Solution
= Solution
Apply the <triangle inequality> to the path from $x_1$ through $y_2$ to $y_1$:
$$
\|x_1-y_1\|-\|x_1-y_2\|\leq\|y_1-y_2\|.
$$
This is one direction of the <reverse triangle inequality>. A second application of the <triangle inequality>, this time through $x_2$, gives
$$
\boxed{\|x_1-y_1\|-\|x_1-y_2\|\leq\|y_1-y_2\|\leq\|x_2-y_1\|+\|x_2-y_2\|.}
$$
The resulting <four-point triangle inequality> follows without any assumptions on coincidences among the four points.