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 Th31: :: Corollary 1.12 p. 106
  sup_op L is jointly_Scott-continuous implies L is sober
proof
  assume
A1: sup_op L is jointly_Scott-continuous;
  let S be irreducible Subset of L;
A2: sigma L = the topology of ConvergenceSpace Scott-Convergence L & the
  TopStruct of L = ConvergenceSpace Scott-Convergence L by WAYBEL11:32,def 12;
A3: S is non empty closed by YELLOW_8:def 3;
  then the carrier of InclPoset sigma L = sigma L & S` is open by YELLOW_1:1;
  then reconsider V = S` as Element of InclPoset sigma L by A2,PRE_TOPC:def 2;
  S` <> the carrier of L by Th2;
  then consider p being Element of L such that
A4: V = (downarrow p)` by A1,A2,Th17,Th29;
A5: L is T_0 by WAYBEL11:10;
  take p;
A6: Cl {p} = downarrow p by WAYBEL11:9;
A7: S = downarrow p by A4,TOPS_1:1;
  hence p is_dense_point_of S by A6,YELLOW_8:18;
  let q be Point of L;
  assume q is_dense_point_of S;
  then Cl {q} = S by A3,YELLOW_8:16;
  hence thesis by A7,A6,A5,YELLOW_8:23;
end;
