General Associativity Theorem/Formulation 2

Theorem
Let $n \in \N_{>0}$ and let $a_1, \ldots, a_n$ be elements of a set $S$.

Let $\circ$ be an associative operation on $S$.

Let the set $\map {P_n} {a_1, a_2, \ldots, a_n}$ be defined inductively by:


 * $\map {P_1} {a_1} = \set {a_1}$


 * $\map {P_2} {a_1, a_2} = \set {a_1 \circ a_2}$


 * $\map {P_n} {a_1, a_2, \ldots, a_n} = \set {x \circ y: x \in \map {P_r} {a_1, a_2, \ldots, a_r} \land y \in \map {P_s} {a_{r + 1}, a_{r + 2}, \ldots, a_{r + s} }, n = r + s}$

Then $\map {P_n} {a_1, a_2, \ldots, a_n}$ consists of a unique entity which we can denote $a_1 \circ a_2 \circ \ldots \circ a_n$.