 Moreover, if "x" and "y" are contained in a connected (respectively, path connected) subset "A" and "y" and "z" are connected in a connected (respectively, path connected) subset "B", then the Lemma implies that formula_18 is a connected (respectively, path connected) subset containing "x", "y" and "z".Thus each relation is an equivalence relation, and defines a partition of "X" into equivalence classes.We consider these two partitions in turn.