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.

Also see

 * Associativity on Four Elements

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

and the same with intersection.

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

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.