# Real Multiplication Identity is One/Corollary

## Theorem

$\forall x \in \R_{\ne 0}: x \times y = x \implies y = 1$

## Proof

 $\displaystyle x \times y$ $=$ $\displaystyle x$ $\displaystyle \leadsto \ \$ $\displaystyle \frac 1 x \times \paren {x \times y}$ $=$ $\displaystyle \frac 1 x \times x$ as long as $x \ne 0$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {\frac 1 x \times x} \times y$ $=$ $\displaystyle \frac 1 x \times x$ Real Number Axioms: $\R M 1$: Associativity $\displaystyle \leadsto \ \$ $\displaystyle 1 \times y$ $=$ $\displaystyle 1$ Real Number Axioms: $\R M 4$: Inverses $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle 1$ Real Multiplication Identity is One

$\blacksquare$