# Lemma (mathematics)

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]} 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.^{[3]} The word "lemma" derives from the Ancient Greek λῆμμα ("anything which is received",^{[2]} such as a gift, profit, or a bribe).

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.^{[3]}

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.

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