[Coda]

I, for one, had never questioned it. I just followed the properties handed to me and they all seemed to work, so I didn't bother thinking any harder about it. That proof there is actually something I came up with on the fly.

Although technically, that's not a proof of correctness -- step (F) does not follow from the rest. The definition of exponentiation that was asserted at the beginning implies that only natural numbers are valid, so if this were being treated as a rigorous proof this would be an error that disproves it, or at least it would establish that it's only true if b > c.

Instead, this is a proof that if you extend exponentiation so negative integers are allowed in addition to positive ones, then this is what a negative exponent must be in order to be consistent with the property we want it to have. You have to follow a similar procedure if you want to define an extension that allows fractional exponents. (Extending it from fractions to real numbers, on the other hand, can't be done with this procedure, and you have to introduce logarithms.)