
theorem Th34:
  for I being non empty set for J being RelStr-yielding non-Empty
reflexive-yielding ManySortedSet of I for X being directed Subset of product J
  for i being Element of I holds pi(X,i) is directed
proof
  let I be non empty set;
  let J be RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I;
  let X be directed Subset of product J;
  let i be Element of I;
  let x,y be Element of J.i;
  assume x in pi(X,i);
  then consider f being Function such that
A1: f in X and
A2: x = f.i by CARD_3:def 6;
  assume y in pi(X,i);
  then consider g being Function such that
A3: g in X and
A4: y = g.i by CARD_3:def 6;
  reconsider f,g as Element of product J by A1,A3;
  consider h being Element of product J such that
A5: h in X and
A6: f <= h and
A7: g <= h by A1,A3,WAYBEL_0:def 1;
  take h.i;
  thus h.i in pi(X,i) by A5,CARD_3:def 6;
  thus thesis by A2,A4,A6,A7,WAYBEL_3:28;
end;
