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how do you prove that -(-a)=a using the basic axioms? I know that -a is the adatative inverse of a but i'm not sure where to go from that.

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Clearly $(-a)+a=a+(-a)=0$ since $-a$ is the additive inverse of $a$. Since additive inverses are unique, $a$ must also be the additive inverse of $(-a)$, which is (by the definition of the notation) $-(-a)$

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