
theorem Th24:
  for P being pcs-Str, a being set, p being Element of P,
  p1, q1 being Element of pcs-extension(P,a)
  st p = p1 & p <> a & p1 <= q1 & not a in the carrier of P holds
  q1 in the carrier of P & q1 <> a
proof
  let P be pcs-Str, a be set, p be Element of P,
  p1, q1 be Element of pcs-extension(P,a) such that
A1: p = p1 and
A2: p <> a and
A3: p1 <= q1 and
A4: not a in the carrier of P;
  set R = pcs-extension(P,a);
A5: the InternalRel of R = [:{a},the carrier of R:] \/ the InternalRel of P
  by Def39;
  [p1,q1] in the InternalRel of R by A3;
  then [p1,q1] in [:{a},the carrier of R:] or [p1,q1] in the InternalRel of P
  by A5,XBOOLE_0:def 3;
  hence thesis by A1,A2,A4,ZFMISC_1:87,105;
end;
