General Associative Law for Ordinal Sum

Theorem
Let $x$ be a member of the minimal infinite successor set.

Let $\langle a_i \rangle$ be a sequence of ordinals.

Then:


 * $\displaystyle \sum_{i \mathop = 1}^{x + 1} a_i = a_1 + \sum_{i \mathop = 1}^x a_{i + 1}$

Proof
The proof shall proceed by induction on $x$.

Basis for the Induction
If $x = 0$, then:

This proves the basis for the induction.

Induction Step
Suppose that $\displaystyle \sum_{i \mathop = 1}^{x + 1} a_i = a_1 + \sum_{i \mathop = 1}^x a_{i + 1}$.

Then:

This proves the induction step.