General Associativity Theorem

Theorem
If an operation is associative on 3 entities, then it is associative on any number of them.

Also known as
Also known as the general (or generalized) associative law.

Comment
This 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: $R \cup \left({S \cup T}\right) = \left({R \cup S}\right) \cup T$ and the same with intersection.

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

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.