Antilexicographic Product of Totally Ordered Sets is Totally Ordered/General Result

Theorem
Let $S_1, S_2, \ldots, S_n$ all be tosets.

Let $T_n$ be the ordered product of $S_1, S_2, \ldots, S_n$:


 * $\forall n \in \N_{>0}: T_n = \begin{cases}

S_1 & : n = 1 \\ T_{n-1} \cdot S_n & : n > 1 \end{cases}$

Then $T_n$ is a toset.

Proof
From Ordered Product of Tosets is Totally Ordered Set, $S_1 \cdot S_2$ is a toset.

Suppose $T_{n-1}$ is a toset.

Given that $S_n$ is a toset, it follows from Ordered Product of Tosets is Totally Ordered Set that $T_{n-1} \cdot S_n$ is also a toset.

The result follows by induction.