Non-Zero Real Numbers under Multiplication form Abelian Group/Proof 3

Proof
From Non-Zero Real Numbers under Multiplication form Group, $\struct {\R_{\ne 0}, \times}$ forms a group.

From Real Multiplication is Commutative it follows that $\struct {\R_{\ne 0}, \times}$ is abelian.

From Real Numbers are Uncountably Infinite it follows that $\struct {\R_{\ne 0}, \times}$ is an uncountable abelian group.