
theorem Th9:
  for X being non empty TopSpace for L being Scott up-complete
non empty TopPoset holds ContMaps(X, L) is directed-sups-inheriting SubRelStr
  of L|^the carrier of X
proof
  let X be non empty TopSpace;
  let L be Scott up-complete non empty TopPoset;
  reconsider XL = ContMaps(X, L) as non empty full SubRelStr of L|^the carrier
  of X by WAYBEL24:def 3;
  XL is directed-sups-inheriting
  proof
    let A be directed Subset of XL;
    assume that
A1: A <> {} and
    ex_sup_of A, L|^the carrier of X;
    reconsider A as directed non empty Subset of XL by A1;
    "\/"(A, L|^the carrier of X) is continuous Function of X, L by Th8;
    hence thesis by WAYBEL24:def 3;
  end;
  hence thesis;
end;
