Exercise 4.1ΒΆ

This follows from the previous theorem we proved, that \((-x)y = -(xy)\):

\[\begin{split}(-x)(-y) &= -(x(-y)) \\ &= -(-(xy)) \\ &= xy\end{split}\]

In the first and second steps, we are just applying the previous theorem. In the final step, we are using a property of groups, which is that the inverse of the inverse of some element is just that element. In other words, if you invert an element twice, you end up with what you started with.