
theorem
  for S, T being antisymmetric up-complete non empty reflexive RelStr,
X being Subset of S, Y being Subset of T st [:X,Y:] is directly_closed holds (Y
  <> {} implies X is directly_closed) & (X <> {} implies 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,T:] st D c= [:X,Y:]
  holds sup D in [:X,Y:];
  hereby
    assume
A2: Y <> {};
    thus X is directly_closed
    proof
      consider t being object such that
A3:   t in Y by A2,XBOOLE_0:def 1;
      reconsider t9 = {t} as non empty directed Subset of T by A3,WAYBEL_0:5;
A4:   t9 c= Y by A3,ZFMISC_1:31;
      let D be non empty directed Subset of S;
      assume D c= X;
      then
A5:   sup [:D,t9:] in [:X,Y:] by A1,A4,ZFMISC_1:96;
      ex_sup_of [:D,t9:],[:S,T:] by WAYBEL_0:75;
      then sup [:D,t9:] = [sup proj1 [:D,t9:], sup proj2 [:D,t9:]] by
YELLOW_3:46
        .= [sup D, sup proj2 [:D,t9:]] by FUNCT_5:9
        .= [sup D,sup t9] by FUNCT_5:9
        .= [sup D,t] by A3,YELLOW_0:39;
      hence thesis by A5,ZFMISC_1:87;
    end;
  end;
  assume X <> {};
  then consider s being object such that
A6: s in X by XBOOLE_0:def 1;
  reconsider s9 = {s} as non empty directed Subset of S by A6,WAYBEL_0:5;
  let D be non empty directed Subset of T such that
A7: D c= Y;
  ex_sup_of [:s9,D:],[:S,T:] by WAYBEL_0:75;
  then
A8: sup [:s9,D:] = [sup proj1 [:s9,D:], sup proj2 [:s9,D:]] by YELLOW_3:46
    .= [sup s9,sup proj2 [:s9,D:]] by FUNCT_5:9
    .= [sup s9,sup D] by FUNCT_5:9
    .= [s,sup D] by A6,YELLOW_0:39;
  s9 c= X by A6,ZFMISC_1:31;
  then sup [:s9,D:] in [:X,Y:] by A1,A7,ZFMISC_1:96;
  hence thesis by A8,ZFMISC_1:87;
end;
