
theorem Th23:
  for L being complemented' join-commutative meet-commutative
  lower-bounded' upper-bounded' join-idempotent distributive distributive' non
  empty LattStr holds L is complemented
proof
  let L be complemented' join-commutative meet-commutative lower-bounded'
upper-bounded' join-idempotent distributive distributive' 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 Def7;
    take a;
A2: b "/\" a = Bot' L by A1;
    b "\/" a = Top' L by A1;
    hence a"\/"b = Top L & b"\/"a = Top L &
    a"/\"b = Bottom L & b"/\"a = Bottom L by A2,Th18,Th19;
  end;
  hence thesis;
end;
