Let $m$ and $n$ be positive integers. A child walks the Cartesian plane taking a few steps. The child begins its journey at the point $(0, n)$ and ends at the point $(m, 0)$ in such a way that: $\bullet$ Each step has length $1$ and is parallel to either the $X$ or $Y$ axis. $\bullet$ For each point $(x, y)$ of its path it is true that $x\ge 0$ and $y\ge 0$. For each step of the child, the distance between the child and the axis to which said step is parallel is calculated. If the step causes the child to be further from the point $(0, 0)$ than before, we consider that distance as positive, otherwise, we consider that distance as negative. Prove that at the end of the boy's journey, the sum of all the distances is $0$.