, Greendlinger's lemma, Itō's lemma, Jordan's lemma, Nakayama's lemma, Poincaré's lemma, Riesz's lemma, Schur's lemma, Schwarz's lemma, Sperner's lemma, Urysohn's lemma, Vitali covering lemma, Yoneda's lemma and Zorn's lemma. While these results originally seemed too simple or too technical to warrant independent interest, they have turned out to be central to the theories in which they occur.

See also[edit]


  1. ^ Higham, Nicholas J. (1998). Handbook of Writing for the Mathematical Sciences. Society for Industrial and Applied Mathematics. p. 16. ISBN 0-89871-420-6.
  2. ^ "What is the difference between a theorem, a lemma, and a corollary?"
  3. ^ Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1176. ISBN 1-57955-008-8.

External links[edit]

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