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 Th39: :: Theorem 1.14 (3) iff (5) p. 107
:: The conjunct InclPoset sigma L is continuous is dropped
  (for x ex B being Basis of x st for Y st Y in B holds Y is open
filtered) iff for V ex VV st V = sup VV & for W st W in VV holds W is co-prime
proof
  set IPs = InclPoset the topology of L;
A1: sigma L = the topology of L by Th23;
  then
A2: the carrier of IPs = sigma L by YELLOW_1:1;
  hereby
    assume
A3: for x being Element of L ex X being Basis of x st for Y being
    Subset of L st Y in X holds Y is open filtered;
    let V be Element of InclPoset sigma L;
    set X = {Y where Y is Subset of L : Y c= V & ex x being Element of L, bas
being Basis of x st x in V & Y in bas & for Yx being Subset of L st Yx in bas
    holds Yx is open filtered};
    now
      let YY be object;
      assume YY in X;
      then consider Y being Subset of L such that
A4:   Y = YY and
      Y c= V and
A5:   ex x being Element of L, bas being Basis of x st x in V & Y in
      bas & for Yx being Subset of L st Yx in bas holds Yx is open filtered;
      Y is open by A5;
      then Y in sigma L by Th24;
      hence YY in the carrier of InclPoset sigma L by A4,YELLOW_1:1;
    end;
    then reconsider X as Subset of InclPoset sigma L by TARSKI:def 3;
    take X;
    V in sigma L by A1,A2;
    then reconsider Vl = V as Subset of L;
A6: Vl is open by A1,A2,Th24;
    now
      let x be object;
      hereby
        assume
A7:     x in V;
        Vl = V;
        then reconsider d = x as Element of L by A7;
        consider bas being Basis of d such that
A8:     for Y being Subset of L st Y in bas holds Y is open filtered by A3;
        consider Y being Subset of L such that
A9:     Y in bas and
A10:    Y c= V by A6,A7,YELLOW_8:13;
A11:    x in Y by A9,YELLOW_8:12;
        Y in X by A7,A8,A9,A10;
        hence x in union X by A11,TARSKI:def 4;
      end;
      assume x in union X;
      then consider YY being set such that
A12:  x in YY and
A13:  YY in X by TARSKI:def 4;
      ex Y being Subset of L st Y = YY & Y c= V & ex x being Element of L,
bas being Basis of x st x in V & Y in bas & for Yx being Subset of L st Yx in
      bas holds Yx is open filtered by A13;
      hence x in V by A12;
    end;
    then V = union X by TARSKI:2;
    hence V = sup X by A1,YELLOW_1:22;
    let Yp be Element of InclPoset sigma L;
    assume Yp in X;
    then consider Y being Subset of L such that
A14: Y = Yp and
    Y c= V and
A15: ex x being Element of L, bas being Basis of x st x in V & Y in
    bas & for Yx being Subset of L st Yx in bas holds Yx is open filtered;
A16: Y is open filtered by A15;
    then Y is upper by WAYBEL11:def 4;
    hence Yp is co-prime by A14,A16,Th27;
  end;
  assume
A17: for V being Element of InclPoset sigma L ex X being Subset of
InclPoset sigma L st V = sup X & for x being Element of InclPoset sigma L st x
  in X holds x is co-prime;
  let x be Element of L;
  set bas = {V where V is Element of InclPoset sigma L : x in V & V is
  co-prime};
  bas c= bool the carrier of L
  proof
    let VV be object;
    assume VV in bas;
    then ex V being Element of InclPoset sigma L st VV= V & x in V & V is
    co-prime;
    then VV in sigma L by A1,A2;
    hence thesis;
  end;
  then reconsider bas as Subset-Family of L;
  reconsider bas as Subset-Family of L;
  bas is Basis of x
  proof
A18: bas is open
    proof
      let VV be Subset of L;
      assume VV in bas;
      then ex V being Element of InclPoset sigma L st VV= V & x in V & V is
      co-prime;
      hence thesis by A1,A2,PRE_TOPC:def 2;
    end;
    bas is x-quasi_basis
    proof
    now
      per cases;
      suppose
        bas is empty;
        then Intersect bas = the carrier of L by SETFAM_1:def 9;
        hence x in Intersect bas;
      end;
      suppose
A19:    bas is non empty;
A20:    now
          let Y be set;
          assume Y in bas;
          then ex V being Element of InclPoset sigma L st Y = V & x in V & V
          is co-prime;
          hence x in Y;
        end;
        Intersect bas = meet bas by A19,SETFAM_1:def 9;
        hence x in Intersect bas by A19,A20,SETFAM_1:def 1;
      end;
    end;
    hence x in Intersect bas;
    let S be Subset of L;
    assume that
A21: S is open and
A22: x in S;
    reconsider S9 = S as Element of IPs by A2,A21,Th24;
    consider X being Subset of IPs such that
A23: S9 = sup X and
A24: for x being Element of IPs st x in X holds x is co-prime by A1,A17;
    S9 = union X by A23,YELLOW_1:22;
    then consider V being set such that
A25: x in V and
A26: V in X by A22,TARSKI:def 4;
    V in sigma L by A2,A26;
    then reconsider V as Subset of L;
    reconsider Vp = V as Element of IPs by A26;
    take V;
    Vp is co-prime by A24,A26;
    hence V in bas by A1,A25;
    sup X is_>=_than X by YELLOW_0:32;
    then Vp <= sup X by A26,LATTICE3:def 9;
    hence thesis by A23,YELLOW_1:3;
  end;
  hence thesis by A18;
  end;
  then reconsider bas as Basis of x;
  take bas;
  let V be Subset of L;
  assume V in bas;
  then ex Vp being Element of InclPoset sigma L st V = Vp & x in Vp & Vp is
  co-prime;
  hence thesis by A1,A2,Th24,Th27;
end;
