Zero Element is Unique

Theorem
Let $$\left({S, \circ}\right)$$ be an algebraic structure that has a zero element $$z \in S$$.

Then $$z$$ is unique.

When discussing an algebraic structure $$S$$ which has a zero element, then this zero is usually denoted $$z_S$$, $$n_S$$ or $$0_S$$.

If it is clearly understood what structure is being discussed, then $$z$$, $$n$$ or $$0$$ are usually used.

Proof
Suppose $$z_1$$ and $$z_2$$ are both zeroes of $$\left({S, \circ}\right)$$.

Then by the definition of zero element:
 * $$z_2 \circ z_1 = z_1$$ by dint of $$z_1$$ being a zero;
 * $$z_2 \circ z_1 = z_2$$ by dint of $$z_2$$ being a zero.

So $$z_1 = z_2 \circ z_1 = z_2$$.

So $$z_1 = z_2$$ and there is only one zero after all.