Ring Zero is Unique/Proof 3

Proof
Suppose $0$ and $0'$ are both ring zeroes of $\left({R, +, \circ}\right)$.

Then by Ring Product with Zero:
 * $0' \circ 0 = 0$ by dint of $0$ being a ring zero
 * $0' \circ 0 = 0'$ by dint of $0'$ being a ring zero.

So $0 = 0' \circ 0 = 0'$.

So $0 = 0'$ and there is only one ring zero of $\left({R, +, \circ}\right)$ after all.