A relation between $L_{\infty}$ metric and $L_{1}$ metric in $R^{2}$

While solving the Meeting Point problem from hackerrank, I have stumbled upon a relation between $L_{\infty}$ metric and $L_{1}$ metric in $R^{2}$ .

Theorem1

Let $x = (x_{1}, x_{2})$ , consider the map $f : R^{2} \to R^{2}$ , $f (x) = (x_{1} - x_{2}, x_{1} + x_{2})$ then $d_{\infty} (x, y) = \frac{d _{1} ( f ( x ) , f ( y ))}{2}$