[EM] Bah Humbug... can someone prove/disprove a relationship for me? [g is rational (II)(n) on X implies g is rational (I)(n) on X] (fwd)

[David Catchpole]

Well, I tried with matlab and discovered its boolean systems are just
rubbish. I also tried to use some didactic software from various
philosophy departments, but none of them had any real automatic use.

Could someone please help with this? I'm running out of ideas save
spending a month expanding the bloody thing out... Here it is, as
corrected...

General conditions:

Let X be some set. Let g be some function which returns some set over the
domain of all subsets of X, including X. Let f be some function such that
for any of the subsets Y of X, including X, f(Y) is equal to the
intersection of Y and g(Y) [that is, the set of all elements of Y which
are also elements of g(Y)]. Let n be some natural number.
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Rational (I)(n):

g is rational (I)(n) on X if and only if;

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For any of the subsets Y of X, including X, for any of the subsets Z of X,
including X, for any of the subsets W of X, including W;

If;

f(Y) is any of the subsets of Z, including Z, AND W is any of the
subsets of Y, including Y, AND the cardinality of W [number of elements in
W] is greater than or equal to n, AND the cardinality of Z is greater
than or equal to n, AND;

f(V) is any of the subsets of W, including W, OR f(V) is any of the
subsets of f(Z), including f(Z), where V is equal to the union of W and
f(Z) [that is, the set of all elements of W or f(Z)];

Then it must be that;

f(V) is equal to f(Z), where V is equal to the union of W and f(Z).
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Rational (II)(n):

g is rational (II)(n) on X if and only if;

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For any of the subsets Y of X, including X, for any of the subsets Z of X,
including X;

If;

f(Y) is any of the subsets of Z, including Z, AND the cardinality of Y 
[number of elements in Y] is greater than or equal to n, AND the 
cardinality of Z is greater than or equal to n, AND;

f(V) is any of the subsets of Y, including Y, OR f(V) is any of the
subsets of f(Z), including f(Z), where V is equal to the union of Y and
f(Z) [that is, the set of all elements of Y or f(Z)];

Then it must be that;

f(V) is equal to f(Z), where V is equal to the union of Y and
f(Z).
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THEOREM:

g is rational (II)(n) on X implies g is rational (I)(n) on X
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I'm having BIG problems proving/disproving this theorem... Could someone
give me a hand?

Thanks,

David