> [!NOTE] Theorem (Product With Ring Negatives) > Let $(R,+,\times)$ be a [[Rings|ring]]. Then $\forall x,y \in R: (-x)\times(-y)=x \times y$ Proof. By [[Product With Ring Negative]], $\begin{align} (-x)\times(-y) &= -(x \times (-y)) \\ & = -(-(x \times y)) \\ & = x \times y &\text{since $a$ is the inverse $-a$} \end{align}$