Definition:Lemma

Definition

A lemma is a statement which is proven during the course of reaching the proof of a theorem.

Logically there is no qualitative difference between a lemma and a theorem.

They are both statements whose value is either true or false.

However, a lemma is seen more as a stepping-stone than a theorem in itself (and frequently takes a lot more work to prove than the theorem to which it leads).

Some lemmata are famous enough to be named after the mathematician who proved them (for example: Abel's Lemma and Urysohn's Lemma), but they are still categorised as second-class citizens in the aristocracy of mathematics.

Always the lemma, never the theorem.

Also see

• Definition:Corollary
• Definition:Proof

• Results about lemmata can be found here.

Linguistic Note

The plural of lemma is either lemmas or lemmata.

The form lemmata is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Sources

• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Axiom systems
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): lemma
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): lemma
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): theorem
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): lemma
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): theorem
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): lemma