I hope you were talking to me, cause now I'm drunk enough for real math. Dunno why I included the less than, that overcomplicates shit and the proof is simple, it's just assuming a negative times a negative is or isn't positive, it's a simple proof. A is any nonzero number.
-1(a - a) = 0
-a + a = 0 👍
-a - a = 0 🖕
QED
The one in my head was the ancient chinese method of simple algebra, say 3x = 9. They set the problem equal to 0 and make two guesses, say 2 and 4, yes I'm making this easy, but large and small guesses were the common practice. So they take the guesses and multiply them by the answer of the opposite problem, subtract those results, divide that by the same relation in guesses, an arrive at the correct answer. So, y = mx + b, and y is 0, and we solve for x. M = (y2 - y1)/(x2 - x1). So x = -b[(x2 - x1)/(y2 - y1)].
So what is happening in ancient Chinese math, is (y2x1 - y1x2)/(x2 -x1) = x. So how? Y = mx + b. So you have [(mx2 + b)x1 - (mx1 + b)x2]/(x2 - x1).
Oh damn, I started working this out on paper and I am not fucking phone typing this.