Since $\bi$ has the constraint that all tuples from the same block are mutually exclusive from one another, it is the case that there exist query polynomials $\poly$ such that $\rpoly$ will cancel out monomials that violate this condition. Let us assume that we have the following $\poly=\poly_1\cdot\poly_2$, where $\poly_1=\sum_{i =1}^\numvar\tup^{1_i}$ and $\poly_2=\sum{j =1}^\numvar\tup^{2_j}$, and $\tup^{a_i}$ is a monomial as defined in \Cref{def:monomial}, i.e., every term in $\tup^{a_i}$ is a single variable factor of the monomial as opposed to allowing product of sums. Note that each $\tup^{a_i}$ has at most a degree of $k$ and that each of its variables are associated with a particular block $\block$. We can assume WLOG that each monomial $\tup^{a_i}$ has at most one variable from each block since any $\tup^{a_i}$ having non-identitcal variables from the same $\block$ can easily be pruned in a $O(\numvar)$ scan.