Inverse of Multiplicative Inverse/Proof 1

Theorem

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$.

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

Let $a^{-1}$ be the multiplicative inverse of $a$.

Then $\paren {a^{-1} }^{-1} = a$.

Proof

The multiplicative inverse is, by definition of a field, the inverse element of $a$ in the multiplicative group $\struct {F^*, \times}$.

The result then follows from Inverse of Group Inverse.

$\blacksquare$