Difference of Two Squares

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. $$\mathbb{Z}, \mathbb{Q}, \mathbb{R}$$ etc., then this translates into:

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

Proof
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