General Associativity Theorem/Formulation 2/Proof 1

Proof
The cases where $n = 1$ and $n = 2$ are clear.

Let $a = x \circ y \in P_n: x \in P_r, y \in P_s$.

If $r > 1$ then we write $x = a_1 \circ z$ where $z = a_2 \circ a_3 \circ \ldots \circ a_r$ by induction.

Then $x \circ y = \left({a_1 \circ z}\right) \circ y = a_1 \circ \left({z \circ y}\right) = a_1 \circ \left({a_2 \circ a_3 \circ \ldots \circ a_n}\right)$ (again by induction).

If $r=1$, then by induction $x \circ y = a_1 \circ y = a_1 \circ \left({a_2 \circ a_3 \circ \ldots \circ a_n}\right)$.

Thus in either case, $x \circ y = a_1 \circ \left({a_2 \circ a_3 \circ \ldots \circ a_n}\right)$ which is a single element of $P_n$.

Hence we see that $P_n \left({a_1, a_2, \ldots, a_n}\right)$ consists of a single element.