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 Th30: :: Proposition 1.11 (iv) p. 105
  L is continuous implies sup_op L is jointly_Scott-continuous
proof
  assume
A1: L is continuous;
  set Tsup = sup_op L;
  let T be non empty TopSpace such that
A2: the TopStruct of T = ConvergenceSpace Scott-Convergence L;
A3: the carrier of [:T, T:] = [:the carrier of T, the carrier of T:] by
BORSUK_1:def 2;
A4: the carrier of T = the carrier of L by A2,YELLOW_6:def 24;
  then the carrier of [:T,T:] = the carrier of [:L,L:] by A3,YELLOW_3:def 2;
  then reconsider Tsup as Function of [:T, T:], T by A4;
  take Tsup;
  thus Tsup = sup_op L;
A5: the TopStruct of L = ConvergenceSpace Scott-Convergence L by WAYBEL11:32;
  for x being Point of [:T, T:] holds Tsup is_continuous_at x
  proof
    reconsider Lc = L as continuous complete Scott TopLattice by A1;
    let ab be Point of [:T, T:];
    reconsider Tsab = Tsup.ab as Point of T;
    consider a, b being Point of T such that
A6: ab = [a,b] by A3,DOMAIN_1:1;
    reconsider a9 = a, b9 = b as Element of L by A2,YELLOW_6:def 24;
    set D1 = waybelow a9, D2 = waybelow b9;
    set D = D1"\/"D2;
    reconsider Tsab9 = Tsab as Element of L by A2,YELLOW_6:def 24;
    let G be a_neighborhood of Tsup.ab;
A7: Tsab in Int G by CONNSP_2:def 1;
    reconsider basab = { wayabove q where q is Element of L: q << Tsab9 } as
    Basis of Tsab9 by A1,WAYBEL11:44;
    basab is Basis of Tsab by A2,A5,Th21;
    then consider V being Subset of T such that
A8: V in basab and
A9: V c= Int G by A7,YELLOW_8:def 1;
    consider u being Element of L such that
A10: V = wayabove u and
A11: u << Tsab9 by A8;
A12: D = { x "\/" y where x, y is Element of L : x in D1 & y in D2 } by
YELLOW_4:def 3;
    Lc = L;
    then
A13: a9 = sup waybelow a9 & b9 = sup waybelow b9 by WAYBEL_3:def 5;
    Tsab9 = Tsup.(a,b) by A6
      .= a9"\/"b9 by WAYBEL_2:def 5
      .= sup ((waybelow a9)"\/"(waybelow b9)) by A13,WAYBEL_2:4;
    then consider xy being Element of L such that
A14: xy in D and
A15: u << xy by A1,A11,WAYBEL_4:53;
    consider x, y being Element of L such that
A16: xy = x"\/"y and
A17: x in D1 and
A18: y in D2 by A14,A12;
    reconsider H = [:wayabove x, wayabove y:] as Subset of [:T, T:] by A4,A3,
YELLOW_3:def 2;
    y << b9 by A18,WAYBEL_3:7;
    then
A19: b9 in wayabove y;
    Int G c= G by TOPS_1:16;
    then
A20: V c= G by A9;
    reconsider wx = wayabove x, wy = wayabove y as Subset of T by A2,
YELLOW_6:def 24;
    wayabove y is open by A1,WAYBEL11:36;
    then wayabove y in the topology of L by PRE_TOPC:def 2;
    then
A21: wy is open by A2,A5,PRE_TOPC:def 2;
    wayabove x is open by A1,WAYBEL11:36;
    then wayabove x in the topology of L by PRE_TOPC:def 2;
    then wx is open by A2,A5,PRE_TOPC:def 2;
    then H is open by A21,BORSUK_1:6;
    then
A22: H = Int H by TOPS_1:23;
    x << a9 by A17,WAYBEL_3:7;
    then a9 in wayabove x;
    then [a9,b9] in H by A19,ZFMISC_1:87;
    then reconsider H as a_neighborhood of ab by A6,A22,CONNSP_2:def 1;
    take H;
A23: Tsup.:H = (wayabove x)"\/"(wayabove y) & (wayabove x)"\/"(wayabove y)
    c= uparrow (x"\/"y) by Th12,WAYBEL_2:35;
    uparrow (x"\/"y) c= wayabove u by A15,A16,Th7;
    then Tsup.:H c= V by A10,A23;
    hence thesis by A20;
  end;
  hence thesis by TMAP_1:44;
end;
