> You are assuming that additive inverse is unique
Suppose that some number a has two additive inverses,x and y.
a + x = 0 = x + a
a + y = 0 = y + a
Consider
x + a + y = (x + a) + y = 0 + y = y
x + a + y = x + (a + y) = x + 0 = x
Thus x=y, and so the additive inverse is unique.
> you are assuming that -1(-1) is some value at all;
We can define it to have a value and then derive the properties of the value. We can show it to be consistent by creating an explicit model. Such things are relatively easy to do, but require a level of detail inappropriate for this context.
> you could write an extremely similar proof that
> leaves you with 0/0 = 1 and it would be faulty to
> conclude that 0/0 actually is 1.
