Multiple of Ring Product

Theorem
Let $\left({R, +, \circ}\right)$ be a ring.

Let $x, y \in \left({R, +, \circ}\right)$.

Then:
 * $\forall n \in \Z_{> 0}: \left({n \cdot x} \right) \circ y = n \cdot \left({x \circ y}\right) = x \circ \left({n \cdot y}\right)$

where $n \cdot x$ denotes the $n$th multiple of $x$.

Proof
By definition:
 * $\displaystyle n \cdot x := \sum_{j \mathop = 1}^n x$

Thus: