We　generalize　an　important　theorem　of　Fred　Galvin　from　the　Stone-Čech　compactification　βT　of　any　discrete　semigroup　T　to　any　compact　Hausdorff　right-topological　semigroup　with　a　dense　topological　center;　and　then　apply　it　to　Ellis’　semigroups　to　prove　that　a　point　is　distal　if　and　only　if　it　is　IP*-recurrent,　for　any　semiflow　(T;X)　with　arbitrary　compact　Hausdorff　phase　space　X　not　necessarily　metrizable　and　with　arbitrary　phase　semigroup　T　not　necessarily　cancelable.　©　2017,　Science　China　Press　and　Springer-Verlag　GmbH　Germany.