
theorem Th27:
  for P being pcs-Str, a being set,
  p, q being Element of pcs-extension(P,a) st p = a holds p (--) q & q (--) p
proof
  let P be pcs-Str, a be set;
  set R = pcs-extension(P,a);
  let p, q be Element of R such that
A1: p = a;
  the ToleranceRel of R = [:{a},the carrier of R:] \/
  [:the carrier of R,{a}:] \/ the ToleranceRel of P by Def39;
  then
A2: the ToleranceRel of R = [:{a},the carrier of R:] \/
  ([:the carrier of R,{a}:] \/ the ToleranceRel of P) by XBOOLE_1:4;
A3: [a,q] in [:{a},the carrier of R:] by ZFMISC_1:105;
  [q,a] in [:the carrier of R,{a}:] by ZFMISC_1:106;
  then [q,a] in [:the carrier of R,{a}:] \/ the ToleranceRel of P
  by XBOOLE_0:def 3;
  hence [p,q] in the ToleranceRel of R & [q,p] in the ToleranceRel of R
  by A1,A2,A3,XBOOLE_0:def 3;
end;
