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 (1) p. 108
  (ex B being Basis of L st B = {uparrow x :x in the carrier of
  CompactSublatt L}) implies L is algebraic
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};
  thus for x being Element of L holds compactbelow x is non empty directed;
  thus L is up-complete;
  let x be Element of L;
  set y = sup compactbelow x;
  set cx = compactbelow x;
  set dx = downarrow x;
  set dy = downarrow y;
  now
    for z be Element of L st z in dx holds z <= x by WAYBEL_0:17;
    then x is_>=_than dx by LATTICE3:def 9;
    then
A2: sup dx <= x by YELLOW_0:32;
    set GB = { G where G is Subset of L: G in B & G c= dy`};
A3: cx = dx /\ the carrier of CompactSublatt L by WAYBEL_8:5;
A4: y is_>=_than cx by YELLOW_0:32;
    ex_sup_of cx, L & ex_sup_of dx, L by YELLOW_0:17;
    then sup compactbelow x <= sup dx by A3,XBOOLE_1:17,YELLOW_0:34;
    then
A5: y <= x by A2,ORDERS_2:3;
    assume
A6: y <> x;
    now
      assume x in dy;
      then x <= y by WAYBEL_0:17;
      hence contradiction by A6,A5,ORDERS_2:2;
    end;
    then
A7: x in dy` by XBOOLE_0:def 5;
    dy` = union GB by WAYBEL11:12,YELLOW_8:9;
    then consider X being set such that
A8: x in X and
A9: X in GB by A7,TARSKI:def 4;
    consider G being Subset of L such that
A10: G = X and
A11: G in B and
A12: G c= dy` by A9;
    consider k being Element of L such that
A13: G = uparrow k and
A14: k in the carrier of CompactSublatt L by A1,A11;
A15: k is compact by A14,WAYBEL_8:def 1;
    k <= x by A8,A10,A13,WAYBEL_0:18;
    then k in cx by A15;
    then k <= y by A4,LATTICE3:def 9;
    then y in uparrow k by WAYBEL_0:18;
    then y <= y & not y in dy by A12,A13,XBOOLE_0:def 5;
    hence contradiction by WAYBEL_0:17;
  end;
  hence thesis;
end;
