reserve x,y,X,X1,Y,Z for set;
reserve L for Lattice;
reserve F,H for Filter of L;
reserve p,q,r for Element of L;
reserve L1, L2 for Lattice;
reserve a1,b1 for Element of L1;
reserve a2 for Element of L2;
reserve f for Homomorphism of L1,L2;
reserve B for Element of Fin the carrier of L;
reserve DL for distributive Lattice;
reserve f for Homomorphism of DL,L2;
reserve 0L for lower-bounded Lattice,
  B,B1,B2 for Element of Fin the carrier of 0L,
  b for Element of 0L;

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;
