Negative of Sum of Real Numbers/Corollary

Theorem

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

Proof

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

\(=\)

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

Definition of Real Subtraction

\(\ds \)

\(=\)

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

Negative of Sum of Real Numbers

\(\ds \)

\(=\)

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

Definition of Real Subtraction

\(\ds \)

\(=\)

\(\ds -x + y\)

Negative of Negative Real Number

$\blacksquare$

Sources

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