
theorem Th19:
  for L being complete co-noetherian Lattice for a being Element
  of L st a <> Bottom L holds a is completely-join-irreducible iff a% is
  join-irreducible
proof
  let L be complete co-noetherian Lattice;
  let a be Element of L;
  assume a <> Bottom L;
  then consider b being Element of L such that
A1: b is-lower-neighbour-of a by Th8;
A2: a <> b by A1;
  now
    assume
A3: a% is join-irreducible;
    for d being Element of L holds d is-lower-neighbour-of a implies d = b
    proof
      let d be Element of L;
A4:   a% = a by LATTICE3:def 3;
A5:   d% = d & b% = b by LATTICE3:def 3;
      assume
A6:   d is-lower-neighbour-of a;
      then
A7:   a <> d;
      assume d <> b;
      then a = d "\/" b by A1,A6,Th2;
      then a% = d% "\/" b% by A4,Lm8;
      hence thesis by A2,A3,A7,A4,A5,WAYBEL_6:def 3;
    end;
    hence a is completely-join-irreducible by A1,Th15;
  end;
  hence thesis by Th18;
end;
