Unique Representation in Polynomial Forms

Theorem
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $\left({D, +, \circ}\right)$ be an integral subdomain of $R$.

Let $X \in R$ be transcendental over $D$.

Let $D \left[{X}\right]$ be the ring of polynomials in $X$ over $D$.

Then each non-zero member of $D \left[{X}\right]$ can be expressed in just one way in the form:
 * $\displaystyle f \in D \left[{X}\right]: f = \sum_{k \mathop = 0}^n {a_k \circ X^k}$

Proof
Suppose $f \in D \left[{X}\right] \setminus \left\{{0_R}\right\}$ has more than one way of being expressed in the above form.

Then you would be able to subtract one from the other and get a polynomial in $D \left[{X}\right]$ equal to zero.

As $f$ is transcendental, the result follows.