- and a
simplex !i(q)(_a) must be in general position. By Lemma 14.1, this condition is*
* automatically
satisfied (with no restriction on z0) by those pairs for which the affine span *
*of Ol[ !i(q)(_a)
is all of Rm . For the remaining pairs, it suffices by Lemma 14.3 that !i(q)(z*
*0) should not
be in the affine span of Ol[ !i(q)(_a) (note that Oland !i(q)(_a) are in genera*
*l position by
the inductive hypothesis). Since this affine span is nowhere dense, the set of *
*allowable z0for
each such pair contains an open dense subset of a neighborhood of vp. If we now*
* let l, a and
oe vary through all relevant choices, we still have an open dense subset U of a*
* neighborhood
of vp for which property (*), and hence property (4a), is valid.
A similar argument shows that there is an open dense subset V of a neighborh*
*ood of vp
for which property (4b) is valid and an open dense subset W of a neighborhood o*
*f vp for
which property (4c) is valid. If z0 is chosen in the intersection of U, V and W*
* , all parts of
property (4) are valid. This concludes the proof.
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