Negative of Field Negative

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:

$-\paren {-a} = a$

$\ds \paren {-a} + a$

$=$

$\ds a + \paren {-a}$

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

$\blacksquare$

1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Theorem $2 \ \text {(i)}$