Negative of Negative Real Number

Theorem

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

$\ds 0$

$=$

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

Real Number Axiom $\R \text A4$: Inverses for Addition

$\ds \leadsto \ \ $

$\ds -\paren {-x} + 0$

$=$

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

adding $-\paren {-x}$ to both sides

$\ds \leadsto \ \ $

$\ds -\paren {-x} + 0$

$=$

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

Real Number Axiom $\R \text A1$: Associativity of Addition

$\ds \leadsto \ \ $

$\ds -\paren {-x} + 0$

$=$

$\ds 0 + x$

Real Number Axiom $\R \text A4$: Inverses for Addition

$\ds \leadsto \ \ $

$\ds -\paren {-x}$

$=$

$\ds x$

Real Number Axiom $\R \text A3$: Identity for Addition

$\blacksquare$

2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(d)}$