theorem Th12:
  for f being Homomorphism of 0L,L2 st f is onto holds
  L2 is lower-bounded & f preserves_bottom
proof
  let f be Homomorphism of 0L,L2;
  set r = f.(Bottom 0L);
  assume
A1: f is onto;
A2: now
    let a2 be Element of L2;
    consider a1 be Element of 0L such that
A3: f.a1 = a2 by A1,Th6;
    thus r"/\"a2 = f.(Bottom 0L "/\" a1) by A3,D2
      .= r;
    hence a2"/\"r = r;
  end;
  thus L2 is lower-bounded
  by A2;
  then Bottom L2=r by A2,LATTICES:def 16;
  hence thesis;
end;
