 reserve L for Lattice;
 reserve I,P for non empty ClosedSubset of L;
reserve L for lower-bounded pseudocomplemented Lattice;

theorem ThE:
  for L being Boolean Lattice, x being Element of L holds x` = x*
  proof
    let L be Boolean Lattice, x be Element of L;
a1: x* [= x`
    proof
      x* "/\" x = Bottom L by ThD;
      hence thesis by LATTICES:25;
    end;
    x` [= x*
    proof
a1:   (x`) "/\" x = Bottom L by LATTICES:20;
      x* is_a_pseudocomplement_of x by def3;
      hence thesis by a1;
    end;
    hence thesis by a1,LATTICES:8;
  end;
