
theorem Th26:
  for L being with_infima Poset, X being Subset of L st X is Open
  lower holds X is filtered
proof
  let L be with_infima Poset, X be Subset of L such that
A1: X is Open lower;
  let x, y be Element of L such that
A2: x in X and
  y in X;
A3: x "/\" y <= x by YELLOW_0:23;
  then x "/\" y in X by A1,A2;
  then consider z being Element of L such that
A4: z in X and
A5: z << x "/\" y by A1;
  take z;
  x "/\" y <= y & z <= x "/\" y by A5,WAYBEL_3:1,YELLOW_0:23;
  hence z in X & z <= x & z <= y by A3,A4,ORDERS_2:3;
end;
