Unity of Integral Domain is Unique

Theorem
Let $\left({D, +, \times}\right)$ be an integral domain.

Then the unity of $\left({D, +, \times}\right)$ is unique.

Proof
From the definition of an integral domain, $\left({D, +, \times}\right)$ is a commutative ring such that $\left({D^*, \circ}\right)$ is a monoid.

Again by definition of a monoid, $\left({D^*, \circ}\right)$ is a semigroup with an identity.

The result follows from Identity of Semigroup is Unique.