
theorem
  for I being non empty set, C being pcs-Str-yielding ManySortedSet of I
  for p, q being Element of pcs-union C holds p <= q iff
  ex i being Element of I, p9, q9 being Element of C.i st
  p9 = p & q9 = q & p9 <= q9
proof
  let I be non empty set, C be pcs-Str-yielding ManySortedSet of I;
  let p, q be Element of pcs-union C;
  thus
  p <= q implies ex i being Element of I, p9, q9 being Element of C.i st
  p9 = p & q9 = q & p9 <= q9
  proof
    assume p <= q;
    then ex i being object, P being pcs-Str, p9, q9 being Element of P st
    i in I & P = C.i & p9 = p & q9 = q & p9 <= q9 by Th10;
    hence thesis;
  end;
  thus thesis by Th10;
end;
