The theorem of this paper has appeared under stronger hypotheses and sometimes with weaker conclusions a number of times [1; 2; 3 ], and was known to Koch (in approximately the form of [2]) in 1959 (unpublished). Since it is a useful tool in the study of topological semigroups, and the proof here is simpler, and the theorem stronger than those mentioned, it seems advisable to have it in the literature.