Zero of Field is Unique/Proof 2

Proof
Let $0_1$ and $0_2$ both be elements of $F$ such that:


 * $\forall a \in F: a + 0_1 = a$
 * $\forall a \in F: a + 0_2 = a$

Then:
 * $0_1 + 0_2 = 0_2$

because $0_1$ is a zero element


 * $0_1 + 0_2 = 0_1$

because $0_2$ is a zero element

Hence:
 * $0_1 = 0_2$

and the two zero elements are the same.