diff --git a/app_hard_lem-mult-p.tex b/app_hard_lem-mult-p.tex
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--- a/app_hard_lem-mult-p.tex
+++ b/app_hard_lem-mult-p.tex
@@ -13,8 +13,8 @@ Given that we then have $2\kElem + 1$ distinct values of $\rpoly_{G}^\kElem(\pro
 Next, we show why we can compute $\numocc{G}{\kmatch}$ from $c_{2k}$ in $O(1)$ additional time.
 We claim that $c_{2\kElem}$ is $\kElem! \cdot \numocc{G}{\kmatch}$.  This can be seen intuitively by looking at the expansion of the original factorized representation 
 \[\poly_{G}^\kElem(\vct{X}) = \sum_{\substack{(i_1, j_1),\cdots,(i_\kElem, j_\kElem) \in E}}X_{i_1}X_{j_1}\cdots X_{i_\kElem}X_{j_\kElem},\] 
-where a unique $\kElem$-matching in the multi-set of product terms can be selected $\prod_{i = 1}^\kElem i = \kElem!$ times.  
-Indeed, note that each $\kElem$-matching $(i_1, j_1)\ldots$ $(i_k, j_k)$ in $G$ corresponds to the monomial $\prod_{\ell = 1}^\kElem X_{i_\ell}X_{j_\ell}$ in $\poly_{G}^\kElem(\vct{X})$, with distinct indexes, and this implies that each distinct $\kElem$-matching appears the exact number of permutations that exist for its particular set of $\kElem$ edges, or $k!$.
+where a unique $\kElem$-matching in the multi-set of product terms has $\kElem$ distinct $\inparen{i_\ell, j_\ell}$ index pairs.  Further, any monomial composed of such a distinct set of $2\kElem$ variables will be produced $\kElem!$ times in a $\kElem$-wise product of the sum of a set of elements with itself.  This is true because each (identical) product term contains each of the $\kElem$ distinct elements, giving us $\kElem\cdot\kElem-1\cdots 1 = \kElem!$ permutations of a distinct $\kElem$-matching.  %Such must be the case since each product term has  since this is the number of permutations for a given monomial.can be selected $\prod_{i = 1}^\kElem i = \kElem!$ times.  
+%Indeed, note that each $\kElem$-matching $(i_1, j_1)\ldots$ $(i_k, j_k)$ in $G$ corresponds to the monomial $\prod_{\ell = 1}^\kElem X_{i_\ell}X_{j_\ell}$ in $\poly_{G}^\kElem(\vct{X})$, with distinct indexes, and this implies that each distinct $\kElem$-matching appears the exact number of permutations that exist for its particular set of $\kElem$ edges, or $k!$.
   
 Since, as noted earlier, $c_{2\kElem}$ represents the number of monomials with $2\kElem$ distinct variables, then it must be that $c_{2\kElem}$ is the overall number of $\kElem$-matchings.  And since we have $\kElem!$ copies of each distinct $\kElem$-matching, it follows that
 $c_{2\kElem}= \kElem! \cdot \numocc{G}{\kmatch}$.
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