
theorem
  for L being non empty Poset for S being directed-sups-inheriting non
  empty full SubRelStr of L for X be non empty set holds S|^X is
  directed-sups-inheriting SubRelStr of L|^X
proof
  let L be non empty Poset;
  let S be directed-sups-inheriting non empty full SubRelStr of L;
  let X be non empty set;
  reconsider SX = S|^X as full non empty SubRelStr of L|^X by Th38;
  defpred P[set, non empty reflexive RelStr] means $1 is directed non empty
  Subset of $2;
A1: now
    let A be Subset of S|^X;
    assume P[A,S|^X];
    then reconsider B = A as directed non empty Subset of S|^X;
    let i be Element of X;
    (X --> S).i = S;
    then pi(B, i) is directed non empty Subset of S by Th34;
    hence P[pi(A, i),S];
  end;
A2: now
    let X be Subset of S;
    assume P[X,S];
    then ex_sup_of X,L implies ex_sup_of X, S & sup X = "\/"(X, L) by
WAYBEL_0:7;
    hence S inherits_sup_of X, L;
  end;
  SX is directed-sups-inheriting
  proof
    let A be directed Subset of SX;
    for A being Subset of S|^X st P[A,S|^X] holds S|^X inherits_sup_of A,
    L|^X from PowerSupsInheriting(A1,A2);
    then A <> {} implies S|^X inherits_sup_of A, L|^X;
    hence thesis;
  end;
  hence thesis;
end;
