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 :: Corollary 1.13 p. 106
  L is continuous implies L is compact locally-compact sober Baire
proof
  assume
A1: L is continuous;
A2: uparrow Bottom L = the carrier of L & [#]L = the carrier of L by Th10;
A3: for X being Subset of L st X is open holds X is upper by WAYBEL11:def 4;
  then uparrow Bottom L is compact by Th22;
  hence L is compact by A2;
  thus
A4: L is locally-compact
  proof
    let x be Point of L, X be Subset of L such that
A5: x in X and
A6: X is open;
    reconsider x9 = x as Element of L;
    consider y being Element of L such that
A7: y << x9 and
A8: y in X by A1,A5,A6,WAYBEL11:43;
    set Y = uparrow y;
    set bas = { wayabove q where q is Element of L: q << x9 };
A9: bas is Basis of x by A1,WAYBEL11:44;
    wayabove y in bas by A7;
    then wayabove y is open by A9,YELLOW_8:12;
    then
A10: wayabove y c= Int Y by TOPS_1:24,WAYBEL_3:11;
    take Y;
    x in wayabove y by A7;
    hence x in Int Y by A10;
    X is upper by A6,WAYBEL11:def 4;
    hence Y c= X by A8,WAYBEL11:42;
    thus thesis by A3,Th22;
  end;
  sup_op L is jointly_Scott-continuous by A1,Th30;
  hence L is sober by Th31;
  hence thesis by A4,WAYBEL12:44;
end;
