What are examples of theorems get extensions based on simple observation?

Here are some examples illustrate what I meant:

Bonnet-Myers:Bonnet in 1855 proved n=2 case, Myers in 1941 extended to any dimension using the same idea.

Bishop-Gromov Volume comparison: Bishop knew the result inside cut locus in 1963, Gromov made a simple observation that beyond cut locus it is still true.

Ok, you got what I mean.

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This seems unhelpfully broad -- most important theorems get extended by others, and it is very subjective as to what counts as a simple observation (especially in retrospect). Can you be more specific about what you want to know? – Pete L. Clark Jan 2 '10 at 4:27
Observations that the first one proved the theorem could think of by a more thorough thought or whatever. In some sense, it is unlucky they passed those thoughts. – lemega Jan 2 '10 at 4:37
Will you please change this question to community wiki mode (by clicking the check-box when editing)? That is usually preferred for questions that ask for a bunch of examples. – Jonas Meyer Jan 2 '10 at 5:05
I've converted the question to wiki, but please make such questions (anything with the [big-list] tag) community wiki in the future. – Anton Geraschenko Jan 2 '10 at 7:25

Perhaps Euler's polyhedral formula:

V(vertices) + F(faces) - E(edges) = 2

provides an example of what you mean?

Euler did not give a proper proof but shortly thereafter this result inspired huge advances that had dramatic effects on the evolution of geometry, topology, convexity, and what today is called graph theory.

One measure of how rich this topic is can be seen from the many types and styles of proofs that can be found for this result collected below by David Eppstein:

http://www.ics.uci.edu/~eppstein/junkyard/euler/

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