# Inverse of Multiplicative Inverse/Proof 2

## 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

 $\ds \paren {a^{-1} } \times a$ $=$ $\ds a \times \paren {a^{-1} }$ Field Axiom $\text M2$: Commutativity of Product $\ds$ $=$ $\ds 1_F$ Field Axiom $\text M4$: Inverses for Product $\ds \leadsto \ \$ $\ds a$ $=$ $\ds \paren {a^{-1} }^{-1}$ Definition of Multiplicative Inverse in Field

$\blacksquare$