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 Th37: :: Theorem 1.14 (3) implies (4) p. 107
  (for x ex B being Basis of x st for Y st Y in B holds Y is open
filtered) & InclPoset sigma L is continuous implies x = "\/" ({inf X : x in X &
  X in sigma L}, L)
proof
  assume that
A1: for x being Element of L ex B being Basis of x st for Y being Subset
  of L st Y in B holds Y is open filtered and
A2: InclPoset sigma L is continuous;
A3: sigma L = the topology of L by Th23;
  set IU = {inf V where V is Subset of L : x in V & V in sigma L};
  set IPs = InclPoset the topology of L;
A4: the carrier of IPs = the topology of L by YELLOW_1:1;
  set y = "\/"(IU,L);
  set VVl = (downarrow y)`;
  now
    let b be Element of L;
    assume b in IU;
    then consider V being Subset of L such that
A5: b = inf V and
A6: x in V and
    V in sigma L;
    b is_<=_than V by A5,YELLOW_0:33;
    hence b <= x by A6,LATTICE3:def 8;
  end;
  then x is_>=_than IU by LATTICE3:def 9;
  then
A7: y <= x by YELLOW_0:32;
  assume
A8: x <> y;
  now
    assume x in downarrow y;
    then x <= y by WAYBEL_0:17;
    hence contradiction by A7,A8,ORDERS_2:2;
  end;
  then
A9: x in VVl by XBOOLE_0:def 5;
  VVl is open by WAYBEL11:12;
  then reconsider VVp = VVl as Element of IPs by A3,A4,Th24;
  VVp = sup waybelow VVp by A2,A3,WAYBEL_3:def 5;
  then VVp = union waybelow VVp by YELLOW_1:22;
  then consider Vp being set such that
A10: x in Vp and
A11: Vp in waybelow VVp by A9,TARSKI:def 4;
  reconsider Vp as Element of IPs by A11;
  Vp in sigma L by A3,A4;
  then reconsider Vl = Vp as Subset of L;
A12: Vp << VVp by A11,WAYBEL_3:7;
  consider bas being Basis of x such that
A13: for Y being Subset of L st Y in bas holds Y is open filtered by A1;
A14: y is_>=_than IU by YELLOW_0:32;
  Vl is open by A4,PRE_TOPC:def 2;
  then consider Ul being Subset of L such that
A15: Ul in bas and
A16: Ul c= Vl by A10,YELLOW_8:def 1;
  set F = {downarrow u where u is Element of L : u in Ul};
A17: x in Ul by A15,YELLOW_8:12;
  then
A18: downarrow x in F;
  F c= bool the carrier of L
  proof
    let X be object;
    assume X in F;
    then ex u being Element of L st X = downarrow u & u in Ul;
    hence thesis;
  end;
  then reconsider F as non empty Subset-Family of L by A18;
  COMPLEMENT F c= the topology of L
  proof
    let X be object;
    assume
A19: X in COMPLEMENT F;
    then reconsider X9 = X as Subset of L;
    X9` in F by A19,SETFAM_1:def 7;
    then consider u being Element of L such that
A20: X9` = downarrow u and
    u in Ul;
    X9 = (downarrow u)` by A20;
    then X9 is open by WAYBEL11:12;
    hence thesis by PRE_TOPC:def 2;
  end;
  then reconsider CF = COMPLEMENT F as Subset of IPs by YELLOW_1:1;
  Ul is filtered by A13,A15;
  then
A21: CF is directed by A3,Lm2;
  Ul is open by A15,YELLOW_8:12;
  then Ul in sigma L by A3,PRE_TOPC:def 2;
  then inf Ul in IU by A17;
  then inf Ul <= y by A14,LATTICE3:def 9;
  then downarrow inf Ul c= downarrow y by WAYBEL_0:21;
  then
A22: (downarrow y)` c= (downarrow inf Ul)` by SUBSET_1:12;
  downarrow inf Ul = meet F by A17,Th15;
  then (downarrow inf Ul)` = union COMPLEMENT F by TOPS_2:7;
  then VVp c= sup CF by A22,YELLOW_1:22;
  then
A23: VVp <= sup CF by YELLOW_1:3;
  (downarrow x)` in COMPLEMENT F by A18,YELLOW_8:5;
  then consider d being Element of IPs such that
A24: d in CF and
A25: Vp << d by A2,A3,A12,A21,A23,WAYBEL_4:53;
  Vp <= d by A25,WAYBEL_3:1;
  then
A26: Vp c= d by YELLOW_1:3;
  d in sigma L by A3,A4;
  then reconsider d9 = d as Subset of L;
  d9` in F by A24,SETFAM_1:def 7;
  then consider u being Element of L such that
A27: d9` = downarrow u and
A28: u in Ul;
  u <= u;
  then u in downarrow u by WAYBEL_0:17;
  then not u in Vp by A27,A26,XBOOLE_0:def 5;
  hence contradiction by A16,A28;
end;
