Multiplicative Inverse in Field is Unique/Proof 2

Theorem
Let $\left({F, +, \times}\right)$ be a field whose zero is $0_F$.

Let $a \in F$ such that $a \ne 0_F$.

Then the multiplicative inverse $a^{-1}$ of $a$ is unique.

Proof
From the definition of a field as a division ring, every element of $F^*$ is a unit.

The result follows from Product Inverse in Ring is Unique.