
theorem Th14:
  for L being noetherian complete Lattice for a being Element of L
  holds a is completely-meet-irreducible iff ex b being Element of L st b
  is-upper-neighbour-of a & for c being Element of L holds c
  is-upper-neighbour-of a implies c = b
proof
  let L be noetherian complete Lattice;
  let a be Element of L;
  set X = { x where x is Element of L : a [= x & x <> a};
  hereby
    assume a is completely-meet-irreducible;
    then a*' is-upper-neighbour-of a & for c being Element of L holds c
    is-upper-neighbour-of a implies c = a*' by Th12;
    hence ex b being Element of L st b is-upper-neighbour-of a & for c being
    Element of L holds c is-upper-neighbour-of a implies c = b;
  end;
  given b being Element of L such that
A1: b is-upper-neighbour-of a and
A2: for c being Element of L holds c is-upper-neighbour-of a implies c = b;
A3: a <> b by A1;
  for q being Element of L st q in X holds b [= q
  proof
    let q be Element of L;
    assume q in X;
    then ex q9 being Element of L st q9 = q & a [= q9 & q9 <> a;
    then ex c being Element of L st c [= q & c is-upper-neighbour-of a by Th3;
    hence thesis by A2;
  end;
  then
A4: b is_less_than X by LATTICE3:def 16;
  a [= b by A1;
  then b in X by A3;
  hence a <> a*' by A3,A4,LATTICE3:41;
end;
