\(\displaystyle -0 + 0\) \(=\) \(\displaystyle 0\) $\quad$ Real Number Axioms: $\R A 4$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle -0\) \(=\) \(\displaystyle 0\) $\quad$ Real Addition Identity is Zero: Corollary $\quad$

$\blacksquare$

>>32096
You use the fact that the vectors with addition (+) build a group with neutral element 0 (that is, the equality -0 = 0 holds for groups in general).

0 is neutral element:
for all a : 0 + a = a => 0 + (-0) = -0 (1)

existence of the inverse element:
for each b there is a (-b) with b + (-b) = 0
=>
0 + (-0) = 0 (2)

Since + is an operation,
(1), (2) => -0 = 0

(corrected a mistake)