exists a ${\mathrm{GHD}}_k(s_i+e,g_i+u)$ and this design contains a ${\mathrm{GHD}}_k(e,u)$ as a subdesign for $1 \leq i \leq m-1$. Then there exists a where $s=\sum_{i=1}^m s_i$ and $v=\sum_{i=1}^m g_i$.

Suppose the ${\mathrm{GHF}}_k$ of type $\Pi (s_i,g_i)$ has point set $X$ and groups $G_i$, $i=1,2,\ldots,m$. Let $U$ be a set of size $u$, disjoint from $X$, and take our new point set to be $X \cup U$. Now, add $e$ new rows and columns. For each group $G_i$ with $1 \leq i \leq m-1$, fill the $s_i \times s_i$ subsquare corresponding to $G_i$, together with the $e$ new rows and columns, with a copy of the ${\mathrm{GHD}}_k(s_i+e,g_i+u)$, with the sub-${\mathrm{GHD}}_k(e,u)$ (containing the points in $U$) over the $e \times e$ subsquare which forms the intersection of the new rows and columns. See Figure \[BasicFrameFigure\]. We then delete the blocks of the sub-${\mathrm{GHD}}_k(e,u)$. Now, fill the $s_m \times s_m$ subsquare corresponding to $G_m$, together with the $e$ new rows and columns with the ${\mathrm{GHD}}_k(s_m+e,g_m+u)$.

$$\begin{array}{|c|c|c|c|}  \hline
{\cellcolor[gray]{.8}}U & \ldots & {\cellcolor[gray]{.8}}- & \ldots \\ \hline
\vdots &  \ddots & \vdots &  \ddots \\  \hline
{\cellcolor[gray]{.8}}- & \ldots & {\cellcolor[gray]{.8}}G_i & \ldots  \\  \hline
\vdots &  \ddots & \vdots & \ddots \\  \hline
\end{array}$$

