# Real Addition Identity is Zero/Corollary

## Corollary to Real Addition Identity is Zero

$\forall x, y \in \R: x + y = x \implies y = 0$

## Proof

 $\displaystyle x + y$ $=$ $\displaystyle x$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {-x} + \paren {x + y}$ $=$ $\displaystyle \paren {-x} + x$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {\paren {-x} + x} + y$ $=$ $\displaystyle \paren {-x} + x$ Real Number Axioms: $\R A 1$ $\displaystyle \leadsto \ \$ $\displaystyle 0 + y$ $=$ $\displaystyle 0$ Real Number Axioms: $\R A 4$ $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle 0$ Real Addition Identity is Zero

$\blacksquare$