PROOF:Suppose that *T* and
are both quasi-parity structures for
a finite set *S*. Then for any even subset *b* of *S*, the number of
odd subsets of *b* that lie in *T* is even, as is the number that lie
in
.
Thus the same holds for their symmetric difference.
And since the image under
of the symmetric difference of *T*
and
is the sum of their images under ,
it suffices
to check the result for some collection of parity structures whose
images under
form an
-basis for *R*_{S}.
Hence by the previous theorem, it is enough to show that parity
structures of the form
,
with *t* an even-degree
monomial (in distinct indeterminants) are quasi-parity structures.

Such a monomial *t* equals
for some even subset *c* of *S*.
>From the definition of ,
the parity
structure
consists of *c* together
with all subsets of *S* obtained by adding one new element to *c*.
Now take any even subset *b* of *S*. If *c* is not a subset of *b*,
then the number of odd subsets of *b* that lie in *T* is zero, so we
may assume that *c* is a subset of *b*. Then the odd subsets of *b*
that lie in *T* are exactly those subsets of *b* obtained by adding a
new element to *c*. But there are |*b*| - |*c*| such subsets, and since
both |*b*| and |*c*| are even, the number of such subsets is even.
Thus *T* is a quasi-parity structure.
This completes the proof of the theorem.

It is worth noting that the converse of Theorem 2 is false.
Since the definition of a quasi-parity structure refers only to those
elements of *T* that are odd subsets of *S*, any collection of even
subsets of *S* will be a quasi-parity structure. On the other hand,
the collection consisting of just the empty set will not be a parity
structure, unless *S* is empty. More generally, since quasi-parity
structures are closed under taking symmetric differences, the symmetric
difference of any parity structure and any collection of even subsets
of *S* will be a quasi-parity structure, and it can be shown that all
quasi-parity structures are of this form. Moreover, for ,
there are exactly
quasi-parity structures for *S*.