given to . When $a_i^j=0$ for all $i,j$, this reduces to $$\label{hi1}
      M={Y_1(r-1)}^{(p_1,0)}{Y_2(r-2)}^{(p_2,0)}.$$ We denote by $\M(p_1,p_2;r;\infty)$ the set of all monomials of this form and write the monomial of  as $M_{(p_1,p_2;r;\infty)}$.

A result similar to Proposition \[propnk\] may be obtained for $M_{(p_1,p_2;r;\infty)}$.

Each element of $\M(p_1,p_2;r;\infty)$ may be written uniquely in the form $$~\label{equ:jin6}
\begin{aligned}
      M=& X_1(r-1)^{(p_1+p_2,-b^1_2-b^1_3-b^1_0-
                      b^1_{\bar3}-b^1_{\bar2}-b^1_{\bar1})}
          X_2(r-1)^{(0,b_2^1)}
          X_3(r-1)^{(0,b_3^1)}\\
        & \cdot
          X_0(r-1)^{(0,b_0^1)}
          X_{\bar3}(r-1)^{(0,b_{\bar3}^1)}
          X_{\bar2}(r-1)^{(0,b_{\bar2}^1)}
          X_{\bar1}(r-1)^{(0,b_{\bar1}^1)}\\
        & \cdot
          X_2(r-2)^{(p_2,-b_3^2)}
          X_3(r-2)^{(0,b_3^2)}
\end{aligned}$$ where $b_i^j\ge 0$ for all $i,j$ and $b_0^1\le 1$. Conversely, any element in $\Me$ of this form is an element of $\M(p_1,p_2;r;\infty)$.

We believe the readers can easily write down the process for *change of variable* similar to that given by  and  for $\M(p_1,p_2;r;\infty)$.

The set $\M(\infty)$ is a special case of this set $\M(p_1,p_2;r;\infty)$ corresponding to $r=0$ and $p_i=1$ for all $i\in
I$.

\[lem6\] It is possible to obtain the result of Lemma \[anaysiskashi2\] also for the case $\M(p_1,p_2;r;\infty)$. Thus we can state that the set $\M(p_1,p_2;r;\infty)$ forms a $\uq(G_2)$-subcrystal of $\Me$.

\[cor4\] The set $\M(p_1,p_2;r;\infty)$ forms a subcrystal of $\Me$ isomorphic to $\M(\infty)$ as a $\uq(G_2)$-crystal.

As mentioned in Remark \[lem6\], we can show that the set $\M(p_1,p_2;r;\infty)$ forms a $\uq(G_2)$-subcrystal of $\Me$. Let us show 