Multiplicative Inverse in Field is Unique

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 1
From the definition of multiplicative inverse, $a^{-1}$ is the inverse element of the multiplicative group $\left({F^*, \times}\right)$.

The result follows from Inverses in Group are Unique.

Proof 2
From the definition of a field as a division ring, every element of $F^*$ is a unit.

The result follows from Product Inverses in Ring are Unique.