reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;
reserve L for complete LATTICE;
reserve AR for Relation of L;
reserve x, y, z for Element of L;

theorem
  for L being RelStr, X being Subset of L st
  X is_directed_wrt (the InternalRel of L) holds X is directed
proof
  let L be RelStr, X be Subset of L;
  assume
A1: X is_directed_wrt (the InternalRel of L);
  let x,y be Element of L;
  assume that
A2: x in X and
A3: y in X;
  consider z being Element of L such that
A4: z in X and
A5: [x,z] in the InternalRel of L and
A6: [y,z] in the InternalRel of L by A1,A2,A3;
  take z;
  thus z in X by A4;
  thus thesis by A5,A6,ORDERS_2:def 5;
end;
