 reserve L for Lattice;
 reserve I,P for non empty ClosedSubset of L;

theorem Th1:
  for L being Boolean Lattice holds L is pseudocomplemented
  proof
    let L be Boolean Lattice;
    L is pseudocomplemented
    proof
      let x be Element of L;
      take y = x`;
      for z being Element of L st x "/\" z = Bottom L holds z [= y
      proof
        let z be Element of L;
        assume E1: x "/\" z = Bottom L;
        z = z "/\" Top L .= z "/\" (x "\/" y) by LATTICES:21
         .= (z "/\" x) "\/" (z "/\" y) by LATTICES:def 11 .= z "/\" y by E1;
        hence z [= y by LATTICES:4;
      end;
      hence thesis by LATTICES:20;
    end;
    hence thesis;
  end;
