
theorem Th4:
  for L being up-complete non empty Poset for S being
directed-sups-inheriting full non empty SubRelStr of L holds S is up-complete
proof
  let L be up-complete non empty Poset;
  let S be directed-sups-inheriting full non empty SubRelStr of L;
  now
    let X be non empty directed Subset of S;
    reconsider Y = X as non empty directed Subset of L by YELLOW_2:7;
    ex_sup_of Y,L by WAYBEL_0:75;
    hence ex_sup_of X,S by WAYBEL_0:7;
  end;
  hence thesis by WAYBEL_0:75;
end;
