reserve A, B, X, Y for set;

theorem
  for L1 being up-complete antisymmetric non empty reflexive RelStr,
L2 being non empty reflexive RelStr, X being Subset of L1, Y being Subset of L2
  st the RelStr of L1 = the RelStr of L2 & X = Y & X is directly_closed holds Y
  is directly_closed
proof
  let L1 be up-complete antisymmetric non empty reflexive RelStr, L2 be non
  empty reflexive RelStr, X be Subset of L1, Y be Subset of L2 such that
A1: the RelStr of L1 = the RelStr of L2 and
A2: X = Y and
A3: for D being non empty directed Subset of L1 st D c= X holds sup D in X;
  let E be non empty directed Subset of L2 such that
A4: E c= Y;
  reconsider D = E as non empty directed Subset of L1 by A1,WAYBEL_0:3;
  ex_sup_of D, L1 & sup D in X by A2,A3,A4,WAYBEL_0:75;
  hence thesis by A1,A2,YELLOW_0:26;
end;
