reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;

theorem Th9:
  for L1,L2 being 0_Lattice st the LattStr of L1 = the LattStr of
  L2 holds Bottom L1 = Bottom L2
proof
  let L1,L2 be 0_Lattice;
  assume
A1: the LattStr of L1 = the LattStr of L2;
  then reconsider c = Bottom L1 as Element of L2;
  now
    let a be Element of L2;
    reconsider b = a as Element of L1 by A1;
    Bottom L1"/\"b = Bottom L1;
    hence c"/\"a = c by A1;
    hence a"/\"c = c;
  end;
  hence thesis by LATTICES:def 16;
end;
