
theorem Th24:
  for L being Boolean lower-bounded' upper-bounded' distributive'
  Lattice-like non empty LattStr holds L is complemented'
proof
  let L be Boolean lower-bounded' upper-bounded' distributive' Lattice-like
  non empty LattStr;
  for b being Element of L ex a being Element of L st a is_a_complement'_of b
  proof
    let b be Element of L;
    consider a being Element of L such that
A1: a is_a_complement_of b by LATTICES:def 19;
    take a;
A2: b "/\" a = Bottom L by A1;
    b "\/" a = Top L by A1;
    hence b "\/" a = Top' L & a "\/" b = Top' L &
    b "/\" a = Bot' L & a "/\" b = Bot' L by A2,Th20,Th21;
  end;
  hence thesis;
end;
