If I start with the triangle inequality:
$\displaystyle ||x-z|+|y-z|| \ge |x-y|$
and then I switch the sign on the LHS, which is true:
$\displaystyle ||x-z|-|y-z|| \le |x-y|$
or
$\displaystyle ||x-z|-|y-z|| = |x-y|$
If I start with the triangle inequality:
$\displaystyle ||x-z|+|y-z|| \ge |x-y|$
and then I switch the sign on the LHS, which is true:
$\displaystyle ||x-z|-|y-z|| \le |x-y|$
or
$\displaystyle ||x-z|-|y-z|| = |x-y|$