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.:;

theorem Th36:
  <.D.:.) = (.D.> & <.D.) = (.D.:.> & <..:D9.) = (.D9.> & <.D9.) = (..: D9.>
proof
A1: for L,D holds <.D.:.) = (.D.>
  proof
    let L,D;
    (.D.>.: = (.D.>;
    then
A2: D c= (.D.> implies <.D.:.) c= (.D.> by FILTER_0:def 4;
    .:<.D.:.) = <.D.:.);
    then D c= <.D.:.) implies (.D.> c= <.D.:.) by Def9;
    hence thesis by A2,Def9,FILTER_0:def 4;
  end;
  <.D .: .:.) = <.D.) & <.( .:D9) .: .:.) = <..:D9.) by Lm2;
  hence thesis by A1;
end;
