(Try really, really hard to understand Fermat's Theorem AND the Binomial Expansion)
rem(a,b,n) is simply everything that is not
in the Binomial Expansion.
c < (a+b) merely says
which is a true expression, but is a metric expression, not a proof of Fermat's Theorem, so I don't have to prove it, trivial as it is. It is irrelevant to the proof. (In particular, it is not true for all integers a, b, c, in particular for c > (a + b) - c cannot satisfy both equations.
c=(a+b) sets the foundation for the proof (by contradiction)
My proof is true for all integers (and numbers) in Fermat's equation. And I don't really need all the analysis in terms of vectors, the Binomial Theorem is just fine as it is. (I had been trying to justify the Binomial Theorem in terms of vectors by including Pythagorean triples for the case n = 2, but Fermat specifies n>2, so that analysis is also irrelevant to the actual proof).