Cancellation Law for Ring Product of Integral Domain

Theorem
Let $$\left({D, +, \circ}\right)$$ be an integral domain whose zero is $$0_D$$.

Let $$a \in D: a \ne 0_D$$.

Then:
 * $$\forall x, y \in D: a \circ x = a \circ y \implies x = y$$

That is, all elements of $$D^*$$ are cancellable for ring product.

Proof
From the definition of integral domain, no elements of $$D^*$$ are zero divisors.

From Zero Divisor Not Cancellable, it follows that all elements of $$D^*$$ are cancellable for ring product.