I’m studying section 3.2 in Herbet Enderton’s "A Mathematical Introduction to Logic" and I can’t seem to find a justification for something in a proof. The domain of the structures we are dealing is the Natural numbers, however, when substracting terms, one may end up with negative values and it is not justified why this can be written nonetheless.
You can see this for example here:
Here, terms $r_i$ and $s_i$ are terms not containing $x$, but nothing assures us that $r_i-s_i$ is not negative, hence going out of our domain.
The writer kinda acknowledges this later but he only seems to find it problematic if every $r_i-s_i$ are negative, in this case he provides a solution to this.
My question here is why is it only a problem if they are all negative since we don't have negative numbers at all.
Thank you all in advance and sorry if the question is not detailed enough. I will make improvents if needed.