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In
mathematics, the '''Nine Lemma''' is a statement about
commutative diagrams and
exact sequences valid in any
abelian category, as well as in the category of
groups. It states: if
:
Image:Nine lemma.png
is a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well. Likewise, if all columns as well as the two top rows are exact, then the bottom row is exact as well.
The Nine Lemma can be proved by direct
diagram chasing, or by applying the
snake lemma (to the two bottom rows in the first case, and to the two top rows in the second case).
