Real Numbers form Ring

Theorem
The set of real numbers $\R$ forms a ring under addition and multiplication: $\struct {\R, +, \times}$.

Proof
From Real Numbers under Addition form Abelian Group, $\struct {\R, +}$ is an abelian group.

We also have that:
 * Real Multiplication is Closed:
 * $\forall x, y \in \R: x \times y \in \R$


 * Real Multiplication is Associative:
 * $\forall x, y, z \in \R: x \times \paren {y \times z} = \paren {x \times y} \times z$

Thus $\struct {\R, +}$ is a semigroup.

Finally we have that Real Multiplication Distributes over Addition:

Hence the result, by definition of ring.