Every proof just shows a shortcut to get an answer more quickly. The need for the shortcut in a real world situation is often a motivation to prove something. Equally, though, many proofs are purely motivated by mathematical curiosity.

The proof is to show that the shortcut follows from the axioms without breaking a logical progression.

E.g. If you have a data set generated by a random variable and you want to describe it... you ultimately want to know that the way you think you're describing it is actually how you're describing it.

Suppose you want to know the EV of the next value to be collected to the data. Should you use the mean? The median? The mode?

We use a formal language to say, we want the unbiased estimator of the data set's EV. This is a complicated proof, but ultimately it shows that the mean is the one we want. So, knowing that, we don't need to go through the whole derivation of what formula gives us the EV, we can just use the mean and know that we're justified.

The same is true for the variance and standard deviation, and plenty of other stats. So we now have a small amount of data (these stats) which allow us to describe large amounts of data (the actual random variable). These are the shortcuts, and we know that we are justified to use them because we proved it.

Proving it once is enough. Just make sure that the axioms of your proof are in line with the axioms of your usage. If your usage doesn't match the assumptions of your proof, then you've got a bad model.