\begin{definition}[Definition:Internal Group Direct Product/General Definition/Definition by Subset Product]
Let $\sequence {H_n} = \struct {H_1, \circ {\restriction_{H_1} } }, \ldots, \struct {H_n, \circ {\restriction_{H_n} } }$ be a (finite) sequence of subgroups of a group $\struct {G, \circ}$
where $\circ {\restriction_{H_1} }, \ldots, \circ {\restriction_{H_n} }$ are the restrictions of $\circ$ to $H_1, \ldots, H_n$ respectively.
The group $\struct {G, \circ}$ is the '''internal group direct product of $\sequence {H_n}$''' {{iff}}:
:$(1): \quad$ Each $H_1, H_2, \ldots, H_n$ is a normal subgroup of $G$
:$(2): \quad G$ is the subset product of $H_1, H_2, \ldots, H_k$, that is: $G = H_1 \circ H_2 \circ \cdots \circ H_n$
:$(3): \quad$ For all $k \in \set {1, 2, \ldots, n}$: $H_k \cap \paren {H_1 \circ H_2 \circ \cdots \circ H_{k - 1} \circ H_{k + 1} \circ \cdots \circ H_n} =  H_k \set e$ where $e$ is the identity element of $G$.
\end{definition}