Real Numbers form Algebra

Theorem
The set of real numbers $\R$ forms an algebra over the Field of Real Numbers.

Proof
We have that $\left({\R, +, \times}\right)$ is a field.

Let this be expressed as $\left({\R, +_\R, \times_\R}\right)$ in order to call attention to the precise scope of the operators.

From Real Vector Space, we have that $\left({\R, +, \times}\right)_\R$