
theorem Th27:
  for L being complete LATTICE, U1 being Subset of L holds U1 in
  xi(L) implies U1 is property(S)
proof
  let L be complete LATTICE;
  let U1 be Subset of L;
  assume U1 in xi(L);
  then
  U1 in { V where V is Subset of L: for p being Element of L st p in V for
N being net of L st [N,p] in lim_inf-Convergence L holds N is_eventually_in V}
  by YELLOW_6:def 24;
  then
A1: ex V being Subset of L st U1=V & for p being Element of L st p in V for N
  being net of L st [N,p] in lim_inf-Convergence L holds N is_eventually_in V;
  let D be non empty directed Subset of L;
  assume
A2: sup D in U1;
  [Net-Str D,sup D] in lim_inf-Convergence L by Th26;
  then (Net-Str D) is_eventually_in U1 by A1,A2;
  then consider y being Element of (Net-Str D) such that
A3: for x being Element of (Net-Str D) st y <= x holds (Net-Str D).x in
  U1 by WAYBEL_0:def 11;
A4: y in the carrier of Net-Str D;
  then y in D by WAYBEL21:32;
  then reconsider y1=y as Element of L;
  reconsider y1 as Element of L;
  take y1;
  thus y1 in D by A4,WAYBEL21:32;
  let x1 be Element of L;
  assume that
A5: x1 in D and
A6: x1 >= y1;
A7: Net-Str D is full SubRelStr of L by WAYBEL21:32;
  reconsider x=x1 as Element of Net-Str D by A5,WAYBEL21:32;
  reconsider x as Element of Net-Str D;
  (Net-Str D).x = (the mapping of Net-Str D).x by WAYBEL_0:def 8
    .= (id D).x by WAYBEL21:32
    .= x by A5,FUNCT_1:18;
  hence thesis by A3,A6,A7,YELLOW_0:60;
end;
