
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 iff p1 <= p2 & 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];
  thus p <= q implies p1 <= p2 & q1 <= q2
  proof
    assume p <= q;
    then [p,q] in [" the InternalRel of P, the InternalRel of Q "];
    then consider a, b, s, t being object such that
A3: p = [a,b] and
A4: q = [s,t] and
A5: [a,s] in the InternalRel of P and
A6: [b,t] in the InternalRel of Q by YELLOW_3:def 1;
A7: a = p1 by A1,A3,XTUPLE_0:1;
A8: b = q1 by A1,A3,XTUPLE_0:1;
    thus [p1,p2] in the InternalRel of P by A2,A4,A5,A7,XTUPLE_0:1;
    thus [q1,q2] in the InternalRel of Q by A2,A4,A6,A8,XTUPLE_0:1;
  end;
  assume that
A9: p1 <= p2 and
A10: q1 <= q2;
A11: [p1,p2] in the InternalRel of P by A9;
  [q1,q2] in the InternalRel of Q by A10;
  hence [p,q] in the InternalRel of P pcs-times Q by A1,A2,A11,YELLOW_3:def 1;
end;
