theorem
  f is onto implies CL is Boolean & f preserves_complement
proof
  assume
A1: f is onto;
  then
A2: f preserves_top by Th18;
  thus CL is bounded complemented;
  thus CL is distributive by A1,Th11;
  then reconsider CL as Boolean Lattice;
A3: f preserves_bottom by A1,Th12;
  reconsider f as Homomorphism of BL,CL;
  now
    let a1;
A4: f.(a1`)"/\"f.a1 = f.(a1` "/\" a1) by D2
      .=f.(Bottom BL) by LATTICES:20
      .= Bottom CL by A3;
A5: f.(a1`)"\/"f.a1 = f.a1"\/"f.(a1`) & f.(a1`)"/\"f.a1 = f.a1"/\"f.(a1`);
    f.(a1`)"\/"f.a1 =f.(a1` "\/" a1) by D1
      .=f.(Top BL) by LATTICES:21
      .= Top CL by A2;
    then f.(a1`) is_a_complement_of f.a1 by A4,A5;
    hence (f.a1)` = f.(a1`) by LATTICES:def 21;
  end;
  hence thesis;
end;
