Taxicab Metric is Metric/Proof 1

Proof
From the definition, the taxicab metric is as follows:

Let $M_{1'} = \struct {A_{1'}, d_{1'} }, M_{2'} = \struct {A_{2'}, d_{2'} }, \ldots, M_{n'} = \struct {A_{n'}, d_{n'} }$ be a finite number of metric spaces.

Let $\AA$ be the Cartesian product $\ds \prod_{i \mathop = 1}^n A_{i'}$.

The taxicab metric on $\AA$ is:
 * $\ds \map {d_1} {x, y} = \sum_{i \mathop = 1}^n \map {d_{i'} } {x_{i'}, y_{i'} }$

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

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 $d_1$.