
theorem Th11:
  for I being non empty set for J being Poset-yielding non-Empty
ManySortedSet of I st for i being Element of I holds J.i is up-complete holds I
  -POS_prod J is up-complete
proof
  let I be non empty set;
  let J be Poset-yielding non-Empty ManySortedSet of I such that
A1: for i being Element of I holds J.i is up-complete;
  set L = I-POS_prod J;
  now
    let A be non empty directed Subset of L;
    now
      let x be Element of I;
      J.x is up-complete non empty Poset & pi(A,x) is directed non empty
      by A1,YELLOW16:35;
      hence ex_sup_of pi(A,x), J.x by WAYBEL_0:75;
    end;
    hence ex_sup_of A,L by YELLOW16:31;
  end;
  hence thesis by WAYBEL_0:75;
end;
