Euclidean Metric on Real Number Line is Metric/Proof 2

Proof
Consider the real number line under the Euclidean metric:
 * $M = \struct {\R, d}$

where $d$ is the distance function given by:
 * $\map d {x, y} = \size {x - y}$

Proof of $\text M 1$
So axiom $\text M 1$ holds for $d$.

Proof of $\text M 2$
So axiom $M2$ holds for $d$.

Proof of $\text M 3$
So axiom $\text M 3$ holds for $d$.

Proof of $\text M 4$
So axiom $\text M 4$ holds for $d$.