Prove |a|^2=a^2 with cases

Ok, so basically we can just prove this with -1 * -1 = 1, and that's about all we need to know. Knowing this, we can say that, as long as a is a real number (let's just say a positive real number) a = a^2. Also knowing that if a is less than 0 (i.e. negative) than |a|= -1*a, so we can say that a^2 * -1^2 is a positive number... I guess. Here's a few cases :D

-3^2 = 9 = 3^2

-4^2 = 16 = 4^2

-1^2 = 1 = 1^2