# Negative of Negative Real Number

## 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

$\blacksquare$