A simple proof that -1*-1 = 1

Carlos Scheidegger sent this proof which is better than my attempt at why -1 * -1 = 1...

The definition of multiplication for whole numbers is:

x * y = y + y + y + ... + y + y, where y appears x times.

Using this, it is easy to prove that, being (succ x) the successor of x, 

if x * y = z, then (succ x) * y = z + y, and vice-versa.

By definition, 0 is the successor of -1. Also by definition,

0 * x = 0, 

and so, 0 * -1 = 0.

(succ -1) * -1 = 0
(succ -1) * -1 = 1 + -1

Now, we apply the property:

(succ -1) * -1 = 1 + -1 ->

-1 * -1 = 1

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This proof's only assumption is that -n + n = 0, which is easily
provable. (Very easy using peano arithmetic)