Lemma 1

In E $\times$ S' the relation R$\mathscr{R}$x, y$\mathscr{R}$ defined by:

"there exist $a, b \in E$ and $p, q, s \in S'$ such that $x = (a, p), y = (b, q)$ and $aqs = bps$"

is an equivalence relation compatible with the law on the product monoid E $\times$ S'.

It is immediate that R is reflexive and symmetric. Let $x = (a, p), y = (b, q)$ and $z = (c, r)$ be elements of E $\times$ S' such that R$\mathscr{R}$x, y$\mathscr{R}$ and R$\mathscr{R}$y, z$\mathscr{R}$ hold. Then there exist two elements s and t of S' such that

whence it follows that

and hence $R \{ x, z \}$ holds, for $stq$ belongs to $S'$. The relation $R$ is therefore transitive.

Further, let $x = (a, p), y = (b, q), x' = (a', p'),$ and $y' = (b', q')$ be elements of $E \times S'$ such that $R \{ x, y \}$ and $R \{ x', y' \}$ hold. There exist $s$ and $s'$ in $S'$ such that

whence it follows that $(aa')(qq')(ss') = (bb')(pp')(ss')$ and hence $R \{ xx', yy' \}$ for $ss' \in S'$. The equivalence relation $R$ is therefore compatible with the law of composition on $E \times S'$.

The quotient magma $(E \times S')/R$ is a commutative monoid.