Because a lot of logical implications are one way, writing things backwards can be confusing. We work backwards to know where we're going, but we write forwards to make sure everything actually works.

However, it is not always the case that proofs proceed from assumptions to goals. Here are two typical exceptions to the rule of start at the beginning and end at the end:

**Theorem:** XXX

*proof.* First, we observe that to prove XXX, it suffices to prove YYY, and proving YYY is equivalent to proving ZZZ....

or

**Theorem:** XXX

First, we have the following lemma:

**Lemma** YYY

With the lemma, we can prove the theorem as follows....

*Proof of lemma.* (proof goes here)

In both cases, the first step in the proof is showing we can move our goal to something simpler.

However, there are a few caveats to this style of proof. First, because lots of logical implications go only one way, you need to make sure that you are writing down things which imply your conclusion and NOT just things that follow from your conclusion. Second, because you are not proceeding in a simple order from things you know to things you don't, it is much easier to make mistakes with circular reasoning.

Third, and perhaps most important, while working backwards can make things easier for discovering a proof, it is difficult to read a long proof that is written entirely backwards. The decision to put part of the end at the beginning (or in general, to do anything out of the standard forwards order) must only be done when it improves clarity of exposition. The main reason it might improve clarity is because you have to spend a significant amount of time working towards something that seems off topic, unmotivated, or intermediate. Putting the end of the proof first in these cases means that the reader knows what they are working towards and why they are working towards it.

Please note that putting the end of a proof at the beginning and then jumping to the beginning is very different from doing the proof backwards. Until you appreciate the difference, and until you are sure that you have a very good reason for doing so and have seen enough examples to know how to do so clearly, this is not a proof-writing technique that I would recommend. Yes, if done right, it makes things clearer. However, if done wrong, it either makes things more complicated or introduces logical errors.