Taxicab Metric on Real Number Plane is Translation Invariant

Theorem
Let $\tau_{\mathbf t}: \R^2 \to \R^2$ denote the translation of the Euclidean plane by the vector $\mathbf t = \begin {pmatrix} a \\ b \end {pmatrix}$.

Let $d_1$ denote the taxicab metric on $\R^2$.

Then $d_1$ is unchanged by application of $\tau$:


 * $\forall x, y \in \R^2: \map {d_1} {\map \tau x, \map \tau y} = \map {d_1} {x, y}$

Proof
Let $x = \tuple {x_1, x_2}$ and $y = \tuple {y_1, y_2}$ be arbitrary points in $\R^2$.

Then: