epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

4 The Nuts and Bolts of Proof, Third Edition
that means "to grasp." Indeed, in a lemma one "grasps" some truth to be
used in the proof of a larger result. The proofs of some theorems are long
and difficult to follow. In these cases, it is common for one or more of the
intermediate steps to be isolated as lemmas and to be proved ahead. Then, in
the proof of the theorem we can refer to the lemmas already established and
use them to move to the next step. Often the results stated in lemmas are not
very interesting by themselves, but they play key roles in the proof of more
important results. On the other hand, some lemmas are used in so many
different cases and are so important that they are named after famous
mathematicians.
Corollary: A corollary is a theorem that follows logically and easily from
a theorem already proved. Corollaries can be important theorems. The
name, which derives from the Latin word for "Uttle garland," underlines the
fact that the result stated in a corollary follows naturally from another
theorem. The James & James Mathematics Dictionary defines a corollary as
a "by-product of another theorem."