"If a, b [are real numbers] then one and only one of the following statements is true: a>b, a=b, a<b. For if we apply part (2) of the order property to the number a-b then exactly one of three possibilities holds, a-b [is in R+], a-b=0, or b-a [is in R+], which are the three cases of the assertion O1." (see ita-2-2-6-pg19)
The first of the above possibilities is a direct application of the definition of the greater than symbol. The other two cases are examined in detail below. Please note that some of the results from the field properties section have been translated into different wording. This is a useful technique for avoiding confusion when the symbols have changed.
Consequently, b-a is in R+ so that b>a, which we may also write as a<b.