
theorem Th25:
  for L being non empty LattStr holds L is Boolean Lattice iff L
is lower-bounded' upper-bounded' join-commutative meet-commutative distributive
  distributive' complemented'
proof
  let L be non empty LattStr;
  thus L is Boolean Lattice implies L is lower-bounded' upper-bounded'
  join-commutative meet-commutative distributive distributive' complemented'
  proof
    assume
A1: L is Boolean Lattice;
    then reconsider L9 = L as Boolean Lattice;
    ex c being Element of L9 st for a being Element of L9 holds c "\/" a =
    a & a "\/" c = a
    proof
      take Bottom L9;
      thus thesis;
    end;
    hence
A2: L is lower-bounded';
    ex c being Element of L9 st for a being Element of L9 holds c "/\" a =
    a & a "/\" c = a
    proof
      take Top L9;
      thus thesis;
    end;
    hence
A3: L is upper-bounded';
    thus L is join-commutative meet-commutative by A1;
    for a,b,c being Element of L9 holds a "/\" (b "\/" c) = (a "/\" b)
    "\/" (a "/\" c) by LATTICES:def 11;
    then for a, b, c being Element of L9 holds a "\/" (b "/\" c) = (a "\/" b)
    "/\" (a "\/" c) by LATTICES:3;
    hence L is distributive distributive';
    hence thesis by A1,A2,A3,Th24;
  end;
  thus L is lower-bounded' upper-bounded' join-commutative meet-commutative
  distributive distributive' complemented' implies L is Boolean Lattice
  proof
    assume L is lower-bounded' upper-bounded' join-commutative
    meet-commutative distributive distributive' complemented';
    then reconsider L9 = L as lower-bounded' upper-bounded' complemented'
join-commutative meet-commutative join-idempotent distributive distributive'
    non empty LattStr by Th9;
A4: L9 is complemented by Th23;
A5: L9 is lower-bounded & L9 is upper-bounded by Th12,Th14;
A6: L9 is meet-absorbing join-absorbing by Th10,Th11;
    then L9 is join-associative & L9 is meet-associative by Th16,Th17;
    hence thesis by A5,A4,A6;
  end;
end;
