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PROOF. In [4] it was shown that ,s <,ut for any two positive integers s and t for which s <t. Hence the lemma is true if y and 8 are each integers. Since, for s finite, ,us<,(s+1) ?MW ?,(co) and ,us<,u(s +1) iw*<,u(X*+), it follows that the lemma is true if one of the elements y or 8 is finite, and the other is not. Suppose that .u1o and Aco* are demonstrated to be incomparable order types.2 This will imply that jiw <,u(co* +co) and that ,u* <1uz(w* +w). Consequently the lemma will be true in all cases. In order to see that ,co and ,uco* are incomparable order types let

Uniqueness in the left division of order types