Borel reducibility as an additive property of domains

We prove that under certain requirements if $\mathrm E$ and $\mathrm F$ are Borel equivalence relations, $X=\bigcup_nX_n$ is a countable union of Borel sets, and $\mathrm E\upharpoonright X_n$ is Borel reducible to $\mathrm F$ for all $n$ then $\mathrm E\upharpoonright X$ is Borel reducible to $\mathrm F$. Thus the property of Borel reducibility to $\mathrm F$ is countably additive as a property of domains. Bibl. – 18 titles.