Negative of Sum of Real Numbers

Theorem

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

Proof

\(\ds -\paren {x + y}\)

\(=\)

\(\ds \paren {-1} \times \paren {x + y}\)

Multiplication by Negative Real Number: Corollary

\(\ds \)

\(=\)

\(\ds \paren {\paren {-1} \times x} + \paren {\paren {-1} \times y}\)

Real Number Axiom $\R \text D$: Distributivity of Multiplication over Addition

\(\ds \)

\(=\)

\(\ds \paren {-x} + \paren {-y}\)

Multiplication by Negative Real Number: Corollary

\(\ds \)

\(=\)

\(\ds -x - y\)

Definition of Real Subtraction

$\blacksquare$

Sources

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