Can someone please verify any mistakes in my proof of this theorem. Is this a correct proof of the corollary of $(-1)\times (-1) = 1$ that the product of two negative numbers is positive?
Theorem: The product of two real negative numbers is positive
Proof: Let $x,y\in\mathbb{R^+}$. We have
\begin{equation} \begin{alignedat}{2} (-1) \times0=0\quad \Rightarrow\quad && (-1) \times (-xy+xy) &= 0 \\ \Rightarrow\quad && (-1)\times((-1)\times xy+xy)&= 0 \\ \Rightarrow\quad && (-1)\times(-1)\times xy+(-1)\times xy &= 0 \\ \Rightarrow\quad && (-1)\times x \times (-1) \times y +(-1)\times xy&=0 \\ \Rightarrow\quad &&(-x)\times (-y)- xy&=0 \\ \Rightarrow\quad &&(-x)\times (-y)&=xy. \end{alignedat} \end{equation}
Yes it is correct, but perhaps this would be faster. Take any two negative number $x$ and $y$. Then $x=-a$ and $y=-b$ where $a,b$ are positive. Now we have: \begin{eqnarray} xy &=& (-a)(-b)\\ &=& (-1)a(-1)b\\ &=& (-1)(-1)ab \\ &=& 1\cdot ab \\&=& ab>0\end{eqnarray}