Multiplicative Inverse in Field is Unique/Proof 1

Theorem

Let $\struct {F, +, \times}$ 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 multiplicative inverse, $a^{-1}$ is the inverse element of the multiplicative group $\struct {F^*, \times}$.

The result follows from Inverse in Group is Unique.

$\blacksquare$