# Real Numbers under Multiplication form Monoid

## Theorem

The set of real numbers under multiplication $\struct {\R, \times}$ forms a monoid.

## Proof

Taking the monoid axioms in turn:

### Monoid Axiom $\text S 0$: Closure

$\Box$

### Monoid Axiom $\text S 1$: Associativity

$\Box$

### Monoid Axiom $\text S 2$: Identity

$\Box$

Hence the result.

$\blacksquare$