
theorem
  for P, Q being pcs-Str, p, q being Element of P --> Q
  for p1, p2 being Element of P, q1, q2 being Element of Q st
  p = [p1,q1] & q = [p2,q2] holds p (--) q implies not p1 (--) p2 or q1 (--) q2
proof
  let P, Q be pcs-Str, p, q be Element of P --> Q;
  let p1, p2 be Element of P, q1, q2 be Element of Q such that
A1: p = [p1,q1] and
A2: q = [p2,q2];
  reconsider r1 = p1, r2 = p2 as Element of pcs-reverse P by Def40;
  assume [p,q] in the ToleranceRel of P --> Q;
  then consider a, b, s, t being set such that
A3: p = [a,b] and
A4: q = [s,t] and
  a in the carrier of pcs-reverse P and b in the carrier of Q
  and s in the carrier of pcs-reverse P
  and t in the carrier of Q and
A5: [a,s] in the ToleranceRel of pcs-reverse P or
  [b,t] in the ToleranceRel of Q by Def2;
A6: a = p1 by A1,A3,XTUPLE_0:1;
A7: b = q1 by A1,A3,XTUPLE_0:1;
A8: s = p2 by A2,A4,XTUPLE_0:1;
  t = q2 by A2,A4,XTUPLE_0:1;
  then r1 (--) r2 or q1 (--) q2 by A5,A6,A7,A8;
  hence thesis by Th34;
end;
