General Associativity Theorem/Motivation

Motivation for General Associativity Theorem

The General Associativity Theorem answers the following question:

It has been proved that, for example, union and intersection are associative in Union is Associative and Intersection is Associative.

That is:

$R \cup \paren {S \cup T} = \paren {R \cup S} \cup T$

and the same with intersection.

However, are we sure that there is only one possible answer to $\ds \bigcup_{i \mathop = 1}^n S_i$ and $\ds \bigcap_{i \mathop = 1}^n S_i$?

That is, is it completely immaterial where we put the brackets in an expression containing an arbitrary number of multiple instances of one of these operations?

The question is a larger one than that: given any associative operation, is it completely associative?

This result shows that it is. Always.