
theorem
  for I being non empty set for J being RelStr-yielding non-Empty
  ManySortedSet of I st for i being Element of I holds J.i is upper-bounded
  antisymmetric RelStr holds product J is upper-bounded
proof
  let I be non empty set, J be RelStr-yielding non-Empty ManySortedSet of I
  such that
A1: for i being Element of I holds J.i is upper-bounded antisymmetric RelStr;
  deffunc F(Element of I) = Top (J.$1);
  consider f being ManySortedSet of I such that
A2: for i being Element of I holds f.i = F(i) from PBOOLE:sch 5;
A3: now
    let i be Element of I;
    f.i = Top (J.i) by A2;
    hence f.i is Element of J.i;
  end;
  dom f = I by PARTFUN1:def 2;
  then reconsider f as Element of product J by A3,WAYBEL_3:27;
  take f;
  let b be Element of product J such that
  b in the carrier of product J;
  now
    let i be Element of I;
    f.i = Top (J.i) & J.i is upper-bounded antisymmetric non empty RelStr
    by A1,A2;
    hence f.i >= b.i by YELLOW_0:45;
  end;
  hence thesis by WAYBEL_3:28;
end;
