Here, on page 158, how do I see that (12.4) and (12.5) does not depend on the choice of representatives from the equivalence classes $[f_1],...,[f_n]$ ?
2026-03-30 00:21:26.1774830086
Independence of choice of represepresentatives, Filter,Reduced product
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Well, you have to verify that and said verification is a bit tedious - but essentially trivial.
It all comes down to two things:
Using those, the proof boils down to intersecting finitely many sets - each of which are in the filter - to see that a certain superset of this intersection ends up in the filter and witnesses that in fact our interpretation of predicates and functions is independent of our choice of representatives.
I'll leave the mechanical verification of this fact to you - it's a useful exercise which, while tedious and a bit boring, should help you get more comfortable with reduced products and the way their elements are formed.