Order Type Addition is Associative

Theorem
Let $\alpha$, $\beta$ and $\gamma$ be order types of ordered sets.

Then:
 * $\paren {\alpha + \beta} + \gamma = \alpha + \paren {\beta + \gamma}$

where $+$ denotes order type addition.

Proof
Let $\struct {S_1, \preccurlyeq_1}$, $\struct {S_2, \preccurlyeq_2}$ and $\struct {S_3, \preccurlyeq_3}$ be ordered structures such that:

where $\ot$ denotes order type.

Thus by definition of order type we are required to show that:


 * $\paren {\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2} } \oplus \struct {S_3, \preccurlyeq_3} \cong \struct {S_1, \preccurlyeq_1} \oplus \paren {\struct {S_2, \preccurlyeq_2} \oplus \struct {S_3, \preccurlyeq_3} }$

where:
 * $\oplus$ denotes order sum
 * $\cong$ denotes order isomorphism.

Let: