# Negative of Sum of Real Numbers

## Theorem

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

## Proof

 $\displaystyle -\paren {x + y}$ $=$ $\displaystyle \paren {-1} \times \paren {x + y}$ Multiplication by Negative Real Number: Corollary $\displaystyle$ $=$ $\displaystyle \paren {\paren {-1} \times x} + \paren {\paren {-1} \times y}$ Real Number Axioms: $\R D$: Distributivity $\displaystyle$ $=$ $\displaystyle \paren {-x} + \paren {-y}$ Multiplication by Negative Real Number: Corollary $\displaystyle$ $=$ $\displaystyle -x - y$ Definition of Real Subtraction

$\blacksquare$