I thought the concept of "negative" was defined by reference to an operation. "Negative 5" is whatever value Q satisfies the equation 5 + Q = 0.
That definition immediately tells you that the negative of a negative is a positive. Once we know 5 + Q = 0, we ask what the negative of Q is. It's the value V such that Q + V = 0. But by the definition of Q (and the commutativity of addition), we already know V = 5.
Once you define negatives this way, it's trivial to show that negatives obey the standard ordering. But that ordering wasn't necessary in order to define them.
Summing up, the product of negatives is positive because negation is a kind of inversion (additive inversion), and two successive inversions always cancel in any context.
