Remark

Let $P\{x, x'\}$ be a relation in which the letters $y$, $y'$ do not occur. P is said to be compatible with the equivalence relations $R\{x, y\}$ and $R'\{x', y'\}$ (with respect to $x$ and $x'$) if the relation $(P\{x, x'\}$ and $R\{x, y\}$ and $R'\{x', y'\})$ implies $P\{y, y'\}$. Let $Q\{u\}$ be the relation $(\exists x)(\exists x')(u = (x, x')$ and $P\{x, x'\})$; then it comes to the same thing to say that $Q\{u\}$ is compatible (with respect to $u$) with the equivalence relation $S\{u, v\}$, the product of $R$ and $R'$.