reserve L for complete Scott TopLattice,
  x for Element of L,
  X, Y for Subset of L,
  V, W for Element of InclPoset sigma L,
  VV for Subset of InclPoset sigma L;

theorem :: Proposition 1.11 (ii) p. 105
  (V = X & ex x st X = (downarrow x)`) implies V is prime & V <> the
  carrier of L
proof
  assume
A1: V = X;
A2: sigma L = the topology of ConvergenceSpace Scott-Convergence L & the
  TopStruct of L = ConvergenceSpace Scott-Convergence L by WAYBEL11:32,def 12;
  given u being Element of L such that
A3: X = (downarrow u)`;
  Cl {u} = downarrow u & Cl {u} is irreducible by WAYBEL11:9,YELLOW_8:17;
  hence V is prime by A1,A2,A3,Th17;
  assume V = the carrier of L;
  hence contradiction by A1,A3,Th2;
end;
