Field/Uniqueness of negative and inverse element/Fact/Proof2

Proof

Let ${}x$ be given and suppose that ${}y$ and ${}y'$ are elements fulfilling ${}x+y=0=x+y'$. Then

${}y=y+0=y+(x+y')=(y+x)+y'=(x+y)+y'=0+y'=y'\,,$

which means altogether ${}y=y'$. For the second part see exercise.

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