= Solution
Suppose $A\cup B=U\cup V$ were a <separation>. By <connectedness>, all of $A$ lies on one side, say $U$: otherwise $A\cap U,A\cap V$ would separate $A$. A point of $A\cap B$ then lies in $B\cap U$, so <connectedness> of $B$ forces all of $B$ into $U$ as well. This makes $V$ empty, a contradiction.
Therefore \b[the union of two intersecting <connected subsets> is <connected>]. The argument also proves the <connected union with a connected hub> rule for arbitrarily many connected pieces meeting one connected subset.
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