recently I was reading the article

by Bhattacharya, R. N., and Lee, O.

The authors consider the $\leq$ relation defined by: $$ (x_1, \ldots, x_n)\leq (a_1, \dots a_n)~\text{iff}~x_1\leq a_1, \ldots, x_n\leq a_n. $$ I was reading the proof of Theorem 3.1, and I'm stuck in the conclusion at the end of the equation (3.5). It seems to me that the authors are using the fact that

$$\{{y\leq x_{0}\}}\cup \{{y>x_{0}\}}=\mathbb{R}^n$$ is the whole space, which is not true for $n\geq 2$. As an example, for $n=2$, $\{(x,y)\leq (0,0)\}$ equals the first quadrant and $\{(x,y)> (0,0)\}$ equals the third quadrant.

**Obs 1:** In the original paper the relation "$\leq$" is not clearly specified, however in this Corrigendum (of another part of the paper) the authors set the relation defined as above

It seems to me that the argument of the authors works for $n=1$ where $\leq$ is a total order. However for $n\geq 2$ we have the above reasoning. Has someone already attempt for this question? Has the authors another corrigendum?