
theorem Th3:
  for L1,L2 being RelStr st the RelStr of L1 = the RelStr of L2
  for X1 being Subset of L1, X2 being Subset of L2 st X1 = X2 & X1 is directed
  holds X2 is directed
proof
  let L1,L2 be RelStr such that
A1: the RelStr of L1 = the RelStr of L2;
  let X1 be Subset of L1, X2 be Subset of L2 such that
A2: X1 = X2;
  assume
A3: for x,y being Element of L1 st x in X1 & y in X1
  ex z being Element of L1 st z in X1 & x <= z & y <= z;
  let x,y be Element of L2;
  reconsider x9 = x, y9 = y as Element of L1 by A1;
  assume that
A4: x in X2 and
A5: y in X2;
  consider z9 being Element of L1 such that
A6: z9 in X1 and
A7: x9 <= z9 and
A8: y9 <= z9 by A2,A3,A4,A5;
  reconsider z = z9 as Element of L2 by A1;
  take z;
  thus thesis by A1,A2,A6,A7,A8,YELLOW_0:1;
end;
