
theorem
  for S, T being antisymmetric up-complete non empty reflexive RelStr,
  X being Subset of S, Y being Subset of T st X is directly_closed & Y is
  directly_closed holds [:X,Y:] is directly_closed
proof
  let S, T be antisymmetric up-complete non empty reflexive RelStr, X be
  Subset of S, Y be Subset of T such that
A1: for D being non empty directed Subset of S st D c= X holds sup D in X and
A2: for D being non empty directed Subset of T st D c= Y holds sup D in Y;
  let D be non empty directed Subset of [:S,T:];
  assume
A3: D c= [:X,Y:];
  proj2 D is non empty directed by YELLOW_3:21,22;
  then
A4: sup proj2 D in Y by A2,A3,FUNCT_5:11;
  ex_sup_of D,[:S,T:] by WAYBEL_0:75;
  then
A5: sup D = [sup proj1 D,sup proj2 D] by YELLOW_3:46;
  proj1 D is non empty directed by YELLOW_3:21,22;
  then sup proj1 D in X by A1,A3,FUNCT_5:11;
  hence thesis by A4,A5,ZFMISC_1:87;
end;
