
theorem
  for L being lower-bounded antisymmetric non empty RelStr for X being
  non empty Subset of L holds Bottom L in downarrow X
proof
  let L be lower-bounded antisymmetric non empty RelStr, X be non empty Subset
  of L;
  consider y being object such that
A1: y in X by XBOOLE_0:def 1;
  reconsider y as Element of X by A1;
  downarrow X = {x where x is Element of L: ex y being Element of L st x
  <= y & y in X} & Bottom L <= y by WAYBEL_0:14,YELLOW_0:44;
  hence thesis;
end;
