Taxicab Metric on Real Vector Space is Metric/Proof 2

Proof
The taxicab metric on $\R^n$ is:
 * $\ds \map {d_1} {x, y} = \sum_{i \mathop = 1}^n \size {x_i - y_i}$

for $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.

Proof of $\text M 1$
So axiom $\text M 1$ holds for $d_1$.

Proof of $\text M 2$
So axiom $\text M 2$ holds for $d_1$.

Proof of $\text M 3$
So axiom $\text M 3$ holds for $d_1$.

Proof of $\text M 4$
So axiom $\text M 4$ holds for $d_1$.