Difference of Two Squares/Algebraic Proof 1

Theorem
Let $\left({R, +, \circ}\right)$ be a commutative ring whose zero is $0_R$.

Let $x, y \in R$.

Then:
 * $x \circ x + \left({- \left({y \circ y}\right)}\right) = \left({x + y}\right) \circ \left({x + \left({- y}\right)}\right)$

When $R$ is one of the standard sets of numbers, i.e. $\Z, \Q, \R$ etc., then this translates into:
 * $x^2 - y^2 = \left({x + y}\right) \left({x - y}\right)$