Proposition

$$(-a)(-a) = a^2$$

Proof:

(1)

$$a(-a)+aa = a(-a+a) = a0 = 0 \implies a(-a) = -(aa) = -(a^2)$$
This step uses distributive law and then the definition of inverse.

(2)

$$(-a)(-a) + a(-a) = (-a+a)(-a) = 0(-a) = 0 \implies (-a)(-a) = -(a(-a))$$

This step uses distributive law and then the definition of inverse.

(3)

$$(-a)(-a) = -(a(-a))) = -(-(a^2)) = a^2$$
This step uses the definition of inverse and $-(-(x)) = x$.