Internal Direct Product Theorem/General Result/Proof 2

Proof
It is to be shown that:

by definition 2 of Internal Group Direct Product.

Condition $(3)$ already gives that $H_i$ is normal for all $i \in \set {1, 2, \ldots, n}$.

Condition $(1)$ gives us that each element $g$ of $G$ can be expressed in the form:
 * $g = h_1 h_2 \dotsm h_n$ with $h_i \in H_i$ for all $i \in \set {1, 2, \ldots, n}$.

It is now shown that this expression is unique.

Suppose that:
 * $g = h_1 h_2 \dotsm h_n = k_1 k_2 \dotsm k_n$

where $h_i, k_i \in H_i$ for all $i \in \set {1, 2, \ldots, n}$ and $h_j \ne k_j$ for at least one $j$.

Let $j$ be the largest integer such that $h_j \ne k_j$, so that $h_i = k_i$ for $i > j$.

Cancelling $h_i$ for $i > j$ gives:
 * $h_i h_2 \dotsm h_j = k_1 k_2 \dotsm k_j$

and so:
 * $h_j {k_j}^{-1} = \paren {h_1 h_2 \dotsm h_{j - 1} }^{-1} \paren {k_1 k_2 \dotsm k_{j - 1} } \in \paren {H_1 H_2 \dotsm H_{j - 1} } \cap G_j$

But by condition $(2)$:
 * $\paren {H_1 H_2 \dotsm H_{j - 1} } \cap G_j = \set e$

by definition of independent subgroups.

Thus $h_j = k_j$, which contradicts our assertion that $h_j \ne k_j$.

Hence the decomposition is unique.