Inverse of Field Product

Theorem
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $a, b \in F$ such that $a \ne 0$ and $b \ne 0$.

Then:
 * $\paren {a \times b}^{-1} = b^{-1} \times a^{-1}$

Proof
We are given that $a \ne 0$ and $b \ne 0$.

From Field has no Proper Zero Divisors and Rule of Transposition, we have:
 * $a \times b \ne 0$

By we have that $\paren {a \times b}^{-1}$ exists.

Then we have:

Hence the result by definition of multiplicative inverse.