
theorem
  for L being RelStr, A being Subset-Family of L st
  (for X being Subset of L st X in A holds X is directed) &
  (for X,Y being Subset of L st X in A & Y in A
  ex Z being Subset of L st Z in A & X \/ Y c= Z)
  for X being Subset of L st X = union A holds X is directed
proof
  let L be RelStr, A be Subset-Family of L such that
A1: for X being Subset of L st X in A holds X is directed and
A2: for X,Y being Subset of L st X in A & Y in A
  ex Z being Subset of L st Z in A & X \/ Y c= Z;
  let Z be Subset of L;
  assume
A3: Z = union A;
  let x,y be Element of L;
  assume x in Z;
  then consider X being set such that
A4: x in X and
A5: X in A by A3,TARSKI:def 4;
  assume y in Z;
  then consider Y being set such that
A6: y in Y and
A7: Y in A by A3,TARSKI:def 4;
  reconsider X,Y as Subset of L by A5,A7;
  consider W being Subset of L such that
A8: W in A and
A9: X \/ Y c= W by A2,A5,A7;
A10: X c= X \/ Y by XBOOLE_1:7;
A11: Y c= X \/ Y by XBOOLE_1:7;
A12: x in X \/ Y by A4,A10;
A13: y in X \/ Y by A6,A11;
  W is directed by A1,A8;
  then consider z being Element of L such that
A14: z in W and
A15: x <= z and
A16: y <= z by A9,A12,A13;
  take z;
  thus thesis by A3,A8,A14,A15,A16,TARSKI:def 4;
end;
