Real Multiplication Distributes over Addition/Algebraic Proof

Proof
From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.

Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}, z = \eqclass {\sequence {z_n} } {}$, where $\eqclass {\sequence {x_n} } {}$, $\eqclass {\sequence {y_n} } {}$ and $\eqclass {\sequence {z_n} } {}$ are such equivalence classes.

From the definition of real multiplication, $x \times y$ is defined as:
 * $\eqclass {\sequence {x_n} } {} \times \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n \times y_n} } {}$

From the definition of real addition, $x + y$ is defined as:
 * $\eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n + y_n} } {}$

Thus:

By Real Addition is Commutative and Real Multiplication is Commutative, it follows that:
 * $\paren {y + z} \times x = \paren {y \times x} + \paren {z \times x}$