reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem Th18: ::3.10, p.70
  for L be lower-bounded continuous LATTICE, X be Subset of L st X
  = IRR L \ { Top L} holds X is order-generating
proof
  let L be lower-bounded continuous LATTICE, X be Subset of L;
  assume
A1: X = IRR L \ {Top L};
  for l1,l2 be Element of L st not l2 <= l1 ex p be Element of L st p in X
  & l1 <= p & not l2 <= p
  proof
    let x,y be Element of L;
    assume not y <= x;
    then consider p be Element of L such that
A2: p is irreducible and
A3: x <= p and
A4: not y <= p by Th14;
    p <> Top L & p in IRR L by A2,A4,Def4,YELLOW_0:45;
    then p in X by A1,ZFMISC_1:56;
    hence thesis by A3,A4;
  end;
  hence thesis by Th17;
end;
