
theorem Th39:
  for L being lower-bounded continuous LATTICE for D being non
  empty countable dense Subset of L, u being Element of L st u <> Bottom L ex p
  being irreducible Element of L st p <> Top L & not p in uparrow ({u} "/\" D)
proof
  let L be lower-bounded continuous LATTICE, D be non empty countable dense
  Subset of L, u be Element of L such that
A1: u <> Bottom L;
A2: for d, y being Element of L st not y <= Bottom L & d in D holds not y
  "/\" d <= Bottom L
  proof
    let d, y be Element of L such that
A3: not y <= Bottom L;
    assume d in D;
    then d is dense by Def5;
    then
A4: y "/\" d <> Bottom L by A3;
    Bottom L <= y "/\" d by YELLOW_0:44;
    hence thesis by A4,ORDERS_2:2;
  end;
  Bottom L <= u by YELLOW_0:44;
  then not u <= Bottom L by A1,ORDERS_2:2;
  then consider p being irreducible Element of L such that
  Bottom L <= p and
A5: not p in uparrow ({u} "/\" D) by A2,Th36;
  take p;
  thus p <> Top L by A5,Th9;
  thus thesis by A5;
end;
