
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 iff p2 <= p1 & 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;
  thus p <= q implies p2 <= p1 & q1 <= q2
  proof
    assume p <= q;
    then [p,q] in ["the InternalRel of pcs-reverse 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 pcs-reverse 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;
    s = p2 by A2,A4,XTUPLE_0:1;
    then r1 <= r2 by A5,A7;
    hence p2 <= p1 by Th33;
    thus [q1,q2] in the InternalRel of Q by A2,A4,A6,A8,XTUPLE_0:1;
  end;
  assume that
A9: p2 <= p1 and
A10: q1 <= q2;
  r1 <= r2 by A9,Th33;
  then
A11: [r1,r2] in the InternalRel of pcs-reverse P;
  [q1,q2] in the InternalRel of Q by A10;
  hence [p,q] in the InternalRel of P --> Q by A1,A2,A11,YELLOW_3:def 1;
end;
