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O6: *a·a* is greater than or equal to zero.

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.

Converted [ in part ] by *Mathematica*
December 4, 2007