Theorem

A vector space has a unique additive identity.

Proof

Suppose 0 and 0’ are both additive identities for some vector space V. Then ${latex.inline0' = 0' + 0 = 0 + 0' = 0} . The left equality holds beacuse 0 is the additive identity. The middle one holds because of the commutative property of the vector space. The right equality holds because 0’ is also an additive identity. Thus, we get ${latex.inline0' = 0}.