Non-Zero Real Numbers under Multiplication form Group

Theorem
Let $\R_{\ne 0}$ be the set of real numbers without zero:
 * $\R_{\ne 0} = \R \setminus \set 0$

The structure $\struct {\R_{\ne 0}, \times}$ forms a group.

Proof
Taking the group axioms in turn:

From Non-Zero Real Numbers Closed under Multiplication: Proof 2, $\R_{\ne 0}$ is closed under multiplication.

Note that proof 2 needs to be used specifically here, as proof 1 rests on this result.

Real Multiplication is Associative.