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 (1) implies (2) p. 108
  L is algebraic implies ex B being Basis of L st B = {uparrow x : x in
  the carrier of CompactSublatt L}
proof
  set P = {uparrow k where k is Element of L : k in the carrier of
  CompactSublatt L};
  P c= bool the carrier of L
  proof
    let x be object;
    assume x in P;
    then ex k being Element of L st x = uparrow k & k in the carrier of
    CompactSublatt L;
    hence thesis;
  end;
  then reconsider P as Subset-Family of L;
  reconsider P as Subset-Family of L;
A1: P c= the topology of L
  proof
    let x be object;
    assume x in P;
    then consider k being Element of L such that
A2: x = uparrow k and
A3: k in the carrier of CompactSublatt L;
    k is compact by A3,WAYBEL_8:def 1;
    then uparrow k is Open by WAYBEL_8:2;
    then uparrow k is open by WAYBEL11:41;
    hence thesis by A2,PRE_TOPC:def 2;
  end;
  assume
A4: L is algebraic;
  now
    let x be Point of L;
    set B = {uparrow k where k is Element of L : uparrow k in P & x in uparrow
    k};
    B c= bool the carrier of L
    proof
      let y be object;
      assume y in B;
      then ex k being Element of L st y = uparrow k & uparrow k in P & x in
      uparrow k;
      hence thesis;
    end;
    then reconsider B as Subset-Family of L;
    reconsider B as Subset-Family of L;
    B is Basis of x
    proof
A5:  B is open
      proof
        let y be Subset of L;
        assume y in B;
        then ex k being Element of L st y = uparrow k & uparrow k in P & x in
        uparrow k;
        hence thesis by A1,PRE_TOPC:def 2;
      end;
      B is x-quasi_basis
      proof
      now
        per cases;
        suppose
          B is empty;
          then Intersect B = the carrier of L by SETFAM_1:def 9;
          hence x in Intersect B;
        end;
        suppose
A6:       B is non empty;
A7:       now
            let Y be set;
            assume Y in B;
            then
            ex k being Element of L st Y = uparrow k & uparrow k in P & x
            in uparrow k;
            hence x in Y;
          end;
          Intersect B = meet B by A6,SETFAM_1:def 9;
          hence x in Intersect B by A6,A7,SETFAM_1:def 1;
        end;
      end;
      hence x in Intersect B;
      reconsider x9 = x as Element of L;
      let S be Subset of L such that
A8:   S is open and
A9:   x in S;
A10:   x = sup compactbelow x9 by A4,WAYBEL_8:def 3;
      S is inaccessible by A8,WAYBEL11:def 4;
      then (compactbelow x9) meets S by A9,A10,WAYBEL11:def 1;
      then consider k being object such that
A11:  k in compactbelow x9 and
A12:  k in S by XBOOLE_0:3;
      reconsider k as Element of L by A11;
A13:  compactbelow x9 = downarrow x9 /\ the carrier of CompactSublatt L
      by WAYBEL_8:5;
      then k in downarrow x9 by A11,XBOOLE_0:def 4;
      then k <= x9 by WAYBEL_0:17;
      then
A14:  x in uparrow k by WAYBEL_0:18;
      take V = uparrow k;
      k in the carrier of CompactSublatt L by A11,A13,XBOOLE_0:def 4;
      then uparrow k in P;
      hence V in B by A14;
      S is upper by A8,WAYBEL11:def 4;
      hence thesis by A12,WAYBEL11:42;
    end;
    hence thesis by A5;
    end;
    then reconsider B as Basis of x;
    take B;
    thus B c= P
    proof
      let y be object;
      assume y in B;
      then ex k being Element of L st y = uparrow k & uparrow k in P & x in
      uparrow k;
      hence thesis;
    end;
  end;
  then P is Basis of L by A1,YELLOW_8:14;
  hence thesis;
end;
