A dichotomy theorem for regular types

Theorem 1.(a) LetT be a superstable theory without the omitting type order property. Then every regular type is either locally modular, or non-orthogonal to a strongly regular type. In the latter case, a realization of the strongly regular type can be found algebraically in any realization of the given one.(b) LetT be a superstable theory with NOTOP and NDOP. Then every regular type is either locally modular or strongly regular.Theorem 2.(a) Letp be a nontrivial regular type. Thenp-weight is continuous and definable inside some definable setD of positivep-weight. Ifp is non-orthogonal toB, thenD can be chosen definable overB.(b) Letp be a nontrivial regular type of depth 0. Let stp(a/B) bep-semi-regular. Thena lies in some acl(B)-definable setD such thatp-weight is continuousand definable insideD.