Negative of Sum of Real Numbers/Corollary

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

$\blacksquare$