Linear Transformation Maps Zero Vector to Zero Vector/Proof 2

Theorem
Let $\mathbf V$ be a vector space, with zero $\mathbf 0$.

Likewise let $\mathbf V\,'$ be another vector space, with zero $\mathbf 0\,'$.

Let $T: \mathbf V \to \mathbf V\,'$ be a linear transformation.

Then:
 * $T: \mathbf 0 \mapsto \mathbf 0\,'$

Proof
From the vector space axioms we have that $\exists \mathbf 0 \in \mathbf V$.

What remains is to prove that $T\left({\mathbf 0}\right) = \mathbf 0\,'$: