Vector Space has Unique Additive Identity/Proof 1

Proof
Let $V$ be a vector space.

Let $0$ and $0'$ both be additive identities for $V$.

Because $0$ is an additive identity:
 * $0' = 0' + 0$

From commutativity of vector space addition:
 * $0' + 0 = 0 + 0'$

Because $0'$ is an additive identity:
 * $0 + 0 = 0$

Hence:
 * $0' = 0$

Thus $V$ has a unique additive identity.