# What are various proofs good for?

There are plenty of questions around here, which are proven to be right or wrong in various ways. I wonder, what one can learn from these differing ways of how to prove something, despite the fact that: The more proofs, the better.

Let's say a statement is something like a way $A\to Z$. One proof might then break this down to $$A\to B\to \cdots \to W\to Z,$$ while the other proof takes another route $$A\to \beta \to \cdots \to \omega\to Z.$$

Is there way to morph between the various ways and by that learn something about the general structure?

EDIT What is it worth to have plenty of proofs for the "$\Rightarrow$" direction, if a have only one proof for "$\Leftarrow$"?

The question is very general. Examples, are welcome.

-
You may be interested in this and this. –  Bruno Stonek Feb 19 '12 at 23:33
I think the question is too general. –  lhf Mar 23 '12 at 0:58
@draks Different proofs might be related to different areas of math. The origin of various proofs is either genious (like Gauss' MO) or different people that do maths their way. I personally prefer a variety of proofs since you might not get one and understand another, or because they provide different insights. Think about $$1 = \cos(x-x) = \cos(x)\cos(-x)-\sin(x)\sin(-x)=\cos ^2 x+\sin ^2 x$$ It is a purely analitical proof a the Pythagorean Theorem, which I like the most over any other. –  Pedro Tamaroff Mar 23 '12 at 1:32