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 Th29: :: Proposition 1.11 (iii) p. 105
  V = X & sup_op L is jointly_Scott-continuous & V is prime & V <>
  the carrier of L implies ex x st X = (downarrow x)`
proof
  assume that
A1: V = X and
A2: sup_op L is jointly_Scott-continuous and
A3: V is prime and
A4: V <> the carrier of L;
A5: the TopStruct of L = ConvergenceSpace Scott-Convergence L by WAYBEL11:32;
  set A = X`;
A6: sigma L = the topology of ConvergenceSpace Scott-Convergence L by
WAYBEL11:def 12;
A7: the carrier of InclPoset sigma L = sigma L by YELLOW_1:1;
  then
A8: X is open by A1,A6,A5,PRE_TOPC:def 2;
  then A is closed;
  then
A9: A is directly_closed lower by WAYBEL11:7;
A10: A is directed
  proof
    set LxL = [:L qua non empty TopSpace, L:];
    given a, b being Element of L such that
A11: a in A & b in A and
A12: for z being Element of L holds not (z in A & a <= z & b <= z);
    a <= a"\/"b & b <= a"\/"b by YELLOW_0:22;
    then not a"\/"b in A by A12;
    then
A13: a"\/"b in X by XBOOLE_0:def 5;
    consider Tsup being Function of LxL, L such that
A14: Tsup = sup_op L and
A15: Tsup is continuous by A2,A5;
A16: Tsup.(a,b) = a"\/"b by A14,WAYBEL_2:def 5;
    [#]L <> {};
    then Tsup"X is open by A8,A15,TOPS_2:43;
    then consider AA being Subset-Family of LxL such that
A17: Tsup"X = union AA and
A18: for e being set st e in AA ex X1 being Subset of L, Y1 being
    Subset of L st e = [:X1,Y1:] & X1 is open & Y1 is open by BORSUK_1:5;
A19: the carrier of LxL = [:the carrier of L, the carrier of L:] by
BORSUK_1:def 2;
    then [a,b] in the carrier of LxL by ZFMISC_1:def 2;
    then [a,b] in Tsup"X by A13,A16,FUNCT_2:38;
    then consider AAe being set such that
A20: [a,b] in AAe and
A21: AAe in AA by A17,TARSKI:def 4;
    consider Va, Vb being Subset of L such that
A22: AAe = [:Va, Vb:] and
A23: Va is open and
A24: Vb is open by A18,A21;
A25: a in Va & b in Vb by A20,A22,ZFMISC_1:87;
    reconsider Va9 = Va, Vb9 = Vb as Subset of L;
    now
      let x be object;
      hereby
        assume x in Tsup.:AAe;
        then consider cd being object such that
A26:    cd in the carrier of LxL and
A27:    cd in AAe and
A28:    Tsup.cd = x by FUNCT_2:64;
        consider c, d being Element of L such that
A29:    cd = [c,d] by A19,A26,DOMAIN_1:1;
        reconsider c, d as Element of L;
A30:    x = Tsup.(c,d) by A28,A29
          .= c"\/"d by A14,WAYBEL_2:def 5;
A31:    d <= c"\/"d & Vb9 is upper by A24,WAYBEL11:def 4,YELLOW_0:22;
        d in Vb by A22,A27,A29,ZFMISC_1:87;
        then
A32:    x in Vb by A30,A31;
A33:    c <= c"\/"d & Va9 is upper by A23,WAYBEL11:def 4,YELLOW_0:22;
        c in Va by A22,A27,A29,ZFMISC_1:87;
        then x in Va by A30,A33;
        hence x in Va/\Vb by A32,XBOOLE_0:def 4;
      end;
      assume
A34:  x in Va/\Vb;
      then reconsider c = x as Element of L;
      x in Va & x in Vb by A34,XBOOLE_0:def 4;
      then
A35:  [c,c] in AAe by A22,ZFMISC_1:87;
      c <= c;
      then c = c"\/"c by YELLOW_0:24;
      then
A36:  c = Tsup.(c,c) by A14,WAYBEL_2:def 5;
      [c,c] in the carrier of LxL by A19,ZFMISC_1:87;
      hence x in Tsup.:AAe by A35,A36,FUNCT_2:35;
    end;
    then
A37: Tsup.:AAe = Va/\Vb by TARSKI:2;
A38: Tsup.:(Tsup"X) c= X by FUNCT_1:75;
    Tsup.:AAe c= Tsup.:(Tsup"X) by A17,A21,RELAT_1:123,ZFMISC_1:74;
    then
A39: Tsup.:AAe c= X by A38;
    Va in sigma L & Vb in sigma L by A6,A5,A23,A24,PRE_TOPC:def 2;
    then Va c= X or Vb c= X by A1,A3,A6,A7,A37,A39,Th19;
    hence contradiction by A11,A25,XBOOLE_0:def 5;
  end;
  take u = sup A;
  now
    assume A = {};
    then A` = the carrier of L;
    hence contradiction by A1,A4;
  end;
  then u in A by A9,A10,WAYBEL11:def 2;
  then A = downarrow u by A9,Th5;
  hence thesis;
end;
