
theorem Th6:
  for L being noetherian upper-bounded Lattice for a being Element
of L holds a = Top L iff not(ex b being Element of L st b is-upper-neighbour-of
  a)
proof
  let L be noetherian upper-bounded Lattice;
  let a be Element of L;
  now
    assume
A1: not(ex b being Element of L st b is-upper-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:   a [= c by A2,LATTICES:5;
      per cases;
      suppose
        a <> c;
        then ex d being Element of L st d [= c & d is-upper-neighbour-of a by
A3,Th3;
        hence thesis by A1;
      end;
      suppose
        a = c;
        hence thesis by A2;
      end;
    end;
    hence a = Top L by LATTICES:def 17;
  end;
  hence thesis by Th5;
end;
