clarification on the definition of meaningful product

I am studying Hungerford's book "Algebra". In the page 27 he defines the meaningful product as follows.

Given any sequence of elements of a semigroup $G, > \{a_{1},a_{2},\dots\}$ define inductively a meaningful product (in this order) as follows. If $n=1$, then the only meaningful product is $a_{1}$. If $n>1$, then a meaningful product is defined to be any product of the form $(a_{1}\cdots a_{m})(a_{m+1}\cdots a_{n})$ where $m< n$ and $(a_{1}\cdots a_{m})$ and $(a_{m+1}\cdots a_{n})$ are meaningful products of $m$ and $n-m$ elements respectively.

He notes next the following:

To show that this definition is in the fact well defined requires a stronger version of Recursion Theorem 6.2 of the Introduction; see C.W. Burril: Foundations of Real Numbers.

I don't have access to this book, so I would like to know this version and see how to use it, or a reference if possible.

I've never seen this definition before. Is it really necessary to define a meaningful product in order to prove that Generalized Associative law holds on a semigroup?

Thanks for your help.

What is often done for semigroups is defining positive integer powers of elements, and defining products using associativity of multiplication in the semigroup.