Polynomial Addition is Associative

Theorem
Addition of polynomials is an associative operation.

Proof
Let $(R, +, \circ)$ be a commutative ring with unity.

Let $\left\{{X_j: j \in J}\right\}$ be a set of indeterminates.

Let $Z$ be the set of all multiindices indexed by $\left\{{X_j: j \in J}\right\}$.

Let


 * $\displaystyle f = \sum_{k\in Z} a_k \mathbf X^k$


 * $\displaystyle g = \sum_{k\in Z} b_k \mathbf X^k$


 * $\displaystyle h = \sum_{k\in Z} c_k \mathbf X^k$

be arbitrary polynomials in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$.

Then