Inequality involving the min function I'm trying to prove the following inequality: 
$$
\left|y_{1}\land x_{1}-y_{2}\land x_{2}\right|\leq\left|y_{1}-y_{2}\right|+\left|x_{1}-x_{2}\right|,
$$
where $x\land y=\mbox{min}(x,y)$. By considering different cases
it seems to hold but I haven't been able to find a simple way of proving
it. I'd appreciate any suggestions. 

Here is an exotic, one-line-r based on $$\min\left ( x,y \right )=\frac{1}{2}\left ( x+y - \left | x-y \right | \right )$$
$$\left | \min\left ( x_1,y_1 \right ) - \min\left ( x_2,y_2 \right ) \right |=\left | \frac{1}{2}\left ( x_1+y_1-\left | x_1-y_1 \right | \right ) - \frac{1}{2}\left ( x_2+y_2-\left | x_2-y_2 \right | \right ) \right |=$$
$$=\left | \frac{1}{2}\left ( x_1-x_2 \right ) + \frac{1}{2}\left ( y_1-y_2 \right ) + \frac{1}{2}(\left | x_2-y_2 \right | - \left | x_1-y_1 \right |) \right | \leq$$ 
$$\leq \frac{1}{2}\left | x_1-x_2 \right | + \frac{1}{2}\left | y_1-y_2 \right | + \frac{1}{2}\left | \left | x_2-y_2 \right | - \left | x_1-y_1 \right | \right |\leq $$
$$\leq \frac{1}{2}\left | x_1-x_2 \right | + \frac{1}{2}\left | y_1-y_2 \right | + \frac{1}{2}\left | \left ( x_2-y_2 \right ) - \left ( x_1-y_1 \right ) \right |= $$
$$= \frac{1}{2}\left | x_1-x_2 \right | + \frac{1}{2}\left | y_1-y_2 \right | + \frac{1}{2}\left | \left ( x_2-x_1 \right ) + \left ( y_1-y_2 \right ) \right |\leq$$
$$\leq \frac{1}{2}\left | x_1-x_2 \right | + \frac{1}{2}\left | y_1-y_2 \right | + \frac{1}{2} \left | x_2-x_1 \right | + \frac{1}{2} \left | y_1-y_2 \right | = \left | x_1-x_2 \right | + \left | y_1-y_2 \right | $$
