
theorem Th15:
  for L being co-noetherian complete Lattice for a being Element
  of L holds a is completely-join-irreducible iff ex b being Element of L st b
  is-lower-neighbour-of a & for c being Element of L holds c
  is-lower-neighbour-of a implies c = b
proof
  let L be co-noetherian complete Lattice;
  let a be Element of L;
  set X = { x where x is Element of L : x [= a & x <> a};
A1: now
    given b being Element of L such that
A2: b is-lower-neighbour-of a and
A3: for c being Element of L holds c is-lower-neighbour-of a implies c = b;
A4: a <> b by A2;
    for q being Element of L st q in X holds q [= b
    proof
      let q be Element of L;
      assume q in X;
      then ex q9 being Element of L st q9 = q & q9 [= a & q9 <> a;
      then ex c being Element of L st q [= c & c is-lower-neighbour-of a by Th4
;
      hence thesis by A3;
    end;
    then
A5: b is_greater_than X by LATTICE3:def 17;
    b [= a by A2;
    then b in X by A4;
    then a <> *'a by A4,A5,LATTICE3:40;
    hence a is completely-join-irreducible;
  end;
  now
    assume a is completely-join-irreducible;
    then *'a is-lower-neighbour-of a & for c being Element of L holds c
    is-lower-neighbour-of a implies c = *'a by Th13;
    hence ex b being Element of L st b is-lower-neighbour-of a & for c being
    Element of L holds c is-lower-neighbour-of a implies c = b;
  end;
  hence thesis by A1;
end;
