
theorem
  for L being with_infima Poset, X being Subset of L holds
  X c= uparrow fininfs X &
  for F being Filter of L st X c= F holds uparrow fininfs X c= F
proof
  let L be with_infima Poset, X be Subset of L;
A1: X c= fininfs X by Th50;
  fininfs X c= uparrow fininfs X by Th16;
  hence X c= uparrow fininfs X by A1;
  let I be Filter of L such that
A2: X c= I;
  let x be object;
  assume
A3: x in uparrow fininfs X;
  then reconsider x as Element of L;
  consider y being Element of L such that
A4: x >= y and
A5: y in fininfs X by A3,Def16;
  consider Y being finite Subset of X such that
A6: y = "/\"(Y,L) and
A7: ex_inf_of Y,L by A5;
  set i = the Element of I;
  reconsider i as Element of L;
A8: ex_inf_of {i}, L by YELLOW_0:38;
A9: inf {i} = i by YELLOW_0:39;
A10: now
    assume ex_inf_of {},L;
    then "/\"({},L) >= inf {i} by A8,XBOOLE_1:2,YELLOW_0:35;
    hence "/\"({},L) in I by A9,Def20;
  end;
  Y c= I by A2;
  then y in I by A6,A7,A10,Th43;
  hence thesis by A4,Def20;
end;
