We call an orthomodular lattice £ m-orthocomplete for an infinite cardinal m if every orthogonal family of Sm elements from £ has a join in £, and we call £ m-complete if every family, orthogonal or not, of %m elements from £ has a join in £. We prove that an »z-orthocomplete orthomodular lattice is mcomplete. Since a Boolean algebra is a distributive orthomodular lattice, we obtain as a special case the Smith-Tarski theorem: An m-orthocomplete Boolean algebra is m-complete. © On-i -JA p + K 8)

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... ^ V (ya; a < 8). We need therefore prove only the statement P(B): V(ya; a<8) = © (yp+i-%; P + l<)3)-P(2) is the assertion yi^yi-yo which is true because yo = 0. We use transfinite induction. Assume thatPCy) Received by the editors June 30, 1969. A MS Subject Classifications. Primary 0635, 0660.