Polynomials Closed under Addition/Polynomials over Integral Domain/Proof 2

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

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

Then $\forall x \in R$, the set $D \left[{x}\right]$ of polynomials in $x$ over $D$ is closed under the operation $+$.

Proof
A commutative ring with unity is a ring.

An integral subdomain of a commutative ring with unity $R$ is a subring of $R$.

The result then follows as a special case of Polynomials Closed under Addition: Polynomials over Ring