PROOF. We shall first prove that every complete graph of power t$i can be split up into the countable sum of trees. Let G be a complete graph of cardinal number ML Let {xa}, a n. I t is clear that G = U*=3]Gn and that for each Gn, for every j8 i, there exists one and only one a such that (xa, ^ ) GGM and ce , is a well ordered set of cardinal number tn. We shall first decompose each Tn into four parts Tn,i, i = l, 2, 3, 4, such that Tn,i and rw,2 satisfy the condition: (1) Any two consecutive segments of the graphs Tn,i and Tn,2 are of the form: (xa, xp), (xa, xy), a 4i satisfy: (2) Any two consecutive segments of the graphs are of the form: (Xp, Xa), (Xy, Xa)9 j8 < « , J < « , JST^Y-