General Associative Law for Ordinal Sum/Proof 2

Theorem

Let $x$ be a finite ordinal.

Let $\sequence {a_i}$ be a sequence of ordinals.

Then:

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

Proof

From Ordinal Addition is Associative we have that:

$\forall a, b, c \in \On: a + \paren {b + c} = \paren {a + b} + c$

The result follows directly from the General Associativity Theorem.

$\blacksquare$