
theorem
  for L being Lattice holds L is noetherian iff L.: is co-noetherian
proof
  let L be Lattice;
  set R = LattPOSet L;
  set Ri = (LattPOSet L)~;
A1: now
A2: (LattPOSet L)~ = LattPOSet (L.:) by LATTICE3:20;
    assume L.: is co-noetherian;
    then (Ri)~ is well_founded by A2;
    then R is well_founded by LATTICE3:8;
    hence L is noetherian;
  end;
  now
    assume L is noetherian;
    then R is well_founded;
    then
A3: (Ri)~ is well_founded by LATTICE3:8;
    (LattPOSet L)~ = LattPOSet (L.:) by LATTICE3:20;
    hence L.: is co-noetherian by A3;
  end;
  hence thesis by A1;
end;
