
theorem Th8:
  for L being co-noetherian lower-bounded Lattice for a being
  Element of L holds a = Bottom L iff not(ex b being Element of L st b
  is-lower-neighbour-of a)
proof
  let L be co-noetherian lower-bounded Lattice;
  let a be Element of L;
  now
    assume
A1: not(ex b being Element of L st b is-lower-neighbour-of a);
    for b being Element of L holds a "/\" b = a & b "/\" a = a
    proof
      let b be Element of L;
      consider c being Element of L such that
A2:   c = a "/\" b;
A3:   c [= a by A2,LATTICES:6;
      per cases;
      suppose
        a <> c;
        then ex d being Element of L st c [= d & d is-lower-neighbour-of a by
A3,Th4;
        hence thesis by A1;
      end;
      suppose
        a = c;
        hence thesis by A2;
      end;
    end;
    hence a = Bottom L by LATTICES:def 16;
  end;
  hence thesis by Th7;
end;
