reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;
reserve D for non empty Subset of L;
reserve D1,D2 for non empty Subset of L;

theorem Th23:
  p in D implies <.p.) c= <.D.)
proof
  assume
A1: p in D;
  let x be object;
A2: D c= <.D.) by Def4;
  assume x in <.p.);
  then ex q st x = q & p [= q;
  hence thesis by A1,A2,Th9;
end;
