
theorem
  for P, Q being pcs-Str, p, q being Element of pcs-sum(P,Q) holds p (--) q iff
  (ex p9, q9 being Element of P st p9 = p & q9 = q & p9 (--) q9) or
  ex p9, q9 being Element of Q st p9 = p & q9 = q & p9 (--) q9
proof
  let P, Q be pcs-Str;
  set R = pcs-sum(P,Q);
  let p, q be Element of R;
A1: the ToleranceRel of R =
  (the ToleranceRel of P) \/ the ToleranceRel of Q by Th14;
  thus p (--) q implies
  (ex p9, q9 being Element of P st p9 = p & q9 = q & p9 (--) q9) or
  ex p9, q9 being Element of Q st p9 = p & q9 = q & p9 (--) q9
  proof
    assume
A2: [p,q] in the ToleranceRel of R;
    per cases by A1,A2,XBOOLE_0:def 3;
    suppose
A3:   [p,q] in the ToleranceRel of P;
      then reconsider p9 = p, q9 = q as Element of P by ZFMISC_1:87;
      p9 (--) q9 by A3;
      hence thesis;
    end;
    suppose
A4:   [p,q] in the ToleranceRel of Q;
      then reconsider p9 = p, q9 = q as Element of Q by ZFMISC_1:87;
      p9 (--) q9 by A4;
      hence thesis;
    end;
  end;
  assume
A5: (ex p9, q9 being Element of P st p9 = p & q9 = q & p9 (--) q9) or
  ex p9, q9 being Element of Q st p9 = p & q9 = q & p9 (--) q9;
  per cases by A5;
  suppose ex p9, q9 being Element of P st p9 = p & q9 = q & p9 (--) q9;
    then consider p9, q9 being Element of P such that
A6: p9 = p and
A7: q9 = q and
A8: p9 (--) q9;
    [p9,q9] in the ToleranceRel of P by A8;
    hence [p,q] in the ToleranceRel of R by A1,A6,A7,XBOOLE_0:def 3;
  end;
  suppose ex p9, q9 being Element of Q st p9 = p & q9 = q & p9 (--) q9;
    then consider p9, q9 being Element of Q such that
A9: p9 = p and
A10: q9 = q and
A11: p9 (--) q9;
    [p9,q9] in the ToleranceRel of Q by A11;
    hence [p,q] in the ToleranceRel of R by A1,A9,A10,XBOOLE_0:def 3;
  end;
end;
