Ordered Sum of Tosets is Totally Ordered Set/General Result

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Theorem

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

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

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

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

Then $T_n$ is a toset.

Proof

From Ordered Sum of Tosets is Totally Ordered Set, $S_1 + S_2$ is a toset.

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

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

The result follows by induction.

$\blacksquare$