- a)
- For every odd subset
*b*of*S*and every even natural number*k*, there are an even number of subsets of*b*of size*k*or*k*+ 1 that lie in*T*. - b)
- For every even subset
*b*of*S*and every odd natural number*k*, there are an even number of subsets of*b*of size*k*that lie in*T*.

PROOF:As in the proof of Theorem 2, it suffices to consider the
case when *T* is of the form
for some
even-degree monomial *t*. But then parts a) and b) follow from
Theorems 1 and 2, respectively,
for *k* equal to the degree of *t* or this degree plus 1,
and they hold trivially for all other values of *k*.

Of course, given the fact that every quasi-parity structure is the symmetric difference of a parity structure and a collection of even sets, it easily follows that quasi-parity structures also satisfy the condition in part b) of Corollary 2.