Note first that is read as 'a squared' and that means that 'a squared' is defined to be a·a. Also note that means that a is not equal to b.
"For any a in R we have , with equality holding only if a=0;" (see ita-2-2-12-pg20)
"[...]more generally the sum of the squares of several elements of R is always greater than or equal to zero, with equality only if all the elements in question are zero." (see ita-2-2-12-pg20)
"For by O5, the statement implies , and a sum of positive elements is positive." (see ita-2-2-12-pg20)
"Note the special consequence that ." (see ita-2-2-12-pg20)
It seems to me that the 1>0 is another one of those deceptively profound results, and therefore I will discuss it in more detail. Recall from the preliminary section that the terms positive and negative were defined as follows: Numbers that are in R+ are called positive. Numbers, a, whose additive inverse, -a, is in R+ are called negative. Also recall that the greater than symbol, >, was defined in the preliminary section as follows: a>b means that a-b is in R+. Consequently, 1>0 means that 1-0=1+(-0)=1+0=1 is in R+. But this is an assertion, and not a proof, that 1 is in R+. A more detailed proof follows.
Also note, in the assertion above that 1 is in R+, it was assumed that -0=0, and this is so because -0=-1·0=0 by F10 and F5.