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What exactly does this assumption mean?

Here is an example of where I get caught up. One of the first exersizes is this:

"Prove that for all intervals I, 0 is conatined in I - I."

My proof would be something like this,

Let I = [r, s]

by definition I - I = [r - s, s - r]

since s >= r, subtracting s from both sides yields 0 >= r - s

also since s>= r, subtracting r from both sides yields s - r >= 0

we now see that r - s <= 0 <= s - r, which by definition implies 0 is contained in [r-s , s-r] = I - I, as required.

I am guessing that this is good, if properties of inequalities are covered in the assumed "arithmetic and order" properties??