Real Multiplication Distributes over Addition/Algebraic Proof

Theorem
The operation of multiplication on the set of real numbers $\R$ is distributive over the operation of addition:
 * $\forall x, y, z \in \R:$
 * $x \times \left({y + z}\right) = x \times y + x \times z$
 * $\left({y + z}\right) \times x = y \times x + z \times x$

Proof
From the definition, the real numbers are the set of all equivalence classes $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$ of Cauchy sequences of rational numbers.

Let $x = \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right], y = \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right], z = \left[\!\left[{\left \langle {z_n} \right \rangle}\right]\!\right]$, where $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$, $\left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]$ and $\left[\!\left[{\left \langle {z_n} \right \rangle}\right]\!\right]$ are such equivalence classes.

From the definition of real multiplication, $x \times y$ is defined as:
 * $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] = \left[\!\left[{\left \langle {x_n \times y_n} \right \rangle}\right]\!\right]$

From the definition of real addition, $x + y$ is defined as:
 * $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] + \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] = \left[\!\left[{\left \langle {x_n + y_n} \right \rangle}\right]\!\right]$

Thus:

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