Negative of Negative Real Number

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Theorem

$\forall x \in \R: -\paren {-x} = x$


Proof

\(\displaystyle 0\) \(=\) \(\displaystyle \paren {-x} + x\) Real Number Axioms: $\R A 4$: Inverse for Addition
\(\displaystyle \leadsto \ \ \) \(\displaystyle -\paren {-x} + 0\) \(=\) \(\displaystyle -\paren {-x} + \paren {-x} + x\) adding $-\paren {-x}$ to both sides
\(\displaystyle \leadsto \ \ \) \(\displaystyle -\paren {-x} + 0\) \(=\) \(\displaystyle \paren {-\paren {-x} + \paren {-x} } + x\) Real Number Axioms: $\R A 1$: Associativity
\(\displaystyle \leadsto \ \ \) \(\displaystyle -\paren {-x} + 0\) \(=\) \(\displaystyle 0 + x\) Real Number Axioms: $\R A 4$: Inverse for Addition
\(\displaystyle \leadsto \ \ \) \(\displaystyle -\paren {-x}\) \(=\) \(\displaystyle x\) Real Number Axioms: $\R A 3$: Identity for Addition

$\blacksquare$


Sources