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 1.15 (2) implies (3) p. 108
  (ex B being Basis of L st B = {uparrow x :x in the carrier of
CompactSublatt L}) implies InclPoset sigma L is algebraic & for V ex VV st V =
  sup VV & for W st W in VV holds W is co-prime
proof
  given B being Basis of L such that
A1: B = {uparrow k where k is Element of L : k in the carrier of
  CompactSublatt L};
  set IPs = InclPoset sigma L;
  set IPt = InclPoset the topology of L;
A2: the carrier of IPs = sigma L by YELLOW_1:1;
A3: sigma L = the topology of L by Th23;
A4: IPs = IPt by Th23;
  thus InclPoset sigma L is algebraic
  proof
    thus for X being Element of IPs holds compactbelow X is non empty directed
    by A3;
    thus IPs is up-complete by A4;
    let X be Element of IPs;
    set cX = compactbelow X;
    set GB = { G where G is Subset of L: G in B & G c= X };
    X in sigma L by A2;
    then reconsider X9 = X as Subset of L;
    X9 is open by A2,Th24;
    then
A5: X = union GB by YELLOW_8:9;
A6: now
      let x be object;
      hereby
        assume x in X;
        then consider GG being set such that
A7:     x in GG and
A8:     GG in GB by A5,TARSKI:def 4;
        consider G being Subset of L such that
A9:     G = GG and
A10:    G in B and
A11:    G c= X by A8;
        consider k being Element of L such that
A12:    G = uparrow k and
A13:    k in the carrier of CompactSublatt L by A1,A10;
        k is compact by A13,WAYBEL_8:def 1;
        then uparrow k is Open by WAYBEL_8:2;
        then uparrow k is open by WAYBEL11:41;
        then reconsider G as Element of IPs by A3,A2,A12,PRE_TOPC:def 2;
        for X being Subset of L st X is open holds X is upper by WAYBEL11:def 4
;
        then uparrow k is compact by Th22;
        then
A14:    G is compact by A3,A12,WAYBEL_3:36;
        G <= X by A11,YELLOW_1:3;
        then G in cX by A14;
        hence x in union cX by A7,A9,TARSKI:def 4;
      end;
      assume x in union cX;
      then consider G being set such that
A15:  x in G and
A16:  G in cX by TARSKI:def 4;
      reconsider G as Element of IPs by A16;
      G <= X by A16,WAYBEL_8:4;
      then G c= X by YELLOW_1:3;
      hence x in X by A15;
    end;
    sup cX = union cX by A3,YELLOW_1:22;
    hence thesis by A6,TARSKI:2;
  end;
  let V be Element of InclPoset sigma L;
  V in sigma L by A2;
  then reconsider V9 = V as Subset of L;
  set GB = { G where G is Subset of L: G in B & G c= V };
  GB c= the carrier of IPs
  proof
    let x be object;
    assume x in GB;
    then consider G being Subset of L such that
A17: G = x and
A18: G in B and
    G c= V;
    G is open by A18,YELLOW_8:10;
    hence thesis by A2,A17,Th24;
  end;
  then reconsider GB as Subset of InclPoset sigma L;
  take GB;
  V9 is open by A2,Th24;
  then V = union GB by YELLOW_8:9;
  hence V = sup GB by A3,YELLOW_1:22;
  let x be Element of InclPoset sigma L;
  assume x in GB;
  then consider G being Subset of L such that
A19: G = x and
A20: G in B and
  G c= V;
  ex k being Element of L st G = uparrow k & k in the carrier of
  CompactSublatt L by A1,A20;
  hence thesis by A19,Th27;
end;
