
theorem
  for P, Q being pcs-Str, p, q being Element of P pcs-times 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 p1 (--) p2 or q1 (--) q2
proof
  let P, Q be pcs-Str, p, q be Element of P pcs-times 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];
  assume p (--) q;
  then [p,q] in [^ the ToleranceRel of P, the ToleranceRel of Q ^];
  then consider a, b, c, d being set such that
A3: p = [a,b] and
A4: q = [c,d] and a in the carrier of P
  and b in the carrier of Q
  and c in the carrier of P
  and d in the carrier of Q and
A5: [a,c] in the ToleranceRel of P or [b,d] 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: c = p2 by A2,A4,XTUPLE_0:1;
  d = q2 by A2,A4,XTUPLE_0:1;
  hence thesis by A5,A6,A7,A8;
end;
