gn.general topology - Connectedness of a union regading a proximity

Consider $Xcap A$ and $Ycap A$, starting from a partition $lbrace X,Yrbrace$ of $Acup B$. If both intersections are nonempty we are done, as $(Xcap A)delta(Ycap A)$. Otherwise, $Asubseteq X$, say, but then $Xcap B$ and $Ycap B$ are nonempty and we find $(Xcap B)delta(Ycap B)$.
In either case $Xdelta Y$.