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Let (E,ξ)=indlim (En,ξn) be an inductive limit of a sequence of locally convex spaces,For brevity,denote by (DS) each set Bbounded in (E,ξ) is contained in some En; and (DST) each set B bounded in (E,ξ) is contained and bounded in some (En,ξn). Theovem 1.