Difference of Two Squares/Algebraic Proof 2

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:
 * $x^2 - y^2 = \left({x + y}\right) \left({x - y}\right)$

Proof
This is a special case of Difference of Two Powers:


 * $\displaystyle a^n - b^n = \left({a - b}\right) \left({a^{n-1} + a^{n-2} b + a^{n-3} b^2 + \ldots + a b^{n-2} + b^{n-1}}\right) = \left({a - b}\right) \sum_{j \mathop = 0}^{n-1} a^{n-j-1} b^j$

The result follows by setting $n = 2$.