Taxicab Metric is Metric

Theorem
The taxicab metric is a metric.

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

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

Let $\mathcal A$ be the Cartesian product $\displaystyle \prod_{i \mathop = 1}^n A_{i'}$.

The taxicab metric on $\displaystyle \mathcal A$ is:
 * $\displaystyle d_1 \left({x, y}\right) = \sum_{i \mathop = 1}^n d_{i'} \left({x_{i'}, y_{i'}}\right)$

for $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$.

Proof of $M1$
So axiom $M1$ holds for $d_1$.

Proof of $M2$
So axiom $M2$ holds for $d_1$.

Proof of $M3$
So axiom $M3$ holds for $d_1$.

Proof of $M4$
So axiom $M4$ holds for $d_1$.