
theorem
  for P being pcs-Str, D being non empty Subset-Family of P
  for p, q being Element of pcs-general-power(D) st
  for p9 being Element of P st p9 in p
  ex q9 being Element of P st q9 in q & p9 <= q9 holds p <= q
proof
  let P be pcs-Str, D be non empty Subset-Family of P;
  set R = pcs-general-power(D);
  let p, q be Element of R;
  assume
A1: for p9 being Element of P st p9 in p
  ex q9 being Element of P st q9 in q & p9 <= q9;
A2: p in D;
  for a being set st a in p ex b being set st b in q &
  [a,b] in the InternalRel of P
  proof
    let a be set;
    assume
A3: a in p;
    then reconsider a as Element of P by A2;
    consider q9 being Element of P such that
A4: q9 in q and
A5: a <= q9 by A1,A3;
    take q9;
    thus thesis by A4,A5;
  end;
  hence [p,q] in the InternalRel of R by Def45;
end;
