Taxicab Metric on Real Number Plane is not Rotation Invariant

Theorem
Let $r_\alpha: \R^2 \to \R^2$ denote the rotation of the Euclidean plane about the origin through an angle of $\alpha$.

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

Then it is not necessarily the case that:


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

Proof
Proof by Counterexample:

Let $x = \tuple {0, 0}$ and $y = \tuple {0, 1}$ be arbitrary points in $\R^2$.

Then:

Now let $\alpha = \dfrac \pi 4 = 45 \degrees$.