# Product of Field Negatives

## Theorem

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

Let $a, b \in F$.

Then:

$-\paren {a \times b} = \paren {-a} \times \paren {-b} = a \times b$

## Proof

 $\ds \paren {-a} \times \paren {-b}$ $=$ $\ds -\paren {a \times \paren {-b} }$ Product with Field Negative $\ds$ $=$ $\ds -\paren {-\paren {a \times b} }$ Product with Field Negative $\ds$ $=$ $\ds a \times b$ Negative of Field Negative

$\blacksquare$