Proposition 2

Let E be an associative magma and X a subset of E. Let X' be the set of $x_1 \top x_2 \top \cdots \top x_n$, where $n \geq 1$ and where $x_i \in X$ for $1 \leq i \leq n$. The stable subset generated by X is equal to X'.

It follows immediately by induction on n that the composition of an ordered sequence of n terms belonging to X belongs to the stable subset generated by X; it is therefore sufficient to verify that X' is stable. Now if u and v are elements of X' they are of the form $u = x_0 \top x_1 \top \cdots \top x_{n-1}$, $v = x_n \top x_{n+1} \top \cdots \top x_{n+p}$ with $x_i \in X$ for $0 \leq i \leq n + p$; then (Theorem 1) $u \top v = x_0 \top x_1 \top \cdots \top x_{n+p}$ belongs to X'.