Reverse triangle inequality

Step 1 — Two applications of the triangle inequality. By (M3), $d(x,z) \le d(x,y) + d(y,z)$, which rearranges to $d(x,z) - d(y,z) \le d(x,y)$. Swapping the roles of $x$ and $y$ in the same axiom gives $d(y,z) \le d(y,x) + d(x,z) = d(x,y) + d(x,z)$ (using symmetry $d(y,x)=d(x,y)$), which rearranges to $d(y,z) - d(x,z) \le d(x,y)$, i.e. $-(d(x,z)-d(y,z)) \le d(x,y)$.

Step 2 — Combine both bounds. The two inequalities say $d(x,z)-d(y,z) \le d(x,y)$ and $-(d(x,z)-d(y,z)) \le d(x,y)$ simultaneously, which is exactly the statement $|d(x,z)-d(y,z)| \le d(x,y)$ by the definition of absolute value: a real number $u$ satisfies $|u| \le c$ precisely when both $u \le c$ and $-u \le c$ hold.