Negative of Sum of Real Numbers/Corollary

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Theorem

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


Proof

\(\displaystyle -\paren {x - y}\) \(=\) \(\displaystyle -\paren {x + \paren {-y} }\) Definition of Real Subtraction
\(\displaystyle \) \(=\) \(\displaystyle -x - \paren {-y}\) Negative of Sum of Real Numbers
\(\displaystyle \) \(=\) \(\displaystyle -x + \paren {-\paren {-y} }\) Definition of Real Subtraction
\(\displaystyle \) \(=\) \(\displaystyle -x + y\) Negative of Negative Real Number

$\blacksquare$


Sources