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 Th24:
  D = {p} implies <.D.) = <.p.)
proof
  assume
A1: D = {p};
  D c= <.p.)
  proof
    let x be object;
    assume x in D;
    then x = p by A1,TARSKI:def 1;
    hence thesis;
  end;
  hence <.D.) c= <.p.) by Def4;
  p in D by A1,TARSKI:def 1;
  hence thesis by Th23;
end;
