
theorem Th44:
  for P being pcs-Str, D be set
  for p, q being Element of pcs-general-power(P,D) holds p <= q
  implies for p9 being Element of P st p9 in p
  ex q9 being Element of P st q9 in q & p9 <= q9
proof
  let P be pcs-Str, D be set;
  set R = pcs-general-power(P,D);
  let p, q be Element of R;
  assume
A1: [p,q] in the InternalRel of R;
  let p9 be Element of P;
  assume p9 in p;
  then consider b being set such that
A2: b in q and
A3: [p9,b] in the InternalRel of P by A1,Def45;
  reconsider b as Element of P by A3,ZFMISC_1:87;
  take b;
  thus b in q & [p9,b] in the InternalRel of P by A2,A3;
end;
