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 property(S) holds Y is
  property(S)
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 sup D in X ex y being
Element of L1 st y in D & for x being Element of L1 st x in D & x >= y holds x
  in X;
  let E be non empty directed Subset of L2 such that
A4: sup E in Y;
  reconsider D = E as non empty directed Subset of L1 by A1,WAYBEL_0:3;
  ex_sup_of D, L1 by WAYBEL_0:75;
  then sup D in X by A1,A2,A4,YELLOW_0:26;
  then consider y being Element of L1 such that
A5: y in D and
A6: for x being Element of L1 st x in D & x >= y holds x in X by A3;
  reconsider y2 = y as Element of L2 by A1;
  take y2;
  thus y2 in E by A5;
  let x2 be Element of L2;
  assume x2 in E & x2 >= y2;
  hence thesis by A1,A2,A6,YELLOW_0:1;
end;
