Formal Composition

Given Formal Power Series in a Ring :

with , define the composition

as a Formal Infinite Sum.
Note that was necessary but also sufficient for convergence.

Lemma

Suppose and .
Let be a Structure with
and a Structure with .
Let be the number of ways to split into disjoint nonempty intervals
and build in each interval
and build on the set of intervals.
Set .
Then