**(for grapes): It is your inability to understand simple vectors w.r.t. sets that is ridiculous..**

When one makes the statement "for all positive integers a and b", then to make a distinction between them, they must belong to disjoint (distinct) sets. That is, when the sets are disjoint, they do not "interact"; there is no relation between the elements. This is the case for Pythagorean triples, where the legs of the triangle are perpendicular, so the dot product vanishes.

**(for grapes): For the case n=2 (Binomial Theorem)** 2ab does not vanish because the legs of the triangle are not perpendicular. However, 2ab is an area, and is in the same plane as the ordered pair (a,b). So Rem(a,b,2)=2ab

0.

**(for grapes): The case of 2 is not relevant to the proof of Fermat's Theorem (n > 2)**

For n > 2, the terms in rem(a,b,n) has the general form of the Binomial theorem (e.g.

).

I use "absolute value" of cross products merely to indicate positive integer multiplication, but also that these products can never be in either {a} or {b}, so therefore could never be in the plane (a,b). Therefore they can be represented as a "non-interacting" vector w.r.t. (a,b) by a third vector

The sets in the Binomial Theorem are then related by vector addition, with the resultant vector not orthogonal to either i, j, or k, but a mixture (e.g., the hypotenuse of the right triangle for n = 2 (Pythagorean) - or the longest leg of the triangle for Binomial case n = 2 .

**(for grapes): Once again, the case of 2 is not relevant to the proof of Fermat's Theorem (n > 2)**

**(Once again, for grapes) (sigh):**
For n > 2, then the full expression for the Binomial Theorem is represented by:

,

which holds for all members of the sets (I couldn't use squiggly brackets for the powers because tex wouldn't let me).

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