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It is known that a theorem is a crucial result that solves many problems in a given field, and a lemma is a claim that holds for proving other important results.

We all know the Pythagoras Theorem, and we know that this theorem helps us prove a lot of other more complex claims. Then why don't we call it the Pythagorean Lemma but the Pythagorean Theorem?

Emo
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    In my experience, the distinction between lemmas and theorems tends to be more historical than anything else, cf. Zorn's Lemma vs. the Well-ordering Theorem. – Ben Millwood Mar 08 '14 at 20:16
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    In Book I of Euclid's Elements, the Triangle Inequality is the 20th proposition and is used as a lemma for proving the 47th proposition, the Pythagorean Theorem. There is no formal distinction between lemmas and theorems. The only difference is intent. – David H Mar 08 '14 at 20:23

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A theorem is an important statement in mathematics on its own, while a lemma's main/sole purpose is proving a theorem.

The Pythagorean Theorem is an important result in mathematics, and we should not call it a lemma just because it has many profound consequences. Quite the contrary, in fact, that is a reason we should call it a theorem.

William Chang
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