Probably the same thing as the principle. You find it discussed in the old advanced calculus texts but never modern. You need it to justify, among other things, the formula for arc length, which these days is just defined as an integral (without giving any idea of where it came from). It's similar to the situation for the complex integral, which is just defined in a certain way while the old authors such as Titchmarsh and Copson would actually explain where it came from.
I started a thread perhaps a year back asking what advanced calculus texts people were using -- but no-one came up with titles. The modern books are mostly no good.
But don't provide any kind of proof. The problem is that you don't quite have a Riemann sum. If you had one, then as the norm partition goes to zero, you get a Riemann integral, which is the definition of arc length. You don't quite have a Riemann sum because for a curve in the plane you have two functions x(t) and y(t), and you are using the mean value theorem on both. The points t' and t" inside an arc segment may not (and usually will not) coincide, so you can't get a Riemann sum for the formula for arc length.[/quote]