Unity of Integral Domain is Unique

Theorem
Let $\struct {D, +, \times}$ be an integral domain.

Then the unity of $\struct {D, +, \times}$ is unique.

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

The result follows from Identity of Monoid is Unique.