Euclidean Metric on Real Number Line is Metric/Proof 2

Theorem
The Euclidean metric on the real number line $\R$ is a metric.

Proof
Consider the Euclidean space $M = \left({\R, d}\right)$ where $d$ is the distance function given by:
 * $d \left({x, y}\right) = \left\lvert{x - y}\right\rvert$

Proof of $M1$
So axiom $M1$ holds for $d$.

Proof of $M2$
So axiom $M2$ holds for $d$.

Proof of $M3$
So axiom $M3$ holds for $d$.

Proof of $M4$
So axiom $M4$ holds for $d$.