# Product of Ring Negatives

## Theorem

Let $\struct {R, +, \circ}$ be a ring.

Then:

$\forall x, y \in \struct {R, +, \circ}: \paren {-x} \circ \paren {-y} = x \circ y$

where $\paren {-x}$ denotes the negative of $x$.

## Proof

We have:

 $\displaystyle \paren {-x} \circ \paren {-y}$ $=$ $\displaystyle -\paren {x \circ \paren {-y} }$ Product with Ring Negative $\displaystyle$ $=$ $\displaystyle -\paren {-\paren {x \circ y} }$ Product with Ring Negative $\displaystyle$ $=$ $\displaystyle x \circ y$ Negative of Ring Negative

$\blacksquare$