In  several conditions are described which imply the closedness of linear images of convex sets. Here we combine two such theorems into a common generalization. We give a proof of the general theorem which is simpler than the proof obtained by combining the proofs of the two theorems. The paper is almost self-contained as we give transparent proofs of the separation theorems used. Also we present an application of our general closedness theorem in the theory of duality in convex programming.  R. T. Rockafellar, Convex analysis, Princeton University Press, Princeton, 1970.
Operation Research Report 2005-2, Eötvös Loránd University Budapest, H-1117 Budapest, Pázmány Péter 1/c, Hungary