reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;

theorem
  (.D1 \/ D2.> = (.(.D1.> \/ D2.> & (.D1 \/ D2.> = (.D1 \/ (.D2.>.>
proof
A1: (.D1 \/ (.D2.>.> = <.(D1 \/ (.D2.>).:.) & (.D2.> = <.D2.:.) by Th36;
A2: (.D1 \/ D2.> = <.(D1 \/ D2).:.) by Th36;
  (.(.D1.> \/ D2.> = <.((.D1.> \/ D2).:.) & (.D1.> = <.D1.:.) by Th36;
  hence thesis by A1,A2,FILTER_0:34;
end;
