
theorem
  for L being non empty transitive RelStr for S being
  directed-sups-inheriting non empty full SubRelStr of L for R being
  directed-sups-inheriting non empty SubRelStr of S holds R is
  directed-sups-inheriting SubRelStr of L
proof
  let L be non empty transitive RelStr;
  let S be directed-sups-inheriting non empty full SubRelStr of L;
  let R be directed-sups-inheriting non empty SubRelStr of S;
  reconsider T = R as SubRelStr of L by YELLOW_6:7;
  T is directed-sups-inheriting
  proof
    let X be directed Subset of T;
    reconsider Y = X as directed Subset of S by YELLOW_2:7;
    assume
A1: X <> {};
    assume
A2: ex_sup_of X,L;
    then
A3: ex_sup_of Y, S by A1,WAYBEL_0:7;
    sup Y = "\/"(X,L) by A1,A2,WAYBEL_0:7;
    hence thesis by A1,A3,WAYBEL_0:def 4;
  end;
  hence thesis;
end;
