
theorem Th13:
  for L being complete Lattice for a being Element of L st a is
  completely-join-irreducible holds *'a is-lower-neighbour-of a & for c being
  Element of L holds c is-lower-neighbour-of a implies c = *'a
proof
  let L be complete Lattice;
  let a be Element of L;
  set X = { x where x is Element of L : x [= a & x <> a};
A1: for c being Element of L st *'a [= c & c [= a holds c = a or c = *'a
  proof
    let c be Element of L;
    assume that
A2: *'a [= c and
A3: c [= a;
    assume c <> a;
    then c in X by A3;
    then c [= *'a by LATTICE3:38;
    hence thesis by A2,LATTICES:8;
  end;
  assume a is completely-join-irreducible;
  then
A4: *'a <> a;
A5: for c being Element of L holds c is-lower-neighbour-of a implies c = *' a
  proof
    let c be Element of L;
    assume
A6: c is-lower-neighbour-of a;
    then a <> c & c [= a;
    then c in X;
    then
A7: c [= *'a by LATTICE3:38;
    *'a [= a by Th9;
    hence thesis by A4,A6,A7;
  end;
  *'a [= a by Th9;
  hence thesis by A4,A1,A5;
end;
