>>14025845I will use properties that the real numbers are known to obey. I will not make the assumption that they apply to supernegative numbers. I will clearly define these properties. When I say "non-supernegative number," I mean a real number which does not happen to be a supernegative number.
Identity property of multiplication:
If x is a non-supernegative number, then x*1 = x.
Definition of additive inverse:
If x is a non-supernegative number, then x+(-x) = 0.
Distributive property:
If x, y, and z are non-supernegative numbers, then x*(y+z) = x*y + x*z.
Zero-product property:
If x is a non-supernegative number, then x*0 = 0.
By the Identity property of multiplication (with x = -1): -1 = (-1)*1.
So -1 + (-1)(-1) = (-1)*1 + (-1)(-1).
By the Distributive property (with x = -1, y = 1, and z = -1): (-1)(1+(-1)) = (-1)*1 + (-1)(-1).
So (-1)*1 + (-1)(-1) = (-1)(1+(-1)).
By the Definition of additive inverse (with x = 1): 1+(-1) = 0.
So (-1)(1+(-1)) = (-1)*0.
By the Zero-product property (with x = -1): (-1)*0 = 0.
So (-1)*0 = 0.
Taking everything together, we've got
-1 + (-1)(-1) = (-1)*1 + (-1)(-1) = (-1)(1+(-1)) = (-1)*0 = 0,
so -1 + (-1)(-1) = 0.
The final leap would be to add 1 to both sides. If you still reject that (-1)(-1) = 1, then you must accept one of the following:
0 + (-1)(-1) does not equal (-1)(-1).
1 + (-1 + (-1)(-1)) does not equal (1 + (-1)) + (-1)(-1).
