This is LaTeX code:

[latex]
 
$\begin{align*}
\| f(x) - f(y) \| & = \| ((x_1 - x_3) - (y_1 - y_3), (x_1 + x_2) - (y_1 + y_2)) \| \\
 & = \| ((x_1 - y_1) - (x_3 - y_3), (x_1 - y_1) + (x_2 - y_2)) \| \\
 & = \| (x_1 - y_1, 0) - (x_3 - y_3, 0) + (0, x_1 - y_1) + (0, x_2 - y_2) \| \\
 & \leq \| (x_1 - y_1, 0) \| + \| (x_3 - y_3, 0) \| + \| (0, x_1 - y_1) \| + \| (0, x_2 - y_2) \| \\
 & = |x_1 - y_1| + |x_3 - y_3| + |x_1 - y_1| + |x_2 - y_2| \\
 & < \delta + \delta + \delta + \delta = 4 \delta
\end{align*}$
 
[/latex]