
theorem Th29:
  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 & q <> 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: q <> a and
A5: p1 (--) q1;
  set R = pcs-extension(P,a);
A6: the ToleranceRel of R = [:{a},the carrier of R:] \/
  [:the carrier of R,{a}:] \/ the ToleranceRel of P by Def39;
  [p1,q1] in the ToleranceRel of R by A5;
  then [p1,q1] in [:{a},the carrier of R:] \/ [:the carrier of R,{a}:] or
  [p1,q1] in the ToleranceRel of P by A6,XBOOLE_0:def 3;
  then [p1,q1] in [:{a},the carrier of R:] or
  [p1,q1] in [:the carrier of R,{a}:] or
  [p1,q1] in the ToleranceRel of P by XBOOLE_0:def 3;
  hence [p,q] in the ToleranceRel of P by A1,A2,A3,A4,ZFMISC_1:105,106;
end;
