
theorem
  for L being up-complete non empty Poset for S1,S2 being
directed-sups-inheriting full non empty SubRelStr of L st S1 is SubRelStr of S2
  holds S1 is directed-sups-inheriting full SubRelStr of S2
proof
  let L be up-complete non empty Poset;
  let S1,S2 be directed-sups-inheriting full non empty SubRelStr of L;
  assume S1 is SubRelStr of S2;
  then reconsider S = S1 as SubRelStr of S2;
  S is directed-sups-inheriting
  proof
    let X be directed Subset of S;
    assume X <> {};
    then reconsider Y1 = X as non empty directed Subset of S1;
    reconsider Y2 = Y1 as non empty directed Subset of S2 by YELLOW_2:7;
    reconsider Y = Y1 as non empty directed Subset of L by YELLOW_2:7;
A1: ex_sup_of Y, L by WAYBEL_0:75;
    then
A2: sup Y = sup Y1 by WAYBEL_0:7;
    sup Y2 = sup Y by A1,WAYBEL_0:7;
    hence thesis by A2;
  end;
  hence thesis by Th25;
end;
