
theorem Th26:
  for P being pcs-Str, a being set, p, q being Element of P,
  p1, q1 being Element of pcs-extension(P,a) st p = p1 & q = q1 &
  p <> a & p1 <= q1 holds p <= q
proof
  let P be pcs-Str, a be set, p, q be Element of P,
  p1, q1 be Element of pcs-extension(P,a) such that
A1: p = p1 and
A2: q = q1 and
A3: p <> a and
A4: p1 <= q1;
  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 A4;
  then [p1,q1] in [:{a},the carrier of R:] or [p1,q1] in the InternalRel of P
  by A5,XBOOLE_0:def 3;
  hence [p,q] in the InternalRel of P by A1,A2,A3,ZFMISC_1:105;
end;
