Unique Representation in Polynomial Forms

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

Let $\struct {D, +, \circ}$ be an integral subdomain of $R$.

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

Let $D \sqbrk X$ 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:
 * $\displaystyle f \in D \sqbrk X: f = \sum_{k \mathop = 0}^n {a_k \circ X^k}$

Proof
Suppose $f \in D \sqbrk X \setminus \set {0_R}$ 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 $D \sqbrk X$ equal to zero.

As $f$ is transcendental, the result follows.