Negative of Sum of Real Numbers

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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

$\blacksquare$


Sources