
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 for i being Element of I holds Top (product J).i =
  Top (J.i)
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;
  let i be Element of I;
A4: {} is_>=_than f;
A5: now
    let c be Element of product J such that
    {} is_>=_than c and
A6: for b being Element of product J st {} is_>=_than b holds b <= c;
    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 >= c.i by YELLOW_0:45;
    end;
    then
A7: f >= c by WAYBEL_3:28;
    for i being Element of I holds J.i is antisymmetric by A1;
    then
A8: product J is antisymmetric by WAYBEL_3:30;
    c >= f by A6,YELLOW_0:6;
    hence c = f by A8,A7;
  end;
A9: now
    let a be Element of product J such that
    {} is_>=_than a;
    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 >= a.i by YELLOW_0:45;
    end;
    hence f >= a by WAYBEL_3:28;
  end;
  now
    let b be Element of product J such that
    {} is_>=_than b;
    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 f >= b by WAYBEL_3:28;
  end;
  then ex_inf_of {}, product J by A4,A5;
  then f = "/\"({}, product J) by A4,A9,YELLOW_0:def 10;
  hence thesis by A2;
end;
