Negative of Field Negative

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Theorem

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

Let $a \in F$ and let $-a$ be the field negative of $a$.


Then:

$-\paren {-a} = a$


Proof

\(\ds \paren {-a} + a\) \(=\) \(\ds a + \paren {-a}\) Field Axiom $\text A2$: Commutativity of Addition
\(\ds \) \(=\) \(\ds 0_F\) Field Axiom $\text A4$: Inverses for Addition
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds -\paren {-a}\) Definition of Field Negative

$\blacksquare$


Sources